src/HOL/Ring_and_Field.thy
author nipkow
Sat Jun 23 19:33:22 2007 +0200 (2007-06-23)
changeset 23477 f4b83f03cac9
parent 23413 5caa2710dd5b
child 23482 2f4be6844f7c
permissions -rw-r--r--
tuned and renamed group_eq_simps and ring_eq_simps
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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_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> pordered_cancel_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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class ordered_ring_strict = ring + ordered_semiring_strict + abs_if + lordered_ab_group
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instance ordered_ring_strict \<subseteq> lordered_ring
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  by intro_classes (simp add: abs_if sup_eq_if)
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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 + abs_if + lordered_ab_group
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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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class ordered_field = field + ordered_idom
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lemmas linorder_neqE_ordered_idom =
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 linorder_neqE[where 'a = "?'b::ordered_idom"]
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lemma eq_add_iff1:
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  "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
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by (simp add: ring_simps)
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lemma eq_add_iff2:
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  "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
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   323
by (simp add: ring_simps)
paulson@14272
   324
paulson@14272
   325
lemma less_add_iff1:
nipkow@23477
   326
  "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
nipkow@23477
   327
by (simp add: ring_simps)
paulson@14272
   328
paulson@14272
   329
lemma less_add_iff2:
nipkow@23477
   330
  "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
nipkow@23477
   331
by (simp add: ring_simps)
paulson@14272
   332
paulson@14272
   333
lemma le_add_iff1:
nipkow@23477
   334
  "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
nipkow@23477
   335
by (simp add: ring_simps)
paulson@14272
   336
paulson@14272
   337
lemma le_add_iff2:
nipkow@23477
   338
  "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
nipkow@23477
   339
by (simp add: ring_simps)
paulson@14272
   340
wenzelm@23389
   341
paulson@14270
   342
subsection {* Ordering Rules for Multiplication *}
paulson@14270
   343
paulson@14348
   344
lemma mult_left_le_imp_le:
nipkow@23477
   345
  "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
nipkow@23477
   346
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
paulson@14348
   347
 
paulson@14348
   348
lemma mult_right_le_imp_le:
nipkow@23477
   349
  "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
nipkow@23477
   350
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
paulson@14348
   351
paulson@14348
   352
lemma mult_left_less_imp_less:
nipkow@23477
   353
  "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
nipkow@23477
   354
by (force simp add: mult_left_mono linorder_not_le [symmetric])
paulson@14348
   355
 
paulson@14348
   356
lemma mult_right_less_imp_less:
nipkow@23477
   357
  "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
nipkow@23477
   358
by (force simp add: mult_right_mono linorder_not_le [symmetric])
paulson@14348
   359
paulson@14265
   360
lemma mult_strict_left_mono_neg:
nipkow@23477
   361
  "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
paulson@14265
   362
apply (drule mult_strict_left_mono [of _ _ "-c"])
paulson@14265
   363
apply (simp_all add: minus_mult_left [symmetric]) 
paulson@14265
   364
done
paulson@14265
   365
obua@14738
   366
lemma mult_left_mono_neg:
nipkow@23477
   367
  "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
obua@14738
   368
apply (drule mult_left_mono [of _ _ "-c"])
obua@14738
   369
apply (simp_all add: minus_mult_left [symmetric]) 
obua@14738
   370
done
obua@14738
   371
paulson@14265
   372
lemma mult_strict_right_mono_neg:
nipkow@23477
   373
  "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
paulson@14265
   374
apply (drule mult_strict_right_mono [of _ _ "-c"])
paulson@14265
   375
apply (simp_all add: minus_mult_right [symmetric]) 
paulson@14265
   376
done
paulson@14265
   377
obua@14738
   378
lemma mult_right_mono_neg:
nipkow@23477
   379
  "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
obua@14738
   380
apply (drule mult_right_mono [of _ _ "-c"])
obua@14738
   381
apply (simp)
obua@14738
   382
apply (simp_all add: minus_mult_right [symmetric]) 
obua@14738
   383
done
paulson@14265
   384
wenzelm@23389
   385
paulson@14265
   386
subsection{* Products of Signs *}
paulson@14265
   387
avigad@16775
   388
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
paulson@14265
   389
by (drule mult_strict_left_mono [of 0 b], auto)
paulson@14265
   390
avigad@16775
   391
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
obua@14738
   392
by (drule mult_left_mono [of 0 b], auto)
obua@14738
   393
obua@14738
   394
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   395
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   396
avigad@16775
   397
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
obua@14738
   398
by (drule mult_left_mono [of b 0], auto)
obua@14738
   399
obua@14738
   400
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
obua@14738
   401
by (drule mult_strict_right_mono[of b 0], auto)
obua@14738
   402
avigad@16775
   403
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
obua@14738
   404
by (drule mult_right_mono[of b 0], auto)
obua@14738
   405
avigad@16775
   406
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
paulson@14265
   407
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   408
avigad@16775
   409
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
obua@14738
   410
by (drule mult_right_mono_neg[of a 0 b ], auto)
obua@14738
   411
paulson@14341
   412
lemma zero_less_mult_pos:
obua@14738
   413
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   414
apply (cases "b\<le>0") 
paulson@14265
   415
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   416
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   417
 apply (auto dest: order_less_not_sym)
paulson@14265
   418
done
paulson@14265
   419
obua@14738
   420
lemma zero_less_mult_pos2:
obua@14738
   421
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   422
apply (cases "b\<le>0") 
obua@14738
   423
 apply (auto simp add: order_le_less linorder_not_less)
obua@14738
   424
apply (drule_tac mult_pos_neg2 [of a b]) 
obua@14738
   425
 apply (auto dest: order_less_not_sym)
obua@14738
   426
done
obua@14738
   427
paulson@14265
   428
lemma zero_less_mult_iff:
obua@14738
   429
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
   430
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos 
avigad@16775
   431
  mult_neg_neg)
paulson@14265
   432
apply (blast dest: zero_less_mult_pos) 
obua@14738
   433
apply (blast dest: zero_less_mult_pos2)
paulson@14265
   434
done
paulson@14265
   435
huffman@22990
   436
lemma mult_eq_0_iff [simp]:
huffman@22990
   437
  fixes a b :: "'a::ring_no_zero_divisors"
huffman@22990
   438
  shows "(a * b = 0) = (a = 0 \<or> b = 0)"
huffman@22990
   439
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors)
huffman@22990
   440
huffman@22990
   441
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
huffman@22990
   442
apply intro_classes
paulson@14265
   443
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   444
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   445
done
paulson@14265
   446
paulson@14265
   447
lemma zero_le_mult_iff:
obua@14738
   448
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   449
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   450
                   zero_less_mult_iff)
paulson@14265
   451
paulson@14265
   452
lemma mult_less_0_iff:
obua@14738
   453
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   454
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   455
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   456
done
paulson@14265
   457
paulson@14265
   458
lemma mult_le_0_iff:
obua@14738
   459
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   460
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   461
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   462
done
paulson@14265
   463
obua@14738
   464
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
   465
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14738
   466
obua@14738
   467
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
   468
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
obua@14738
   469
obua@23095
   470
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a"
obua@23095
   471
by (simp add: zero_le_mult_iff linorder_linear)
obua@23095
   472
obua@23095
   473
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
obua@23095
   474
by (simp add: not_less)
paulson@14265
   475
obua@14738
   476
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
obua@14738
   477
      theorems available to members of @{term ordered_idom} *}
obua@14738
   478
obua@14738
   479
instance ordered_idom \<subseteq> ordered_semidom
paulson@14421
   480
proof
paulson@14421
   481
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
paulson@14430
   482
  thus "(0::'a) < 1" by (simp add: order_le_less) 
paulson@14421
   483
qed
paulson@14421
   484
obua@14738
   485
instance ordered_idom \<subseteq> idom ..
