src/HOL/Ring_and_Field.thy
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer, Steven Obua, Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel,
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             with contributions by Jeremy Avigad
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*)
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header {* (Ordered) Rings and Fields *}
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theory Ring_and_Field
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imports OrderedGroup
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begin
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text {*
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  The theory of partially ordered rings is taken from the books:
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  \begin{itemize}
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  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
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  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
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  \end{itemize}
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  Most of the used notions can also be looked up in 
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  \begin{itemize}
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  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
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  \item \emph{Algebra I} by van der Waerden, Springer.
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  \end{itemize}
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*}
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class semiring = ab_semigroup_add + semigroup_mult +
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  assumes left_distrib: "(a \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* c"
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  assumes right_distrib: "a \<^loc>* (b \<^loc>+ c) = a \<^loc>* b \<^loc>+ a \<^loc>* c"
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "\<^loc>0 \<^loc>* a = \<^loc>0"
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  assumes mult_zero_right [simp]: "a \<^loc>* \<^loc>0 = \<^loc>0"
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class semiring_0 = semiring + comm_monoid_add + mult_zero
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class semiring_0_cancel = semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance semiring_0_cancel \<subseteq> semiring_0
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proof
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  fix a :: 'a
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  have "0 * a + 0 * a = 0 * a + 0"
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    by (simp add: left_distrib [symmetric])
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  thus "0 * a = 0"
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    by (simp only: add_left_cancel)
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  have "a * 0 + a * 0 = a * 0 + 0"
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    by (simp add: right_distrib [symmetric])
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  thus "a * 0 = 0"
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    by (simp only: add_left_cancel)
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qed
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a \<^loc>+ b) \<^loc>* c = a \<^loc>* c \<^loc>+ b \<^loc>* c"
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instance comm_semiring \<subseteq> semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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instance comm_semiring_0 \<subseteq> semiring_0 ..
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class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
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instance comm_semiring_0_cancel \<subseteq> comm_semiring_0 ..
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "\<^loc>0 \<noteq> \<^loc>1"
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class semiring_1 = zero_neq_one + semiring_0 + monoid_mult
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class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
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  (*previously almost_semiring*)
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instance comm_semiring_1 \<subseteq> semiring_1 ..
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class no_zero_divisors = zero + times +
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  assumes no_zero_divisors: "a \<noteq> \<^loc>0 \<Longrightarrow> b \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* b \<noteq> \<^loc>0"
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class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one
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  + cancel_ab_semigroup_add + monoid_mult
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instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
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instance semiring_1_cancel \<subseteq> semiring_1 ..
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class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult
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  + zero_neq_one + cancel_ab_semigroup_add
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instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_1 ..
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class ring = semiring + ab_group_add
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instance ring \<subseteq> semiring_0_cancel ..
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class comm_ring = comm_semiring + ab_group_add
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instance comm_ring \<subseteq> ring ..
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instance comm_ring \<subseteq> comm_semiring_0_cancel ..
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class ring_1 = ring + zero_neq_one + monoid_mult
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instance ring_1 \<subseteq> semiring_1_cancel ..
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class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
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  (*previously ring*)
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instance comm_ring_1 \<subseteq> ring_1 ..
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instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..
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class ring_no_zero_divisors = ring + no_zero_divisors
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class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
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class idom = comm_ring_1 + no_zero_divisors
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instance idom \<subseteq> ring_1_no_zero_divisors ..
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class division_ring = ring_1 + inverse +
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  assumes left_inverse [simp]:  "a \<noteq> \<^loc>0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1"
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  assumes right_inverse [simp]: "a \<noteq> \<^loc>0 \<Longrightarrow> a \<^loc>* inverse a = \<^loc>1"
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instance division_ring \<subseteq> ring_1_no_zero_divisors
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proof
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  fix a b :: 'a
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  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
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  show "a * b \<noteq> 0"
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  proof
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    assume ab: "a * b = 0"
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    hence "0 = inverse a * (a * b) * inverse b"
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      by simp
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    also have "\<dots> = (inverse a * a) * (b * inverse b)"
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      by (simp only: mult_assoc)
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    also have "\<dots> = 1"
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      using a b by simp
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    finally show False
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      by simp
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  qed
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qed
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class field = comm_ring_1 + inverse +
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  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a \<^loc>* a = \<^loc>1"
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  assumes divide_inverse: "a \<^loc>/ b = a \<^loc>* inverse b"
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instance field \<subseteq> division_ring
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proof
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  fix a :: 'a
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  assume "a \<noteq> 0"
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  thus "inverse a * a = 1" by (rule field_inverse)
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  thus "a * inverse a = 1" by (simp only: mult_commute)
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qed
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instance field \<subseteq> idom ..
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class division_by_zero = zero + inverse +
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  assumes inverse_zero [simp]: "inverse \<^loc>0 = \<^loc>0"
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subsection {* Distribution rules *}
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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     "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
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by (simp add: left_distrib add_ac)
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lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: left_distrib [symmetric]) 
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done
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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: right_distrib [symmetric]) 
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done
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lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
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by (simp add: right_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
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by (simp add: left_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemmas ring_distribs =
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  right_distrib left_distrib left_diff_distrib right_diff_distrib
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text{*This list of rewrites simplifies ring terms by multiplying
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everything out and bringing sums and products into a canonical form
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(by ordered rewriting). As a result it decides ring equalities but
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also helps with inequalities. *}
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lemmas ring_simps = group_simps ring_distribs
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class mult_mono = times + zero + ord +
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  assumes mult_left_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b"
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  assumes mult_right_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> a \<^loc>* c \<sqsubseteq> b \<^loc>* c"
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class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
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class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
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  + semiring + comm_monoid_add + cancel_ab_semigroup_add
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instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
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instance pordered_cancel_semiring \<subseteq> pordered_semiring .. 
