src/HOL/Ring_and_Field.thy
author huffman
Thu May 17 18:32:17 2007 +0200 (2007-05-17)
changeset 22990 775e9de3db48
parent 22987 550709aa8e66
child 22993 838c66e760b5
permissions -rw-r--r--
added classes ring_no_zero_divisors and dom (non-commutative version of idom);
generalized several cancellation lemmas to use the new classes
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel,
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             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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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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theorems ring_distrib = right_distrib left_distrib
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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     "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
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by (simp add: left_distrib add_ac)
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lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: left_distrib [symmetric]) 
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done
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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: right_distrib [symmetric]) 
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done
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lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
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by (simp add: right_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
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by (simp add: left_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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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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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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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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apply (simp add: diff_minus left_distrib)
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric])
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done
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lemma eq_add_iff2:
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     "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
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apply (simp add: diff_minus left_distrib add_ac)
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apply (simp add: compare_rls minus_mult_left [symmetric]) 
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done
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lemma less_add_iff1:
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     "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
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apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   320
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   321
done
paulson@14272
   322
paulson@14272
   323
lemma less_add_iff2:
obua@14738
   324
     "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
paulson@14272
   325
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   326
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   327
done
paulson@14272
   328
paulson@14272
   329
lemma le_add_iff1:
obua@14738
   330
     "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
paulson@14272
   331
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   332
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   333
done
paulson@14272
   334
paulson@14272
   335
lemma le_add_iff2:
obua@14738
   336
     "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
paulson@14272
   337
apply (simp add: diff_minus left_distrib add_ac)
paulson@14272
   338
apply (simp add: compare_rls minus_mult_left [symmetric]) 
paulson@14272
   339
done
paulson@14272
   340
paulson@14270
   341
subsection {* Ordering Rules for Multiplication *}
paulson@14270
   342
paulson@14348
   343
lemma mult_left_le_imp_le:
obua@14738
   344
     "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
paulson@14348
   345
  by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
paulson@14348
   346
 
paulson@14348
   347
lemma mult_right_le_imp_le:
obua@14738
   348
     "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
paulson@14348
   349
  by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
paulson@14348
   350
paulson@14348
   351
lemma mult_left_less_imp_less:
obua@14738
   352
     "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
paulson@14348
   353
  by (force simp add: mult_left_mono linorder_not_le [symmetric])
paulson@14348
   354
 
paulson@14348
   355
lemma mult_right_less_imp_less:
obua@14738
   356
     "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
paulson@14348
   357
  by (force simp add: mult_right_mono linorder_not_le [symmetric])
paulson@14348
   358
paulson@14265
   359
lemma mult_strict_left_mono_neg:
obua@14738
   360
     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
paulson@14265
   361
apply (drule mult_strict_left_mono [of _ _ "-c"])
paulson@14265
   362
apply (simp_all add: minus_mult_left [symmetric]) 
paulson@14265
   363
done
paulson@14265
   364
obua@14738
   365
lemma mult_left_mono_neg:
obua@14738
   366
     "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
obua@14738
   367
apply (drule mult_left_mono [of _ _ "-c"])
obua@14738
   368
apply (simp_all add: minus_mult_left [symmetric]) 
obua@14738
   369
done
obua@14738
   370
paulson@14265
   371
lemma mult_strict_right_mono_neg:
obua@14738
   372
     "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
paulson@14265
   373
apply (drule mult_strict_right_mono [of _ _ "-c"])
paulson@14265
   374
apply (simp_all add: minus_mult_right [symmetric]) 
paulson@14265
   375
done
paulson@14265
   376
obua@14738
   377
lemma mult_right_mono_neg:
obua@14738
   378
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
obua@14738
   379
apply (drule mult_right_mono [of _ _ "-c"])
obua@14738
   380
apply (simp)
obua@14738
   381
apply (simp_all add: minus_mult_right [symmetric]) 
obua@14738
   382
done
paulson@14265
   383
paulson@14265
   384
subsection{* Products of Signs *}
paulson@14265
   385
avigad@16775
   386
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
paulson@14265
   387
by (drule mult_strict_left_mono [of 0 b], auto)
paulson@14265
   388
avigad@16775
   389
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
obua@14738
   390
by (drule mult_left_mono [of 0 b], auto)
obua@14738
   391
obua@14738
   392
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
paulson@14265
   393
by (drule mult_strict_left_mono [of b 0], auto)
paulson@14265
   394
avigad@16775
   395
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
obua@14738
   396
by (drule mult_left_mono [of b 0], auto)
obua@14738
   397
obua@14738
   398
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
obua@14738
   399
by (drule mult_strict_right_mono[of b 0], auto)
obua@14738
   400
avigad@16775
   401
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
obua@14738
   402
by (drule mult_right_mono[of b 0], auto)
obua@14738
   403
avigad@16775
   404
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
paulson@14265
   405
by (drule mult_strict_right_mono_neg, auto)
paulson@14265
   406
avigad@16775
   407
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
obua@14738
   408
by (drule mult_right_mono_neg[of a 0 b ], auto)
obua@14738
   409
paulson@14341
   410
lemma zero_less_mult_pos:
obua@14738
   411
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   412
apply (cases "b\<le>0") 
paulson@14265
   413
 apply (auto simp add: order_le_less linorder_not_less)
paulson@14265
   414
apply (drule_tac mult_pos_neg [of a b]) 
paulson@14265
   415
 apply (auto dest: order_less_not_sym)
paulson@14265
   416
done
paulson@14265
   417
obua@14738
   418
lemma zero_less_mult_pos2:
obua@14738
   419
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
haftmann@21328
   420
apply (cases "b\<le>0") 
obua@14738
   421
 apply (auto simp add: order_le_less linorder_not_less)
obua@14738
   422
apply (drule_tac mult_pos_neg2 [of a b]) 
obua@14738
   423
 apply (auto dest: order_less_not_sym)
obua@14738
   424
done
obua@14738
   425
paulson@14265
   426
lemma zero_less_mult_iff:
obua@14738
   427
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
avigad@16775
   428
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos 
avigad@16775
   429
  mult_neg_neg)
paulson@14265
   430
apply (blast dest: zero_less_mult_pos) 
obua@14738
   431
apply (blast dest: zero_less_mult_pos2)
paulson@14265
   432
done
paulson@14265
   433
huffman@22990
   434
lemma mult_eq_0_iff [simp]:
huffman@22990
   435
  fixes a b :: "'a::ring_no_zero_divisors"
huffman@22990
   436
  shows "(a * b = 0) = (a = 0 \<or> b = 0)"
huffman@22990
   437
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors)
huffman@22990
   438
huffman@22990
   439
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
huffman@22990
   440
apply intro_classes
paulson@14265
   441
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
paulson@14265
   442
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
paulson@14265
   443
done
paulson@14265
   444
paulson@14265
   445
lemma zero_le_mult_iff:
obua@14738
   446
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
paulson@14265
   447
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
paulson@14265
   448
                   zero_less_mult_iff)
paulson@14265
   449
paulson@14265
   450
lemma mult_less_0_iff:
obua@14738
   451
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
paulson@14265
   452
apply (insert zero_less_mult_iff [of "-a" b]) 
paulson@14265
   453
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   454
done
paulson@14265
   455
paulson@14265
   456
lemma mult_le_0_iff:
obua@14738
   457
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
paulson@14265
   458
apply (insert zero_le_mult_iff [of "-a" b]) 
paulson@14265
   459
apply (force simp add: minus_mult_left[symmetric]) 
paulson@14265
   460
done
paulson@14265
   461
obua@14738
   462
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
   463
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14738
   464
obua@14738
   465
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
   466
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
obua@14738
   467
obua@14738
   468
lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"
paulson@14265
   469
by (simp add: zero_le_mult_iff linorder_linear) 
paulson@14265
   470
obua@14738
   471
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
obua@14738
   472
      theorems available to members of @{term ordered_idom} *}
obua@14738
   473
obua@14738
   474
instance ordered_idom \<subseteq> ordered_semidom
paulson@14421
   475
proof
paulson@14421
   476
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
paulson@14430
   477
  thus "(0::'a) < 1" by (simp add: order_le_less) 
paulson@14421
   478
qed
paulson@14421
   479
obua@14738
   480
instance ordered_idom \<subseteq> idom ..
