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 + b) * c = a * c + b * c"
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  assumes right_distrib: "a * (b + c) = a * b + a * c"
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begin
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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"
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  by (simp add: left_distrib add_ac)
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end
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class mult_zero = times + zero +
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  assumes mult_zero_left [simp]: "0 * a = 0"
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  assumes mult_zero_right [simp]: "a * 0 = 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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begin
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subclass semiring_0
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proof unfold_locales
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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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next
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  fix a :: 'a
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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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end
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class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
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  assumes distrib: "(a + b) * c = a * c + b * c"
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begin
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subclass semiring
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proof unfold_locales
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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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end
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class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
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begin
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subclass semiring_0 ..
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end
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class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add
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begin
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subclass semiring_0_cancel ..
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end
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class zero_neq_one = zero + one +
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  assumes zero_neq_one [simp]: "0 \<noteq> 1"
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begin
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lemma one_neq_zero [simp]: "1 \<noteq> 0"
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  by (rule not_sym) (rule zero_neq_one)
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end
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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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begin
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subclass semiring_1 ..
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end
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class no_zero_divisors = zero + times +
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  assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 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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begin
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subclass semiring_0_cancel ..
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subclass semiring_1 ..
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end
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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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begin
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subclass semiring_1_cancel ..
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subclass comm_semiring_0_cancel ..
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subclass comm_semiring_1 ..
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end
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class ring = semiring + ab_group_add
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begin
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subclass semiring_0_cancel ..
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text {* Distribution rules *}
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lemma minus_mult_left: "- (a * b) = - a * b"
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  by (rule equals_zero_I) (simp add: left_distrib [symmetric]) 
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lemma minus_mult_right: "- (a * b) = a * - b"
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  by (rule equals_zero_I) (simp add: right_distrib [symmetric]) 
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lemma minus_mult_minus [simp]: "- a * - b = a * b"
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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"
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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"
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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"
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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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lemmas ring_simps =
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  add_ac
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  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
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  diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
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  ring_distribs
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   171
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lemma eq_add_iff1:
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  "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
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  by (simp add: ring_simps)
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lemma eq_add_iff2:
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  "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
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  by (simp add: ring_simps)
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end
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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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class comm_ring = comm_semiring + ab_group_add
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begin
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subclass ring ..
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subclass comm_semiring_0 ..
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end
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class ring_1 = ring + zero_neq_one + monoid_mult
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begin
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subclass semiring_1_cancel ..
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end
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   199
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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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begin
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subclass ring_1 ..
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subclass comm_semiring_1_cancel ..
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end
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   208
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class ring_no_zero_divisors = ring + no_zero_divisors
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begin
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   211
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lemma mult_eq_0_iff [simp]:
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  shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
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proof (cases "a = 0 \<or> b = 0")
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  case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
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    then show ?thesis using no_zero_divisors by simp
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next
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  case True then show ?thesis by auto
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qed
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   220
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text{*Cancellation of equalities with a common factor*}
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lemma mult_cancel_right [simp, noatp]:
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  "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
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   224
proof -
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   225
  have "(a * c = b * c) = ((a - b) * c = 0)"
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    by (simp add: ring_distribs right_minus_eq)
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   227
  thus ?thesis
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    by (simp add: disj_commute right_minus_eq)
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   229
qed
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   230
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lemma mult_cancel_left [simp, noatp]:
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  "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
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   233
proof -
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  have "(c * a = c * b) = (c * (a - b) = 0)"
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    by (simp add: ring_distribs right_minus_eq)
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  thus ?thesis
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    by (simp add: right_minus_eq)
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qed
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diff changeset
   239
25230
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   240
end
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
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   241
23544
4b4165cb3e0d rename class dom to ring_1_no_zero_divisors
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class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors
26274
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begin
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   244
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lemma mult_cancel_right1 [simp]:
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  "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
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  by (insert mult_cancel_right [of 1 c b], force)
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   248
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lemma mult_cancel_right2 [simp]:
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  "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
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  by (insert mult_cancel_right [of a c 1], simp)
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   252
 
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lemma mult_cancel_left1 [simp]:
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  "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
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  by (insert mult_cancel_left [of c 1 b], force)
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   256
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lemma mult_cancel_left2 [simp]:
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  "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
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  by (insert mult_cancel_left [of c a 1], simp)
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   260
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   261
end
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   262
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class idom = comm_ring_1 + no_zero_divisors
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begin
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   265
27516
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subclass ring_1_no_zero_divisors ..
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   267
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   268
end
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   269
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class division_ring = ring_1 + inverse +
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  assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
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  assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
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   273
begin
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   274
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subclass ring_1_no_zero_divisors
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   276
proof unfold_locales
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  fix a b :: 'a
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  assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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  show "a * b \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   280
  proof
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   281
    assume ab: "a * b = 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   282
    hence "0 = inverse a * (a * b) * inverse b"
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   283
      by simp
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   284
    also have "\<dots> = (inverse a * a) * (b * inverse b)"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   285
      by (simp only: mult_assoc)
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   286
    also have "\<dots> = 1"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   287
      using a b by simp
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   288
    finally show False
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   289
      by simp
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   290
  qed
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
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   291
qed
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diff changeset
   292
26274
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   293
lemma nonzero_imp_inverse_nonzero:
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   294
  "a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
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   295
proof
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   296
  assume ianz: "inverse a = 0"
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   297
  assume "a \<noteq> 0"
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   298
  hence "1 = a * inverse a" by simp
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diff changeset
   299
  also have "... = 0" by (simp add: ianz)
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   300
  finally have "1 = 0" .
