src/HOL/Ring_and_Field.thy
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Cleaning up HOL/Matrix
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(*  Title:   HOL/Ring_and_Field.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson and Markus Wenzel
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*)
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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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axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult
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  left_distrib: "(a + b) * c = a * c + b * c"
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  right_distrib: "a * (b + c) = a * b + a * c"
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axclass semiring_0 \<subseteq> semiring, comm_monoid_add
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axclass semiring_0_cancel \<subseteq> semiring_0, cancel_ab_semigroup_add
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axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult  
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  mult_commute: "a * b = b * a"
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  distrib: "(a + b) * c = a * c + b * c"
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instance comm_semiring \<subseteq> semiring
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proof
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  fix a b c :: 'a
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  show "(a + b) * c = a * c + b * c" by (simp add: distrib)
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  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
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  also have "... = b * a + c * a" by (simp only: distrib)
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  also have "... = a * b + a * c" by (simp add: mult_ac)
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  finally show "a * (b + c) = a * b + a * c" by blast
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qed
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axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add
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instance comm_semiring_0 \<subseteq> semiring_0 ..
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axclass comm_semiring_0_cancel \<subseteq> comm_semiring_0, cancel_ab_semigroup_add
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instance comm_semiring_0_cancel \<subseteq> semiring_0_cancel ..
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axclass axclass_0_neq_1 \<subseteq> zero, one
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  zero_neq_one [simp]: "0 \<noteq> 1"
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axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult
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axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *)
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instance comm_semiring_1 \<subseteq> semiring_1 ..
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axclass axclass_no_zero_divisors \<subseteq> zero, times
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  no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
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axclass semiring_1_cancel \<subseteq> semiring_1, cancel_ab_semigroup_add
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instance semiring_1_cancel \<subseteq> semiring_0_cancel ..
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axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)
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instance comm_semiring_1_cancel \<subseteq> semiring_1_cancel ..
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instance comm_semiring_1_cancel \<subseteq> comm_semiring_0_cancel ..
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axclass ring \<subseteq> semiring, ab_group_add
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instance ring \<subseteq> semiring_0_cancel ..
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axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add
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instance comm_ring \<subseteq> ring ..
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instance comm_ring \<subseteq> comm_semiring_0_cancel ..
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axclass ring_1 \<subseteq> ring, semiring_1
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instance ring_1 \<subseteq> semiring_1_cancel ..
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axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *)
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instance comm_ring_1 \<subseteq> ring_1 ..
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instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..
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axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors
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axclass field \<subseteq> comm_ring_1, inverse
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  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
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  divide_inverse:      "a / b = a * inverse b"
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lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring_0_cancel)"
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proof -
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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 ?thesis 
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    by (simp only: add_left_cancel)
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qed
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lemma mult_zero_right [simp]: "a * 0 = (0::'a::semiring_0_cancel)"
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proof -
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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 ?thesis 
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    by (simp only: add_left_cancel)
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qed
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lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
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proof cases
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  assume "a=0" thus ?thesis by simp
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next
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  assume anz [simp]: "a\<noteq>0"
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  { assume "a * b = 0"
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    hence "inverse a * (a * b) = 0" by simp
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    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
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  thus ?thesis by force
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qed
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instance field \<subseteq> idom
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by (intro_classes, simp)
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axclass division_by_zero \<subseteq> zero, inverse
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  inverse_zero [simp]: "inverse 0 = 0"
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subsection {* Distribution rules *}
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theorems ring_distrib = right_distrib left_distrib
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text{*For the @{text combine_numerals} simproc*}
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lemma combine_common_factor:
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     "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"
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by (simp add: left_distrib add_ac)
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lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: left_distrib [symmetric]) 
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done
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lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"
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apply (rule equals_zero_I)
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apply (simp add: right_distrib [symmetric]) 
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done
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lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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   157
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3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
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lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"
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  by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])
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   160
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lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"
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by (simp add: right_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"
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by (simp add: left_distrib diff_minus 
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              minus_mult_left [symmetric] minus_mult_right [symmetric]) 
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axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add 
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  mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
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  mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c"
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axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add
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instance pordered_cancel_semiring \<subseteq> semiring_0_cancel ..