obua@14738
   481
paulson@14387
   482
text{*All three types of comparision involving 0 and 1 are covered.*}
paulson@14387
   483
paulson@17085
   484
lemmas one_neq_zero = zero_neq_one [THEN not_sym]
paulson@17085
   485
declare one_neq_zero [simp]
paulson@14387
   486
obua@14738
   487
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
paulson@14268
   488
  by (rule zero_less_one [THEN order_less_imp_le]) 
paulson@14268
   489
obua@14738
   490
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
obua@14738
   491
by (simp add: linorder_not_le) 
paulson@14387
   492
obua@14738
   493
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
obua@14738
   494
by (simp add: linorder_not_less) 
paulson@14268
   495
paulson@14268
   496
subsection{*More Monotonicity*}
paulson@14268
   497
paulson@14268
   498
text{*Strict monotonicity in both arguments*}
paulson@14268
   499
lemma mult_strict_mono:
obua@14738
   500
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
haftmann@21328
   501
apply (cases "c=0")
avigad@16775
   502
 apply (simp add: mult_pos_pos) 
paulson@14268
   503
apply (erule mult_strict_right_mono [THEN order_less_trans])
paulson@14268
   504
 apply (force simp add: order_le_less) 
paulson@14268
   505
apply (erule mult_strict_left_mono, assumption)
paulson@14268
   506
done
paulson@14268
   507
paulson@14268
   508
text{*This weaker variant has more natural premises*}
paulson@14268
   509
lemma mult_strict_mono':
obua@14738
   510
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
paulson@14268
   511
apply (rule mult_strict_mono)
paulson@14268
   512
apply (blast intro: order_le_less_trans)+
paulson@14268
   513
done
paulson@14268
   514
paulson@14268
   515
lemma mult_mono:
paulson@14268
   516
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
obua@14738
   517
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
paulson@14268
   518
apply (erule mult_right_mono [THEN order_trans], assumption)
paulson@14268
   519
apply (erule mult_left_mono, assumption)
paulson@14268
   520
done
paulson@14268
   521
huffman@21258
   522
lemma mult_mono':
huffman@21258
   523
     "[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] 
huffman@21258
   524
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
huffman@21258
   525
apply (rule mult_mono)
huffman@21258
   526
apply (fast intro: order_trans)+
huffman@21258
   527
done
huffman@21258
   528
obua@14738
   529
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
paulson@14387
   530
apply (insert mult_strict_mono [of 1 m 1 n]) 
paulson@14430
   531
apply (simp add:  order_less_trans [OF zero_less_one]) 
paulson@14387
   532
done
paulson@14387
   533
avigad@16775
   534
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
avigad@16775
   535
    c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
avigad@16775
   536
  apply (subgoal_tac "a * c < b * c")
avigad@16775
   537
  apply (erule order_less_le_trans)
avigad@16775
   538
  apply (erule mult_left_mono)
avigad@16775
   539
  apply simp
avigad@16775
   540
  apply (erule mult_strict_right_mono)
avigad@16775
   541
  apply assumption
avigad@16775
   542
done
avigad@16775
   543
avigad@16775
   544
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
avigad@16775
   545
    c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
avigad@16775
   546
  apply (subgoal_tac "a * c <= b * c")
avigad@16775
   547
  apply (erule order_le_less_trans)
avigad@16775
   548
  apply (erule mult_strict_left_mono)
avigad@16775
   549
  apply simp
avigad@16775
   550
  apply (erule mult_right_mono)
avigad@16775
   551
  apply simp
avigad@16775
   552
done
avigad@16775
   553
paulson@14268
   554
subsection{*Cancellation Laws for Relationships With a Common Factor*}
paulson@14268
   555
paulson@14268
   556
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
paulson@14268
   557
   also with the relations @{text "\<le>"} and equality.*}
paulson@14268
   558
paulson@15234
   559
text{*These ``disjunction'' versions produce two cases when the comparison is
paulson@15234
   560
 an assumption, but effectively four when the comparison is a goal.*}
paulson@15234
   561
paulson@15234
   562
lemma mult_less_cancel_right_disj:
obua@14738
   563
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   564
apply (cases "c = 0")
paulson@14268
   565
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
paulson@14268
   566
                      mult_strict_right_mono_neg)
paulson@14268
   567
apply (auto simp add: linorder_not_less 
paulson@14268
   568
                      linorder_not_le [symmetric, of "a*c"]
paulson@14268
   569
                      linorder_not_le [symmetric, of a])
paulson@14268
   570
apply (erule_tac [!] notE)
paulson@14268
   571
apply (auto simp add: order_less_imp_le mult_right_mono 
paulson@14268
   572
                      mult_right_mono_neg)
paulson@14268
   573
done
paulson@14268
   574
paulson@15234
   575
lemma mult_less_cancel_left_disj:
obua@14738
   576
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
haftmann@21328
   577
apply (cases "c = 0")
obua@14738
   578
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
obua@14738
   579
                      mult_strict_left_mono_neg)
obua@14738
   580
apply (auto simp add: linorder_not_less 
obua@14738
   581
                      linorder_not_le [symmetric, of "c*a"]
obua@14738
   582
                      linorder_not_le [symmetric, of a])
obua@14738
   583
apply (erule_tac [!] notE)
obua@14738
   584
apply (auto simp add: order_less_imp_le mult_left_mono 
obua@14738
   585
                      mult_left_mono_neg)
obua@14738
   586
done
paulson@14268
   587
paulson@15234
   588
paulson@15234
   589
text{*The ``conjunction of implication'' lemmas produce two cases when the
paulson@15234
   590
comparison is a goal, but give four when the comparison is an assumption.*}
paulson@15234
   591
paulson@15234
   592
lemma mult_less_cancel_right:
paulson@15234
   593
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   594
  shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   595
by (insert mult_less_cancel_right_disj [of a c b], auto)
paulson@15234
   596
paulson@15234
   597
lemma mult_less_cancel_left:
paulson@15234
   598
  fixes c :: "'a :: ordered_ring_strict"
paulson@15234
   599
  shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
paulson@15234
   600
by (insert mult_less_cancel_left_disj [of c a b], auto)
paulson@15234
   601
paulson@14268
   602
lemma mult_le_cancel_right:
obua@14738
   603
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   604
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
paulson@14268
   605
paulson@14268
   606
lemma mult_le_cancel_left:
obua@14738
   607
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
paulson@15234
   608
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
paulson@14268
   609
paulson@14268
   610
lemma mult_less_imp_less_left:
paulson@14341
   611
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
obua@14738
   612
      shows "a < (b::'a::ordered_semiring_strict)"
paulson@14377
   613
proof (rule ccontr)
paulson@14377
   614
  assume "~ a < b"
paulson@14377
   615
  hence "b \<le> a" by (simp add: linorder_not_less)
paulson@14377
   616
  hence "c*b \<le> c*a" by (rule mult_left_mono)
paulson@14377
   617
  with this and less show False 
paulson@14377
   618
    by (simp add: linorder_not_less [symmetric])
paulson@14377
   619
qed
paulson@14268
   620
paulson@14268
   621
lemma mult_less_imp_less_right:
obua@14738
   622
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
obua@14738
   623
  shows "a < (b::'a::ordered_semiring_strict)"
obua@14738
   624
proof (rule ccontr)
obua@14738
   625
  assume "~ a < b"
obua@14738
   626
  hence "b \<le> a" by (simp add: linorder_not_less)
obua@14738
   627
  hence "b*c \<le> a*c" by (rule mult_right_mono)
obua@14738
   628
  with this and less show False 
obua@14738
   629
    by (simp add: linorder_not_less [symmetric])
obua@14738
   630
qed  
paulson@14268
   631
paulson@14268
   632
text{*Cancellation of equalities with a common factor*}
paulson@14268
   633
lemma mult_cancel_right [simp]:
huffman@22990
   634
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   635
  shows "(a * c = b * c) = (c = 0 \<or> a = b)"
huffman@22990
   636
proof -
huffman@22990
   637
  have "(a * c = b * c) = ((a - b) * c = 0)"
huffman@22990
   638
    by (simp add: left_diff_distrib)
huffman@22990
   639
  thus ?thesis
huffman@22990
   640
    by (simp add: disj_commute)
huffman@22990
   641
qed
paulson@14268
   642
paulson@14268
   643
lemma mult_cancel_left [simp]:
huffman@22990
   644
  fixes a b c :: "'a::ring_no_zero_divisors"
huffman@22990
   645
  shows "(c * a = c * b) = (c = 0 \<or> a = b)"
huffman@22990
   646
proof -
huffman@22990
   647
  have "(c * a = c * b) = (c * (a - b) = 0)"
huffman@22990
   648
    by (simp add: right_diff_distrib)
huffman@22990
   649
  thus ?thesis
huffman@22990
   650
    by simp
huffman@22990
   651
qed
paulson@14268
   652
paulson@15234
   653
paulson@15234
   654
subsubsection{*Special Cancellation Simprules for Multiplication*}
paulson@15234
   655
paulson@15234
   656
text{*These also produce two cases when the comparison is a goal.*}
paulson@15234
   657
paulson@15234
   658
lemma mult_le_cancel_right1:
paulson@15234
   659
  fixes c :: "'a :: ordered_idom"
paulson@15234
   660
  shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   661
by (insert mult_le_cancel_right [of 1 c b], simp)
paulson@15234
   662
paulson@15234
   663
lemma mult_le_cancel_right2:
paulson@15234
   664
  fixes c :: "'a :: ordered_idom"
paulson@15234
   665
  shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   666
by (insert mult_le_cancel_right [of a c 1], simp)
paulson@15234
   667
paulson@15234
   668
lemma mult_le_cancel_left1:
paulson@15234
   669
  fixes c :: "'a :: ordered_idom"
paulson@15234
   670
  shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
paulson@15234
   671
by (insert mult_le_cancel_left [of c 1 b], simp)
paulson@15234
   672
paulson@15234
   673
lemma mult_le_cancel_left2:
paulson@15234
   674
  fixes c :: "'a :: ordered_idom"
paulson@15234
   675
  shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
paulson@15234
   676
by (insert mult_le_cancel_left [of c a 1], simp)
paulson@15234
   677
paulson@15234
   678
lemma mult_less_cancel_right1:
paulson@15234
   679
  fixes c :: "'a :: ordered_idom"
paulson@15234
   680
  shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   681
by (insert mult_less_cancel_right [of 1 c b], simp)
paulson@15234
   682
paulson@15234
   683
lemma mult_less_cancel_right2:
paulson@15234
   684
  fixes c :: "'a :: ordered_idom"
paulson@15234
   685
  shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   686
by (insert mult_less_cancel_right [of a c 1], simp)
paulson@15234
   687
paulson@15234
   688
lemma mult_less_cancel_left1:
paulson@15234
   689
  fixes c :: "'a :: ordered_idom"
paulson@15234
   690
  shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
paulson@15234
   691
by (insert mult_less_cancel_left [of c 1 b], simp)
paulson@15234
   692
paulson@15234
   693
lemma mult_less_cancel_left2:
paulson@15234
   694
  fixes c :: "'a :: ordered_idom"
paulson@15234
   695
  shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
paulson@15234
   696
by (insert mult_less_cancel_left [of c a 1], simp)
paulson@15234
   697
paulson@15234
   698
lemma mult_cancel_right1 [simp]:
huffman@22990
   699
  fixes c :: "'a :: dom"
paulson@15234
   700
  shows "(c = b*c) = (c = 0 | b=1)"
paulson@15234
   701
by (insert mult_cancel_right [of 1 c b], force)
paulson@15234
   702
paulson@15234
   703
lemma mult_cancel_right2 [simp]:
huffman@22990
   704
  fixes c :: "'a :: dom"
paulson@15234
   705
  shows "(a*c = c) = (c = 0 | a=1)"
paulson@15234
   706
by (insert mult_cancel_right [of a c 1], simp)
paulson@15234
   707
 
paulson@15234
   708
lemma mult_cancel_left1 [simp]:
huffman@22990
   709
  fixes c :: "'a :: dom"
paulson@15234
   710
  shows "(c = c*b) = (c = 0 | b=1)"
paulson@15234
   711
by (insert mult_cancel_left [of c 1 b], force)
paulson@15234
   712
paulson@15234
   713
lemma mult_cancel_left2 [simp]:
huffman@22990
   714
  fixes c :: "'a :: dom"
paulson@15234
   715
  shows "(c*a = c) = (c = 0 | a=1)"
paulson@15234
   716
by (insert mult_cancel_left [of c a 1], simp)
paulson@15234
   717
paulson@15234
   718
paulson@15234
   719
text{*Simprules for comparisons where common factors can be cancelled.*}
paulson@15234
   720
lemmas mult_compare_simps =
paulson@15234
   721
    mult_le_cancel_right mult_le_cancel_left
paulson@15234
   722
    mult_le_cancel_right1 mult_le_cancel_right2
paulson@15234
   723
    mult_le_cancel_left1 mult_le_cancel_left2
paulson@15234
   724
    mult_less_cancel_right mult_less_cancel_left
paulson@15234
   725
    mult_less_cancel_right1 mult_less_cancel_right2
paulson@15234
   726
    mult_less_cancel_left1 mult_less_cancel_left2
paulson@15234
   727
    mult_cancel_right mult_cancel_left
paulson@15234
   728
    mult_cancel_right1 mult_cancel_right2
paulson@15234
   729
    mult_cancel_left1 mult_cancel_left2
paulson@15234
   730
paulson@15234
   731
obua@14738
   732
text{*This list of rewrites decides ring equalities by ordered rewriting.*}
obua@15178
   733
lemmas ring_eq_simps =  
obua@15178
   734
(*  mult_ac*)
obua@14738
   735
  left_distrib right_distrib left_diff_distrib right_diff_distrib
obua@15178
   736
  group_eq_simps
obua@15178
   737
(*  add_ac
obua@14738
   738
  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
obua@15178
   739
  diff_eq_eq eq_diff_eq *)
obua@14738
   740
    
paulson@14265
   741
subsection {* Fields *}
paulson@14265
   742
paulson@14288
   743
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
paulson@14288
   744
proof
paulson@14288
   745
  assume neq: "b \<noteq> 0"
paulson@14288
   746
  {
paulson@14288
   747
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
paulson@14288
   748
    also assume "a / b = 1"
paulson@14288
   749
    finally show "a = b" by simp
paulson@14288
   750
  next
paulson@14288
   751
    assume "a = b"
paulson@14288
   752
    with neq show "a / b = 1" by (simp add: divide_inverse)
paulson@14288
   753
  }
paulson@14288
   754
qed
paulson@14288
   755
paulson@14288
   756
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
paulson@14288
   757
by (simp add: divide_inverse)
paulson@14288
   758
paulson@15228
   759
lemma divide_self: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
paulson@14288
   760
  by (simp add: divide_inverse)
paulson@14288
   761
paulson@14430
   762
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
paulson@14430
   763
by (simp add: divide_inverse)
paulson@14277
   764
paulson@15228
   765
lemma divide_self_if [simp]:
paulson@15228
   766
     "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
paulson@15228
   767
  by (simp add: divide_self)
paulson@15228
   768
paulson@14430
   769
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
paulson@14430
   770
by (simp add: divide_inverse)
paulson@14277
   771
paulson@14430
   772
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
paulson@14430
   773
by (simp add: divide_inverse)
paulson@14277
   774
paulson@14430
   775
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
paulson@14293
   776
by (simp add: divide_inverse left_distrib) 
paulson@14293
   777
paulson@14293
   778
paulson@14270
   779
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
paulson@14270
   780
      of an ordering.*}
huffman@20496
   781
lemma field_mult_eq_0_iff [simp]:
huffman@20496
   782
  "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
huffman@22990
   783
by simp
paulson@14270
   784
paulson@14268
   785
text{*Cancellation of equalities with a common factor*}
paulson@14268
   786
lemma field_mult_cancel_right_lemma:
huffman@20496
   787
      assumes cnz: "c \<noteq> (0::'a::division_ring)"
huffman@20496
   788
         and eq:  "a*c = b*c"
huffman@20496
   789
        shows "a=b"
paulson@14377
   790
proof -
paulson@14268
   791
  have "(a * c) * inverse c = (b * c) * inverse c"
paulson@14268
   792
    by (simp add: eq)
paulson@14268
   793
  thus "a=b"
paulson@14268
   794
    by (simp add: mult_assoc cnz)
paulson@14377
   795
qed
paulson@14268
   796
paulson@14348
   797
lemma field_mult_cancel_right [simp]:
huffman@20496
   798
     "(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   799
by simp
paulson@14268
   800
paulson@14348
   801
lemma field_mult_cancel_left [simp]:
huffman@20496
   802
     "(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
huffman@22990
   803
by simp
paulson@14268
   804
huffman@20496
   805
lemma nonzero_imp_inverse_nonzero:
huffman@20496
   806
  "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
paulson@14377
   807
proof
paulson@14268
   808
  assume ianz: "inverse a = 0"
paulson@14268
   809
  assume "a \<noteq> 0"
paulson@14268
   810
  hence "1 = a * inverse a" by simp
paulson@14268
   811
  also have "... = 0" by (simp add: ianz)
huffman@20496
   812
  finally have "1 = (0::'a::division_ring)" .
paulson@14268
   813
  thus False by (simp add: eq_commute)
paulson@14377
   814
qed
paulson@14268
   815
paulson@14277
   816
paulson@14277
   817
subsection{*Basic Properties of @{term inverse}*}
paulson@14277
   818
huffman@20496
   819
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   820
apply (rule ccontr) 
paulson@14268
   821
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   822
done
paulson@14268
   823
paulson@14268
   824
lemma inverse_nonzero_imp_nonzero:
huffman@20496
   825
   "inverse a = 0 ==> a = (0::'a::division_ring)"
paulson@14268
   826
apply (rule ccontr) 
paulson@14268
   827
apply (blast dest: nonzero_imp_inverse_nonzero) 
paulson@14268
   828
done
paulson@14268
   829
paulson@14268
   830
lemma inverse_nonzero_iff_nonzero [simp]:
huffman@20496
   831
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
paulson@14268
   832
by (force dest: inverse_nonzero_imp_nonzero) 
paulson@14268
   833
paulson@14268
   834
lemma nonzero_inverse_minus_eq:
huffman@20496
   835
      assumes [simp]: "a\<noteq>0"
huffman@20496
   836
      shows "inverse(-a) = -inverse(a::'a::division_ring)"
paulson@14377
   837
proof -
paulson@14377
   838
  have "-a * inverse (- a) = -a * - inverse a"
paulson@14377
   839
    by simp
paulson@14377
   840
  thus ?thesis 
paulson@14377
   841
    by (simp only: field_mult_cancel_left, simp)
paulson@14377
   842
qed
paulson@14268
   843
paulson@14268
   844
lemma inverse_minus_eq [simp]:
huffman@20496
   845
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
paulson@14377
   846
proof cases
paulson@14377
   847
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
paulson@14377
   848
next
paulson@14377
   849
  assume "a\<noteq>0" 
paulson@14377
   850
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
paulson@14377
   851
qed
paulson@14268
   852
paulson@14268
   853
lemma nonzero_inverse_eq_imp_eq:
paulson@14269
   854
      assumes inveq: "inverse a = inverse b"
paulson@14269
   855
	  and anz:  "a \<noteq> 0"
paulson@14269
   856
	  and bnz:  "b \<noteq> 0"
huffman@20496
   857
	 shows "a = (b::'a::division_ring)"
paulson@14377
   858
proof -
paulson@14268
   859
  have "a * inverse b = a * inverse a"
paulson@14268
   860
    by (simp add: inveq)
paulson@14268
   861
  hence "(a * inverse b) * b = (a * inverse a) * b"
paulson@14268
   862
    by simp
paulson@14268
   863
  thus "a = b"
paulson@14268
   864
    by (simp add: mult_assoc anz bnz)
paulson@14377
   865
qed
paulson@14268
   866
paulson@14268
   867
lemma inverse_eq_imp_eq:
huffman@20496
   868
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
haftmann@21328
   869
apply (cases "a=0 | b=0") 
paulson@14268
   870
 apply (force dest!: inverse_zero_imp_zero
paulson@14268
   871
              simp add: eq_commute [of "0::'a"])
paulson@14268
   872
apply (force dest!: nonzero_inverse_eq_imp_eq) 
paulson@14268
   873
done
paulson@14268
   874
paulson@14268
   875
lemma inverse_eq_iff_eq [simp]:
huffman@20496
   876
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
huffman@20496
   877
by (force dest!: inverse_eq_imp_eq)
paulson@14268
   878
paulson@14270
   879
lemma nonzero_inverse_inverse_eq:
huffman@20496
   880
      assumes [simp]: "a \<noteq> 0"
huffman@20496
   881
      shows "inverse(inverse (a::'a::division_ring)) = a"
paulson@14270
   882
  proof -
paulson@14270
   883
  have "(inverse (inverse a) * inverse a) * a = a" 
paulson@14270
   884
    by (simp add: nonzero_imp_inverse_nonzero)
paulson@14270
   885
  thus ?thesis
paulson@14270
   886
    by (simp add: mult_assoc)
paulson@14270
   887
  qed
paulson@14270
   888
paulson@14270
   889
lemma inverse_inverse_eq [simp]:
huffman@20496
   890
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
paulson@14270
   891
  proof cases
paulson@14270
   892
    assume "a=0" thus ?thesis by simp
paulson@14270
   893
  next
paulson@14270
   894
    assume "a\<noteq>0" 
paulson@14270
   895
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
paulson@14270
   896
  qed
paulson@14270
   897
huffman@20496
   898
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
paulson@14270
   899
  proof -
huffman@20496
   900
  have "inverse 1 * 1 = (1::'a::division_ring)" 
paulson@14270
   901
    by (rule left_inverse [OF zero_neq_one [symmetric]])
paulson@14270
   902
  thus ?thesis  by simp
paulson@14270
   903
  qed
paulson@14270
   904
paulson@15077
   905
lemma inverse_unique: 
paulson@15077
   906
  assumes ab: "a*b = 1"
huffman@20496
   907
  shows "inverse a = (b::'a::division_ring)"
paulson@15077
   908
proof -
paulson@15077
   909
  have "a \<noteq> 0" using ab by auto
paulson@15077
   910
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