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   301
  thus False by (simp add: eq_commute)
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   302
qed
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diff changeset
   303
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diff changeset
   304
lemma inverse_zero_imp_zero:
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   305
  "inverse a = 0 \<Longrightarrow> a = 0"
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   306
apply (rule classical)
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   307
apply (drule nonzero_imp_inverse_nonzero)
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   308
apply auto
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   309
done
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diff changeset
   310
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diff changeset
   311
lemma nonzero_inverse_minus_eq:
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   312
  assumes "a \<noteq> 0"
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   313
  shows "inverse (- a) = - inverse a"
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diff changeset
   314
proof -
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diff changeset
   315
  have "- a * inverse (- a) = - a * - inverse a"
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parents: 26234
diff changeset
   316
    using assms by simp
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diff changeset
   317
  then show ?thesis unfolding mult_cancel_left using assms by simp 
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diff changeset
   318
qed
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diff changeset
   319
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diff changeset
   320
lemma nonzero_inverse_inverse_eq:
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diff changeset
   321
  assumes "a \<noteq> 0"
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diff changeset
   322
  shows "inverse (inverse a) = a"
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parents: 26234
diff changeset
   323
proof -
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diff changeset
   324
  have "(inverse (inverse a) * inverse a) * a = a" 
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parents: 26234
diff changeset
   325
    using assms by (simp add: nonzero_imp_inverse_nonzero)
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parents: 26234
diff changeset
   326
  then show ?thesis using assms by (simp add: mult_assoc)
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diff changeset
   327
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   328
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diff changeset
   329
lemma nonzero_inverse_eq_imp_eq:
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diff changeset
   330
  assumes inveq: "inverse a = inverse b"
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parents: 26234
diff changeset
   331
    and anz:  "a \<noteq> 0"
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parents: 26234
diff changeset
   332
    and bnz:  "b \<noteq> 0"
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haftmann
parents: 26234
diff changeset
   333
  shows "a = b"
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parents: 26234
diff changeset
   334
proof -
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   335
  have "a * inverse b = a * inverse a"
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parents: 26234
diff changeset
   336
    by (simp add: inveq)
2bdb61a28971 continued localization
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diff changeset
   337
  hence "(a * inverse b) * b = (a * inverse a) * b"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   338
    by simp
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diff changeset
   339
  then show "a = b"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   340
    by (simp add: mult_assoc anz bnz)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   341
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   342
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   343
lemma inverse_1 [simp]: "inverse 1 = 1"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   344
proof -
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   345
  have "inverse 1 * 1 = 1" 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   346
    by (rule left_inverse) (rule one_neq_zero)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   347
  then show ?thesis by simp
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   348
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   349
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   350
lemma inverse_unique: 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   351
  assumes ab: "a * b = 1"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   352
  shows "inverse a = b"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   353
proof -
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   354
  have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   355
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   356
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   357
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   358
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   359
lemma nonzero_inverse_mult_distrib: 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   360
  assumes anz: "a \<noteq> 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   361
    and bnz: "b \<noteq> 0"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   362
  shows "inverse (a * b) = inverse b * inverse a"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   363
proof -
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   364
  have "inverse (a * b) * (a * b) * inverse b = inverse b" 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   365
    by (simp add: anz bnz)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   366
  hence "inverse (a * b) * a = inverse b" 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   367
    by (simp add: mult_assoc bnz)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   368
  hence "inverse (a * b) * a * inverse a = inverse b * inverse a" 
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   369
    by simp
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   370
  thus ?thesis
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   371
    by (simp add: mult_assoc anz)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   372
qed
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   373
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   374
lemma division_ring_inverse_add:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   375
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   376
  by (simp add: ring_simps mult_assoc)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   377
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   378
lemma division_ring_inverse_diff:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   379
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   380
  by (simp add: ring_simps mult_assoc)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   381
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   382
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   383
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   384
class field = comm_ring_1 + inverse +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   385
  assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   386
  assumes divide_inverse: "a / b = a * inverse b"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   387
begin
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
   388
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   389
subclass division_ring
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   390
proof unfold_locales
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   391
  fix a :: 'a
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   392
  assume "a \<noteq> 0"
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   393
  thus "inverse a * a = 1" by (rule field_inverse)
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   394
  thus "a * inverse a = 1" by (simp only: mult_commute)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   395
qed
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   396
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   397
subclass idom ..