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class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
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instance ordered_semiring \<subseteq> pordered_cancel_semiring ..
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class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
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  assumes mult_strict_left_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b"
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  assumes mult_strict_right_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> a \<^loc>* c \<sqsubset> b \<^loc>* c"
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instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..
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instance ordered_semiring_strict \<subseteq> ordered_semiring
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proof
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  fix a b c :: 'a
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  assume A: "a \<le> b" "0 \<le> c"
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  from A show "c * a \<le> c * b"
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    unfolding order_le_less
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    using mult_strict_left_mono by auto
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  from A show "a * c \<le> b * c"
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    unfolding order_le_less
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    using mult_strict_right_mono by auto
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qed
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class mult_mono1 = times + zero + ord +
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  assumes mult_mono: "a \<sqsubseteq> b \<Longrightarrow> \<^loc>0 \<sqsubseteq> c \<Longrightarrow> c \<^loc>* a \<sqsubseteq> c \<^loc>* b"
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class pordered_comm_semiring = comm_semiring_0
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  + pordered_ab_semigroup_add + mult_mono1
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class pordered_cancel_comm_semiring = comm_semiring_0_cancel
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  + pordered_ab_semigroup_add + mult_mono1
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instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
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class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
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  assumes mult_strict_mono: "a \<sqsubset> b \<Longrightarrow> \<^loc>0 \<sqsubset> c \<Longrightarrow> c \<^loc>* a \<sqsubset> c \<^loc>* b"
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instance pordered_comm_semiring \<subseteq> pordered_semiring
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proof
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  fix a b c :: 'a
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  assume "a \<le> b" "0 \<le> c"
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  thus "c * a \<le> c * b" by (rule mult_mono)
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  thus "a * c \<le> b * c" by (simp only: mult_commute)
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qed
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instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
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instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict
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proof
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  fix a b c :: 'a
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  assume "a < b" "0 < c"
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  thus "c * a < c * b" by (rule mult_strict_mono)
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  thus "a * c < b * c" by (simp only: mult_commute)
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qed
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instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
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proof
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  fix a b c :: 'a
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  assume "a \<le> b" "0 \<le> c"
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  thus "c * a \<le> c * b"
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   285
    unfolding order_le_less
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   286
    using mult_strict_mono by auto
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
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   287
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
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   288
22390
378f34b1e380 now using "class"
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   289
class pordered_ring = ring + pordered_cancel_semiring 
14270
342451d763f9 More re-organising of numerical theorems
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   290
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
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parents: 14603
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   291
instance pordered_ring \<subseteq> pordered_ab_group_add ..
14270
342451d763f9 More re-organising of numerical theorems
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parents: 14269
diff changeset
   292
22452
8a86fd2a1bf0 adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents: 22422
diff changeset
   293
class lordered_ring = pordered_ring + lordered_ab_group_abs
14270
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parents: 14269
diff changeset
   294
14940
b9ab8babd8b3 Further development of matrix theory
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parents: 14770
diff changeset
   295
instance lordered_ring \<subseteq> lordered_ab_group_meet ..
b9ab8babd8b3 Further development of matrix theory
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parents: 14770
diff changeset
   296
b9ab8babd8b3 Further development of matrix theory
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parents: 14770
diff changeset
   297
instance lordered_ring \<subseteq> lordered_ab_group_join ..
b9ab8babd8b3 Further development of matrix theory
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parents: 14770
diff changeset
   298
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   299
class abs_if = minus + ord + zero +
378f34b1e380 now using "class"
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parents: 21328
diff changeset
   300
  assumes abs_if: "abs a = (if a \<sqsubset> 0 then (uminus a) else a)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   301
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   302
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   303
   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   304
 *)
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   305
class ordered_ring = ring + ordered_semiring + lordered_ab_group + abs_if
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   306
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   307
instance ordered_ring \<subseteq> lordered_ring
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   308
proof
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   309
  fix x :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   310
  show "\<bar>x\<bar> = sup x (- x)"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   311
    by (simp only: abs_if sup_eq_if)
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   312
qed
23521
195fe3fe2831 added ordered_ring and ordered_semiring
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parents: 23496
diff changeset
   313
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   314
class ordered_ring_strict = ring + ordered_semiring_strict + lordered_ab_group + abs_if
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   315
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   316
instance ordered_ring_strict \<subseteq> ordered_ring ..