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axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add
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  mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
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  mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"
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a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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instance ordered_semiring_strict \<subseteq> semiring_0_cancel ..
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instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring
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apply intro_classes
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apply (case_tac "a < b & 0 < c")
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apply (auto simp add: mult_strict_left_mono order_less_le)
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apply (auto simp add: mult_strict_left_mono order_le_less)
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apply (simp add: mult_strict_right_mono)
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   189
done
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   190
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axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add
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  mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"
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   193
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axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add
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   195
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instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..
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   197
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   198
axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add
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   199
  mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
14265
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instance pordered_comm_semiring \<subseteq> pordered_semiring
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by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)
14265
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instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..
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   206
instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict
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   207
by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)
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   209
instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring
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   210
apply (intro_classes)
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apply (case_tac "a < b & 0 < c")
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apply (auto simp add: mult_strict_left_mono order_less_le)
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   213
apply (auto simp add: mult_strict_left_mono order_le_less)
14272
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done
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   215
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axclass pordered_ring \<subseteq> ring, pordered_semiring 
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   217
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instance pordered_ring \<subseteq> pordered_ab_group_add ..
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   219
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instance pordered_ring \<subseteq> pordered_cancel_semiring ..
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   221
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   222
axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs
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   223
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instance lordered_ring \<subseteq> lordered_ab_group_meet ..
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b9ab8babd8b3 Further development of matrix theory
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instance lordered_ring \<subseteq> lordered_ab_group_join ..
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   227
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axclass axclass_abs_if \<subseteq> minus, ord, zero
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  abs_if: "abs a = (if (a < 0) then (-a) else a)"
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   230
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   231
axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if
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   233
instance ordered_ring_strict \<subseteq> lordered_ab_group ..
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   234
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   235
instance ordered_ring_strict \<subseteq> lordered_ring
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   236
by (intro_classes, simp add: abs_if join_eq_if)
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   237
14738
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   238
axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring
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   239
14738
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   240
axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *)
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   241
  zero_less_one [simp]: "0 < 1"
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   242
14738
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   243
axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *)
14270
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   244
14738
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   245
instance ordered_idom \<subseteq> ordered_ring_strict ..
14272
5efbb548107d Tidying of the integer development; towards removing the
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   246
14738
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   247
axclass ordered_field \<subseteq> field, ordered_idom
14272
5efbb548107d Tidying of the integer development; towards removing the
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   248
5efbb548107d Tidying of the integer development; towards removing the
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   249
lemma eq_add_iff1:
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   250
     "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
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diff changeset
   251
apply (simp add: diff_minus left_distrib)
14272
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   252
apply (simp add: diff_minus left_distrib add_ac)
14738
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   253
apply (simp add: compare_rls minus_mult_left [symmetric])
14272
5efbb548107d Tidying of the integer development; towards removing the
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   254
done
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diff changeset
   255
5efbb548107d Tidying of the integer development; towards removing the
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   256
lemma eq_add_iff2:
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   257
     "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"
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diff changeset
   258
apply (simp add: diff_minus left_distrib add_ac)
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   259
apply (simp add: compare_rls minus_mult_left [symmetric]) 
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   260
done
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diff changeset
   261
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   262
lemma less_add_iff1:
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83f1a514dcb4 changes made due to new Ring_and_Field theory
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   263
     "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"
14272
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   264
apply (simp add: diff_minus left_distrib add_ac)
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diff changeset
   265
apply (simp add: compare_rls minus_mult_left [symmetric]) 
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   266
done
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diff changeset
   267
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   268
lemma less_add_iff2:
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83f1a514dcb4 changes made due to new Ring_and_Field theory
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   269
     "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"
14272
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   270
apply (simp add: diff_minus left_distrib add_ac)