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   398
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   399
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   400
proof
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   401
  assume neq: "b \<noteq> 0"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   402
  {
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   403
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   404
    also assume "a / b = 1"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   405
    finally show "a = b" by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   406
  next
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   407
    assume "a = b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   408
    with neq show "a / b = 1" by (simp add: divide_inverse)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   409
  }
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   410
qed
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   411
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   412
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   413
  by (simp add: divide_inverse)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   414
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   415
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   416
  by (simp add: divide_inverse)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   417
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   418
lemma divide_zero_left [simp]: "0 / a = 0"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   419
  by (simp add: divide_inverse)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   420
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   421
lemma inverse_eq_divide: "inverse a = 1 / a"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   422
  by (simp add: divide_inverse)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   423
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   424
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   425
  by (simp add: divide_inverse ring_distribs) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   426
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   427
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   428
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   429
class division_by_zero = zero + inverse +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   430
  assumes inverse_zero [simp]: "inverse 0 = 0"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   431
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   432
lemma divide_zero [simp]:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   433
  "a / 0 = (0::'a::{field,division_by_zero})"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   434
  by (simp add: divide_inverse)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   435
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   436
lemma divide_self_if [simp]:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   437
  "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   438
  by (simp add: divide_self)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   439
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   440
class mult_mono = times + zero + ord +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   441
  assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   442
  assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   443
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   444
class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add 
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   445
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   446
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   447
lemma mult_mono:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   448
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   449
     \<Longrightarrow> a * c \<le> b * d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   450
apply (erule mult_right_mono [THEN order_trans], assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   451
apply (erule mult_left_mono, assumption)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   452
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   453
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   454
lemma mult_mono':
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   455
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   456
     \<Longrightarrow> a * c \<le> b * d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   457
apply (rule mult_mono)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   458
apply (fast intro: order_trans)+
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   459
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   460
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   461
end
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   462
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   463
class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
22987
550709aa8e66 instance division_ring < no_zero_divisors; clean up field instance proofs
huffman
parents: 22842
diff changeset
   464
  + semiring + comm_monoid_add + cancel_ab_semigroup_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   465
begin
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   466
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   467
subclass semiring_0_cancel ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   468
subclass pordered_semiring ..
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   469
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   470
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   471
  by (drule mult_left_mono [of zero b], auto)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   472
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   473
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   474
  by (drule mult_left_mono [of b zero], auto)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   475
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   476
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0" 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   477
  by (drule mult_right_mono [of b zero], auto)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   478
26234
1f6e28a88785 clarified proposition
haftmann
parents: 26193
diff changeset
   479
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0" 
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   480
  by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   481
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   482
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   483
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   484
class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   485
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   486
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   487
subclass pordered_cancel_semiring ..
25512
4134f7c782e2 using intro_locales instead of unfold_locales if appropriate
haftmann
parents: 25450
diff changeset
   488
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   489
subclass pordered_comm_monoid_add ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   490
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   491
lemma mult_left_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   492
  "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   493
  by (force simp add: mult_left_mono not_le [symmetric])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   494
 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   495
lemma mult_right_less_imp_less:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   496
  "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   497
  by (force simp add: mult_right_mono not_le [symmetric])
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   498
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   499
end
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   500
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   501
class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   502
  assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   503
  assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   504
begin
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   505
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   506
subclass semiring_0_cancel ..
14940
b9ab8babd8b3 Further development of matrix theory
obua
parents: 14770
diff changeset
   507
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   508
subclass ordered_semiring
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   509
proof unfold_locales
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   510
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   511
  assume A: "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   512
  from A show "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   513
    unfolding le_less
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   514
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   515
  from A show "a * c \<le> b * c"
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   516
    unfolding le_less
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   517
    using mult_strict_right_mono by (cases "c = 0") auto
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   518
qed
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   519
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   520
lemma mult_left_le_imp_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   521
  "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   522
  by (force simp add: mult_strict_left_mono _not_less [symmetric])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   523
 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   524
lemma mult_right_le_imp_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   525
  "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   526
  by (force simp add: mult_strict_right_mono not_less [symmetric])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   527
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   528
lemma mult_pos_pos:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   529
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   530
  by (drule mult_strict_left_mono [of zero b], auto)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   531
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   532
lemma mult_pos_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   533
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   534
  by (drule mult_strict_left_mono [of b zero], auto)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   535
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   536
lemma mult_pos_neg2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   537