14270
342451d763f9 More re-organising of numerical theorems
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parents: 14269
diff changeset
   317
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   318
class pordered_comm_ring = comm_ring + pordered_comm_semiring
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   319
23527
c1d2fa4b76df instance pordered_comm_ring < pordered_ring
huffman
parents: 23521
diff changeset
   320
instance pordered_comm_ring \<subseteq> pordered_ring ..
c1d2fa4b76df instance pordered_comm_ring < pordered_ring
huffman
parents: 23521
diff changeset
   321
23073
d810dc04b96d add missing instance declarations
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parents: 22993
diff changeset
   322
instance pordered_comm_ring \<subseteq> pordered_cancel_comm_semiring ..
d810dc04b96d add missing instance declarations
huffman
parents: 22993
diff changeset
   323
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   324
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   325
  (*previously ordered_semiring*)
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   326
  assumes zero_less_one [simp]: "\<^loc>0 \<sqsubset> \<^loc>1"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   327
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   328
class ordered_idom = comm_ring_1 + ordered_comm_semiring_strict + lordered_ab_group + abs_if
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   329
  (*previously ordered_ring*)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   330
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   331
instance ordered_idom \<subseteq> ordered_ring_strict ..
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   332
23073
d810dc04b96d add missing instance declarations
huffman
parents: 22993
diff changeset
   333
instance ordered_idom \<subseteq> pordered_comm_ring ..
d810dc04b96d add missing instance declarations
huffman
parents: 22993
diff changeset
   334
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   335
class ordered_field = field + ordered_idom
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   336
15923
01d5d0c1c078 fixed lin.arith
nipkow
parents: 15769
diff changeset
   337
lemmas linorder_neqE_ordered_idom =
01d5d0c1c078 fixed lin.arith
nipkow
parents: 15769
diff changeset
   338
 linorder_neqE[where 'a = "?'b::ordered_idom"]
01d5d0c1c078 fixed lin.arith
nipkow
parents: 15769
diff changeset
   339
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   340
lemma eq_add_iff1:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   341
  "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   342
by (simp add: ring_simps)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   343
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   344
lemma eq_add_iff2:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   345
  "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   346
by (simp add: ring_simps)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   347
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   348
lemma less_add_iff1:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   349
  "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   350
by (simp add: ring_simps)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   351
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   352
lemma less_add_iff2:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   353
  "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   354
by (simp add: ring_simps)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   355
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   356
lemma le_add_iff1:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   357
  "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   358
by (simp add: ring_simps)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   359
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   360
lemma le_add_iff2:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   361
  "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   362
by (simp add: ring_simps)
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   363
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   364
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   365
subsection {* Ordering Rules for Multiplication *}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   366
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   367
lemma mult_left_le_imp_le:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   368
  "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   369
by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   370
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   371
lemma mult_right_le_imp_le:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   372
  "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   373
by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   374
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   375
lemma mult_left_less_imp_less:
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   376
  "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   377
by (force simp add: mult_left_mono linorder_not_le [symmetric])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   378
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   379
lemma mult_right_less_imp_less:
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   380
  "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   381
by (force simp add: mult_right_mono linorder_not_le [symmetric])
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   382
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   383
lemma mult_strict_left_mono_neg:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   384
  "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   385
apply (drule mult_strict_left_mono [of _ _ "-c"])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   386
apply (simp_all add: minus_mult_left [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   387
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   388
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   389
lemma mult_left_mono_neg:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   390
  "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   391
apply (drule mult_left_mono [of _ _ "-c"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   392
apply (simp_all add: minus_mult_left [symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   393
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   394
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   395
lemma mult_strict_right_mono_neg:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   396
  "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   397
apply (drule mult_strict_right_mono [of _ _ "-c"])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   398
apply (simp_all add: minus_mult_right [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   399
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   400
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   401
lemma mult_right_mono_neg:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   402
  "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   403
apply (drule mult_right_mono [of _ _ "-c"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   404
apply (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   405
apply (simp_all add: minus_mult_right [symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   406
done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   407
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   408
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   409
subsection{* Products of Signs *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   410
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   411
lemma mult_pos_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   412
by (drule mult_strict_left_mono [of 0 b], auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   413
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   414
lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   415
by (drule mult_left_mono [of 0 b], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   416
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   417
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   418
by (drule mult_strict_left_mono [of b 0], auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   419
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   420
lemma mult_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   421
by (drule mult_left_mono [of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   422
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   423
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   424
by (drule mult_strict_right_mono[of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   425
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   426
lemma mult_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   427
by (drule mult_right_mono[of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   428
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   429
lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   430
by (drule mult_strict_right_mono_neg, auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   431
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   432
lemma mult_nonpos_nonpos: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   433
by (drule mult_right_mono_neg[of a 0 b ], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   434
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   435
lemma zero_less_mult_pos:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   436
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   437
apply (cases "b\<le>0") 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   438
 apply (auto simp add: order_le_less linorder_not_less)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   439
apply (drule_tac mult_pos_neg [of a b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   440
 apply (auto dest: order_less_not_sym)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   441
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   442
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   443
lemma zero_less_mult_pos2:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   444
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   445
apply (cases "b\<le>0") 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   446
 apply (auto simp add: order_le_less linorder_not_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   447
apply (drule_tac mult_pos_neg2 [of a b]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   448
 apply (auto dest: order_less_not_sym)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   449
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   450
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   451
lemma zero_less_mult_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   452
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   453
apply (auto simp add: order_le_less linorder_not_less mult_pos_pos 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   454
  mult_neg_neg)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   455
apply (blast dest: zero_less_mult_pos) 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   456
apply (blast dest: zero_less_mult_pos2)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   457
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   458
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   459
lemma mult_eq_0_iff [simp]:
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   460
  fixes a b :: "'a::ring_no_zero_divisors"
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   461
  shows "(a * b = 0) = (a = 0 \<or> b = 0)"
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   462
by (cases "a = 0 \<or> b = 0", auto dest: no_zero_divisors)
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   463
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   464
instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   465
apply intro_classes
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   466
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   467
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   468
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   469
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   470
lemma zero_le_mult_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   471
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   472
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   473
                   zero_less_mult_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   474
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   475
lemma mult_less_0_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   476
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   477
apply (insert zero_less_mult_iff [of "-a" b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   478
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   479
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   480
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   481
lemma mult_le_0_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   482
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   483
apply (insert zero_le_mult_iff [of "-a" b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   484
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   485
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   486
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   487
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   488
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   489
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   490
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)" 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   491
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   492
23095
45f10b70e891 Squared things out.