obua@14738
   486
paulson@14387
   487
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   488
paulson@17085
   489
lemmas one_neq_zero = zero_neq_one [THEN not_sym]
paulson@17085
   490
declare one_neq_zero [simp]
paulson@14387
   491
obua@14738
   492
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
paulson@14268
   493
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   494
obua@14738
   495
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
obua@14738
   496
by (simp add: linorder_not_le) 
paulson@14387
   497
obua@14738
   498
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
obua@14738
   499
by (simp add: linorder_not_less) 
paulson@14268
   500
wenzelm@23389
   501
paulson@14268
   502
subsection{*More Monotonicity*}
paulson@14268
   503
paulson@14268
   504
text{*Strict monotonicity in both arguments*}
paulson@14268
   505
lemma mult_strict_mono:
obua@14738
   506
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
haftmann@21328
   507
apply (cases "c=0")
avigad@16775
   508
 apply (simp add: mult_pos_pos) 
paulson@14268
   509
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   510
 apply (force simp add: order_le_less) 
paulson@14268
   511
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   512
done
paulson@14268
   513
paulson@14268
   514
text{*This weaker variant has more natural premises*}
paulson@14268
   515
lemma mult_strict_mono':
obua@14738
   516
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   517
apply (rule mult_strict_mono)
paulson@14268
   518
apply (blast intro: order_le_less_trans)+
paulson@14268
   519
done
paulson@14268
   520
paulson@14268
   521
lemma mult_mono:
paulson@14268
   522
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
obua@14738
   523
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
paulson@14268
   524
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   525
apply (erule mult_left_mono, assumption)
paulson@14268
   526
done
paulson@14268
   527
huffman@21258
   528
lemma mult_mono':
huffman@21258
   529
     "[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] 
huffman@21258
   530
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
huffman@21258
   531
apply (rule mult_mono)
huffman@21258
   532
apply (fast intro: order_trans)+
huffman@21258
   533
done
huffman@21258
   534
obua@14738
   535
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
paulson@14387
   536
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14430
   537
apply (simp add:  order_less_trans [OF zero_less_one]) 
paulson@14387
   538
done
paulson@14387
   539
avigad@16775
   540
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
avigad@16775
   541
    c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
avigad@16775
   542
  apply (subgoal_tac "a * c < b * c")
avigad@16775
   543
  apply (erule order_less_le_trans)
avigad@16775
   544
  apply (erule mult_left_mono)
avigad@16775
   545
  apply simp
avigad@16775
   546
  apply (erule mult_strict_right_mono)
avigad@16775
   547
  apply assumption
avigad@16775
   548
done
avigad@16775
   549
avigad@16775
   550
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
avigad@16775
   551
    c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
avigad@16775
   552
  apply (subgoal_tac "a * c <= b * c")
avigad@16775
   553
  apply (erule order_le_less_trans)
avigad@16775
   554
  apply (erule mult_strict_left_mono)
avigad@16775
   555
  apply simp
avigad@16775
   556
  apply (erule mult_right_mono)
avigad@16775
   557
  apply simp
avigad@16775
   558
done
avigad@16775
   559
wenzelm@23389
   560
paulson@14268
   561
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   562
paulson@14268
   563
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   564
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   565
paulson@15234
   566
text{*These ``disjunction'' versions produce two cases when the comparison is
paulson@15234
   567
 an assumption, but effectively four when the comparison is a goal.*}
paulson@15234
   568
paulson@15234
   569
lemma mult_less_cancel_right_disj:
obua@14738
   570
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   571
apply (cases "c = 0")
paulson@14268
   572
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   573
                      mult_strict_right_mono_neg)
paulson@14268
   574
apply (auto simp add: linorder_not_less 
paulson@14268
   575
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   576
                      linorder_not_le [symmetric, of a])
paulson@14268
   577
apply (erule_tac [!] notE)
paulson@14268
   578
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   579
                      mult_right_mono_neg)
paulson@14268
   580
done
paulson@14268
   581
paulson@15234
   582
lemma mult_less_cancel_left_disj:
obua@14738
   583
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   584
apply (cases "c = 0")
obua@14738
   585
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
obua@14738
   586
                      mult_strict_left_mono_neg)
obua@14738
   587
apply (auto simp add: linorder_not_less 
obua@14738
   588
                      linorder_not_le [symmetric, of "c*a"]
obua@14738
   589
                      linorder_not_le [symmetric, of a])
obua@14738
   590
apply (erule_tac [!] notE)
obua@14738
   591
apply (auto simp add: order_less_imp_le mult_left_mono 
obua@14738
   592
                      mult_left_mono_neg)
obua@14738
   593
done
paulson@14268
   594
paulson@15234
   595
paulson@15234
   596
text{*The ``conjunction of implication'' lemmas produce two cases when the
paulson@15234
   597
comparison is a goal, but give four when the comparison is an assumption.*}
paulson@15234
   598
paulson@15234
   599
lemma mult_less_cancel_right:
paulson@15234
   600
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   601
  shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   602
by (insert mult_less_cancel_right_disj [of a c b], auto)
paulson@15234
   603
paulson@15234
   604
lemma mult_less_cancel_left:
paulson@15234
   605
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   606
  shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   607
by (insert mult_less_cancel_left_disj [of c a b], auto)
paulson@15234
   608
paulson@14268
   609
lemma mult_le_cancel_right:
obua@14738
   610
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   611
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
paulson@14268
   612
paulson@14268
   613
lemma mult_le_cancel_left:
obua@14738
   614
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   615
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
paulson@14268
   616
paulson@14268
   617
lemma mult_less_imp_less_left:
paulson@14341
   618
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
obua@14738
   619
      shows "a < (b::'a::ordered_semiring_strict)"
paulson@14377
   620
proof (rule ccontr)
paulson@14377
   621
  assume "~ a < b"
paulson@14377
   622
  hence "b \<le> a" by (simp add: linorder_not_less)
wenzelm@23389
   623
  hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
paulson@14377
   624
  with this and less show False 
paulson@14377
   625
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   626
qed
paulson@14268
   627
paulson@14268
   628
lemma mult_less_imp_less_right:
obua@14738
   629
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
obua@14738
   630
  shows "a < (b::'a::ordered_semiring_strict)"
obua@14738
   631
proof (rule ccontr)
obua@14738
   632
  assume "~ a < b"
obua@14738
   633
  hence "b \<le> a" by (simp add: linorder_not_less)
wenzelm@23389
   634
  hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono)
obua@14738
   635
  with this and less show False 
obua@14738
   636
    by (simp add: linorder_not_less [symmetric])
obua@14738
   637
qed  
paulson@14268
   638
paulson@14268
   639
text{*Cancellation of equalities with a common factor*}
paulson@14268
   640
lemma mult_cancel_right [simp]:
huffman@22990
   641
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   642
  shows "(a * c = b * c) = (c = 0 \<or> a = b)"
huffman@22990
   643
proof -
huffman@22990
   644
  have "(a * c = b * c) = ((a - b) * c = 0)"
nipkow@23477
   645
    by (simp add: ring_distribs)
huffman@22990
   646
  thus ?thesis
huffman@22990
   647
    by (simp add: disj_commute)
huffman@22990
   648
qed
paulson@14268
   649
paulson@14268
   650
lemma mult_cancel_left [simp]:
huffman@22990
   651
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   652
  shows "(c * a = c * b) = (c = 0 \<or> a = b)"
huffman@22990
   653
proof -
huffman@22990
   654
  have "(c * a = c * b) = (c * (a - b) = 0)"
nipkow@23477
   655
    by (simp add: ring_distribs)
huffman@22990
   656
  thus ?thesis
huffman@22990
   657
    by simp
huffman@22990
   658
qed
paulson@14268
   659
paulson@15234
   660
paulson@15234
   661
subsubsection{*Special Cancellation Simprules for Multiplication*}
paulson@15234
   662
paulson@15234
   663
text{*These also produce two cases when the comparison is a goal.*}
paulson@15234
   664
paulson@15234
   665
lemma mult_le_cancel_right1:
paulson@15234
   666
  fixes c :: "'a :: ordered_idom"
paulson@15234
   667
  shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   668
by (insert mult_le_cancel_right [of 1 c b], simp)
paulson@15234
   669
paulson@15234
   670
lemma mult_le_cancel_right2:
paulson@15234
   671
  fixes c :: "'a :: ordered_idom"
paulson@15234
   672
  shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   673
by (insert mult_le_cancel_right [of a c 1], simp)
paulson@15234
   674
paulson@15234
   675
lemma mult_le_cancel_left1:
paulson@15234
   676
  fixes c :: "'a :: ordered_idom"
paulson@15234
   677
  shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   678
by (insert mult_le_cancel_left [of c 1 b], simp)
paulson@15234
   679
paulson@15234
   680
lemma mult_le_cancel_left2:
paulson@15234
   681
  fixes c :: "'a :: ordered_idom"
paulson@15234
   682
  shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   683
by (insert mult_le_cancel_left [of c a 1], simp)
paulson@15234
   684
paulson@15234
   685
lemma mult_less_cancel_right1:
paulson@15234
   686
  fixes c :: "'a :: ordered_idom"
paulson@15234
   687
  shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   688
by (insert mult_less_cancel_right [of 1 c b], simp)
paulson@15234
   689
paulson@15234
   690
lemma mult_less_cancel_right2:
paulson@15234
   691
  fixes c :: "'a :: ordered_idom"
paulson@15234
   692
  shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   693
by (insert mult_less_cancel_right [of a c 1], simp)
paulson@15234
   694
paulson@15234
   695
lemma mult_less_cancel_left1:
paulson@15234
   696
  fixes c :: "'a :: ordered_idom"
paulson@15234
   697
  shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   698
by (insert mult_less_cancel_left [of c 1 b], simp)
paulson@15234
   699
paulson@15234
   700
lemma mult_less_cancel_left2:
paulson@15234
   701
  fixes c :: "'a :: ordered_idom"
paulson@15234
   702
  shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   703
by (insert mult_less_cancel_left [of c a 1], simp)
paulson@15234