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   271
apply (simp add: compare_rls minus_mult_left [symmetric]) 
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   272
done
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   273
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   274
lemma le_add_iff1:
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83f1a514dcb4 changes made due to new Ring_and_Field theory
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   275
     "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"
14272
5efbb548107d Tidying of the integer development; towards removing the
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   276
apply (simp add: diff_minus left_distrib add_ac)
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   277
apply (simp add: compare_rls minus_mult_left [symmetric]) 
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   278
done
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diff changeset
   279
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   280
lemma le_add_iff2:
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83f1a514dcb4 changes made due to new Ring_and_Field theory
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   281
     "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"
14272
5efbb548107d Tidying of the integer development; towards removing the
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   282
apply (simp add: diff_minus left_distrib add_ac)
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   283
apply (simp add: compare_rls minus_mult_left [symmetric]) 
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   284
done
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   285
14270
342451d763f9 More re-organising of numerical theorems
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   286
subsection {* Ordering Rules for Multiplication *}
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   287
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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   288
lemma mult_left_le_imp_le:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
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diff changeset
   289
     "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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diff changeset
   290
  by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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diff changeset
   291
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
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parents: 14341
diff changeset
   292
lemma mult_right_le_imp_le:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   293
     "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   294
  by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   295
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   296
lemma mult_left_less_imp_less:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   297
     "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   298
  by (force simp add: mult_left_mono linorder_not_le [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   299
 
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   300
lemma mult_right_less_imp_less:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   301
     "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   302
  by (force simp add: mult_right_mono linorder_not_le [symmetric])
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   303
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   304
lemma mult_strict_left_mono_neg:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   305
     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   306
apply (drule mult_strict_left_mono [of _ _ "-c"])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   307
apply (simp_all add: minus_mult_left [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   308
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   309
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   310
lemma mult_left_mono_neg:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   311
     "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   312
apply (drule mult_left_mono [of _ _ "-c"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   313
apply (simp_all add: minus_mult_left [symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   314
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   315
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   316
lemma mult_strict_right_mono_neg:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   317
     "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   318
apply (drule mult_strict_right_mono [of _ _ "-c"])
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   319
apply (simp_all add: minus_mult_right [symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   320
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   321
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   322
lemma mult_right_mono_neg:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   323
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   324
apply (drule mult_right_mono [of _ _ "-c"])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   325
apply (simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   326
apply (simp_all add: minus_mult_right [symmetric]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   327
done
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   328
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   329
subsection{* Products of Signs *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   330
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   331
lemma mult_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   332
by (drule mult_strict_left_mono [of 0 b], auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   333
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   334
lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   335
by (drule mult_left_mono [of 0 b], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   336
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   337
lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   338
by (drule mult_strict_left_mono [of b 0], auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   339
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   340
lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   341
by (drule mult_left_mono [of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   342
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   343
lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   344
by (drule mult_strict_right_mono[of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   345
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   346
lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   347
by (drule mult_right_mono[of b 0], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   348
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   349
lemma mult_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   350
by (drule mult_strict_right_mono_neg, auto)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   351
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   352
lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   353
by (drule mult_right_mono_neg[of a 0 b ], auto)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   354
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   355
lemma zero_less_mult_pos:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   356
     "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   357
apply (case_tac "b\<le>0") 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   358
 apply (auto simp add: order_le_less linorder_not_less)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   359
apply (drule_tac mult_pos_neg [of a b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   360
 apply (auto dest: order_less_not_sym)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   361
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   362
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   363
lemma zero_less_mult_pos2:
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   364
     "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   365
apply (case_tac "b\<le>0") 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   366
 apply (auto simp add: order_le_less linorder_not_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   367
apply (drule_tac mult_pos_neg2 [of a b]) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   368
 apply (auto dest: order_less_not_sym)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   369
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   370
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   371
lemma zero_less_mult_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   372
     "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   373
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   374
apply (blast dest: zero_less_mult_pos) 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   375