obua
parents: 23073
diff changeset
   493
lemma zero_le_square[simp]: "(0::'a::ordered_ring_strict) \<le> a*a"
45f10b70e891 Squared things out.
obua
parents: 23073
diff changeset
   494
by (simp add: zero_le_mult_iff linorder_linear)
45f10b70e891 Squared things out.
obua
parents: 23073
diff changeset
   495
45f10b70e891 Squared things out.
obua
parents: 23073
diff changeset
   496
lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
45f10b70e891 Squared things out.
obua
parents: 23073
diff changeset
   497
by (simp add: not_less)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   498
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   499
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   500
      theorems available to members of @{term ordered_idom} *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   501
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   502
instance ordered_idom \<subseteq> ordered_semidom
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   503
proof
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   504
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   505
  thus "(0::'a) < 1" by (simp add: order_le_less) 
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   506
qed
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   507
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   508
instance ordered_idom \<subseteq> idom ..
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   509
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   510
text{*All three types of comparision involving 0 and 1 are covered.*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   511
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   512
lemmas one_neq_zero = zero_neq_one [THEN not_sym]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   513
declare one_neq_zero [simp]
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   514
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   515
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   516
  by (rule zero_less_one [THEN order_less_imp_le]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   517
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   518
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   519
by (simp add: linorder_not_le) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   520
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   521
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   522
by (simp add: linorder_not_less) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   523
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   524
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   525
subsection{*More Monotonicity*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   526
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   527
text{*Strict monotonicity in both arguments*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   528
lemma mult_strict_mono:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   529
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   530
apply (cases "c=0")
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   531
 apply (simp add: mult_pos_pos) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   532
apply (erule mult_strict_right_mono [THEN order_less_trans])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   533
 apply (force simp add: order_le_less) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   534
apply (erule mult_strict_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   535
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   536
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   537
text{*This weaker variant has more natural premises*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   538
lemma mult_strict_mono':
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   539
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   540
apply (rule mult_strict_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   541
apply (blast intro: order_le_less_trans)+
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   542
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   543
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   544
lemma mult_mono:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   545
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   546
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   547
apply (erule mult_right_mono [THEN order_trans], assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   548
apply (erule mult_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   549
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   550
21258
62f25a96f0c1 added lemma mult_mono'
huffman
parents: 21199
diff changeset
   551
lemma mult_mono':
62f25a96f0c1 added lemma mult_mono'
huffman
parents: 21199
diff changeset
   552
     "[|a \<le> b; c \<le> d; 0 \<le> a; 0 \<le> c|] 
62f25a96f0c1 added lemma mult_mono'
huffman
parents: 21199
diff changeset
   553
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
62f25a96f0c1 added lemma mult_mono'
huffman
parents: 21199
diff changeset
   554
apply (rule mult_mono)
62f25a96f0c1 added lemma mult_mono'
huffman
parents: 21199
diff changeset
   555
apply (fast intro: order_trans)+
62f25a96f0c1 added lemma mult_mono'
huffman
parents: 21199
diff changeset
   556
done
62f25a96f0c1 added lemma mult_mono'
huffman
parents: 21199
diff changeset
   557
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   558
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   559
apply (insert mult_strict_mono [of 1 m 1 n]) 
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   560
apply (simp add:  order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   561
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   562
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   563
lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   564
    c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   565
  apply (subgoal_tac "a * c < b * c")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   566
  apply (erule order_less_le_trans)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   567
  apply (erule mult_left_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   568
  apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   569
  apply (erule mult_strict_right_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   570
  apply assumption
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   571
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   572
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   573
lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   574
    c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   575
  apply (subgoal_tac "a * c <= b * c")
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   576
  apply (erule order_le_less_trans)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   577
  apply (erule mult_strict_left_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   578
  apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   579
  apply (erule mult_right_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   580
  apply simp
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   581
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   582
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   583
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   584
subsection{*Cancellation Laws for Relationships With a Common Factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   585
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   586
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   587
   also with the relations @{text "\<le>"} and equality.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   588
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   589
text{*These ``disjunction'' versions produce two cases when the comparison is
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   590
 an assumption, but effectively four when the comparison is a goal.*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   591
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   592
lemma mult_less_cancel_right_disj:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   593
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   594
apply (cases "c = 0")
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   595
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   596
                      mult_strict_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   597
apply (auto simp add: linorder_not_less 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   598
                      linorder_not_le [symmetric, of "a*c"]
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   599
                      linorder_not_le [symmetric, of a])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   600
apply (erule_tac [!] notE)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   601
apply (auto simp add: order_less_imp_le mult_right_mono 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   602
                      mult_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   603
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   604
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   605
lemma mult_less_cancel_left_disj:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   606
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   607
apply (cases "c = 0")
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   608
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   609
                      mult_strict_left_mono_neg)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   610
apply (auto simp add: linorder_not_less 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   611
                      linorder_not_le [symmetric, of "c*a"]
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   612
                      linorder_not_le [symmetric, of a])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   613
apply (erule_tac [!] notE)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   614
apply (auto simp add: order_less_imp_le mult_left_mono 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   615
                      mult_left_mono_neg)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   616
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   617
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   618
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   619
text{*The ``conjunction of implication'' lemmas produce two cases when the
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   620
comparison is a goal, but give four when the comparison is an assumption.*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   621
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   622
lemma mult_less_cancel_right:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   623
  fixes c :: "'a :: ordered_ring_strict"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   624
  shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   625
by (insert mult_less_cancel_right_disj [of a c b], auto)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   626
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   627
lemma mult_less_cancel_left:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   628
  fixes c :: "'a :: ordered_ring_strict"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   629
  shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   630
by (insert mult_less_cancel_left_disj [of c a b], auto)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   631
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   632
lemma mult_le_cancel_right:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   633
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   634
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   635
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   636
lemma mult_le_cancel_left:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   637