   704
paulson@15234
   705
lemma mult_cancel_right1 [simp]:
huffman@22990
   706
  fixes c :: "'a :: dom"
paulson@15234
   707
  shows "(c = b*c) = (c = 0 | b=1)"
paulson@15234
   708
by (insert mult_cancel_right [of 1 c b], force)
paulson@15234
   709
paulson@15234
   710
lemma mult_cancel_right2 [simp]:
huffman@22990
   711
  fixes c :: "'a :: dom"
paulson@15234
   712
  shows "(a*c = c) = (c = 0 | a=1)"
paulson@15234
   713
by (insert mult_cancel_right [of a c 1], simp)
paulson@15234
   714
 
paulson@15234
   715
lemma mult_cancel_left1 [simp]:
huffman@22990
   716
  fixes c :: "'a :: dom"
paulson@15234
   717
  shows "(c = c*b) = (c = 0 | b=1)"
paulson@15234
   718
by (insert mult_cancel_left [of c 1 b], force)
paulson@15234
   719
paulson@15234
   720
lemma mult_cancel_left2 [simp]:
huffman@22990
   721
  fixes c :: "'a :: dom"
paulson@15234
   722
  shows "(c*a = c) = (c = 0 | a=1)"
paulson@15234
   723
by (insert mult_cancel_left [of c a 1], simp)
paulson@15234
   724
paulson@15234
   725
paulson@15234
   726
text{*Simprules for comparisons where common factors can be cancelled.*}
paulson@15234
   727
lemmas mult_compare_simps =
paulson@15234
   728
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
   729
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
   730
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
   731
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
   732
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
   733
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
   734
    mult_cancel_right mult_cancel_left
paulson@15234
   735
    mult_cancel_right1 mult_cancel_right2
paulson@15234
   736
    mult_cancel_left1 mult_cancel_left2
paulson@15234
   737
paulson@15234
   738
paulson@14265
   739
subsection {* Fields *}
paulson@14265
   740
paulson@14288
   741
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   742
proof
paulson@14288
   743
  assume neq: "b \<noteq> 0"
paulson@14288
   744
  {
paulson@14288
   745
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   746
    also assume "a / b = 1"
paulson@14288
   747
    finally show "a = b" by simp
paulson@14288
   748
  next
paulson@14288
   749
    assume "a = b"
paulson@14288
   750
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   751
  }
paulson@14288
   752
qed
paulson@14288
   753
paulson@14288
   754
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   755
by (simp add: divide_inverse)
paulson@14288
   756
nipkow@23398
   757
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   758
  by (simp add: divide_inverse)
paulson@14288
   759
paulson@14430
   760
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
paulson@14430
   761
by (simp add: divide_inverse)
paulson@14277
   762
paulson@15228
   763
lemma divide_self_if [simp]:
paulson@15228
   764
     "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
paulson@15228
   765
  by (simp add: divide_self)
paulson@15228
   766
paulson@14430
   767
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
paulson@14430
   768
by (simp add: divide_inverse)
paulson@14277
   769
paulson@14430
   770
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
paulson@14430
   771
by (simp add: divide_inverse)
paulson@14277
   772
paulson@14430
   773
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
nipkow@23477
   774
by (simp add: divide_inverse ring_distribs) 
paulson@14293
   775
paulson@14293
   776
paulson@14270
   777
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   778
      of an ordering.*}
huffman@20496
   779
lemma field_mult_eq_0_iff [simp]:
huffman@20496
   780
  "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
huffman@22990
   781
by simp
paulson@14270
   782
paulson@14268
   783
text{*Cancellation of equalities with a common factor*}
paulson@14268
   784
lemma field_mult_cancel_right_lemma:
huffman@20496
   785
      assumes cnz: "c \<noteq> (0::'a::division_ring)"
huffman@20496
   786
         and eq:  "a*c = b*c"
huffman@20496
   787
        shows "a=b"
paulson@14377
   788
proof -
paulson@14268
   789
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   790
    by (simp add: eq)
paulson@14268
   791
  thus "a=b"
paulson@14268
   792
    by (simp add: mult_assoc cnz)
paulson@14377
   793
qed
paulson@14268
   794
paulson@14348
   795
lemma field_mult_cancel_right [simp]:
huffman@20496
   796
     "(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   797
by simp
paulson@14268
   798
paulson@14348
   799
lemma field_mult_cancel_left [simp]:
huffman@20496
   800
     "(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   801
by simp
paulson@14268
   802
huffman@20496
   803
lemma nonzero_imp_inverse_nonzero:
huffman@20496
   804
  "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
paulson@14377
   805
proof
paulson@14268
   806
  assume ianz: "inverse a = 0"
paulson@14268
   807
  assume "a \<noteq> 0"
paulson@14268
   808
  hence "1 = a * inverse a" by simp
paulson@14268
   809
  also have "... = 0" by (simp add: ianz)
huffman@20496
   810
  finally have "1 = (0::'a::division_ring)" .
paulson@14268
   811
  thus False by (simp add: eq_commute)
paulson@14377
   812
qed
paulson@14268
   813
paulson@14277
   814
paulson@14277
   815
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   816
huffman@20496
   817
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   818
apply (rule ccontr) 
paulson@14268
   819
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   820
done
paulson@14268
   821
paulson@14268
   822
lemma inverse_nonzero_imp_nonzero:
huffman@20496
   823
   "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   824
apply (rule ccontr) 
paulson@14268
   825
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   826
done
paulson@14268
   827
paulson@14268
   828
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
   829
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
paulson@14268
   830
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   831
paulson@14268
   832
lemma nonzero_inverse_minus_eq:
huffman@20496
   833
      assumes [simp]: "a\<noteq>0"
huffman@20496
   834
      shows "inverse(-a) = -inverse(a::'a::division_ring)"
paulson@14377
   835
proof -
paulson@14377
   836
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   837
    by simp
paulson@14377
   838
  thus ?thesis 
paulson@14377
   839
    by (simp only: field_mult_cancel_left, simp)
paulson@14377
   840
qed
paulson@14268
   841
paulson@14268
   842
lemma inverse_minus_eq [simp]:
huffman@20496
   843
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
   844
proof cases
paulson@14377
   845
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   846
next
paulson@14377
   847
  assume "a\<noteq>0" 
paulson@14377
   848
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   849
qed
paulson@14268
   850
paulson@14268
   851
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   852
      assumes inveq: "inverse a = inverse b"
paulson@14269
   853
	  and anz:  "a \<noteq> 0"
paulson@14269
   854
	  and bnz:  "b \<noteq> 0"
huffman@20496
   855
	 shows "a = (b::'a::division_ring)"
paulson@14377
   856
proof -
paulson@14268
   857
  have "a * inverse b = a * inverse a"
paulson@14268
   858
    by (simp add: inveq)
paulson@14268
   859
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   860
    by simp
paulson@14268
   861
  thus "a = b"
paulson@14268
   862
    by (simp add: mult_assoc anz bnz)
paulson@14377
   863
qed
paulson@14268
   864
paulson@14268
   865
lemma inverse_eq_imp_eq:
huffman@20496
   866
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
   867
apply (cases "a=0 | b=0") 
paulson@14268
   868
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   869
              simp add: eq_commute [of "0::'a"])
paulson@14268
   870
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   871
done
paulson@14268
   872
paulson@14268
   873
lemma inverse_eq_iff_eq [simp]:
huffman@20496
   874
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
   875
by (force dest!: inverse_eq_imp_eq)
paulson@14268
   876
paulson@14270
   877
lemma nonzero_inverse_inverse_eq:
huffman@20496
   878
      assumes [simp]: "a \<noteq> 0"
huffman@20496
   879
      shows "inverse(inverse (a::'a::division_ring)) = a"
paulson@14270
   880
  proof -
paulson@14270
   881
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   882
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   883
  thus ?thesis
paulson@14270
   884
    by (simp add: mult_assoc)
paulson@14270
   885
  qed
paulson@14270
   886
paulson@14270
   887
lemma inverse_inverse_eq [simp]:
huffman@20496
   888
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
   889
  proof cases
paulson@14270
   890
    assume "a=0" thus ?thesis by simp
paulson@14270
   891
  next
paulson@14270
   892
    assume "a\<noteq>0" 
paulson@14270
   893
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   894
  qed
paulson@14270
   895
huffman@20496
   896
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
paulson@14270
   897
  proof -
huffman@20496
   898
  have "inverse 1 * 1 = (1::'a::division_ring)" 
paulson@14270
   899
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   900
  thus ?thesis  by simp
paulson@14270
   901
  qed
paulson@14270
   902
paulson@15077
   903
lemma inverse_unique: 
paulson@15077
   904
  assumes ab: "a*b = 1"
huffman@20496
   905
  shows "inverse a = (b::'a::division_ring)"
paulson@15077
   906
proof -
paulson@15077
   907
  have "a \<noteq> 0" using ab by auto
paulson@15077
   908
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
paulson@15077
   909
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
paulson@15077
   910
qed
paulson@15077
   911
paulson@14270
   912
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   913
      assumes anz: "a \<noteq> 0"
paulson@14270
   914
          and bnz: "b \<noteq> 0"
huffman@20496
   915
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
paulson@14270
   916
  proof -
paulson@14270
   917
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
paulson@14270
   918
    by (simp add: field_mult_eq_0_iff anz bnz)
paulson@14270
   919
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   920
    by (simp add: mult_assoc bnz)
paulson@14270
   921
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   922
    by simp
paulson@14270
   923
  thus ?thesis
paulson@14270
   924
    by (simp add: mult_assoc anz)
paulson@14270
   925
  qed
paulson@14270
   926
paulson@14270
   927
text{*This version builds in division by zero while also re-orienting
paulson@14270
   928
      the right-hand side.*}
paulson@14270
   929
lemma inverse_mult_distrib [simp]:
paulson@14270
   930
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   931
  proof cases
paulson@14270
   932
    assume "a \<noteq> 0 & b \<noteq> 0" 
haftmann@22993
   933
    thus ?thesis
haftmann@22993
   934
      by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   935
  next
paulson@14270
   936
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
haftmann@22993
   937
    thus ?thesis
haftmann@22993
   938
      by force
paulson@14270