  "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0" 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   538
  by (drule mult_strict_right_mono [of b zero], auto)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   539
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   540
lemma zero_less_mult_pos:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   541
  "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   542
apply (cases "b\<le>0") 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   543
 apply (auto simp add: le_less not_less)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   544
apply (drule_tac mult_pos_neg [of a b]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   545
 apply (auto dest: less_not_sym)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   546
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   547
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   548
lemma zero_less_mult_pos2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   549
  "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   550
apply (cases "b\<le>0") 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   551
 apply (auto simp add: le_less not_less)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   552
apply (drule_tac mult_pos_neg2 [of a b]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   553
 apply (auto dest: less_not_sym)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   554
done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   555
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   556
text{*Strict monotonicity in both arguments*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   557
lemma mult_strict_mono:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   558
  assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   559
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   560
  using assms apply (cases "c=0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   561
  apply (simp add: mult_pos_pos) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   562
  apply (erule mult_strict_right_mono [THEN less_trans])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   563
  apply (force simp add: le_less) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   564
  apply (erule mult_strict_left_mono, assumption)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   565
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   566
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   567
text{*This weaker variant has more natural premises*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   568
lemma mult_strict_mono':
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   569
  assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   570
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   571
  by (rule mult_strict_mono) (insert assms, auto)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   572
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   573
lemma mult_less_le_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   574
  assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   575
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   576
  using assms apply (subgoal_tac "a * c < b * c")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   577
  apply (erule less_le_trans)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   578
  apply (erule mult_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   579
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   580
  apply (erule mult_strict_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   581
  apply assumption
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   582
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   583
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   584
lemma mult_le_less_imp_less:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   585
  assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   586
  shows "a * c < b * d"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   587
  using assms apply (subgoal_tac "a * c \<le> b * c")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   588
  apply (erule le_less_trans)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   589
  apply (erule mult_strict_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   590
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   591
  apply (erule mult_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   592
  apply simp
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   593
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   594
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   595
lemma mult_less_imp_less_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   596
  assumes less: "c * a < c * b" and nonneg: "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   597
  shows "a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   598
proof (rule ccontr)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   599
  assume "\<not>  a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   600
  hence "b \<le> a" by (simp add: linorder_not_less)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   601
  hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   602
  with this and less show False 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   603
    by (simp add: not_less [symmetric])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   604
qed
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   605
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   606
lemma mult_less_imp_less_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   607
  assumes less: "a * c < b * c" and nonneg: "0 \<le> c"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   608
  shows "a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   609
proof (rule ccontr)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   610
  assume "\<not> a < b"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   611
  hence "b \<le> a" by (simp add: linorder_not_less)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   612
  hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   613
  with this and less show False 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   614
    by (simp add: not_less [symmetric])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   615
qed  
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   616
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   617
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   618
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   619
class mult_mono1 = times + zero + ord +
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   620
  assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   621
22390
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   622
class pordered_comm_semiring = comm_semiring_0
378f34b1e380 now using "class"
haftmann
parents: 21328
diff changeset
   623
  + pordered_ab_semigroup_add + mult_mono1
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   624
begin
25152
bfde2f8c0f63 partially localized
haftmann
parents: 25078
diff changeset
   625
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   626
subclass pordered_semiring
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   627
proof unfold_locales
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   628
  fix a b c :: 'a
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   629
  assume "a \<le> b" "0 \<le> c"
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   630
  thus "c * a \<le> c * b" by (rule mult_mono1)
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   631
  thus "a * c \<le> b * c" by (simp only: mult_commute)
21199
2d83f93c3580 * Added annihilation axioms ("x * 0 = 0") to axclass semiring_0.
krauss
parents: 20633
diff changeset
   632
qed
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   633
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   634
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   635
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   636
class pordered_cancel_comm_semiring = comm_semiring_0_cancel
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   637
  + pordered_ab_semigroup_add + mult_mono1
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   638
begin
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   639
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   640
subclass pordered_comm_semiring ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   641
subclass pordered_cancel_semiring ..
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   642
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   643
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   644
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   645
class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   646
  assumes mult_strict_left_mono_comm: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   647
begin
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   648
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   649
subclass ordered_semiring_strict
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   650
proof unfold_locales
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   651
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   652
  assume "a < b" "0 < c"
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   653
  thus "c * a < c * b" by (rule mult_strict_left_mono_comm)
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   654
  thus "a * c < b * c" by (simp only: mult_commute)
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   655
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   656
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   657
subclass pordered_cancel_comm_semiring
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   658
proof unfold_locales
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   659
  fix a b c :: 'a
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   660
  assume "a \<le> b" "0 \<le> c"
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   661
  thus "c * a \<le> c * b"
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   662
    unfolding le_less
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   663
    using mult_strict_left_mono by (cases "c = 0") auto
23550
d4f1d6ef119c convert instance proofs to Isar style
huffman
parents: 23544
diff changeset
   664
qed
14272
5efbb548107d Tidying of the integer development; towards removing the
paulson
parents: 14270
diff changeset
   665
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   666
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   667
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   668
class pordered_ring = ring + pordered_cancel_semiring 
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   669
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   670
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   671
subclass pordered_ab_group_add ..