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   638
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   639
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   640
lemma mult_less_imp_less_left:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   641
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   642
      shows "a < (b::'a::ordered_semiring_strict)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   643
proof (rule ccontr)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   644
  assume "~ a < b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   645
  hence "b \<le> a" by (simp add: linorder_not_less)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   646
  hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   647
  with this and less show False 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   648
    by (simp add: linorder_not_less [symmetric])
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   649
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   650
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   651
lemma mult_less_imp_less_right:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   652
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   653
  shows "a < (b::'a::ordered_semiring_strict)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   654
proof (rule ccontr)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   655
  assume "~ a < b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   656
  hence "b \<le> a" by (simp add: linorder_not_less)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   657
  hence "b*c \<le> a*c" using nonneg by (rule mult_right_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   658
  with this and less show False 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   659
    by (simp add: linorder_not_less [symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   660
qed  
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   661
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   662
text{*Cancellation of equalities with a common factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   663
lemma mult_cancel_right [simp]:
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   664
  fixes a b c :: "'a::ring_no_zero_divisors"
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   665
  shows "(a * c = b * c) = (c = 0 \<or> a = b)"
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   666
proof -
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   667
  have "(a * c = b * c) = ((a - b) * c = 0)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   668
    by (simp add: ring_distribs)
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   669
  thus ?thesis
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   670
    by (simp add: disj_commute)
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   671
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   672
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   673
lemma mult_cancel_left [simp]:
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   674
  fixes a b c :: "'a::ring_no_zero_divisors"
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   675
  shows "(c * a = c * b) = (c = 0 \<or> a = b)"
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   676
proof -
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   677
  have "(c * a = c * b) = (c * (a - b) = 0)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   678
    by (simp add: ring_distribs)
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   679
  thus ?thesis
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   680
    by simp
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   681
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   682
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   683
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   684
subsubsection{*Special Cancellation Simprules for Multiplication*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   685
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   686
text{*These also produce two cases when the comparison is a goal.*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   687
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   688
lemma mult_le_cancel_right1:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   689
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   690
  shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   691
by (insert mult_le_cancel_right [of 1 c b], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   692
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   693
lemma mult_le_cancel_right2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   694
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   695
  shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   696
by (insert mult_le_cancel_right [of a c 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   697
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   698
lemma mult_le_cancel_left1:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   699
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   700
  shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   701
by (insert mult_le_cancel_left [of c 1 b], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   702
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   703
lemma mult_le_cancel_left2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   704
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   705
  shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   706
by (insert mult_le_cancel_left [of c a 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   707
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   708
lemma mult_less_cancel_right1:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   709
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   710
  shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   711
by (insert mult_less_cancel_right [of 1 c b], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   712
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   713
lemma mult_less_cancel_right2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   714
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   715
  shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   716
by (insert mult_less_cancel_right [of a c 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   717
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   718
lemma mult_less_cancel_left1:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   719
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   720
  shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   721
by (insert mult_less_cancel_left [of c 1 b], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   722
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   723
lemma mult_less_cancel_left2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   724
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   725
  shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   726
by (insert mult_less_cancel_left [of c a 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   727
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   728
lemma mult_cancel_right1 [simp]:
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   729
  fixes c :: "'a :: ring_1_no_zero_divisors"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   730
  shows "(c = b*c) = (c = 0 | b=1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   731
by (insert mult_cancel_right [of 1 c b], force)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   732
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   733
lemma mult_cancel_right2 [simp]:
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   734
  fixes c :: "'a :: ring_1_no_zero_divisors"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   735
  shows "(a*c = c) = (c = 0 | a=1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   736
by (insert mult_cancel_right [of a c 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   737
 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   738
lemma mult_cancel_left1 [simp]:
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   739
  fixes c :: "'a :: ring_1_no_zero_divisors"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   740
  shows "(c = c*b) = (c = 0 | b=1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   741
by (insert mult_cancel_left [of c 1 b], force)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   742
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   743
lemma mult_cancel_left2 [simp]:
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
huffman
parents: 23527
diff changeset
   744
  fixes c :: "'a :: ring_1_no_zero_divisors"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   745
  shows "(c*a = c) = (c = 0 | a=1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   746
by (insert mult_cancel_left [of c a 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   747
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   748
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   749
text{*Simprules for comparisons where common factors can be cancelled.*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   750
lemmas mult_compare_simps =
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   751
    mult_le_cancel_right mult_le_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   752
    mult_le_cancel_right1 mult_le_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   753
    mult_le_cancel_left1 mult_le_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   754
    mult_less_cancel_right mult_less_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   755
    mult_less_cancel_right1 mult_less_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   756
    mult_less_cancel_left1 mult_less_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   757
    mult_cancel_right mult_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   758
    mult_cancel_right1 mult_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   759
    mult_cancel_left1 mult_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   760
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   761
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   762
subsection {* Fields *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   763
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   764
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   765
proof
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   766
  assume neq: "b \<noteq> 0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   767
  {
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   768
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   769
    also assume "a / b = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   770
    finally show "a = b" by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   771
  next
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   772
    assume "a = b"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   773
    with neq show "a / b = 1" by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   774
  }
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   775
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   776
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   777
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   778
by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   779
23398
0b5a400c7595 made divide_self a simp rule
nipkow
parents: 23389
diff changeset
   780
lemma divide_self[simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   781
  by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   782
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   783
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   784
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   785
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   786