paulson@15077
   911
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
paulson@15077
   912
qed
paulson@15077
   913
paulson@14270
   914
lemma nonzero_inverse_mult_distrib: 
paulson@14270
   915
      assumes anz: "a \<noteq> 0"
paulson@14270
   916
          and bnz: "b \<noteq> 0"
huffman@20496
   917
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
paulson@14270
   918
  proof -
paulson@14270
   919
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
paulson@14270
   920
    by (simp add: field_mult_eq_0_iff anz bnz)
paulson@14270
   921
  hence "inverse(a*b) * a = inverse(b)" 
paulson@14270
   922
    by (simp add: mult_assoc bnz)
paulson@14270
   923
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
paulson@14270
   924
    by simp
paulson@14270
   925
  thus ?thesis
paulson@14270
   926
    by (simp add: mult_assoc anz)
paulson@14270
   927
  qed
paulson@14270
   928
paulson@14270
   929
text{*This version builds in division by zero while also re-orienting
paulson@14270
   930
      the right-hand side.*}
paulson@14270
   931
lemma inverse_mult_distrib [simp]:
paulson@14270
   932
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
paulson@14270
   933
  proof cases
paulson@14270
   934
    assume "a \<noteq> 0 & b \<noteq> 0" 
paulson@14270
   935
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
paulson@14270
   936
  next
paulson@14270
   937
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
paulson@14270
   938
    thus ?thesis  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"
huffman@20496
   944
by (simp add: right_distrib left_distrib mult_assoc)
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"
huffman@20496
   949
by (simp add: right_diff_distrib left_diff_distrib mult_assoc)
huffman@20496
   950
paulson@14270
   951
text{*There is no slick version using division by zero.*}
paulson@14270
   952
lemma inverse_add:
paulson@14270
   953
     "[|a \<noteq> 0;  b \<noteq> 0|]
paulson@14270
   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]:
paulson@14365
   958
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
paulson@14430
   959
  by (simp add: divide_inverse mult_commute)
paulson@14365
   960
avigad@16775
   961
subsection {* Calculations with fractions *}
avigad@16775
   962
paulson@14277
   963
lemma nonzero_mult_divide_cancel_left:
paulson@14277
   964
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
paulson@14277
   965
    shows "(c*a)/(c*b) = a/(b::'a::field)"
paulson@14277
   966
proof -
paulson@14277
   967
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
paulson@14277
   968
    by (simp add: field_mult_eq_0_iff divide_inverse 
paulson@14277
   969
                  nonzero_inverse_mult_distrib)
paulson@14277
   970
  also have "... =  a * inverse b * (inverse c * c)"
paulson@14277
   971
    by (simp only: mult_ac)
paulson@14277
   972
  also have "... =  a * inverse b"
paulson@14277
   973
    by simp
paulson@14277
   974
    finally show ?thesis 
paulson@14277
   975
    by (simp add: divide_inverse)
paulson@14277
   976
qed
paulson@14277
   977
paulson@14277
   978
lemma mult_divide_cancel_left:
paulson@14277
   979
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
   980
apply (cases "b = 0")
paulson@14277
   981
apply (simp_all add: nonzero_mult_divide_cancel_left)
paulson@14277
   982
done
paulson@14277
   983
paulson@14321
   984
lemma nonzero_mult_divide_cancel_right:
paulson@14321
   985
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
paulson@14321
   986
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
paulson@14321
   987
paulson@14321
   988
lemma mult_divide_cancel_right:
paulson@14321
   989
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
   990
apply (cases "b = 0")
paulson@14321
   991
apply (simp_all add: nonzero_mult_divide_cancel_right)
paulson@14321
   992
done
paulson@14321
   993
paulson@14277
   994
(*For ExtractCommonTerm*)
paulson@14277
   995
lemma mult_divide_cancel_eq_if:
paulson@14277
   996
     "(c*a) / (c*b) = 
paulson@14277
   997
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
paulson@14277
   998
  by (simp add: mult_divide_cancel_left)
paulson@14277
   999
paulson@14284
  1000
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
paulson@14430
  1001
  by (simp add: divide_inverse)
paulson@14284
  1002
paulson@15234
  1003
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
paulson@14430
  1004
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1005
paulson@14430
  1006
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
paulson@14430
  1007
by (simp add: divide_inverse mult_ac)
paulson@14288
  1008
paulson@14288
  1009
lemma divide_divide_eq_right [simp]:
paulson@14288
  1010
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
paulson@14430
  1011
by (simp add: divide_inverse mult_ac)
paulson@14288
  1012
paulson@14288
  1013
lemma divide_divide_eq_left [simp]:
paulson@14288
  1014
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
paulson@14430
  1015
by (simp add: divide_inverse mult_assoc)
paulson@14288
  1016
avigad@16775
  1017
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1018
    x / y + w / z = (x * z + w * y) / (y * z)"
avigad@16775
  1019
  apply (subgoal_tac "x / y = (x * z) / (y * z)")
avigad@16775
  1020
  apply (erule ssubst)
avigad@16775
  1021
  apply (subgoal_tac "w / z = (w * y) / (y * z)")
avigad@16775
  1022
  apply (erule ssubst)
avigad@16775
  1023
  apply (rule add_divide_distrib [THEN sym])
avigad@16775
  1024
  apply (subst mult_commute)
avigad@16775
  1025
  apply (erule nonzero_mult_divide_cancel_left [THEN sym])
avigad@16775
  1026
  apply assumption
avigad@16775
  1027
  apply (erule nonzero_mult_divide_cancel_right [THEN sym])
avigad@16775
  1028
  apply assumption
avigad@16775
  1029
done
paulson@14268
  1030
paulson@15234
  1031
subsubsection{*Special Cancellation Simprules for Division*}
paulson@15234
  1032
paulson@15234
  1033
lemma mult_divide_cancel_left_if [simp]:
paulson@15234
  1034
  fixes c :: "'a :: {field,division_by_zero}"
paulson@15234
  1035
  shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
paulson@15234
  1036
by (simp add: mult_divide_cancel_left)
paulson@15234
  1037
paulson@15234
  1038
lemma mult_divide_cancel_right_if [simp]:
paulson@15234
  1039
  fixes c :: "'a :: {field,division_by_zero}"
paulson@15234
  1040
  shows "(a*c) / (b*c) = (if c=0 then 0 else a/b)"
paulson@15234
  1041
by (simp add: mult_divide_cancel_right)
paulson@15234
  1042
paulson@15234
  1043
lemma mult_divide_cancel_left_if1 [simp]:
paulson@15234
  1044
  fixes c :: "'a :: {field,division_by_zero}"
paulson@15234
  1045
  shows "c / (c*b) = (if c=0 then 0 else 1/b)"
paulson@15234
  1046
apply (insert mult_divide_cancel_left_if [of c 1 b]) 
paulson@15234
  1047
apply (simp del: mult_divide_cancel_left_if)
paulson@15234
  1048
done
paulson@15234
  1049
paulson@15234
  1050
lemma mult_divide_cancel_left_if2 [simp]:
paulson@15234
  1051
  fixes c :: "'a :: {field,division_by_zero}"
paulson@15234
  1052
  shows "(c*a) / c = (if c=0 then 0 else a)" 
paulson@15234
  1053
apply (insert mult_divide_cancel_left_if [of c a 1]) 
paulson@15234
  1054
apply (simp del: mult_divide_cancel_left_if)
paulson@15234
  1055
done
paulson@15234
  1056
paulson@15234
  1057
lemma mult_divide_cancel_right_if1 [simp]:
paulson@15234
  1058
  fixes c :: "'a :: {field,division_by_zero}"
paulson@15234
  1059
  shows "c / (b*c) = (if c=0 then 0 else 1/b)"
paulson@15234
  1060
apply (insert mult_divide_cancel_right_if [of 1 c b]) 
paulson@15234
  1061
apply (simp del: mult_divide_cancel_right_if)
paulson@15234
  1062
done
paulson@15234
  1063
paulson@15234
  1064
lemma mult_divide_cancel_right_if2 [simp]:
paulson@15234
  1065
  fixes c :: "'a :: {field,division_by_zero}"
paulson@15234
  1066
  shows "(a*c) / c = (if c=0 then 0 else a)" 
paulson@15234
  1067
apply (insert mult_divide_cancel_right_if [of a c 1]) 
paulson@15234
  1068
apply (simp del: mult_divide_cancel_right_if)
paulson@15234
  1069
done
paulson@15234
  1070
paulson@15234
  1071
text{*Two lemmas for cancelling the denominator*}
paulson@15234
  1072
paulson@15234
  1073
lemma times_divide_self_right [simp]: 
paulson@15234
  1074
  fixes a :: "'a :: {field,division_by_zero}"
paulson@15234
  1075
  shows "a * (b/a) = (if a=0 then 0 else b)"
paulson@15234
  1076
by (simp add: times_divide_eq_right)
paulson@15234
  1077
paulson@15234
  1078
lemma times_divide_self_left [simp]: 
paulson@15234
  1079
  fixes a :: "'a :: {field,division_by_zero}"
paulson@15234
  1080
  shows "(b/a) * a = (if a=0 then 0 else b)"
paulson@15234
  1081
by (simp add: times_divide_eq_left)
paulson@15234
  1082
paulson@15234
  1083
paulson@14293
  1084
subsection {* Division and Unary Minus *}
paulson@14293
  1085
paulson@14293
  1086
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
paulson@14293
  1087
by (simp add: divide_inverse minus_mult_left)
paulson@14293
  1088
paulson@14293
  1089
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
paulson@14293
  1090
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
paulson@14293
  1091
paulson@14293
  1092
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
paulson@14293
  1093
by (simp add: divide_inverse nonzero_inverse_minus_eq)
paulson@14293
  1094
paulson@14430
  1095
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
paulson@14430
  1096
by (simp add: divide_inverse minus_mult_left [symmetric])
paulson@14293
  1097
paulson@14293
  1098
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
paulson@14430
  1099
by (simp add: divide_inverse minus_mult_right [symmetric])
paulson@14430
  1100
paulson@14293
  1101
paulson@14293
  1102
text{*The effect is to extract signs from divisions*}
paulson@17085
  1103
lemmas divide_minus_left = minus_divide_left [symmetric]
paulson@17085
  1104
lemmas divide_minus_right = minus_divide_right [symmetric]
paulson@17085
  1105
declare divide_minus_left [simp]   divide_minus_right [simp]
paulson@14293
  1106
paulson@14387
  1107
text{*Also, extract signs from products*}
paulson@17085
  1108
lemmas mult_minus_left = minus_mult_left [symmetric]
paulson@17085
  1109
lemmas mult_minus_right = minus_mult_right [symmetric]
paulson@17085
  1110
declare mult_minus_left [simp]   mult_minus_right [simp]
paulson@14387
  1111
paulson@14293
  1112
lemma minus_divide_divide [simp]:
paulson@14293
  1113
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
haftmann@21328
  1114
apply (cases "b=0", simp) 
paulson@14293
  1115
apply (simp add: nonzero_minus_divide_divide) 
paulson@14293
  1116
done
paulson@14293
  1117
paulson@14430
  1118
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
paulson@14387
  1119
by (simp add: diff_minus add_divide_distrib) 
paulson@14387
  1120
avigad@16775
  1121
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1122
    x / y - w / z = (x * z - w * y) / (y * z)"
avigad@16775
  1123
  apply (subst diff_def)+