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   672
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   673
lemmas ring_simps = ring_simps group_simps
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   674
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   675
lemma less_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   676
  "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   677
  by (simp add: ring_simps)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   678
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   679
lemma less_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   680
  "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   681
  by (simp add: ring_simps)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   682
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   683
lemma le_add_iff1:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   684
  "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   685
  by (simp add: ring_simps)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   686
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   687
lemma le_add_iff2:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   688
  "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   689
  by (simp add: ring_simps)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   690
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   691
lemma mult_left_mono_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   692
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   693
  apply (drule mult_left_mono [of _ _ "uminus c"])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   694
  apply (simp_all add: minus_mult_left [symmetric]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   695
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   696
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   697
lemma mult_right_mono_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   698
  "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   699
  apply (drule mult_right_mono [of _ _ "uminus c"])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   700
  apply (simp_all add: minus_mult_right [symmetric]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   701
  done
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   702
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   703
lemma mult_nonpos_nonpos:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   704
  "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   705
  by (drule mult_right_mono_neg [of a zero b]) auto
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   706
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   707
lemma split_mult_pos_le:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   708
  "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   709
  by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   710
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   711
end
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   712
25762
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25564
diff changeset
   713
class abs_if = minus + uminus + ord + zero + abs +
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25564
diff changeset
   714
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25564
diff changeset
   715
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25564
diff changeset
   716
class sgn_if = minus + uminus + zero + one + ord + sgn +
25186
f4d1ebffd025 localized further
haftmann
parents: 25152
diff changeset
   717
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 24491
diff changeset
   718
25564
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25512
diff changeset
   719
lemma (in sgn_if) sgn0[simp]: "sgn 0 = 0"
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25512
diff changeset
   720
by(simp add:sgn_if)
4ca31a3706a4 R&F: added sgn lemma
nipkow
parents: 25512
diff changeset
   721
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   722
class ordered_ring = ring + ordered_semiring
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   723
  + ordered_ab_group_add + abs_if
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   724
begin
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   725
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   726
subclass pordered_ring ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   727
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   728
subclass pordered_ab_group_add_abs
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   729
proof unfold_locales
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   730
  fix a b
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   731
  show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   732
  by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   733
   (auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   734
     neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   735
      auto intro!: less_imp_le add_neg_neg)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   736
qed (auto simp add: abs_if less_eq_neg_nonpos neg_equal_zero)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   737
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   738
end
23521
195fe3fe2831 added ordered_ring and ordered_semiring
obua
parents: 23496
diff changeset
   739
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   740
(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   741
   Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   742
 *)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   743
class ordered_ring_strict = ring + ordered_semiring_strict
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   744
  + ordered_ab_group_add + abs_if
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   745
begin
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   746
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   747
subclass ordered_ring ..
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   748
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   749
lemma mult_strict_left_mono_neg:
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   750
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   751
  apply (drule mult_strict_left_mono [of _ _ "uminus c"])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   752
  apply (simp_all add: minus_mult_left [symmetric]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   753
  done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   754
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   755
lemma mult_strict_right_mono_neg:
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   756
  "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   757
  apply (drule mult_strict_right_mono [of _ _ "uminus c"])
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   758
  apply (simp_all add: minus_mult_right [symmetric]) 
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   759
  done
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   760
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   761
lemma mult_neg_neg:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   762
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   763
  by (drule mult_strict_right_mono_neg, auto)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   764
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   765
subclass ring_no_zero_divisors
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   766
proof unfold_locales
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   767
  fix a b
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   768
  assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   769
  assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   770
  have "a * b < 0 \<or> 0 < a * b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   771
  proof (cases "a < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   772
    case True note A' = this
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   773
    show ?thesis proof (cases "b < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   774
      case True with A'
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   775
      show ?thesis by (auto dest: mult_neg_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   776
    next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   777
      case False with B have "0 < b" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   778
      with A' show ?thesis by (auto dest: mult_strict_right_mono)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   779
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   780
  next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   781
    case False with A have A': "0 < a" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   782
    show ?thesis proof (cases "b < 0")
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   783
      case True with A'
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   784
      show ?thesis by (auto dest: mult_strict_right_mono_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   785
    next
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   786
      case False with B have "0 < b" by auto
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   787
      with A' show ?thesis by (auto dest: mult_pos_pos)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   788
    qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   789
  qed
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   790
  then show "a * b \<noteq> 0" by (simp add: neq_iff)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   791
qed
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   792
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   793
lemma zero_less_mult_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   794
  "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   795
  apply (auto simp add: mult_pos_pos mult_neg_neg)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   796
  apply (simp_all add: not_less le_less)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   797
  apply (erule disjE) apply assumption defer
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   798
  apply (erule disjE) defer apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   799
  apply (erule disjE) defer apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   800
  apply (erule disjE) apply assumption apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   801
  apply (drule sym) apply simp
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   802
  apply (blast dest: zero_less_mult_pos)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   803