   939
  qed
paulson@14270
   940
huffman@20496
   941
lemma division_ring_inverse_add:
huffman@20496
   942
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
huffman@20496
   943
   ==> inverse a + inverse b = inverse a * (a+b) * inverse b"
nipkow@23477
   944
by (simp add: ring_simps)
huffman@20496
   945
huffman@20496
   946
lemma division_ring_inverse_diff:
huffman@20496
   947
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
huffman@20496
   948
   ==> inverse a - inverse b = inverse a * (b-a) * inverse b"
nipkow@23477
   949
by (simp add: ring_simps)
huffman@20496
   950
paulson@14270
   951
text{*There is no slick version using division by zero.*}
paulson@14270
   952
lemma inverse_add:
nipkow@23477
   953
  "[|a \<noteq> 0;  b \<noteq> 0|]
nipkow@23477
   954
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
huffman@20496
   955
by (simp add: division_ring_inverse_add mult_ac)
paulson@14270
   956
paulson@14365
   957
lemma inverse_divide [simp]:
nipkow@23477
   958
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
nipkow@23477
   959
by (simp add: divide_inverse mult_commute)
paulson@14365
   960
wenzelm@23389
   961
avigad@16775
   962
subsection {* Calculations with fractions *}
avigad@16775
   963
nipkow@23413
   964
text{* There is a whole bunch of simp-rules just for class @{text
nipkow@23413
   965
field} but none for class @{text field} and @{text nonzero_divides}
nipkow@23413
   966
because the latter are covered by a simproc. *}
nipkow@23413
   967
nipkow@23413
   968
lemma nonzero_mult_divide_mult_cancel_left[simp]:
nipkow@23477
   969
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
   970
proof -
paulson@14277
   971
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
paulson@14277
   972
    by (simp add: field_mult_eq_0_iff divide_inverse 
paulson@14277
   973
                  nonzero_inverse_mult_distrib)
paulson@14277
   974
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
   975
    by (simp only: mult_ac)
paulson@14277
   976
  also have "... =  a * inverse b"
paulson@14277
   977
    by simp
paulson@14277
   978
    finally show ?thesis 
paulson@14277
   979
    by (simp add: divide_inverse)
paulson@14277
   980
qed
paulson@14277
   981
nipkow@23413
   982
lemma mult_divide_mult_cancel_left:
nipkow@23477
   983
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
   984
apply (cases "b = 0")
nipkow@23413
   985
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
paulson@14277
   986
done
paulson@14277
   987
nipkow@23413
   988
lemma nonzero_mult_divide_mult_cancel_right:
nipkow@23477
   989
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
nipkow@23413
   990
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
paulson@14321
   991
nipkow@23413
   992
lemma mult_divide_mult_cancel_right:
nipkow@23477
   993
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
   994
apply (cases "b = 0")
nipkow@23413
   995
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
paulson@14321
   996
done
nipkow@23413
   997
paulson@14284
   998
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
nipkow@23477
   999
by (simp add: divide_inverse)
paulson@14284
  1000
paulson@15234
  1001
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
  1002
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1003
paulson@14430
  1004
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
  1005
by (simp add: divide_inverse mult_ac)
paulson@14288
  1006
paulson@14288
  1007
lemma divide_divide_eq_right [simp]:
nipkow@23477
  1008
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1009
by (simp add: divide_inverse mult_ac)
paulson@14288
  1010
paulson@14288
  1011
lemma divide_divide_eq_left [simp]:
nipkow@23477
  1012
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1013
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1014
avigad@16775
  1015
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1016
    x / y + w / z = (x * z + w * y) / (y * z)"
nipkow@23477
  1017
apply (subgoal_tac "x / y = (x * z) / (y * z)")
nipkow@23477
  1018
apply (erule ssubst)
nipkow@23477
  1019
apply (subgoal_tac "w / z = (w * y) / (y * z)")
nipkow@23477
  1020
apply (erule ssubst)
nipkow@23477
  1021
apply (rule add_divide_distrib [THEN sym])
nipkow@23477
  1022
apply (subst mult_commute)
nipkow@23477
  1023
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
nipkow@23477
  1024
apply assumption
nipkow@23477
  1025
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
nipkow@23477
  1026
apply assumption
avigad@16775
  1027
done
paulson@14268
  1028
wenzelm@23389
  1029
paulson@15234
  1030
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1031
nipkow@23413
  1032
lemma mult_divide_mult_cancel_left_if[simp]:
nipkow@23477
  1033
fixes c :: "'a :: {field,division_by_zero}"
nipkow@23477
  1034
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
nipkow@23413
  1035
by (simp add: mult_divide_mult_cancel_left)
nipkow@23413
  1036
nipkow@23413
  1037
lemma nonzero_mult_divide_cancel_right[simp]:
nipkow@23413
  1038
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
nipkow@23413
  1039
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
nipkow@23413
  1040
nipkow@23413
  1041
lemma nonzero_mult_divide_cancel_left[simp]:
nipkow@23413
  1042
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
nipkow@23413
  1043
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
nipkow@23413
  1044
nipkow@23413
  1045
nipkow@23413
  1046
lemma nonzero_divide_mult_cancel_right[simp]:
nipkow@23413
  1047
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
nipkow@23413
  1048
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
nipkow@23413
  1049
nipkow@23413
  1050
lemma nonzero_divide_mult_cancel_left[simp]:
nipkow@23413
  1051
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
nipkow@23413
  1052
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
nipkow@23413
  1053
nipkow@23413
  1054
nipkow@23413
  1055
lemma nonzero_mult_divide_mult_cancel_left2[simp]:
nipkow@23477
  1056
  "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
nipkow@23413
  1057
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
nipkow@23413
  1058
nipkow@23413
  1059
lemma nonzero_mult_divide_mult_cancel_right2[simp]:
nipkow@23477
  1060
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
nipkow@23413
  1061
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
nipkow@23413
  1062
paulson@15234
  1063
paulson@14293
  1064
subsection {* Division and Unary Minus *}
paulson@14293
  1065
paulson@14293
  1066
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
  1067
by (simp add: divide_inverse minus_mult_left)
paulson@14293
  1068
paulson@14293
  1069
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
  1070
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
  1071
paulson@14293
  1072
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
  1073
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1074
paulson@14430
  1075
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
  1076
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
  1077
paulson@14293
  1078
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
  1079
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
  1080
paulson@14293
  1081
paulson@14293
  1082
text{*The effect is to extract signs from divisions*}
paulson@17085
  1083
lemmas divide_minus_left = minus_divide_left [symmetric]
paulson@17085
  1084
lemmas divide_minus_right = minus_divide_right [symmetric]
paulson@17085
  1085
declare divide_minus_left [simp]   divide_minus_right [simp]
paulson@14293
  1086
paulson@14387
  1087
text{*Also, extract signs from products*}
paulson@17085
  1088
lemmas mult_minus_left = minus_mult_left [symmetric]
paulson@17085
  1089
lemmas mult_minus_right = minus_mult_right [symmetric]
paulson@17085
  1090
declare mult_minus_left [simp]   mult_minus_right [simp]
paulson@14387
  1091
paulson@14293
  1092
lemma minus_divide_divide [simp]:
nipkow@23477
  1093
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1094
apply (cases "b=0", simp) 
paulson@14293
  1095
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1096
done
paulson@14293
  1097
paulson@14430
  1098
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
  1099
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1100
avigad@16775
  1101
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1102
    x / y - w / z = (x * z - w * y) / (y * z)"
nipkow@23477
  1103
apply (subst diff_def)+
nipkow@23477
  1104
apply (subst minus_divide_left)
nipkow@23477
  1105
apply (subst add_frac_eq)
nipkow@23477
  1106
apply simp_all
avigad@16775
  1107
done
paulson@14293
  1108
wenzelm@23389
  1109
paulson@14268
  1110
subsection {* Ordered Fields *}
paulson@14268
  1111
paulson@14277
  1112
lemma positive_imp_inverse_positive: 
paulson@14269
  1113
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
paulson@14268
  1114
  proof -
paulson@14268
  1115
  have "0 < a * inverse a" 
paulson@14268
  1116
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1117
  thus "0 < inverse a" 
paulson@14268
  1118
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
paulson@14268
  1119
  qed
paulson@14268
  1120
paulson@14277
  1121
lemma negative_imp_inverse_negative:
paulson@14268
  1122
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
paulson@14277
  1123
  by (insert positive_imp_inverse_positive [of "-a"], 
paulson@14268
  1124
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
paulson@14268
  1125
paulson@14268
  1126
lemma inverse_le_imp_le:
paulson@14269
  1127
      assumes invle: "inverse a \<le> inverse b"
paulson@14269
  1128
	  and apos:  "0 < a"
paulson@14269
  1129
	 shows "b \<le> (a::'a::ordered_field)"
paulson@14268
  1130
  proof (rule classical)
paulson@14268
  1131
  assume "~ b \<le> a"
paulson@14268
  1132
  hence "a < b"
paulson@14268
  1133
    by (simp add: linorder_not_le)
paulson@14268
  1134
  hence bpos: "0 < b"
paulson@14268
  1135
    by (blast intro: apos order_less_trans)
paulson@14268
  1136
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1137
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1138
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1139
    by (simp add: bpos order_less_imp_le mult_right_mono)
paulson@14268
  1140
  thus "b \<le> a"
paulson@14268
  1141
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
paulson@14268
  1142
  qed
paulson@14268
  1143
paulson@14277
  1144
lemma inverse_positive_imp_positive:
wenzelm@23389
  1145
  assumes inv_gt_0: "0 < inverse a"
wenzelm@23389
  1146
    and nz: "a \<noteq> 0"
wenzelm@23389
  1147
  shows "0 < (a::'a::ordered_field)"
wenzelm@23389
  1148
proof -
paulson@14277
  1149
  have "0 < inverse (inverse a)"
wenzelm@23389
  1150
    using inv_gt_0 by (rule positive_imp_inverse_positive)
paulson@14277
  1151
  thus "0 < a"
wenzelm@23389
  1152
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1153
qed
paulson@14277
  1154
paulson@14277
  1155
lemma inverse_positive_iff_positive [simp]:
paulson@14277
  1156
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1157
apply (cases "a = 0", simp)
paulson@14277