avigad@16775
  1124
  apply (subst minus_divide_left)
avigad@16775
  1125
  apply (subst add_frac_eq)
avigad@16775
  1126
  apply simp_all
avigad@16775
  1127
done
paulson@14293
  1128
paulson@14268
  1129
subsection {* Ordered Fields *}
paulson@14268
  1130
paulson@14277
  1131
lemma positive_imp_inverse_positive: 
paulson@14269
  1132
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
paulson@14268
  1133
  proof -
paulson@14268
  1134
  have "0 < a * inverse a" 
paulson@14268
  1135
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
paulson@14268
  1136
  thus "0 < inverse a" 
paulson@14268
  1137
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
paulson@14268
  1138
  qed
paulson@14268
  1139
paulson@14277
  1140
lemma negative_imp_inverse_negative:
paulson@14268
  1141
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
paulson@14277
  1142
  by (insert positive_imp_inverse_positive [of "-a"], 
paulson@14268
  1143
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
paulson@14268
  1144
paulson@14268
  1145
lemma inverse_le_imp_le:
paulson@14269
  1146
      assumes invle: "inverse a \<le> inverse b"
paulson@14269
  1147
	  and apos:  "0 < a"
paulson@14269
  1148
	 shows "b \<le> (a::'a::ordered_field)"
paulson@14268
  1149
  proof (rule classical)
paulson@14268
  1150
  assume "~ b \<le> a"
paulson@14268
  1151
  hence "a < b"
paulson@14268
  1152
    by (simp add: linorder_not_le)
paulson@14268
  1153
  hence bpos: "0 < b"
paulson@14268
  1154
    by (blast intro: apos order_less_trans)
paulson@14268
  1155
  hence "a * inverse a \<le> a * inverse b"
paulson@14268
  1156
    by (simp add: apos invle order_less_imp_le mult_left_mono)
paulson@14268
  1157
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
paulson@14268
  1158
    by (simp add: bpos order_less_imp_le mult_right_mono)
paulson@14268
  1159
  thus "b \<le> a"
paulson@14268
  1160
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
paulson@14268
  1161
  qed
paulson@14268
  1162
paulson@14277
  1163
lemma inverse_positive_imp_positive:
paulson@14277
  1164
      assumes inv_gt_0: "0 < inverse a"
paulson@14277
  1165
          and [simp]:   "a \<noteq> 0"
paulson@14277
  1166
        shows "0 < (a::'a::ordered_field)"
paulson@14277
  1167
  proof -
paulson@14277
  1168
  have "0 < inverse (inverse a)"
paulson@14277
  1169
    by (rule positive_imp_inverse_positive)
paulson@14277
  1170
  thus "0 < a"
paulson@14277
  1171
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
  1172
  qed
paulson@14277
  1173
paulson@14277
  1174
lemma inverse_positive_iff_positive [simp]:
paulson@14277
  1175
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1176
apply (cases "a = 0", simp)
paulson@14277
  1177
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
paulson@14277
  1178
done
paulson@14277
  1179
paulson@14277
  1180
lemma inverse_negative_imp_negative:
paulson@14277
  1181
      assumes inv_less_0: "inverse a < 0"
paulson@14277
  1182
          and [simp]:   "a \<noteq> 0"
paulson@14277
  1183
        shows "a < (0::'a::ordered_field)"
paulson@14277
  1184
  proof -
paulson@14277
  1185
  have "inverse (inverse a) < 0"
paulson@14277
  1186
    by (rule negative_imp_inverse_negative)
paulson@14277
  1187
  thus "a < 0"
paulson@14277
  1188
    by (simp add: nonzero_inverse_inverse_eq)
paulson@14277
  1189
  qed
paulson@14277
  1190
paulson@14277
  1191
lemma inverse_negative_iff_negative [simp]:
paulson@14277
  1192
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1193
apply (cases "a = 0", simp)
paulson@14277
  1194
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
paulson@14277
  1195
done
paulson@14277
  1196
paulson@14277
  1197
lemma inverse_nonnegative_iff_nonnegative [simp]:
paulson@14277
  1198
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1199
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1200
paulson@14277
  1201
lemma inverse_nonpositive_iff_nonpositive [simp]:
paulson@14277
  1202
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
paulson@14277
  1203
by (simp add: linorder_not_less [symmetric])
paulson@14277
  1204
paulson@14277
  1205
paulson@14277
  1206
subsection{*Anti-Monotonicity of @{term inverse}*}
paulson@14277
  1207
paulson@14268
  1208
lemma less_imp_inverse_less:
paulson@14269
  1209
      assumes less: "a < b"
paulson@14269
  1210
	  and apos:  "0 < a"
paulson@14269
  1211
	shows "inverse b < inverse (a::'a::ordered_field)"
paulson@14268
  1212
  proof (rule ccontr)
paulson@14268
  1213
  assume "~ inverse b < inverse a"
paulson@14268
  1214
  hence "inverse a \<le> inverse b"
paulson@14268
  1215
    by (simp add: linorder_not_less)
paulson@14268
  1216
  hence "~ (a < b)"
paulson@14268
  1217
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
paulson@14268
  1218
  thus False
paulson@14268
  1219
    by (rule notE [OF _ less])
paulson@14268
  1220
  qed
paulson@14268
  1221
paulson@14268
  1222
lemma inverse_less_imp_less:
paulson@14268
  1223
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1224
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
paulson@14268
  1225
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
paulson@14268
  1226
done
paulson@14268
  1227
paulson@14268
  1228
text{*Both premises are essential. Consider -1 and 1.*}
paulson@14268
  1229
lemma inverse_less_iff_less [simp]:
paulson@14268
  1230
     "[|0 < a; 0 < b|] 
paulson@14268
  1231
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1232
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
paulson@14268
  1233
paulson@14268
  1234
lemma le_imp_inverse_le:
paulson@14268
  1235
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1236
  by (force simp add: order_le_less less_imp_inverse_less)
paulson@14268
  1237
paulson@14268
  1238
lemma inverse_le_iff_le [simp]:
paulson@14268
  1239
     "[|0 < a; 0 < b|] 
paulson@14268
  1240
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1241
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
paulson@14268
  1242
paulson@14268
  1243
paulson@14268
  1244
text{*These results refer to both operands being negative.  The opposite-sign
paulson@14268
  1245
case is trivial, since inverse preserves signs.*}
paulson@14268
  1246
lemma inverse_le_imp_le_neg:
paulson@14268
  1247
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
paulson@14268
  1248
  apply (rule classical) 
paulson@14268
  1249
  apply (subgoal_tac "a < 0") 
paulson@14268
  1250
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
paulson@14268
  1251
  apply (insert inverse_le_imp_le [of "-b" "-a"])
paulson@14268
  1252
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1253
  done
paulson@14268
  1254
paulson@14268
  1255
lemma less_imp_inverse_less_neg:
paulson@14268
  1256
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
paulson@14268
  1257
  apply (subgoal_tac "a < 0") 
paulson@14268
  1258
   prefer 2 apply (blast intro: order_less_trans) 
paulson@14268
  1259
  apply (insert less_imp_inverse_less [of "-b" "-a"])
paulson@14268
  1260
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1261
  done
paulson@14268
  1262
paulson@14268
  1263
lemma inverse_less_imp_less_neg:
paulson@14268
  1264
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
paulson@14268
  1265
  apply (rule classical) 
paulson@14268
  1266
  apply (subgoal_tac "a < 0") 
paulson@14268
  1267
   prefer 2
paulson@14268
  1268
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
paulson@14268
  1269
  apply (insert inverse_less_imp_less [of "-b" "-a"])
paulson@14268
  1270
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1271
  done
paulson@14268
  1272
paulson@14268
  1273
lemma inverse_less_iff_less_neg [simp]:
paulson@14268
  1274
     "[|a < 0; b < 0|] 
paulson@14268
  1275
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
paulson@14268
  1276
  apply (insert inverse_less_iff_less [of "-b" "-a"])
paulson@14268
  1277
  apply (simp del: inverse_less_iff_less 
paulson@14268
  1278
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
paulson@14268
  1279
  done
paulson@14268
  1280
paulson@14268
  1281
lemma le_imp_inverse_le_neg:
paulson@14268
  1282
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
paulson@14268
  1283
  by (force simp add: order_le_less less_imp_inverse_less_neg)
paulson@14268
  1284
paulson@14268
  1285
lemma inverse_le_iff_le_neg [simp]:
paulson@14268
  1286
     "[|a < 0; b < 0|] 
paulson@14268
  1287
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
paulson@14268
  1288
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
paulson@14265
  1289
paulson@14277
  1290
paulson@14365
  1291
subsection{*Inverses and the Number One*}
paulson@14365
  1292
paulson@14365
  1293
lemma one_less_inverse_iff:
paulson@14365
  1294
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
paulson@14365
  1295
  assume "0 < x"
paulson@14365
  1296
    with inverse_less_iff_less [OF zero_less_one, of x]
paulson@14365
  1297
    show ?thesis by simp
paulson@14365
  1298
next
paulson@14365
  1299
  assume notless: "~ (0 < x)"
paulson@14365
  1300
  have "~ (1 < inverse x)"
paulson@14365
  1301
  proof
paulson@14365
  1302
    assume "1 < inverse x"
paulson@14365
  1303
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
paulson@14365
  1304
    also have "... < 1" by (rule zero_less_one) 
paulson@14365
  1305
    finally show False by auto
paulson@14365
  1306
  qed
paulson@14365
  1307
  with notless show ?thesis by simp
paulson@14365
  1308
qed
paulson@14365
  1309
paulson@14365
  1310
lemma inverse_eq_1_iff [simp]:
paulson@14365
  1311
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
paulson@14365
  1312
by (insert inverse_eq_iff_eq [of x 1], simp) 
paulson@14365
  1313
paulson@14365
  1314
lemma one_le_inverse_iff:
paulson@14365
  1315
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1316
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
paulson@14365
  1317
                    eq_commute [of 1]) 
paulson@14365
  1318
paulson@14365
  1319
lemma inverse_less_1_iff:
paulson@14365
  1320
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1321
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
paulson@14365
  1322
paulson@14365
  1323
lemma inverse_le_1_iff:
paulson@14365
  1324
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
paulson@14365
  1325
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
paulson@14365
  1326
paulson@14288
  1327
subsection{*Simplification of Inequalities Involving Literal Divisors*}
paulson@14288
  1328
paulson@14288
  1329
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
paulson@14288
  1330
proof -
paulson@14288
  1331
  assume less: "0<c"
paulson@14288
  1332
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
paulson@14288
  1333
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1334
  also have "... = (a*c \<le> b)"
paulson@14288
  1335
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1336
  finally show ?thesis .