  apply (blast dest: zero_less_mult_pos2)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   804
  done
22990
775e9de3db48 added classes ring_no_zero_divisors and dom (non-commutative version of idom);
huffman
parents: 22987
diff changeset
   805
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   806
lemma zero_le_mult_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   807
  "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   808
  by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   809
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   810
lemma mult_less_0_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   811
  "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   812
  apply (insert zero_less_mult_iff [of "-a" b]) 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   813
  apply (force simp add: minus_mult_left[symmetric]) 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   814
  done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   815
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   816
lemma mult_le_0_iff:
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   817
  "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   818
  apply (insert zero_le_mult_iff [of "-a" b]) 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   819
  apply (force simp add: minus_mult_left[symmetric]) 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   820
  done
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   821
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   822
lemma zero_le_square [simp]: "0 \<le> a * a"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   823
  by (simp add: zero_le_mult_iff linear)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   824
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   825
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   826
  by (simp add: not_less)
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   827
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   828
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   829
   also with the relations @{text "\<le>"} and equality.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   830
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   831
text{*These ``disjunction'' versions produce two cases when the comparison is
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   832
 an assumption, but effectively four when the comparison is a goal.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   833
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   834
lemma mult_less_cancel_right_disj:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   835
  "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   836
  apply (cases "c = 0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   837
  apply (auto simp add: neq_iff mult_strict_right_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   838
                      mult_strict_right_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   839
  apply (auto simp add: not_less 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   840
                      not_le [symmetric, of "a*c"]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   841
                      not_le [symmetric, of a])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   842
  apply (erule_tac [!] notE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   843
  apply (auto simp add: less_imp_le mult_right_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   844
                      mult_right_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   845
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   846
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   847
lemma mult_less_cancel_left_disj:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   848
  "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and>  b < a"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   849
  apply (cases "c = 0")
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   850
  apply (auto simp add: neq_iff mult_strict_left_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   851
                      mult_strict_left_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   852
  apply (auto simp add: not_less 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   853
                      not_le [symmetric, of "c*a"]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   854
                      not_le [symmetric, of a])
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   855
  apply (erule_tac [!] notE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   856
  apply (auto simp add: less_imp_le mult_left_mono 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   857
                      mult_left_mono_neg)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   858
  done
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   859
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   860
text{*The ``conjunction of implication'' lemmas produce two cases when the
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   861
comparison is a goal, but give four when the comparison is an assumption.*}
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   862
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   863
lemma mult_less_cancel_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   864
  "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   865
  using mult_less_cancel_right_disj [of a c b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   866
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   867
lemma mult_less_cancel_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   868
  "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   869
  using mult_less_cancel_left_disj [of c a b] by auto
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   870
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   871
lemma mult_le_cancel_right:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   872
   "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   873
  by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   874
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   875
lemma mult_le_cancel_left:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   876
  "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   877
  by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   878
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   879
end
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   880
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   881
text{*This list of rewrites simplifies ring terms by multiplying
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   882
everything out and bringing sums and products into a canonical form
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   883
(by ordered rewriting). As a result it decides ring equalities but
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   884
also helps with inequalities. *}
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   885
lemmas ring_simps = group_simps ring_distribs
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   886
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   887
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   888
class pordered_comm_ring = comm_ring + pordered_comm_semiring
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   889
begin
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   890
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   891
subclass pordered_ring ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   892
subclass pordered_cancel_comm_semiring ..
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   893
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25238
diff changeset
   894
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   895
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   896
class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   897
  (*previously ordered_semiring*)
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   898
  assumes zero_less_one [simp]: "0 < 1"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   899
begin
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   900
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   901
lemma pos_add_strict:
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   902
  shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   903
  using add_strict_mono [of zero a b c] by simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   904
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   905
lemma zero_le_one [simp]: "0 \<le> 1"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   906
  by (rule zero_less_one [THEN less_imp_le]) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   907
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   908
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   909
  by (simp add: not_le) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   910
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   911
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   912
  by (simp add: not_less) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   913
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   914
lemma less_1_mult:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   915
  assumes "1 < m" and "1 < n"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   916
  shows "1 < m * n"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   917
  using assms mult_strict_mono [of 1 m 1 n]
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   918
    by (simp add:  less_trans [OF zero_less_one]) 
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   919
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   920
end
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   921
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   922
class ordered_idom = comm_ring_1 +
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   923
  ordered_comm_semiring_strict + ordered_ab_group_add +
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   924
  abs_if + sgn_if
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   925
  (*previously ordered_ring*)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   926
begin
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   927
27516
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   928
subclass ordered_ring_strict ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   929
subclass pordered_comm_ring ..
9a5d4a8d4aac by intro_locales -> ..
huffman
parents: 26274
diff changeset
   930
subclass idom ..