  1158
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1159
done
paulson@14277
  1160
paulson@14277
  1161
lemma inverse_negative_imp_negative:
wenzelm@23389
  1162
  assumes inv_less_0: "inverse a < 0"
wenzelm@23389
  1163
    and nz:  "a \<noteq> 0"
wenzelm@23389
  1164
  shows "a < (0::'a::ordered_field)"
wenzelm@23389
  1165
proof -
paulson@14277
  1166
  have "inverse (inverse a) < 0"
wenzelm@23389
  1167
    using inv_less_0 by (rule negative_imp_inverse_negative)
paulson@14277
  1168
  thus "a < 0"
wenzelm@23389
  1169
    using nz by (simp add: nonzero_inverse_inverse_eq)
wenzelm@23389
  1170
qed
paulson@14277
  1171
paulson@14277
  1172
lemma inverse_negative_iff_negative [simp]:
paulson@14277
  1173
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1174
apply (cases "a = 0", simp)
paulson@14277
  1175
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1176
done
paulson@14277
  1177
paulson@14277
  1178
lemma inverse_nonnegative_iff_nonnegative [simp]:
paulson@14277
  1179
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1180
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1181
paulson@14277
  1182
lemma inverse_nonpositive_iff_nonpositive [simp]:
paulson@14277
  1183
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1184
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1185
chaieb@23406
  1186
lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
chaieb@23406
  1187
proof
chaieb@23406
  1188
  fix x::'a
chaieb@23406
  1189
  have m1: "- (1::'a) < 0" by simp
chaieb@23406
  1190
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1191
  have "(- 1) + x < x" by simp
chaieb@23406
  1192
  thus "\<exists>y. y < x" by blast
chaieb@23406
  1193
qed
chaieb@23406
  1194
chaieb@23406
  1195
lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
chaieb@23406
  1196
proof
chaieb@23406
  1197
  fix x::'a
chaieb@23406
  1198
  have m1: " (1::'a) > 0" by simp
chaieb@23406
  1199
  from add_strict_right_mono[OF m1, where c=x] 
chaieb@23406
  1200
  have "1 + x > x" by simp
chaieb@23406
  1201
  thus "\<exists>y. y > x" by blast
chaieb@23406
  1202
qed
paulson@14277
  1203
paulson@14277
  1204
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1205
paulson@14268
  1206
lemma less_imp_inverse_less:
paulson@14269
  1207
      assumes less: "a < b"
paulson@14269
  1208
	  and apos:  "0 < a"
paulson@14269
  1209
	shows "inverse b < inverse (a::'a::ordered_field)"
paulson@14268
  1210
  proof (rule ccontr)
paulson@14268
  1211
  assume "~ inverse b < inverse a"
paulson@14268
  1212
  hence "inverse a \<le> inverse b"
paulson@14268
  1213
    by (simp add: linorder_not_less)
paulson@14268
  1214
  hence "~ (a < b)"
paulson@14268
  1215
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
  1216
  thus False
paulson@14268
  1217
    by (rule notE [OF _ less])
paulson@14268
  1218
  qed
paulson@14268
  1219
paulson@14268
  1220
lemma inverse_less_imp_less:
paulson@14268
  1221
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1222
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1223
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1224
done
paulson@14268
  1225
paulson@14268
  1226
text{*Both premises are essential. Consider -1 and 1.*}
paulson@14268
  1227
lemma inverse_less_iff_less [simp]:
paulson@14268
  1228
     "[|0 < a; 0 < b|] 
paulson@14268
  1229
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1230
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1231
paulson@14268
  1232
lemma le_imp_inverse_le:
paulson@14268
  1233
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1234
  by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1235
paulson@14268
  1236
lemma inverse_le_iff_le [simp]:
paulson@14268
  1237
     "[|0 < a; 0 < b|] 
paulson@14268
  1238
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1239
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1240
paulson@14268
  1241
paulson@14268
  1242
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1243
case is trivial, since inverse preserves signs.*}
paulson@14268
  1244
lemma inverse_le_imp_le_neg:
paulson@14268
  1245
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
paulson@14268
  1246
  apply (rule classical) 
paulson@14268
  1247
  apply (subgoal_tac "a < 0") 
paulson@14268
  1248
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
paulson@14268
  1249
  apply (insert inverse_le_imp_le [of "-b" "-a"])
paulson@14268
  1250
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1251
  done
paulson@14268
  1252
paulson@14268
  1253
lemma less_imp_inverse_less_neg:
paulson@14268
  1254
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
paulson@14268
  1255
  apply (subgoal_tac "a < 0") 
paulson@14268
  1256
   prefer 2 apply (blast intro: order_less_trans) 
paulson@14268
  1257
  apply (insert less_imp_inverse_less [of "-b" "-a"])
paulson@14268
  1258
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1259
  done
paulson@14268
  1260
paulson@14268
  1261
lemma inverse_less_imp_less_neg:
paulson@14268
  1262
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1263
  apply (rule classical) 
paulson@14268
  1264
  apply (subgoal_tac "a < 0") 
paulson@14268
  1265
   prefer 2
paulson@14268
  1266
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
paulson@14268
  1267
  apply (insert inverse_less_imp_less [of "-b" "-a"])
paulson@14268
  1268
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1269
  done
paulson@14268
  1270
paulson@14268
  1271
lemma inverse_less_iff_less_neg [simp]:
paulson@14268
  1272
     "[|a < 0; b < 0|] 
paulson@14268
  1273
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1274
  apply (insert inverse_less_iff_less [of "-b" "-a"])
paulson@14268
  1275
  apply (simp del: inverse_less_iff_less 
paulson@14268
  1276
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1277
  done
paulson@14268
  1278
paulson@14268
  1279
lemma le_imp_inverse_le_neg:
paulson@14268
  1280
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1281
  by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1282
paulson@14268
  1283
lemma inverse_le_iff_le_neg [simp]:
paulson@14268
  1284
     "[|a < 0; b < 0|] 
paulson@14268
  1285
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1286
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1287
paulson@14277
  1288
paulson@14365
  1289
subsection{*Inverses and the Number One*}
paulson@14365
  1290
paulson@14365
  1291
lemma one_less_inverse_iff:
paulson@14365
  1292
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
paulson@14365
  1293
  assume "0 < x"
paulson@14365
  1294
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1295
    show ?thesis by simp
paulson@14365
  1296
next
paulson@14365
  1297
  assume notless: "~ (0 < x)"
paulson@14365
  1298
  have "~ (1 < inverse x)"
paulson@14365
  1299
  proof
paulson@14365
  1300
    assume "1 < inverse x"
paulson@14365
  1301
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1302
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1303
    finally show False by auto
paulson@14365
  1304
  qed
paulson@14365
  1305
  with notless show ?thesis by simp
paulson@14365
  1306
qed
paulson@14365
  1307
paulson@14365
  1308
lemma inverse_eq_1_iff [simp]:
paulson@14365
  1309
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1310
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1311
paulson@14365
  1312
lemma one_le_inverse_iff:
paulson@14365
  1313
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1314
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1315
                    eq_commute [of 1]) 
paulson@14365
  1316
paulson@14365
  1317
lemma inverse_less_1_iff:
paulson@14365
  1318
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1319
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1320
paulson@14365
  1321
lemma inverse_le_1_iff:
paulson@14365
  1322
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1323
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1324
wenzelm@23389
  1325
paulson@14288
  1326
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1327
paulson@14288
  1328
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1329
proof -
paulson@14288
  1330
  assume less: "0<c"
paulson@14288
  1331
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1332
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1333
  also have "... = (a*c \<le> b)"
paulson@14288
  1334
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1335
  finally show ?thesis .
paulson@14288
  1336
qed
paulson@14288
  1337
paulson@14288
  1338
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1339
proof -
paulson@14288
  1340
  assume less: "c<0"
paulson@14288
  1341
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1342
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1343
  also have "... = (b \<le> a*c)"
paulson@14288
  1344
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1345
  finally show ?thesis .
paulson@14288
  1346
qed
paulson@14288
  1347
paulson@14288
  1348
lemma le_divide_eq:
paulson@14288
  1349
  "(a \<le> b/c) = 
paulson@14288
  1350
   (if 0 < c then a*c \<le> b
paulson@14288
  1351
             else if c < 0 then b \<le> a*c
paulson@14288
  1352
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1353
apply (cases "c=0", simp) 
paulson@14288
  1354
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1355
done
paulson@14288
  1356
paulson@14288
  1357
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1358
proof -
paulson@14288
  1359
  assume less: "0<c"
paulson@14288
  1360
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1361
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1362
  also have "... = (b \<le> a*c)"
paulson@14288
  1363
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1364
  finally show ?thesis .
paulson@14288
  1365
qed
paulson@14288
  1366
paulson@14288
  1367
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1368
proof -
paulson@14288
  1369
  assume less: "c<0"
paulson@14288
  1370
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1371
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1372
  also have "... = (a*c \<le> b)"
paulson@14288
  1373
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1374
  finally show ?thesis .
paulson@14288
  1375
qed
paulson@14288
  1376
paulson@14288
  1377
lemma divide_le_eq:
paulson@14288
  1378
  "(b/c \<le> a) = 
paulson@14288
  1379
   (if 0 < c then b \<le> a*c
paulson@14288
  1380
             else if c < 0 then a*c \<le> b
paulson@14288
  1381
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1382
apply (cases "c=0", simp) 
paulson@14288
  1383
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1384
done
paulson@14288
  1385
paulson@14288
  1386
lemma pos_less_divide_eq:
paulson@14288
  1387
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1388
proof -
paulson@14288
  1389
  assume less: "0<c"
paulson@14288
  1390
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1391
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1392
  also have "... = (a*c < b)"
paulson@14288
  1393
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1394
  finally show ?thesis .