paulson@14288
  1337
qed
paulson@14288
  1338
paulson@14288
  1339
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
paulson@14288
  1340
proof -
paulson@14288
  1341
  assume less: "c<0"
paulson@14288
  1342
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
paulson@14288
  1343
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1344
  also have "... = (b \<le> a*c)"
paulson@14288
  1345
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1346
  finally show ?thesis .
paulson@14288
  1347
qed
paulson@14288
  1348
paulson@14288
  1349
lemma le_divide_eq:
paulson@14288
  1350
  "(a \<le> b/c) = 
paulson@14288
  1351
   (if 0 < c then a*c \<le> b
paulson@14288
  1352
             else if c < 0 then b \<le> a*c
paulson@14288
  1353
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1354
apply (cases "c=0", simp) 
paulson@14288
  1355
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
paulson@14288
  1356
done
paulson@14288
  1357
paulson@14288
  1358
lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
paulson@14288
  1359
proof -
paulson@14288
  1360
  assume less: "0<c"
paulson@14288
  1361
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
paulson@14288
  1362
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1363
  also have "... = (b \<le> a*c)"
paulson@14288
  1364
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1365
  finally show ?thesis .
paulson@14288
  1366
qed
paulson@14288
  1367
paulson@14288
  1368
lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
paulson@14288
  1369
proof -
paulson@14288
  1370
  assume less: "c<0"
paulson@14288
  1371
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
paulson@14288
  1372
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
paulson@14288
  1373
  also have "... = (a*c \<le> b)"
paulson@14288
  1374
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1375
  finally show ?thesis .
paulson@14288
  1376
qed
paulson@14288
  1377
paulson@14288
  1378
lemma divide_le_eq:
paulson@14288
  1379
  "(b/c \<le> a) = 
paulson@14288
  1380
   (if 0 < c then b \<le> a*c
paulson@14288
  1381
             else if c < 0 then a*c \<le> b
paulson@14288
  1382
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1383
apply (cases "c=0", simp) 
paulson@14288
  1384
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
paulson@14288
  1385
done
paulson@14288
  1386
paulson@14288
  1387
lemma pos_less_divide_eq:
paulson@14288
  1388
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
paulson@14288
  1389
proof -
paulson@14288
  1390
  assume less: "0<c"
paulson@14288
  1391
  hence "(a < b/c) = (a*c < (b/c)*c)"
paulson@15234
  1392
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1393
  also have "... = (a*c < b)"
paulson@14288
  1394
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1395
  finally show ?thesis .
paulson@14288
  1396
qed
paulson@14288
  1397
paulson@14288
  1398
lemma neg_less_divide_eq:
paulson@14288
  1399
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
paulson@14288
  1400
proof -
paulson@14288
  1401
  assume less: "c<0"
paulson@14288
  1402
  hence "(a < b/c) = ((b/c)*c < a*c)"
paulson@15234
  1403
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1404
  also have "... = (b < a*c)"
paulson@14288
  1405
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1406
  finally show ?thesis .
paulson@14288
  1407
qed
paulson@14288
  1408
paulson@14288
  1409
lemma less_divide_eq:
paulson@14288
  1410
  "(a < b/c) = 
paulson@14288
  1411
   (if 0 < c then a*c < b
paulson@14288
  1412
             else if c < 0 then b < a*c
paulson@14288
  1413
             else  a < (0::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1414
apply (cases "c=0", simp) 
paulson@14288
  1415
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
paulson@14288
  1416
done
paulson@14288
  1417
paulson@14288
  1418
lemma pos_divide_less_eq:
paulson@14288
  1419
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
paulson@14288
  1420
proof -
paulson@14288
  1421
  assume less: "0<c"
paulson@14288
  1422
  hence "(b/c < a) = ((b/c)*c < a*c)"
paulson@15234
  1423
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1424
  also have "... = (b < a*c)"
paulson@14288
  1425
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
paulson@14288
  1426
  finally show ?thesis .
paulson@14288
  1427
qed
paulson@14288
  1428
paulson@14288
  1429
lemma neg_divide_less_eq:
paulson@14288
  1430
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
paulson@14288
  1431
proof -
paulson@14288
  1432
  assume less: "c<0"
paulson@14288
  1433
  hence "(b/c < a) = (a*c < (b/c)*c)"
paulson@15234
  1434
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
paulson@14288
  1435
  also have "... = (a*c < b)"
paulson@14288
  1436
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
paulson@14288
  1437
  finally show ?thesis .
paulson@14288
  1438
qed
paulson@14288
  1439
paulson@14288
  1440
lemma divide_less_eq:
paulson@14288
  1441
  "(b/c < a) = 
paulson@14288
  1442
   (if 0 < c then b < a*c
paulson@14288
  1443
             else if c < 0 then a*c < b
paulson@14288
  1444
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
haftmann@21328
  1445
apply (cases "c=0", simp) 
paulson@14288
  1446
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
paulson@14288
  1447
done
paulson@14288
  1448
paulson@14288
  1449
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
paulson@14288
  1450
proof -
paulson@14288
  1451
  assume [simp]: "c\<noteq>0"
paulson@14288
  1452
  have "(a = b/c) = (a*c = (b/c)*c)"
paulson@14288
  1453
    by (simp add: field_mult_cancel_right)
paulson@14288
  1454
  also have "... = (a*c = b)"
paulson@14288
  1455
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1456
  finally show ?thesis .
paulson@14288
  1457
qed
paulson@14288
  1458
paulson@14288
  1459
lemma eq_divide_eq:
paulson@14288
  1460
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
paulson@14288
  1461
by (simp add: nonzero_eq_divide_eq) 
paulson@14288
  1462
paulson@14288
  1463
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
paulson@14288
  1464
proof -
paulson@14288
  1465
  assume [simp]: "c\<noteq>0"
paulson@14288
  1466
  have "(b/c = a) = ((b/c)*c = a*c)"
paulson@14288
  1467
    by (simp add: field_mult_cancel_right)
paulson@14288
  1468
  also have "... = (b = a*c)"
paulson@14288
  1469
    by (simp add: divide_inverse mult_assoc) 
paulson@14288
  1470
  finally show ?thesis .
paulson@14288
  1471
qed
paulson@14288
  1472
paulson@14288
  1473
lemma divide_eq_eq:
paulson@14288
  1474
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
paulson@14288
  1475
by (force simp add: nonzero_divide_eq_eq) 
paulson@14288
  1476
avigad@16775
  1477
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
avigad@16775
  1478
    b = a * c ==> b / c = a"
avigad@16775
  1479
  by (subst divide_eq_eq, simp)
avigad@16775
  1480
avigad@16775
  1481
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
avigad@16775
  1482
    a * c = b ==> a = b / c"
avigad@16775
  1483
  by (subst eq_divide_eq, simp)
avigad@16775
  1484
avigad@16775
  1485
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
avigad@16775
  1486
    (x / y = w / z) = (x * z = w * y)"
avigad@16775
  1487
  apply (subst nonzero_eq_divide_eq)
avigad@16775
  1488
  apply assumption
avigad@16775
  1489
  apply (subst times_divide_eq_left)
avigad@16775
  1490
  apply (erule nonzero_divide_eq_eq) 
avigad@16775
  1491
done
avigad@16775
  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
paulson@14288
  1571
subsection{*Cancellation Laws for Division*}
paulson@14288
  1572
paulson@14288
  1573
lemma divide_cancel_right [simp]:
paulson@14288
  1574
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
haftmann@21328
  1575
apply (cases "c=0", simp) 
paulson@14430
  1576
apply (simp add: divide_inverse field_mult_cancel_right) 
paulson@14288
  1577
done
paulson@14288
  1578
paulson@14288
  1579
lemma divide_cancel_left [simp]:
paulson@14288
  1580
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
haftmann@21328
  1581
apply (cases "c=0", simp) 
paulson@14430
  1582
apply (simp add: divide_inverse field_mult_cancel_left) 
paulson@14288
  1583
done
paulson@14288
  1584
paulson@14353
  1585
subsection {* Division and the Number One *}
paulson@14353
  1586
paulson@14353
  1587
text{*Simplify expressions equated with 1*}
paulson@14353
  1588
lemma divide_eq_1_iff [simp]:
paulson@14353
  1589
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
haftmann@21328
  1590
apply (cases "b=0", simp) 
paulson@14353
  1591
apply (simp add: right_inverse_eq) 
paulson@14353
  1592
done
paulson@14353
  1593
paulson@14353
  1594
lemma one_eq_divide_iff [simp]:
paulson@14353
  1595
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
paulson@14353
  1596
by (simp add: eq_commute [of 1])  
paulson@14353
  1597
paulson@14353
  1598
lemma zero_eq_1_divide_iff [simp]:
paulson@14353
  1599
     "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
haftmann@21328
  1600
apply (cases "a=0", simp) 
paulson@14353
  1601
apply (auto simp add: nonzero_eq_divide_eq) 
paulson@14353
  1602
done
paulson@14353
  1603
paulson@14353
  1604
lemma one_divide_eq_0_iff [simp]:
paulson@14353
  1605
     "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
haftmann@21328
  1606
apply (cases "a=0", simp) 
paulson@14353
  1607
apply (insert zero_neq_one [THEN not_sym]) 
paulson@14353
  1608
apply (auto simp add: nonzero_divide_eq_eq) 
paulson@14353
  1609
done
paulson@14353
  1610
paulson@14353
  1611
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
paulson@18623
  1612
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
paulson@18623
  1613
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
paulson@18623
  1614
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
paulson@18623
  1615
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
paulson@17085
  1616
paulson@17085
  1617
declare zero_less_divide_1_iff [simp]