apply (blast dest: zero_less_mult_pos2)
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   376
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   377
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   378
text{*A field has no "zero divisors", and this theorem holds without the
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   379
      assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   380
lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   381
apply (case_tac "a < 0")
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   382
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   383
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   384
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   385
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   386
lemma zero_le_mult_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   387
     "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   388
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   389
                   zero_less_mult_iff)
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   390
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   391
lemma mult_less_0_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   392
     "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   393
apply (insert zero_less_mult_iff [of "-a" b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   394
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   395
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   396
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   397
lemma mult_le_0_iff:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   398
     "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   399
apply (insert zero_le_mult_iff [of "-a" b]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   400
apply (force simp add: minus_mult_left[symmetric]) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   401
done
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   402
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   403
lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   404
by (auto simp add: mult_pos_le mult_neg_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   405
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   406
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   407
by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   408
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   409
lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   410
by (simp add: zero_le_mult_iff linorder_linear) 
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   411
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   412
text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   413
      theorems available to members of @{term ordered_idom} *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   414
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   415
instance ordered_idom \<subseteq> ordered_semidom
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   416
proof
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   417
  have "(0::'a) \<le> 1*1" by (rule zero_le_square)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   418
  thus "(0::'a) < 1" by (simp add: order_le_less) 
14421
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   419
qed
ee97b6463cb4 new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents: 14398
diff changeset
   420
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   421
instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   422
by (intro_classes, simp)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   423
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   424
instance ordered_idom \<subseteq> idom ..
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   425
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   426
text{*All three types of comparision involving 0 and 1 are covered.*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   427
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   428
declare zero_neq_one [THEN not_sym, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   429
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   430
lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   431
  by (rule zero_less_one [THEN order_less_imp_le]) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   432
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   433
lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   434
by (simp add: linorder_not_le) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   435
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   436
lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   437
by (simp add: linorder_not_less) 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   438
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   439
subsection{*More Monotonicity*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   440
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   441
text{*Strict monotonicity in both arguments*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   442
lemma mult_strict_mono:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   443
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   444
apply (case_tac "c=0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   445
 apply (simp add: mult_pos) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   446
apply (erule mult_strict_right_mono [THEN order_less_trans])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   447
 apply (force simp add: order_le_less) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   448
apply (erule mult_strict_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   449
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   450
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   451
text{*This weaker variant has more natural premises*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   452
lemma mult_strict_mono':
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   453
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   454
apply (rule mult_strict_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   455
apply (blast intro: order_le_less_trans)+
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   456
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   457
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   458
lemma mult_mono:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   459
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   460
      ==> a * c  \<le>  b * (d::'a::pordered_semiring)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   461
apply (erule mult_right_mono [THEN order_trans], assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   462
apply (erule mult_left_mono, assumption)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   463
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   464
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   465
lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   466
apply (insert mult_strict_mono [of 1 m 1 n]) 
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   467
apply (simp add:  order_less_trans [OF zero_less_one]) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   468
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
   469
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   470
subsection{*Cancellation Laws for Relationships With a Common Factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   471
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   472
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   473
   also with the relations @{text "\<le>"} and equality.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   474
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   475
text{*These ``disjunction'' versions produce two cases when the comparison is
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   476
 an assumption, but effectively four when the comparison is a goal.*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   477
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   478
lemma mult_less_cancel_right_disj:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   479
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   480
apply (case_tac "c = 0")
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   481
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   482
                      mult_strict_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   483
apply (auto simp add: linorder_not_less 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   484
                      linorder_not_le [symmetric, of "a*c"]
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   485
                      linorder_not_le [symmetric, of a])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   486
apply (erule_tac [!] notE)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   487
apply (auto simp add: order_less_imp_le mult_right_mono 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   488