lemma divide_self_if [simp]:
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   787
     "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   788
  by (simp add: divide_self)
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   789
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   790
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   791
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   792
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   793
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   794
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   795
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   796
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   797
by (simp add: divide_inverse ring_distribs) 
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   798
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   799
(* what ordering?? this is a straight instance of mult_eq_0_iff
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   800
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   801
      of an ordering.*}
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   802
lemma field_mult_eq_0_iff [simp]:
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   803
  "(a*b = (0::'a::division_ring)) = (a = 0 | b = 0)"
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   804
by simp
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   805
*)
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
   806
(* subsumed by mult_cancel lemmas on ring_no_zero_divisors
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   807
text{*Cancellation of equalities with a common factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   808
lemma field_mult_cancel_right_lemma:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   809
      assumes cnz: "c \<noteq> (0::'a::division_ring)"
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   810
         and eq:  "a*c = b*c"
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   811
        shows "a=b"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   812
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   813
  have "(a * c) * inverse c = (b * c) * inverse c"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   814
    by (simp add: eq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   815
  thus "a=b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   816
    by (simp add: mult_assoc cnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   817
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   818
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   819
lemma field_mult_cancel_right [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   820
     "(a*c = b*c) = (c = (0::'a::division_ring) | a=b)"
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   821
by simp
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   822
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   823
lemma field_mult_cancel_left [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   824
     "(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   825
by simp
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
   826
*)
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   827
lemma nonzero_imp_inverse_nonzero:
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   828
  "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   829
proof
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   830
  assume ianz: "inverse a = 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   831
  assume "a \<noteq> 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   832
  hence "1 = a * inverse a" by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   833
  also have "... = 0" by (simp add: ianz)
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   834
  finally have "1 = (0::'a::division_ring)" .
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   835
  thus False by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   836
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   837
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   838
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   839
subsection{*Basic Properties of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   840
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   841
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::division_ring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   842
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   843
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   844
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   845
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   846
lemma inverse_nonzero_imp_nonzero:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   847
   "inverse a = 0 ==> a = (0::'a::division_ring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   848
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   849
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   850
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   851
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   852
lemma inverse_nonzero_iff_nonzero [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   853
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   854
by (force dest: inverse_nonzero_imp_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   855
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   856
lemma nonzero_inverse_minus_eq:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   857
      assumes [simp]: "a\<noteq>0"
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   858
      shows "inverse(-a) = -inverse(a::'a::division_ring)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   859
proof -
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   860
  have "-a * inverse (- a) = -a * - inverse a"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   861
    by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   862
  thus ?thesis 
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
   863
    by (simp only: mult_cancel_left, simp)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   864
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   865
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   866
lemma inverse_minus_eq [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   867
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   868
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   869
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   870
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   871
  assume "a\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   872
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   873
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   874
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   875
lemma nonzero_inverse_eq_imp_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   876
      assumes inveq: "inverse a = inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   877
	  and anz:  "a \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   878
	  and bnz:  "b \<noteq> 0"
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   879
	 shows "a = (b::'a::division_ring)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   880
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   881
  have "a * inverse b = a * inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   882
    by (simp add: inveq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   883
  hence "(a * inverse b) * b = (a * inverse a) * b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   884
    by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   885
  thus "a = b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   886
    by (simp add: mult_assoc anz bnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   887
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   888
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   889
lemma inverse_eq_imp_eq:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   890
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   891
apply (cases "a=0 | b=0") 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   892
 apply (force dest!: inverse_zero_imp_zero
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   893
              simp add: eq_commute [of "0::'a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   894
apply (force dest!: nonzero_inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   895
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   896
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   897
lemma inverse_eq_iff_eq [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   898
  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   899
by (force dest!: inverse_eq_imp_eq)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   900
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   901
lemma nonzero_inverse_inverse_eq:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   902
      assumes [simp]: "a \<noteq> 0"
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   903
      shows "inverse(inverse (a::'a::division_ring)) = a"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   904
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   905
  have "(inverse (inverse a) * inverse a) * a = a" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   906
    by (simp add: nonzero_imp_inverse_nonzero)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   907
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   908
    by (simp add: mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   909
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   910
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   911
lemma inverse_inverse_eq [simp]:
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   912
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   913
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   914
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   915
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   916
    assume "a\<noteq>0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   917
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   918
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   919
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   920
lemma inverse_1 [simp]: "inverse 1 = (1::'a::division_ring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   921
  proof -
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   922
  have "inverse 1 * 1 = (1::'a::division_ring)" 
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   923
    by (rule left_inverse [OF zero_neq_one [symmetric]])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   924
  thus ?thesis  by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   925
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   926
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   927
lemma inverse_unique: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   928
  assumes ab: "a*b = 1"
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   929
  shows "inverse a = (b::'a::division_ring)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   930
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   931
  have "a \<noteq> 0" using ab by auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   932
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   933
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   934
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   935
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   936
lemma nonzero_inverse_mult_distrib: 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   937
      assumes anz: "a \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   938
          and bnz: "b \<noteq> 0"
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   939
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   940
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   941