paulson@17085
  1618
declare divide_less_0_1_iff [simp]
paulson@17085
  1619
declare zero_le_divide_1_iff [simp]
paulson@17085
  1620
declare divide_le_0_1_iff [simp]
paulson@14353
  1621
paulson@14293
  1622
subsection {* Ordering Rules for Division *}
paulson@14293
  1623
paulson@14293
  1624
lemma divide_strict_right_mono:
paulson@14293
  1625
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
paulson@14293
  1626
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
paulson@14293
  1627
              positive_imp_inverse_positive) 
paulson@14293
  1628
paulson@14293
  1629
lemma divide_right_mono:
paulson@14293
  1630
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
paulson@14293
  1631
  by (force simp add: divide_strict_right_mono order_le_less) 
paulson@14293
  1632
avigad@16775
  1633
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1634
    ==> c <= 0 ==> b / c <= a / c"
avigad@16775
  1635
  apply (drule divide_right_mono [of _ _ "- c"])
avigad@16775
  1636
  apply auto
avigad@16775
  1637
done
avigad@16775
  1638
avigad@16775
  1639
lemma divide_strict_right_mono_neg:
avigad@16775
  1640
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
avigad@16775
  1641
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 
avigad@16775
  1642
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
avigad@16775
  1643
done
paulson@14293
  1644
paulson@14293
  1645
text{*The last premise ensures that @{term a} and @{term b} 
paulson@14293
  1646
      have the same sign*}
paulson@14293
  1647
lemma divide_strict_left_mono:
paulson@14293
  1648
       "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1649
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 
paulson@14293
  1650
      order_less_imp_not_eq order_less_imp_not_eq2  
paulson@14293
  1651
      less_imp_inverse_less less_imp_inverse_less_neg) 
paulson@14293
  1652
paulson@14293
  1653
lemma divide_left_mono:
paulson@14293
  1654
     "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
paulson@14293
  1655
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1656
   prefer 2 
paulson@14293
  1657
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
haftmann@21328
  1658
  apply (cases "c=0", simp add: divide_inverse)
paulson@14293
  1659
  apply (force simp add: divide_strict_left_mono order_le_less) 
paulson@14293
  1660
  done
paulson@14293
  1661
avigad@16775
  1662
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b 
avigad@16775
  1663
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
avigad@16775
  1664
  apply (drule divide_left_mono [of _ _ "- c"])
avigad@16775
  1665
  apply (auto simp add: mult_commute)
avigad@16775
  1666
done
avigad@16775
  1667
paulson@14293
  1668
lemma divide_strict_left_mono_neg:
paulson@14293
  1669
     "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
paulson@14293
  1670
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
paulson@14293
  1671
   prefer 2 
paulson@14293
  1672
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
paulson@14293
  1673
  apply (drule divide_strict_left_mono [of _ _ "-c"]) 
paulson@14293
  1674
   apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 
paulson@14293
  1675
  done
paulson@14293
  1676
avigad@16775
  1677
text{*Simplify quotients that are compared with the value 1.*}
avigad@16775
  1678
avigad@16775
  1679
lemma le_divide_eq_1:
avigad@16775
  1680
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1681
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
avigad@16775
  1682
by (auto simp add: le_divide_eq)
avigad@16775
  1683
avigad@16775
  1684
lemma divide_le_eq_1:
avigad@16775
  1685
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1686
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
avigad@16775
  1687
by (auto simp add: divide_le_eq)
avigad@16775
  1688
avigad@16775
  1689
lemma less_divide_eq_1:
avigad@16775
  1690
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1691
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
avigad@16775
  1692
by (auto simp add: less_divide_eq)
avigad@16775
  1693
avigad@16775
  1694
lemma divide_less_eq_1:
avigad@16775
  1695
  fixes a :: "'a :: {ordered_field,division_by_zero}"
avigad@16775
  1696
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
avigad@16775
  1697
by (auto simp add: divide_less_eq)
avigad@16775
  1698
avigad@16775
  1699
subsection{*Conditional Simplification Rules: No Case Splits*}
avigad@16775
  1700
avigad@16775
  1701
lemma le_divide_eq_1_pos [simp]:
avigad@16775
  1702
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1703
  shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
avigad@16775
  1704
by (auto simp add: le_divide_eq)
avigad@16775
  1705
avigad@16775
  1706
lemma le_divide_eq_1_neg [simp]:
avigad@16775
  1707
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1708
  shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
avigad@16775
  1709
by (auto simp add: le_divide_eq)
avigad@16775
  1710
avigad@16775
  1711
lemma divide_le_eq_1_pos [simp]:
avigad@16775
  1712
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1713
  shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
avigad@16775
  1714
by (auto simp add: divide_le_eq)
avigad@16775
  1715
avigad@16775
  1716
lemma divide_le_eq_1_neg [simp]:
avigad@16775
  1717
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1718
  shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
avigad@16775
  1719
by (auto simp add: divide_le_eq)
avigad@16775
  1720
avigad@16775
  1721
lemma less_divide_eq_1_pos [simp]:
avigad@16775
  1722
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1723
  shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
avigad@16775
  1724
by (auto simp add: less_divide_eq)
avigad@16775
  1725
avigad@16775
  1726
lemma less_divide_eq_1_neg [simp]:
avigad@16775
  1727
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1728
  shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
avigad@16775
  1729
by (auto simp add: less_divide_eq)
avigad@16775
  1730
avigad@16775
  1731
lemma divide_less_eq_1_pos [simp]:
avigad@16775
  1732
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1733
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
paulson@18649
  1734
by (auto simp add: divide_less_eq)
paulson@18649
  1735
paulson@18649
  1736
lemma divide_less_eq_1_neg [simp]:
paulson@18649
  1737
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1738
  shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
avigad@16775
  1739
by (auto simp add: divide_less_eq)
avigad@16775
  1740
avigad@16775
  1741
lemma eq_divide_eq_1 [simp]:
avigad@16775
  1742
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1743
  shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1744
by (auto simp add: eq_divide_eq)
avigad@16775
  1745
avigad@16775
  1746
lemma divide_eq_eq_1 [simp]:
avigad@16775
  1747
  fixes a :: "'a :: {ordered_field,division_by_zero}"
paulson@18649
  1748
  shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
avigad@16775
  1749
by (auto simp add: divide_eq_eq)
avigad@16775
  1750
avigad@16775
  1751
subsection {* Reasoning about inequalities with division *}
avigad@16775
  1752
avigad@16775
  1753
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1754
    ==> x * y <= x"
avigad@16775
  1755
  by (auto simp add: mult_compare_simps);
avigad@16775
  1756
avigad@16775
  1757
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
avigad@16775
  1758
    ==> y * x <= x"
avigad@16775
  1759
  by (auto simp add: mult_compare_simps);
avigad@16775
  1760
avigad@16775
  1761
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
avigad@16775
  1762
    x / y <= z";
avigad@16775
  1763
  by (subst pos_divide_le_eq, assumption+);
avigad@16775
  1764
avigad@16775
  1765
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
avigad@16775
  1766
    z <= x / y";
avigad@16775
  1767
  by (subst pos_le_divide_eq, assumption+)
avigad@16775
  1768
avigad@16775
  1769
lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
avigad@16775
  1770
    x / y < z"
avigad@16775
  1771
  by (subst pos_divide_less_eq, assumption+)
avigad@16775
  1772
avigad@16775
  1773
lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
avigad@16775
  1774
    z < x / y"
avigad@16775
  1775
  by (subst pos_less_divide_eq, assumption+)
avigad@16775
  1776
avigad@16775
  1777
lemma frac_le: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1778
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
avigad@16775
  1779
  apply (rule mult_imp_div_pos_le)
avigad@16775
  1780
  apply simp;
avigad@16775
  1781
  apply (subst times_divide_eq_left);
avigad@16775
  1782
  apply (rule mult_imp_le_div_pos, assumption)
avigad@16775
  1783
  apply (rule mult_mono)
avigad@16775
  1784
  apply simp_all
paulson@14293
  1785
done
paulson@14293
  1786
avigad@16775
  1787
lemma frac_less: "(0::'a::ordered_field) <= x ==> 
avigad@16775
  1788
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
avigad@16775
  1789
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1790
  apply simp;
avigad@16775
  1791
  apply (subst times_divide_eq_left);
avigad@16775
  1792
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1793
  apply (erule mult_less_le_imp_less)
avigad@16775
  1794
  apply simp_all
avigad@16775
  1795
done
avigad@16775
  1796
avigad@16775
  1797
lemma frac_less2: "(0::'a::ordered_field) < x ==> 
avigad@16775
  1798
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
avigad@16775
  1799
  apply (rule mult_imp_div_pos_less)
avigad@16775
  1800
  apply simp_all
avigad@16775
  1801
  apply (subst times_divide_eq_left);
avigad@16775
  1802
  apply (rule mult_imp_less_div_pos, assumption)
avigad@16775
  1803
  apply (erule mult_le_less_imp_less)
avigad@16775
  1804
  apply simp_all
avigad@16775
  1805
done
avigad@16775
  1806
avigad@16775
  1807
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
avigad@16775
  1808
avigad@16775
  1809
text{*It's not obvious whether these should be simprules or not. 