                      mult_right_mono_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   489
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   490
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   491
lemma mult_less_cancel_left_disj:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   492
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   493
apply (case_tac "c = 0")
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   494
apply (auto simp add: linorder_neq_iff mult_strict_left_mono 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   495
                      mult_strict_left_mono_neg)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   496
apply (auto simp add: linorder_not_less 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   497
                      linorder_not_le [symmetric, of "c*a"]
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   498
                      linorder_not_le [symmetric, of a])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   499
apply (erule_tac [!] notE)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   500
apply (auto simp add: order_less_imp_le mult_left_mono 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   501
                      mult_left_mono_neg)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   502
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   503
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   504
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   505
text{*The ``conjunction of implication'' lemmas produce two cases when the
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   506
comparison is a goal, but give four when the comparison is an assumption.*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   507
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   508
lemma mult_less_cancel_right:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   509
  fixes c :: "'a :: ordered_ring_strict"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   510
  shows      "(a*c < b*c) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   511
by (insert mult_less_cancel_right_disj [of a c b], auto)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   512
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   513
lemma mult_less_cancel_left:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   514
  fixes c :: "'a :: ordered_ring_strict"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   515
  shows      "(c*a < c*b) = ((0 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   516
by (insert mult_less_cancel_left_disj [of c a b], auto)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   517
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   518
lemma mult_le_cancel_right:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   519
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   520
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   521
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   522
lemma mult_le_cancel_left:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   523
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   524
by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   525
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   526
lemma mult_less_imp_less_left:
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14334
diff changeset
   527
      assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   528
      shows "a < (b::'a::ordered_semiring_strict)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   529
proof (rule ccontr)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   530
  assume "~ a < b"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   531
  hence "b \<le> a" by (simp add: linorder_not_less)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   532
  hence "c*b \<le> c*a" by (rule mult_left_mono)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   533
  with this and less show False 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   534
    by (simp add: linorder_not_less [symmetric])
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   535
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   536
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   537
lemma mult_less_imp_less_right:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   538
  assumes less: "a*c < b*c" and nonneg: "0 <= c"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   539
  shows "a < (b::'a::ordered_semiring_strict)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   540
proof (rule ccontr)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   541
  assume "~ a < b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   542
  hence "b \<le> a" by (simp add: linorder_not_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   543
  hence "b*c \<le> a*c" by (rule mult_right_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   544
  with this and less show False 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   545
    by (simp add: linorder_not_less [symmetric])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   546
qed  
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   547
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   548
text{*Cancellation of equalities with a common factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   549
lemma mult_cancel_right [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   550
     "(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   551
apply (cut_tac linorder_less_linear [of 0 c])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   552
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   553
             simp add: linorder_neq_iff)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   554
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   555
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   556
text{*These cancellation theorems require an ordering. Versions are proved
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   557
      below that work for fields without an ordering.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   558
lemma mult_cancel_left [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   559
     "(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   560
apply (cut_tac linorder_less_linear [of 0 c])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   561
apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   562
             simp add: linorder_neq_iff)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   563
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   564
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   565
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   566
subsubsection{*Special Cancellation Simprules for Multiplication*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   567
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   568
text{*These also produce two cases when the comparison is a goal.*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   569
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   570
lemma mult_le_cancel_right1:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   571
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   572
  shows "(c \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   573
by (insert mult_le_cancel_right [of 1 c b], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   574
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   575
lemma mult_le_cancel_right2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   576
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   577
  shows "(a*c \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   578
by (insert mult_le_cancel_right [of a c 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   579
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   580
lemma mult_le_cancel_left1:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   581
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   582
  shows "(c \<le> c*b) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 1))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   583
by (insert mult_le_cancel_left [of c 1 b], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   584
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   585
lemma mult_le_cancel_left2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   586
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   587
  shows "(c*a \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   588
by (insert mult_le_cancel_left [of c a 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   589
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   590
lemma mult_less_cancel_right1:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   591
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   592
  shows "(c < b*c) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   593
by (insert mult_less_cancel_right [of 1 c b], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   594
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   595