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   931
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   932
subclass ordered_semidom
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   933
proof unfold_locales
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   934
  have "0 \<le> 1 * 1" by (rule zero_le_square)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   935
  thus "0 < 1" by (simp add: le_less)
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   936
qed 
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   937
26193
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   938
lemma linorder_neqE_ordered_idom:
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   939
  assumes "x \<noteq> y" obtains "x < y" | "y < x"
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   940
  using assms by (rule neqE)
37a7eb7fd5f7 continued localization
haftmann
parents: 25917
diff changeset
   941
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   942
text {* These cancellation simprules also produce two cases when the comparison is a goal. *}
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   943
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   944
lemma mult_le_cancel_right1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   945
  "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   946
  by (insert mult_le_cancel_right [of 1 c b], simp)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   947
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   948
lemma mult_le_cancel_right2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   949
  "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   950
  by (insert mult_le_cancel_right [of a c 1], simp)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   951
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   952
lemma mult_le_cancel_left1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   953
  "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   954
  by (insert mult_le_cancel_left [of c 1 b], simp)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   955
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   956
lemma mult_le_cancel_left2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   957
  "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   958
  by (insert mult_le_cancel_left [of c a 1], simp)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   959
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   960
lemma mult_less_cancel_right1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   961
  "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   962
  by (insert mult_less_cancel_right [of 1 c b], simp)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   963
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   964
lemma mult_less_cancel_right2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   965
  "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   966
  by (insert mult_less_cancel_right [of a c 1], simp)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   967
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   968
lemma mult_less_cancel_left1:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   969
  "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   970
  by (insert mult_less_cancel_left [of c 1 b], simp)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   971
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   972
lemma mult_less_cancel_left2:
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   973
  "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   974
  by (insert mult_less_cancel_left [of c a 1], simp)
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   975
25917
d6c920623afc further localization
haftmann
parents: 25762
diff changeset
   976
end
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   977
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   978
class ordered_field = field + ordered_idom
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   979
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   980
text {* Simprules for comparisons where common factors can be cancelled. *}
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   981
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   982
lemmas mult_compare_simps =
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   983
    mult_le_cancel_right mult_le_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   984
    mult_le_cancel_right1 mult_le_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   985
    mult_le_cancel_left1 mult_le_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   986
    mult_less_cancel_right mult_less_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   987
    mult_less_cancel_right1 mult_less_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   988
    mult_less_cancel_left1 mult_less_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   989
    mult_cancel_right mult_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   990
    mult_cancel_right1 mult_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   991
    mult_cancel_left1 mult_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   992
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   993
-- {* FIXME continue localization here *}
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   994
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   995
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
   996
   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
26274
2bdb61a28971 continued localization
haftmann
parents: 26234
diff changeset
   997
by (force dest: inverse_zero_imp_zero) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   998
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   999
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
  1000
   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1001
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1002
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1003
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1004
  assume "a\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1005
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
  1006
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1007
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1008
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
  1009
  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1010
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
  1011
 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
  1012
              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
  1013
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
  1014
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1015
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1016
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
  1017
  "(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
  1018
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
  1019
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1020
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
  1021
     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1022
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1023
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1024
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1025
    assume "a\<noteq>0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1026
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1027
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1028
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1029
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1030
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1031
lemma inverse_mult_distrib [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1032
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1033
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1034
    assume "a \<noteq> 0 & b \<noteq> 0" 
22993
haftmann
parents: 22990
diff changeset
  1035
    thus ?thesis
haftmann
parents: 22990
diff changeset
  1036
      by (simp add: nonzero_inverse_mult_distrib mult_commute)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1037
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1038
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
22993
haftmann
parents: 22990
diff changeset
  1039
    thus ?thesis
haftmann
parents: 22990
diff changeset
  1040
      by force
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1041
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1042
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1043
text{*There is no slick version using division by zero.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1044
lemma inverse_add:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1045
  "[|a \<noteq> 0;  b \<noteq> 0|]
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1046
   ==> 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
  1047
by (simp add: division_ring_inverse_add mult_ac)
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
  1048
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1049
lemma inverse_divide [simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1050
  "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
  1051
by (simp add: divide_inverse mult_commute)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1052
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1053
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1054
subsection {* Calculations with fractions *}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1055
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1056
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
  1057
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
  1058
because the latter are covered by a simproc. *}
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1059
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1060
lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1061
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
  1062
proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1063
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1064
    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
  1065
  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
  1066
    by (simp only: mult_ac)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1067
  also have "... =  a * inverse b"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1068
    by simp
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1069
    finally show ?thesis 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1070
    by (simp add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1071
qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1072
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1073
lemma mult_divide_mult_cancel_left:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1074
  "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1075
apply (cases "b = 0")
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1076
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
  1077
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1078
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1079
lemma nonzero_mult_divide_mult_cancel_right [noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1080
  "[|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
  1081
by (simp add: mult_commute [of _ c] nonzero_mult_divide_mult_cancel_left) 
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1082
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1083
lemma mult_divide_mult_cancel_right:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1084
  "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1085
apply (cases "b = 0")
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1086
apply (simp_all add: nonzero_mult_divide_mult_cancel_right)
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
  1087
done
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1088
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
  1089
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
  1090
by (simp add: divide_inverse)
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
  1091
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1092
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
  1093
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1094
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1095
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
  1096
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1097
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1098
lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1099
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
  1100
lemma divide_divide_eq_right [simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1101
  "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
  1102
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1103
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
  1104
lemma divide_divide_eq_left [simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1105
  "(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
  1106
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1107
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1108
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
  1109
    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
  1110
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
  1111
apply (erule ssubst)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1112
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
  1113
apply (erule ssubst)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1114