paulson@14288
  1395
qed
paulson@14288
  1396
paulson@14288
  1397
lemma neg_less_divide_eq:
paulson@14288
  1398
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1399
proof -
paulson@14288
  1400
  assume less: "c<0"
paulson@14288
  1401
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1402
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1403
  also have "... = (b < a*c)"
paulson@14288
  1404
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1405
  finally show ?thesis .
paulson@14288
  1406
qed
paulson@14288
  1407
paulson@14288
  1408
lemma less_divide_eq:
paulson@14288
  1409
  "(a < b/c) = 
paulson@14288
  1410
   (if 0 < c then a*c < b
paulson@14288
  1411
             else if c < 0 then b < a*c
paulson@14288
  1412
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1413
apply (cases "c=0", simp) 
paulson@14288
  1414
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1415
done
paulson@14288
  1416
paulson@14288
  1417
lemma pos_divide_less_eq:
paulson@14288
  1418
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1419
proof -
paulson@14288
  1420
  assume less: "0<c"
paulson@14288
  1421
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1422
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1423
  also have "... = (b < a*c)"
paulson@14288
  1424
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1425
  finally show ?thesis .
paulson@14288
  1426
qed
paulson@14288
  1427
paulson@14288
  1428
lemma neg_divide_less_eq:
paulson@14288
  1429
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1430
proof -
paulson@14288
  1431
  assume less: "c<0"
paulson@14288
  1432
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1433
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1434
  also have "... = (a*c < b)"
paulson@14288
  1435
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1436
  finally show ?thesis .
paulson@14288
  1437
qed
paulson@14288
  1438
paulson@14288
  1439
lemma divide_less_eq:
paulson@14288
  1440
  "(b/c < a) = 
paulson@14288
  1441
   (if 0 < c then b < a*c
paulson@14288
  1442
             else if c < 0 then a*c < b
paulson@14288
  1443
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1444
apply (cases "c=0", simp) 
paulson@14288
  1445
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1446
done
paulson@14288
  1447
paulson@14288
  1448
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
paulson@14288
  1449
proof -
paulson@14288
  1450
  assume [simp]: "c\<noteq>0"
paulson@14288
  1451
  have "(a = b/c) = (a*c = (b/c)*c)"
paulson@14288
  1452
    by (simp add: field_mult_cancel_right)
paulson@14288
  1453
  also have "... = (a*c = b)"
paulson@14288
  1454
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1455
  finally show ?thesis .
paulson@14288
  1456
qed
paulson@14288
  1457
paulson@14288
  1458
lemma eq_divide_eq:
paulson@14288
  1459
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
paulson@14288
  1460
by (simp add: nonzero_eq_divide_eq) 
paulson@14288
  1461
paulson@14288
  1462
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
paulson@14288
  1463
proof -
paulson@14288
  1464
  assume [simp]: "c\<noteq>0"
paulson@14288
  1465
  have "(b/c = a) = ((b/c)*c = a*c)"
paulson@14288
  1466
    by (simp add: field_mult_cancel_right)
paulson@14288
  1467
  also have "... = (b = a*c)"
paulson@14288
  1468
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1469
  finally show ?thesis .
paulson@14288
  1470
qed
paulson@14288
  1471
paulson@14288
  1472
lemma divide_eq_eq:
paulson@14288
  1473
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
paulson@14288
  1474
by (force simp add: nonzero_divide_eq_eq) 
paulson@14288
  1475
avigad@16775
  1476
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
avigad@16775
  1477
    b = a * c ==> b / c = a"
avigad@16775
  1478
  by (subst divide_eq_eq, simp)
avigad@16775
  1479
avigad@16775
  1480
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
avigad@16775
  1481
    a * c = b ==> a = b / c"
avigad@16775
  1482
  by (subst eq_divide_eq, simp)
avigad@16775
  1483
avigad@16775
  1484
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1485
    (x / y = w / z) = (x * z = w * y)"
avigad@16775
  1486
  apply (subst nonzero_eq_divide_eq)
avigad@16775
  1487
  apply assumption
avigad@16775
  1488
  apply (subst times_divide_eq_left)
avigad@16775
  1489
  apply (erule nonzero_divide_eq_eq) 
avigad@16775
  1490
done
avigad@16775
  1491
wenzelm@23389
  1492
avigad@16775
  1493
subsection{*Division and Signs*}
avigad@16775
  1494
avigad@16775
  1495
lemma zero_less_divide_iff:
avigad@16775
  1496
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
  1497
by (simp add: divide_inverse zero_less_mult_iff)
avigad@16775
  1498
avigad@16775
  1499
lemma divide_less_0_iff:
avigad@16775
  1500
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
avigad@16775
  1501
      (0 < a & b < 0 | a < 0 & 0 < b)"
avigad@16775
  1502
by (simp add: divide_inverse mult_less_0_iff)
avigad@16775
  1503
avigad@16775
  1504
lemma zero_le_divide_iff:
avigad@16775
  1505
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
avigad@16775
  1506
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
avigad@16775
  1507
by (simp add: divide_inverse zero_le_mult_iff)
avigad@16775
  1508
avigad@16775
  1509
lemma divide_le_0_iff:
avigad@16775
  1510
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
avigad@16775
  1511
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
avigad@16775
  1512
by (simp add: divide_inverse mult_le_0_iff)
avigad@16775
  1513
avigad@16775
  1514
lemma divide_eq_0_iff [simp]:
avigad@16775
  1515
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
avigad@16775
  1516
by (simp add: divide_inverse field_mult_eq_0_iff)
avigad@16775
  1517
avigad@16775
  1518
lemma divide_pos_pos: "0 < (x::'a::ordered_field) ==> 
avigad@16775
  1519
    0 < y ==> 0 < x / y"
avigad@16775
  1520
  apply (subst pos_less_divide_eq)
avigad@16775
  1521
  apply assumption
avigad@16775
  1522
  apply simp
avigad@16775
  1523
done
avigad@16775
  1524
avigad@16775
  1525
lemma divide_nonneg_pos: "0 <= (x::'a::ordered_field) ==> 0 < y ==> 
avigad@16775
  1526
    0 <= x / y"
avigad@16775
  1527
  apply (subst pos_le_divide_eq)
avigad@16775
  1528
  apply assumption
avigad@16775
  1529
  apply simp
avigad@16775
  1530
done
avigad@16775
  1531
avigad@16775
  1532
lemma divide_neg_pos: "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
avigad@16775
  1533
  apply (subst pos_divide_less_eq)
avigad@16775
  1534
  apply assumption
avigad@16775
  1535
  apply simp
avigad@16775
  1536
done
avigad@16775
  1537
avigad@16775
  1538
lemma divide_nonpos_pos: "(x::'a::ordered_field) <= 0 ==> 
avigad@16775
  1539
    0 < y ==> x / y <= 0"
avigad@16775
  1540
  apply (subst pos_divide_le_eq)
avigad@16775
  1541
  apply assumption
avigad@16775
  1542
  apply simp
avigad@16775
  1543
done
avigad@16775
  1544
avigad@16775
  1545
lemma divide_pos_neg: "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
avigad@16775
  1546
  apply (subst neg_divide_less_eq)
avigad@16775
  1547
  apply assumption
avigad@16775
  1548
  apply simp
avigad@16775
  1549
done
avigad@16775
  1550
avigad@16775
  1551
lemma divide_nonneg_neg: "0 <= (x::'a::ordered_field) ==> 
avigad@16775
  1552
    y < 0 ==> x / y <= 0"
avigad@16775
  1553
  apply (subst neg_divide_le_eq)
avigad@16775
  1554
  apply assumption
avigad@16775
  1555
  apply simp
avigad@16775
  1556
done
avigad@16775
  1557
avigad@16775
  1558
lemma divide_neg_neg: "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
avigad@16775
  1559
  apply (subst neg_less_divide_eq)
avigad@16775
  1560
  apply assumption
avigad@16775
  1561
  apply simp
avigad@16775
  1562
done
avigad@16775
  1563
avigad@16775
  1564
lemma divide_nonpos_neg: "(x::'a::ordered_field) <= 0 ==> y < 0 ==> 
avigad@16775
  1565
    0 <= x / y"
avigad@16775
  1566
  apply (subst neg_le_divide_eq)
avigad@16775
  1567
  apply assumption
avigad@16775
  1568
  apply simp
avigad@16775
  1569
done
paulson@15234
  1570
wenzelm@23389
  1571
paulson@14288
  1572
subsection{*Cancellation Laws for Division*}
paulson@14288
  1573
paulson@14288
  1574
lemma divide_cancel_right [simp]:
paulson@14288
  1575
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
haftmann@21328
  1576
apply (cases "c=0", simp) 
paulson@14430
  1577
apply (simp add: divide_inverse field_mult_cancel_right) 
paulson@14288
  1578
done
paulson@14288
  1579
paulson@14288
  1580
lemma divide_cancel_left [simp]:
paulson@14288
  1581
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
haftmann@21328
  1582
apply (cases "c=0", simp) 
paulson@14430
  1583
apply (simp add: divide_inverse field_mult_cancel_left) 
paulson@14288
  1584
done
paulson@14288
  1585
wenzelm@23389
  1586
paulson@14353
  1587
subsection {* Division and the Number One *}
paulson@14353
  1588
paulson@14353
  1589
text{*Simplify expressions equated with 1*}
paulson@14353
  1590
lemma divide_eq_1_iff [simp]:
paulson@14353
  1591
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
haftmann@21328
  1592
apply (cases "b=0", simp) 
paulson@14353
  1593
apply (simp add: right_inverse_eq) 
paulson@14353
  1594
done
paulson@14353
  1595
paulson@14353
  1596
lemma one_eq_divide_iff [simp]:
paulson@14353
  1597
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1598
by (simp add: eq_commute [of 1])  
paulson@14353
  1599
paulson@14353
  1600
lemma zero_eq_1_divide_iff [simp]:
paulson@14353
  1601
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
haftmann@21328
  1602
apply (cases "a=0", simp) 
paulson@14353
  1603
apply (auto simp add: nonzero_eq_divide_eq) 
paulson@14353
  1604
done
paulson@14353
  1605
paulson@14353
  1606
lemma one_divide_eq_0_iff [simp]:
paulson@14353
  1607
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
haftmann@21328
  1608
apply (cases "a=0", simp) 
paulson@14353
  1609
apply (insert zero_neq_one [THEN not_sym]) 
paulson@14353
  1610
apply (auto simp add: nonzero_divide_eq_eq) 
paulson@14353
  1611
done
paulson@14353
  1612
paulson@14353
  1613
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1614
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1615
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1616
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1617
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1618
paulson@17085
  1619
declare zero_less_divide_1_iff [simp]
paulson@17085
  1620
declare divide_less_0_1_iff [simp]
paulson@17085
  1621
declare zero_le_divide_1_iff [simp]
paulson@17085
  1622
declare divide_le_0_1_iff [simp]
paulson@14353
  1623
wenzelm@23389
  1624
paulson@14293
  1625
subsection {* Ordering Rules for Division *}
paulson@14293
  1626
paulson@14293
  1627
lemma divide_strict_right_mono:
paulson@14293
  1628
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1629