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   942
    by (simp add: anz bnz)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   943
  hence "inverse(a*b) * a = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   944
    by (simp add: mult_assoc bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   945
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   946
    by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   947
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   948
    by (simp add: mult_assoc anz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   949
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   950
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   951
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   952
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   953
lemma inverse_mult_distrib [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   954
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   955
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   956
    assume "a \<noteq> 0 & b \<noteq> 0" 
22993
haftmann
parents: 22990
diff changeset
   957
    thus ?thesis
haftmann
parents: 22990
diff changeset
   958
      by (simp add: nonzero_inverse_mult_distrib mult_commute)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   959
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   960
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
22993
haftmann
parents: 22990
diff changeset
   961
    thus ?thesis
haftmann
parents: 22990
diff changeset
   962
      by force
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   963
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   964
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   965
lemma division_ring_inverse_add:
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   966
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   967
   ==> inverse a + inverse b = inverse a * (a+b) * inverse b"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   968
by (simp add: ring_simps)
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   969
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   970
lemma division_ring_inverse_diff:
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   971
  "[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   972
   ==> inverse a - inverse b = inverse a * (b-a) * inverse b"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   973
by (simp add: ring_simps)
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   974
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   975
text{*There is no slick version using division by zero.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   976
lemma inverse_add:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   977
  "[|a \<noteq> 0;  b \<noteq> 0|]
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   978
   ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
20496
23eb6034c06d added axclass division_ring (like field without commutativity; includes e.g. quaternions) and generalized some theorems from field to division_ring
huffman
parents: 19404
diff changeset
   979
by (simp add: division_ring_inverse_add mult_ac)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   980
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   981
lemma inverse_divide [simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   982
  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   983
by (simp add: divide_inverse mult_commute)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   984
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   985
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   986
subsection {* Calculations with fractions *}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   987
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   988
text{* There is a whole bunch of simp-rules just for class @{text
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   989
field} but none for class @{text field} and @{text nonzero_divides}
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   990
because the latter are covered by a simproc. *}
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   991
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   992
lemma nonzero_mult_divide_mult_cancel_left[simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   993
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/(b::'a::field)"
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   994
proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   995
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   996
    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   997
  also have "... =  a * inverse b * (inverse c * c)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   998
    by (simp only: mult_ac)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   999
  also have "... =  a * inverse b"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1000
    by simp
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1001
    finally show ?thesis 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1002
    by (simp add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1003
qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1004
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1005
lemma mult_divide_mult_cancel_left:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1006
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1007
apply (cases "b = 0")
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1008
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1009
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1010
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1011
lemma nonzero_mult_divide_mult_cancel_right:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1012
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1013
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1014
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1015
lemma mult_divide_mult_cancel_right:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1016
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1017
apply (cases "b = 0")
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1018
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1019
done
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1020
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
  1021
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1022
by (simp add: divide_inverse)
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
  1023
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1024
lemma times_divide_eq_right: "a * (b/c) = (a*b) / (c::'a::field)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1025
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1026
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1027
lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1028
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1029
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1030
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1031
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1032
lemma divide_divide_eq_right [simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1033
  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1034
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1035
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1036
lemma divide_divide_eq_left [simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1037
  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1038
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1039
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1040
lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1041
    x / y + w / z = (x * z + w * y) / (y * z)"
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1042
apply (subgoal_tac "x / y = (x * z) / (y * z)")
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1043
apply (erule ssubst)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1044
apply (subgoal_tac "w / z = (w * y) / (y * z)")
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1045
apply (erule ssubst)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1046
apply (rule add_divide_distrib [THEN sym])
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1047
apply (subst mult_commute)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1048
apply (erule nonzero_mult_divide_mult_cancel_left [THEN sym])
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1049
apply assumption
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1050
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1051
apply assumption
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1052
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1053
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1054
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1055
subsubsection{*Special Cancellation Simprules for Division*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1056
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1057
lemma mult_divide_mult_cancel_left_if[simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1058
fixes c :: "'a :: {field,division_by_zero}"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1059
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1060
by (simp add: mult_divide_mult_cancel_left)
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1061
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1062
lemma nonzero_mult_divide_cancel_right[simp]:
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1063
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1064
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1065
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1066
lemma nonzero_mult_divide_cancel_left[simp]:
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1067
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1068
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1069
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1070
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1071
lemma nonzero_divide_mult_cancel_right[simp]:
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1072
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1073
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1074
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1075
lemma nonzero_divide_mult_cancel_left[simp]:
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1076
  "\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1077
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1078
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1079
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1080
lemma nonzero_mult_divide_mult_cancel_left2[simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1081
  "[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1082
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1083
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1084
lemma nonzero_mult_divide_mult_cancel_right2[simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1085
  "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1086
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1087
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1088
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1089
subsection {* Division and Unary Minus *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1090
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1091
lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1092
by (simp add: divide_inverse minus_mult_left)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1093
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1094
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1095
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1096
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1097
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1098
by (simp add: divide_inverse nonzero_inverse_minus_eq)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1099