apply (rule add_divide_distrib [THEN sym])
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1115
apply (subst mult_commute)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1116
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
  1117
apply assumption
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1118
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
  1119
apply assumption
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1120
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1121
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1122
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1123
subsubsection{*Special Cancellation Simprules for Division*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1124
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1125
lemma mult_divide_mult_cancel_left_if[simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1126
fixes c :: "'a :: {field,division_by_zero}"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1127
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
  1128
by (simp add: mult_divide_mult_cancel_left)
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1129
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1130
lemma nonzero_mult_divide_cancel_right[simp,noatp]:
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1131
  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1132
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
  1133
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1134
lemma nonzero_mult_divide_cancel_left[simp,noatp]:
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1135
  "a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1136
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
  1137
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1138
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1139
lemma nonzero_divide_mult_cancel_right[simp,noatp]:
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1140
  "\<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
  1141
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
  1142
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1143
lemma nonzero_divide_mult_cancel_left[simp,noatp]:
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1144
  "\<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
  1145
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
  1146
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
  1147
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1148
lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1149
  "[|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
  1150
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
  1151
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
  1152
lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1153
  "[|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
  1154
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
  1155
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1156
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1157
subsection {* Division and Unary Minus *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1158
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1159
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
  1160
by (simp add: divide_inverse minus_mult_left)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1161
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1162
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
  1163
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
  1164
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1165
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
  1166
by (simp add: divide_inverse nonzero_inverse_minus_eq)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1167
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1168
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
  1169
by (simp add: divide_inverse minus_mult_left [symmetric])
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1170
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1171
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
  1172
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
  1173
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1174
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1175
text{*The effect is to extract signs from divisions*}
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1176
lemmas divide_minus_left = minus_divide_left [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1177
lemmas divide_minus_right = minus_divide_right [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1178
declare divide_minus_left [simp]   divide_minus_right [simp]
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1179
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1180
text{*Also, extract signs from products*}
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1181
lemmas mult_minus_left = minus_mult_left [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1182
lemmas mult_minus_right = minus_mult_right [symmetric]
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
  1183
declare mult_minus_left [simp]   mult_minus_right [simp]
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1184
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1185
lemma minus_divide_divide [simp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
  1186
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1187
apply (cases "b=0", simp) 
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1188
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1189
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1190
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1191
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
  1192
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1193
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1194
lemma add_divide_eq_iff:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1195
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1196
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
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 divide_add_eq_iff:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1199
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1200
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1201
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1202
lemma diff_divide_eq_iff:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1203
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1204
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1205
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1206
lemma divide_diff_eq_iff:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1207
  "(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1208
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1209
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1210
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
  1211
proof -
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1212
  assume [simp]: "c\<noteq>0"
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
  1213
  have "(a = b/c) = (a*c = (b/c)*c)" by simp
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
  1214
  also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1215
  finally show ?thesis .
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1216
qed
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1217
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1218
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
  1219
proof -
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1220
  assume [simp]: "c\<noteq>0"
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
  1221
  have "(b/c = a) = ((b/c)*c = a*c)"  by simp
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
  1222
  also have "... = (b = a*c)"  by (simp add: divide_inverse mult_assoc) 
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1223
  finally show ?thesis .
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1224
qed
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1225
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1226
lemma eq_divide_eq:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1227
  "((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
  1228
by (simp add: nonzero_eq_divide_eq) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1229
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1230
lemma divide_eq_eq:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1231
  "(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
  1232
by (force simp add: nonzero_divide_eq_eq) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1233
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1234
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1235
    b = a * c ==> b / c = a"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1236
  by (subst divide_eq_eq, simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1237
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1238
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1239
    a * c = b ==> a = b / c"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1240
  by (subst eq_divide_eq, simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1241
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1242
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1243
lemmas field_eq_simps = ring_simps
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1244
  (* pull / out*)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1245
  add_divide_eq_iff divide_add_eq_iff
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1246
  diff_divide_eq_iff divide_diff_eq_iff
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1247
  (* multiply eqn *)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1248
  nonzero_eq_divide_eq nonzero_divide_eq_eq
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1249
(* is added later:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1250
  times_divide_eq_left times_divide_eq_right
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1251
*)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1252
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1253
text{*An example:*}
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1254
lemma fixes a b c d e f :: "'a::field"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1255
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
  1256
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1257
 apply(simp add:field_eq_simps)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1258
apply(simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1259
done
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1260
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1261
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1262
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
  1263
    x / y - w / z = (x * z - w * y) / (y * z)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1264
by (simp add:field_eq_simps times_divide_eq)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1265
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1266
lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1267
    (x / y = w / z) = (x * z = w * y)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1268
by (simp add:field_eq_simps times_divide_eq)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1269
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
  1270
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1271
subsection {* Ordered Fields *}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1272
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1273
lemma positive_imp_inverse_positive: 
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1274
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1275
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1276
  have "0 < a * inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1277
    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
  1278
  thus "0 < inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1279
    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
  1280
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1281
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1282
lemma negative_imp_inverse_negative:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1283
  "a < 0 ==> inverse a < (0::'a::ordered_field)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1284
by (insert positive_imp_inverse_positive [of "-a"], 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1285
    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
  1286
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1287
lemma inverse_le_imp_le:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1288
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1289
shows "b \<le> (a::'a::ordered_field)"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1290
proof (rule classical)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1291
  assume "~ b \<le> a"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1292
  hence "a < b"  by (simp add: linorder_not_le)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1293
  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
  1294
  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
  1295
    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
  1296
  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
  1297
    by (simp add: bpos order_less_imp_le mult_right_mono)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
  1298
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
2f4be6844f7c tuned and used field_simps
ni