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
paulson@14293
  1630
              positive_imp_inverse_positive) 
paulson@14293
  1631
paulson@14293
  1632
lemma divide_right_mono:
paulson@14293
  1633
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
paulson@14293
  1634
  by (force simp add: divide_strict_right_mono order_le_less) 
paulson@14293
  1635
avigad@16775
  1636
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1637
    ==> c <= 0 ==> b / c <= a / c"
avigad@16775
  1638
  apply (drule divide_right_mono [of _ _ "- c"])
avigad@16775
  1639
  apply auto
avigad@16775
  1640
done
avigad@16775
  1641
avigad@16775
  1642
lemma divide_strict_right_mono_neg:
avigad@16775
  1643
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
avigad@16775
  1644
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 
avigad@16775
  1645
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
avigad@16775
  1646
done
paulson@14293
  1647
paulson@14293
  1648
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1649
      have the same sign*}
paulson@14293
  1650
lemma divide_strict_left_mono:
paulson@14293
  1651
       "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1652
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 
paulson@14293
  1653
      order_less_imp_not_eq order_less_imp_not_eq2  
paulson@14293
  1654
      less_imp_inverse_less less_imp_inverse_less_neg) 
paulson@14293
  1655
paulson@14293
  1656
lemma divide_left_mono:
paulson@14293
  1657
     "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
paulson@14293
  1658
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1659
   prefer 2 
paulson@14293
  1660
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
haftmann@21328
  1661
  apply (cases "c=0", simp add: divide_inverse)
paulson@14293
  1662
  apply (force simp add: divide_strict_left_mono order_le_less) 
paulson@14293
  1663
  done
paulson@14293
  1664
avigad@16775
  1665
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1666
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1667
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1668
  apply (auto simp add: mult_commute)
avigad@16775
  1669
done
avigad@16775
  1670
paulson@14293
  1671
lemma divide_strict_left_mono_neg:
paulson@14293
  1672
     "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1673
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1674
   prefer 2 
paulson@14293
  1675
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1676
  apply (drule divide_strict_left_mono [of _ _ "-c"]) 
paulson@14293
  1677
   apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 
paulson@14293
  1678
  done
paulson@14293
  1679
avigad@16775
  1680
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1681
avigad@16775
  1682
lemma le_divide_eq_1:
avigad@16775
  1683
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1684
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1685
by (auto simp add: le_divide_eq)
avigad@16775
  1686
avigad@16775
  1687
lemma divide_le_eq_1:
avigad@16775
  1688
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1689
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1690
by (auto simp add: divide_le_eq)
avigad@16775
  1691
avigad@16775
  1692
lemma less_divide_eq_1:
avigad@16775
  1693
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1694
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1695
by (auto simp add: less_divide_eq)
avigad@16775
  1696
avigad@16775
  1697
lemma divide_less_eq_1:
avigad@16775
  1698
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1699
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1700
by (auto simp add: divide_less_eq)
avigad@16775
  1701
wenzelm@23389
  1702
avigad@16775
  1703
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1704
avigad@16775
  1705
lemma le_divide_eq_1_pos [simp]:
avigad@16775
  1706
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1707
  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1708
by (auto simp add: le_divide_eq)
avigad@16775
  1709
avigad@16775
  1710
lemma le_divide_eq_1_neg [simp]:
avigad@16775
  1711
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1712
  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1713
by (auto simp add: le_divide_eq)
avigad@16775
  1714
avigad@16775
  1715
lemma divide_le_eq_1_pos [simp]:
avigad@16775
  1716
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1717
  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1718
by (auto simp add: divide_le_eq)
avigad@16775
  1719
avigad@16775
  1720
lemma divide_le_eq_1_neg [simp]:
avigad@16775
  1721
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1722
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1723
by (auto simp add: divide_le_eq)
avigad@16775
  1724
avigad@16775
  1725
lemma less_divide_eq_1_pos [simp]:
avigad@16775
  1726
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1727
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1728
by (auto simp add: less_divide_eq)
avigad@16775
  1729
avigad@16775
  1730
lemma less_divide_eq_1_neg [simp]:
avigad@16775
  1731
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1732
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1733
by (auto simp add: less_divide_eq)
avigad@16775
  1734
avigad@16775
  1735
lemma divide_less_eq_1_pos [simp]:
avigad@16775
  1736
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1737
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1738
by (auto simp add: divide_less_eq)
paulson@18649
  1739
paulson@18649
  1740
lemma divide_less_eq_1_neg [simp]:
paulson@18649
  1741
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1742
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1743
by (auto simp add: divide_less_eq)
avigad@16775
  1744
avigad@16775
  1745
lemma eq_divide_eq_1 [simp]:
avigad@16775
  1746
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1747
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1748
by (auto simp add: eq_divide_eq)
avigad@16775
  1749
avigad@16775
  1750
lemma divide_eq_eq_1 [simp]:
avigad@16775
  1751
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1752
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1753
by (auto simp add: divide_eq_eq)
avigad@16775
  1754
wenzelm@23389
  1755
avigad@16775
  1756
subsection {* Reasoning about inequalities with division *}
avigad@16775
  1757
avigad@16775
  1758
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1759
    ==> x * y <= x"
avigad@16775
  1760
  by (auto simp add: mult_compare_simps);
avigad@16775
  1761
avigad@16775
  1762
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1763
    ==> y * x <= x"
avigad@16775
  1764
  by (auto simp add: mult_compare_simps);
avigad@16775
  1765
avigad@16775
  1766
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  1767
    x / y <= z";
avigad@16775
  1768
  by (subst pos_divide_le_eq, assumption+);
avigad@16775
  1769
avigad@16775
  1770
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
avigad@16775
  1771
    z <= x / y";
avigad@16775
  1772
  by (subst pos_le_divide_eq, assumption+)
avigad@16775
  1773
avigad@16775
  1774
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  1775
    x / y < z"
avigad@16775
  1776
  by (subst pos_divide_less_eq, assumption+)
avigad@16775
  1777
avigad@16775
  1778
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  1779
    z < x / y"
avigad@16775
  1780
  by (subst pos_less_divide_eq, assumption+)
avigad@16775
  1781
avigad@16775
  1782
lemma frac_le: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1783
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
avigad@16775
  1784
  apply (rule mult_imp_div_pos_le)
avigad@16775
  1785
  apply simp;
avigad@16775
  1786
  apply (subst times_divide_eq_left);
avigad@16775
  1787
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  1788
  apply (rule mult_mono)
avigad@16775
  1789
  apply simp_all
paulson@14293
  1790
done
paulson@14293
  1791
avigad@16775
  1792
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1793
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  1794
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1795
  apply simp;
avigad@16775
  1796
  apply (subst times_divide_eq_left);
avigad@16775
  1797
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1798
  apply (erule mult_less_le_imp_less)
avigad@16775
  1799
  apply simp_all
avigad@16775
  1800
done
avigad@16775
  1801
avigad@16775
  1802
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  1803
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  1804
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1805
  apply simp_all
avigad@16775
  1806
  apply (subst times_divide_eq_left);
avigad@16775
  1807
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1808
  apply (erule mult_le_less_imp_less)
avigad@16775
  1809
  apply simp_all
avigad@16775
  1810
done
avigad@16775
  1811
avigad@16775
  1812
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
avigad@16775
  1813
avigad@16775
  1814
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  1815
  Their effect is to gather terms into one big fraction, like
avigad@16775
  1816
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  1817
  seem to need them.*}
avigad@16775
  1818
avigad@16775
  1819
declare times_divide_eq [simp]
paulson@14293
  1820
wenzelm@23389
  1821
paulson@14293
  1822
subsection {* Ordered Fields are Dense *}
paulson@14293
  1823
obua@14738
  1824
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
paulson@14293
  1825
proof -
obua@14738
  1826
  have "a+0 < (a+1::'a::ordered_semidom)"
paulson@14365
  1827
    by (blast intro: zero_less_one add_strict_left_mono) 
paulson@14293
  1828
  thus ?thesis by simp
paulson@14293
  1829
qed
paulson@14293
  1830
obua@14738
  1831
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
paulson@14365
  1832
  by (blast intro: order_less_trans zero_less_one less_add_one) 
paulson@14365
  1833
paulson@14293
  1834
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
nipkow@23477
  1835
by (simp add: zero_less_two pos_less_divide_eq ring_distribs) 
paulson@14293
  1836
paulson@14293
  1837
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
nipkow@23477
  1838
by (simp add: zero_less_two pos_divide_less_eq ring_distribs) 
paulson@14293
  1839
paulson@14293
  1840
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
paulson@14293
  1841
by (blast intro!: less_half_sum gt_half_sum)
paulson@14293
  1842
paulson@15234
  1843
paulson@14293
  1844
subsection {* Absolute Value *}
paulson@14293
  1845
obua@14738
  1846
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
nipkow@23477
  1847
by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
paulson@14294
  1848
obua@14738
  1849
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  1850
proof -
obua@14738
  1851
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  1852