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1100
lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1101
by (simp add: divide_inverse minus_mult_left [symmetric])
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1102
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1103
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1104
by (simp add: divide_inverse minus_mult_right [symmetric])
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1105
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1106
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1107
text{*The effect is to extract signs from divisions*}
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1108
lemmas divide_minus_left = minus_divide_left [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1109
lemmas divide_minus_right = minus_divide_right [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1110
declare divide_minus_left [simp]   divide_minus_right [simp]
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1111
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1112
text{*Also, extract signs from products*}
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1113
lemmas mult_minus_left = minus_mult_left [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1114
lemmas mult_minus_right = minus_mult_right [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1115
declare mult_minus_left [simp]   mult_minus_right [simp]
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1116
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1117
lemma minus_divide_divide [simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1118
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1119
apply (cases "b=0", simp) 
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1120
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1121
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1122
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1123
lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1124
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1125
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1126
lemma add_divide_eq_iff:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1127
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1128
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1129
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1130
lemma divide_add_eq_iff:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1131
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1132
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1133
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1134
lemma diff_divide_eq_iff:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1135
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1136
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1137
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1138
lemma divide_diff_eq_iff:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1139
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1140
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1141
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1142
lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1143
proof -
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1144
  assume [simp]: "c\<noteq>0"
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
  1145
  have "(a = b/c) = (a*c = (b/c)*c)" by simp
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
  1146
  also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1147
  finally show ?thesis .
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1148
qed
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1149
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1150
lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1151
proof -
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1152
  assume [simp]: "c\<noteq>0"
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
  1153
  have "(b/c = a) = ((b/c)*c = a*c)"  by simp
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
  1154
  also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1155
  finally show ?thesis .
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1156
qed
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1157
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1158
lemma eq_divide_eq:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1159
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1160
by (simp add: nonzero_eq_divide_eq) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1161
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1162
lemma divide_eq_eq:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1163
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1164
by (force simp add: nonzero_divide_eq_eq) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1165
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1166
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1167
    b = a * c ==> b / c = a"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1168
  by (subst divide_eq_eq, simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1169
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1170
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1171
    a * c = b ==> a = b / c"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1172
  by (subst eq_divide_eq, simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1173
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1174
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1175
lemmas field_eq_simps = ring_simps
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1176
  (* pull / out*)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1177
  add_divide_eq_iff divide_add_eq_iff
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1178
  diff_divide_eq_iff divide_diff_eq_iff
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1179
  (* multiply eqn *)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1180
  nonzero_eq_divide_eq nonzero_divide_eq_eq
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1181
(* is added later:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1182
  times_divide_eq_left times_divide_eq_right
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1183
*)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1184
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1185
text{*An example:*}
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1186
lemma fixes a b c d e f :: "'a::field"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1187
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1188
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1189
 apply(simp add:field_eq_simps)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1190
apply(simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1191
done
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1192
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1193
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1194
lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1195
    x / y - w / z = (x * z - w * y) / (y * z)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1196
by (simp add:field_eq_simps times_divide_eq)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1197
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1198
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1199
    (x / y = w / z) = (x * z = w * y)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1200
by (simp add:field_eq_simps times_divide_eq)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1201
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1202
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1203
subsection {* Ordered Fields *}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1204
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1205
lemma positive_imp_inverse_positive: 
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1206
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1207
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1208
  have "0 < a * inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1209
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1210
  thus "0 < inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1211
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1212
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1213
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1214
lemma negative_imp_inverse_negative:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1215
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1216
by (insert positive_imp_inverse_positive [of "-a"], 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1217
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1218
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1219
lemma inverse_le_imp_le:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1220
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1221
shows "b \<le> (a::'a::ordered_field)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1222
proof (rule classical)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1223
  assume "~ b \<le> a"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1224
  hence "a < b"  by (simp add: linorder_not_le)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1225
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1226
  hence "a * inverse a \<le> a * inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1227
    by (simp add: apos invle order_less_imp_le mult_left_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1228
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1229
    by (simp add: bpos order_less_imp_le mult_right_mono)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1230
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1231
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1232
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1233
lemma inverse_positive_imp_positive:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1234
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1235
shows "0 < (a::'a::ordered_field)"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1236
proof -
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1237
  have "0 < inverse (inverse a)"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1238
    using inv_gt_0 by (rule positive_imp_inverse_positive)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1239
  thus "0 < a"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1240
    using nz by (simp add: nonzero_inverse_inverse_eq)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1241
qed
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1242
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1243
lemma inverse_positive_iff_positive [simp]:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1244
  "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1245
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1246
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1247
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1248
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1249
lemma inverse_negative_imp_negative:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1250
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1251
shows "a < (0::'a::ordered_field)"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1252
proof -
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1253
  have "inverse (inverse a) < 0"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1254
    using inv_less_0 by (rule negative_imp_inverse_negative)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1255
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1256
qed
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1257
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1258
lemma inverse_negative_iff_negative [simp]:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1259
  "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"