avigad@16775
  1810
  Their effect is to gather terms into one big fraction, like
avigad@16775
  1811
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
avigad@16775
  1812
  seem to need them.*}
avigad@16775
  1813
avigad@16775
  1814
declare times_divide_eq [simp]
paulson@14293
  1815
paulson@14293
  1816
subsection {* Ordered Fields are Dense *}
paulson@14293
  1817
obua@14738
  1818
lemma less_add_one: "a < (a+1::'a::ordered_semidom)"
paulson@14293
  1819
proof -
obua@14738
  1820
  have "a+0 < (a+1::'a::ordered_semidom)"
paulson@14365
  1821
    by (blast intro: zero_less_one add_strict_left_mono) 
paulson@14293
  1822
  thus ?thesis by simp
paulson@14293
  1823
qed
paulson@14293
  1824
obua@14738
  1825
lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"
paulson@14365
  1826
  by (blast intro: order_less_trans zero_less_one less_add_one) 
paulson@14365
  1827
paulson@14293
  1828
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
paulson@14293
  1829
by (simp add: zero_less_two pos_less_divide_eq right_distrib) 
paulson@14293
  1830
paulson@14293
  1831
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
paulson@14293
  1832
by (simp add: zero_less_two pos_divide_less_eq right_distrib) 
paulson@14293
  1833
paulson@14293
  1834
lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
paulson@14293
  1835
by (blast intro!: less_half_sum gt_half_sum)
paulson@14293
  1836
paulson@15234
  1837
paulson@14293
  1838
subsection {* Absolute Value *}
paulson@14293
  1839
obua@14738
  1840
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
paulson@14294
  1841
  by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) 
paulson@14294
  1842
obua@14738
  1843
lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 
obua@14738
  1844
proof -
obua@14738
  1845
  let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
obua@14738
  1846
  let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@14738
  1847
  have a: "(abs a) * (abs b) = ?x"
obua@14738
  1848
    by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)
obua@14738
  1849
  {
obua@14738
  1850
    fix u v :: 'a
paulson@15481
  1851
    have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> 
paulson@15481
  1852
              u * v = pprt a * pprt b + pprt a * nprt b + 
paulson@15481
  1853
                      nprt a * pprt b + nprt a * nprt b"
obua@14738
  1854
      apply (subst prts[of u], subst prts[of v])
obua@14738
  1855
      apply (simp add: left_distrib right_distrib add_ac) 
obua@14738
  1856
      done
obua@14738
  1857
  }
obua@14738
  1858
  note b = this[OF refl[of a] refl[of b]]
obua@14738
  1859
  note addm = add_mono[of "0::'a" _ "0::'a", simplified]
obua@14738
  1860
  note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]
obua@14738
  1861
  have xy: "- ?x <= ?y"
obua@14754
  1862
    apply (simp)
obua@14754
  1863
    apply (rule_tac y="0::'a" in order_trans)
nipkow@16568
  1864
    apply (rule addm2)
avigad@16775
  1865
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
nipkow@16568
  1866
    apply (rule addm)
avigad@16775
  1867
    apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
obua@14754
  1868
    done
obua@14738
  1869
  have yx: "?y <= ?x"
nipkow@16568
  1870
    apply (simp add:diff_def)
obua@14754
  1871
    apply (rule_tac y=0 in order_trans)
avigad@16775
  1872
    apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
avigad@16775
  1873
    apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
obua@14738
  1874
    done
obua@14738
  1875
  have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
obua@14738
  1876
  have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
obua@14738
  1877
  show ?thesis
obua@14738
  1878
    apply (rule abs_leI)
obua@14738
  1879
    apply (simp add: i1)
obua@14738
  1880
    apply (simp add: i2[simplified minus_le_iff])
obua@14738
  1881
    done
obua@14738
  1882
qed
paulson@14294
  1883
obua@14738
  1884
lemma abs_eq_mult: 
obua@14738
  1885
  assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
obua@14738
  1886
  shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
obua@14738
  1887
proof -
obua@14738
  1888
  have s: "(0 <= a*b) | (a*b <= 0)"
obua@14738
  1889
    apply (auto)    
obua@14738
  1890
    apply (rule_tac split_mult_pos_le)
obua@14738
  1891
    apply (rule_tac contrapos_np[of "a*b <= 0"])
obua@14738
  1892
    apply (simp)
obua@14738
  1893
    apply (rule_tac split_mult_neg_le)
obua@14738
  1894
    apply (insert prems)
obua@14738
  1895
    apply (blast)
obua@14738
  1896
    done
obua@14738
  1897
  have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
obua@14738
  1898
    by (simp add: prts[symmetric])
obua@14738
  1899
  show ?thesis
obua@14738
  1900
  proof cases
obua@14738
  1901
    assume "0 <= a * b"
obua@14738
  1902
    then show ?thesis
obua@14738
  1903
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1904
      apply (insert prems)
obua@14754
  1905
      apply (auto simp add: 
obua@14754
  1906
	ring_eq_simps 
obua@14754
  1907
	iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]
nipkow@15197
  1908
	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id])
avigad@16775
  1909
	apply(drule (1) mult_nonneg_nonpos[of a b], simp)
avigad@16775
  1910
	apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
obua@14738
  1911
      done
obua@14738
  1912
  next
obua@14738
  1913
    assume "~(0 <= a*b)"
obua@14738
  1914
    with s have "a*b <= 0" by simp
obua@14738
  1915
    then show ?thesis
obua@14738
  1916
      apply (simp_all add: mulprts abs_prts)
obua@14738
  1917
      apply (insert prems)
obua@15580
  1918
      apply (auto simp add: ring_eq_simps)
avigad@16775
  1919
      apply(drule (1) mult_nonneg_nonneg[of a b],simp)
avigad@16775
  1920
      apply(drule (1) mult_nonpos_nonpos[of a b],simp)
obua@14738
  1921
      done
obua@14738
  1922
  qed
obua@14738
  1923
qed
paulson@14294
  1924
obua@14738
  1925
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 
obua@14738
  1926
by (simp add: abs_eq_mult linorder_linear)
paulson@14293
  1927
obua@14738
  1928
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
obua@14738
  1929
by (simp add: abs_if) 
paulson@14294
  1930
paulson@14294
  1931
lemma nonzero_abs_inverse:
paulson@14294
  1932
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
paulson@14294
  1933
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
paulson@14294
  1934
                      negative_imp_inverse_negative)
paulson@14294
  1935
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
paulson@14294
  1936
done
paulson@14294
  1937
paulson@14294
  1938
lemma abs_inverse [simp]:
paulson@14294
  1939
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
paulson@14294
  1940
      inverse (abs a)"
haftmann@21328
  1941
apply (cases "a=0", simp) 
paulson@14294
  1942
apply (simp add: nonzero_abs_inverse) 
paulson@14294
  1943
done
paulson@14294
  1944
paulson@14294
  1945
lemma nonzero_abs_divide:
paulson@14294
  1946
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
paulson@14294
  1947
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
paulson@14294
  1948
paulson@15234
  1949
lemma abs_divide [simp]:
paulson@14294
  1950
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
haftmann@21328
  1951
apply (cases "b=0", simp) 
paulson@14294
  1952
apply (simp add: nonzero_abs_divide) 
paulson@14294
  1953
done
paulson@14294
  1954
paulson@14294
  1955
lemma abs_mult_less:
obua@14738
  1956
     "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"
paulson@14294
  1957
proof -
paulson@14294
  1958
  assume ac: "abs a < c"
paulson@14294
  1959
  hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)
paulson@14294
  1960
  assume "abs b < d"
paulson@14294
  1961
  thus ?thesis by (simp add: ac cpos mult_strict_mono) 
paulson@14294
  1962
qed
paulson@14293
  1963
obua@14738
  1964
lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"
obua@14738
  1965
by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)
obua@14738
  1966
obua@14738
  1967
lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"
obua@14738
  1968
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)
obua@14738
  1969
obua@14738
  1970
lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 
obua@14738
  1971
apply (simp add: order_less_le abs_le_iff)  
obua@14738
  1972
apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)
obua@14738
  1973
apply (simp add: le_minus_self_iff linorder_neq_iff) 
obua@14738
  1974
done
obua@14738
  1975
avigad@16775
  1976
lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==> 
avigad@16775
  1977
    (abs y) * x = abs (y * x)";
avigad@16775
  1978
  apply (subst abs_mult);
avigad@16775
  1979
  apply simp;
avigad@16775
  1980
done;
avigad@16775
  1981
avigad@16775
  1982
lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==> 
avigad@16775
  1983
    abs x / y = abs (x / y)";
avigad@16775
  1984
  apply (subst abs_divide);
avigad@16775
  1985
  apply (simp add: order_less_imp_le);
avigad@16775
  1986
done;
avigad@16775
  1987
obua@19404
  1988
subsection {* Bounds of products via negative and positive Part *}
obua@15178
  1989
obua@15580
  1990
lemma mult_le_prts:
obua@15580
  1991
  assumes
obua@15580
  1992
  "a1 <= (a::'a::lordered_ring)"
obua@15580
  1993
  "a <= a2"
obua@15580
  1994
  "b1 <= b"
obua@15580
  1995
  "b <= b2"
obua@15580
  1996
  shows
obua@15580
  1997
  "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
obua@15580
  1998
proof - 
obua@15580
  1999
  have "a * b = (pprt a + nprt a) * (pprt b + nprt b)" 
obua@15580
  2000
    apply (subst prts[symmetric])+
obua@15580
  2001
    apply simp
obua@15580
  2002
    done
obua@15580
  2003
  then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
obua@15580
  2004
    by (simp add: ring_eq_simps)
obua@15580
  2005
  moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
obua@15580
  2006
    by (simp_all add: prems mult_mono)
obua@15580
  2007
  moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
obua@15580
  2008
  proof -
obua@15580
  2009
    have "pprt a * nprt b <= pprt a * nprt b2"
obua@15580
  2010
      by (simp add: mult_left_mono prems)
obua@15580
  2011
    moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
obua@15580
  2012
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2013
    ultimately show ?thesis
obua@15580
  2014
      by simp
obua@15580
  2015
  qed
obua@15580
  2016
  moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
obua@15580
  2017
  proof - 
obua@15580
  2018
    have "nprt a * pprt b <= nprt a2 * pprt b"
obua@15580
  2019
      by (simp add: mult_right_mono prems)
obua@15580
  2020
    moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
obua@15580
  2021
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2022
    ultimately show ?thesis
obua@15580
  2023
      by simp
obua@15580
  2024
  qed
obua@15580
  2025
  moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
obua@15580
  2026
  proof -
obua@15580
  2027
    have "nprt a * nprt b <= nprt a * nprt b1"
obua@15580
  2028
      by (simp add: mult_left_mono_neg prems)
obua@15580
  2029
    moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
obua@15580
  2030
      by (simp add: mult_right_mono_neg prems)
obua@15580
  2031
    ultimately show ?thesis
obua@15580
  2032
      by simp
obua@15580
  2033
  qed
obua@15580
  2034
  ultimately show ?thesis
obua@15580
  2035
    by - (rule add_mono | simp)+
obua@15580
  2036
qed
obua@19404
  2037
obua@19404
  2038
lemma mult_ge_prts:
obua@15178
  2039
  assumes
obua@19404
  2040
  "a1 <= (a::'a::lordered_ring)"
obua@19404
  2041
  "a <= a2"
obua@19404
  2042
  "b1 <= b"
obua@19404
  2043
  "b <= b2"
obua@15178
  2044
  shows
obua@19404
  2045
  "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
obua@19404
  2046
proof - 
obua@19404
  2047
  from prems have a1:"- a2 <= -a" by auto
obua@19404
  2048
  from prems have a2: "-a <= -a1" by auto
obua@19404
  2049
  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
  2050
  have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp  
obua@19404
  2051
  then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
obua@19404
  2052
    by (simp only: minus_le_iff)
obua@19404
  2053
  then show ?thesis by simp
obua@15178
  2054
qed
obua@15178
  2055
haftmann@22842
  2056
subsection {* Theorems for proof tools *}
haftmann@22842
  2057
haftmann@22842
  2058
lemma add_mono_thms_ordered_semiring:
haftmann@22842
  2059
  fixes i j k :: "'a\<Colon>pordered_ab_semigroup_add"
haftmann@22842
  2060
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2061
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2062
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@22842
  2063
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@22842
  2064
by (rule add_mono, clarify+)+
haftmann@22842
  2065
haftmann@22842
  2066
lemma add_mono_thms_ordered_field:
haftmann@22842
  2067
  fixes i j k :: "'a\<Colon>pordered_cancel_ab_semigroup_add"
haftmann@22842
  2068
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2069
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2070
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2071
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2072
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@22842
  2073
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@22842
  2074
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@22842
  2075
paulson@14265
  2076
end