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  1853
  have a: "(abs a) * (abs b) = ?x"
nipkow@23477
  1854
    by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
obua@14738
  1855
  {
obua@14738
  1856
    fix u v :: 'a
paulson@15481
  1857
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  1858
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  1859
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  1860
      apply (subst prts[of u], subst prts[of v])
nipkow@23477
  1861
      apply (simp add: ring_simps) 
obua@14738
  1862
      done
obua@14738
  1863
  }
obua@14738
  1864
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  1865
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  1866
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  1867
  have xy: "- ?x <= ?y"
obua@14754
  1868
    apply (simp)
obua@14754
  1869
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  1870
    apply (rule addm2)
avigad@16775
  1871
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  1872
    apply (rule addm)
avigad@16775
  1873
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  1874
    done
obua@14738
  1875
  have yx: "?y <= ?x"
nipkow@16568
  1876
    apply (simp add:diff_def)
obua@14754
  1877
    apply (rule_tac y=0 in order_trans)
avigad@16775
  1878
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  1879
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  1880
    done
obua@14738
  1881
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  1882
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  1883
  show ?thesis
obua@14738
  1884
    apply (rule abs_leI)
obua@14738
  1885
    apply (simp add: i1)
obua@14738
  1886
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  1887
    done
obua@14738
  1888
qed
paulson@14294
  1889
obua@14738
  1890
lemma abs_eq_mult: 
obua@14738
  1891
  assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
obua@14738
  1892
  shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
obua@14738
  1893
proof -
obua@14738
  1894
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  1895
    apply (auto)    
obua@14738
  1896
    apply (rule_tac split_mult_pos_le)
obua@14738
  1897
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  1898
    apply (simp)
obua@14738
  1899
    apply (rule_tac split_mult_neg_le)
obua@14738
  1900
    apply (insert prems)
obua@14738
  1901
    apply (blast)
obua@14738
  1902
    done
obua@14738
  1903
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  1904
    by (simp add: prts[symmetric])
obua@14738
  1905
  show ?thesis
obua@14738
  1906
  proof cases
obua@14738
  1907
    assume "0 <= a * b"
obua@14738
  1908
    then show ?thesis
obua@14738
  1909
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1910
      apply (insert prems)
obua@14754
  1911
      apply (auto simp add: 
nipkow@23477
  1912
	ring_simps 
obua@14754
  1913
	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
nipkow@15197
  1914
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
avigad@16775
  1915
	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
avigad@16775
  1916
	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  1917
      done
obua@14738
  1918
  next
obua@14738
  1919
    assume "~(0 <= a*b)"
obua@14738
  1920
    with s have "a*b <= 0" by simp
obua@14738
  1921
    then show ?thesis
obua@14738
  1922
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1923
      apply (insert prems)
nipkow@23477
  1924
      apply (auto simp add: ring_simps)
avigad@16775
  1925
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  1926
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  1927
      done
obua@14738
  1928
  qed
obua@14738
  1929
qed
paulson@14294
  1930
obua@14738
  1931
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
obua@14738
  1932
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  1933
obua@14738
  1934
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
obua@14738
  1935
by (simp add: abs_if) 
paulson@14294
  1936
paulson@14294
  1937
lemma nonzero_abs_inverse:
paulson@14294
  1938
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1939
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  1940
                      negative_imp_inverse_negative)
paulson@14294
  1941
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  1942
done
paulson@14294
  1943
paulson@14294
  1944
lemma abs_inverse [simp]:
paulson@14294
  1945
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  1946
      inverse (abs a)"
haftmann@21328
  1947
apply (cases "a=0", simp) 
paulson@14294
  1948
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1949
done
paulson@14294
  1950
paulson@14294
  1951
lemma nonzero_abs_divide:
paulson@14294
  1952
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  1953
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  1954
paulson@15234
  1955
lemma abs_divide [simp]:
paulson@14294
  1956
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  1957
apply (cases "b=0", simp) 
paulson@14294
  1958
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1959
done
paulson@14294
  1960
paulson@14294
  1961
lemma abs_mult_less:
obua@14738
  1962
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  1963
proof -
paulson@14294
  1964
  assume ac: "abs a < c"
paulson@14294
  1965
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  1966
  assume "abs b < d"
paulson@14294
  1967
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1968
qed
paulson@14293
  1969
obua@14738
  1970
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
obua@14738
  1971
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
obua@14738
  1972
obua@14738
  1973
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
obua@14738
  1974
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
obua@14738
  1975
obua@14738
  1976
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  1977
apply (simp add: order_less_le abs_le_iff)  
obua@14738
  1978
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
obua@14738
  1979
apply (simp add: le_minus_self_iff linorder_neq_iff) 
obua@14738
  1980
done
obua@14738
  1981
avigad@16775
  1982
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
avigad@16775
  1983
    (abs y) * x = abs (y * x)";
avigad@16775
  1984
  apply (subst abs_mult);
avigad@16775
  1985
  apply simp;
avigad@16775
  1986
done;
avigad@16775
  1987
avigad@16775
  1988
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
avigad@16775
  1989
    abs x / y = abs (x / y)";
avigad@16775
  1990
  apply (subst abs_divide);
avigad@16775
  1991
  apply (simp add: order_less_imp_le);
avigad@16775
  1992
done;
avigad@16775
  1993
wenzelm@23389
  1994
obua@19404
  1995
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  1996
obua@15580
  1997
lemma mult_le_prts:
obua@15580
  1998
  assumes
obua@15580
  1999
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  2000
  "a <= a2"
obua@15580
  2001
  "b1 <= b"
obua@15580
  2002
  "b <= b2"
obua@15580
  2003
  shows
obua@15580
  2004
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  2005
proof - 
obua@15580
  2006
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2007
    apply (subst prts[symmetric])+
obua@15580
  2008
    apply simp
obua@15580
  2009
    done
obua@15580
  2010
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
nipkow@23477
  2011
    by (simp add: ring_simps)
obua@15580
  2012
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2013
    by (simp_all add: prems mult_mono)
obua@15580
  2014
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2015
  proof -
obua@15580
  2016
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2017
      by (simp add: mult_left_mono prems)
obua@15580
  2018
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2019
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2020
    ultimately show ?thesis
obua@15580
  2021
      by simp
obua@15580
  2022
  qed
obua@15580
  2023
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2024
  proof - 
obua@15580
  2025
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2026
      by (simp add: mult_right_mono prems)
obua@15580
  2027
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2028
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2029
    ultimately show ?thesis
obua@15580
  2030
      by simp
obua@15580
  2031
  qed
obua@15580
  2032
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2033
  proof -
obua@15580
  2034
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2035
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2036
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2037
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2038
    ultimately show ?thesis
obua@15580
  2039
      by simp
obua@15580
  2040
  qed
obua@15580
  2041
  ultimately show ?thesis
obua@15580
  2042
    by - (rule add_mono | simp)+
obua@15580
  2043
qed
obua@19404
  2044
obua@19404
  2045
lemma mult_ge_prts:
obua@15178
  2046
  assumes
obua@19404
  2047
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2048
  "a <= a2"
obua@19404
  2049
  "b1 <= b"
obua@19404
  2050
  "b <= b2"
obua@15178
  2051
  shows
obua@19404
  2052
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2053
proof - 
obua@19404
  2054
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2055
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2056
  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
  2057
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2058
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2059
    by (simp only: minus_le_iff)
obua@19404
  2060
  then show ?thesis by simp
obua@15178
  2061
qed
obua@15178
  2062
wenzelm@23389
  2063
haftmann@22842
  2064
subsection {* Theorems for proof tools *}
haftmann@22842
  2065
haftmann@22842
  2066
lemma add_mono_thms_ordered_semiring:
haftmann@22842
  2067
  fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
haftmann@22842
  2068
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2069
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2070
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2071
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@22842
  2072
by (rule add_mono, clarify+)+
haftmann@22842
  2073
haftmann@22842
  2074
lemma add_mono_thms_ordered_field:
haftmann@22842
  2075
  fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
haftmann@22842
  2076
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2077
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2078
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2079
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2080
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2081
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@22842
  2082
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@22842
  2083
paulson@14265
  2084
end