lemma mult_less_cancel_right2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   596
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   597
  shows "(a*c < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   598
by (insert mult_less_cancel_right [of a c 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   599
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   600
lemma mult_less_cancel_left1:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   601
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   602
  shows "(c < c*b) = ((0 \<le> c --> 1<b) & (c \<le> 0 --> b < 1))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   603
by (insert mult_less_cancel_left [of c 1 b], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   604
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   605
lemma mult_less_cancel_left2:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   606
  fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   607
  shows "(c*a < c) = ((0 \<le> c --> a<1) & (c \<le> 0 --> 1 < a))"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   608
by (insert mult_less_cancel_left [of c a 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   609
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   610
lemma mult_cancel_right1 [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   611
fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   612
  shows "(c = b*c) = (c = 0 | b=1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   613
by (insert mult_cancel_right [of 1 c b], force)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   614
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   615
lemma mult_cancel_right2 [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   616
fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   617
  shows "(a*c = c) = (c = 0 | a=1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   618
by (insert mult_cancel_right [of a c 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   619
 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   620
lemma mult_cancel_left1 [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   621
fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   622
  shows "(c = c*b) = (c = 0 | b=1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   623
by (insert mult_cancel_left [of c 1 b], force)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   624
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   625
lemma mult_cancel_left2 [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   626
fixes c :: "'a :: ordered_idom"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   627
  shows "(c*a = c) = (c = 0 | a=1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   628
by (insert mult_cancel_left [of c a 1], simp)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   629
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   630
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   631
text{*Simprules for comparisons where common factors can be cancelled.*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   632
lemmas mult_compare_simps =
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   633
    mult_le_cancel_right mult_le_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   634
    mult_le_cancel_right1 mult_le_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   635
    mult_le_cancel_left1 mult_le_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   636
    mult_less_cancel_right mult_less_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   637
    mult_less_cancel_right1 mult_less_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   638
    mult_less_cancel_left1 mult_less_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   639
    mult_cancel_right mult_cancel_left
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   640
    mult_cancel_right1 mult_cancel_right2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   641
    mult_cancel_left1 mult_cancel_left2
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   642
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   643
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   644
text{*This list of rewrites decides ring equalities by ordered rewriting.*}
15178
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
   645
lemmas ring_eq_simps =  
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
   646
(*  mult_ac*)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   647
  left_distrib right_distrib left_diff_distrib right_diff_distrib
15178
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
   648
  group_eq_simps
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
   649
(*  add_ac
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   650
  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
15178
5f621aa35c25 Matrix theory, linear programming
obua
parents: 15140
diff changeset
   651
  diff_eq_eq eq_diff_eq *)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14603
diff changeset
   652
    
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   653
subsection {* Fields *}
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   654
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   655
lemma right_inverse [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   656
      assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   657
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   658
  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   659
  also have "... = 1" using not0 by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   660
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   661
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   662
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   663
lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   664
proof
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   665
  assume neq: "b \<noteq> 0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   666
  {
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   667
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   668
    also assume "a / b = 1"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   669
    finally show "a = b" by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   670
  next
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   671
    assume "a = b"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   672
    with neq show "a / b = 1" by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   673
  }
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   674
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   675
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   676
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   677
by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   678
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   679
lemma divide_self: "a \<noteq> 0 ==> a / (a::'a::field) = 1"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   680
  by (simp add: divide_inverse)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   681
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   682
lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   683
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   684
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   685
lemma divide_self_if [simp]:
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   686
     "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   687
  by (simp add: divide_self)
4d332d10fa3d revised simprules for division
paulson
parents: 15197
diff changeset
   688
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   689
lemma divide_zero_left [simp]: "0/a = (0::'a::field)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   690
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   691
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   692
lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   693
by (simp add: divide_inverse)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   694
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   695
lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   696
by (simp add: divide_inverse left_distrib) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   697
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   698
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   699
text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   700
      of an ordering.*}
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   701
lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   702
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   703
  assume "a=0" thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   704
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   705
  assume anz [simp]: "a\<noteq>0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   706
  { assume "a * b = 0"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   707
    hence "inverse a * (a * b) = 0" by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   708
    hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   709
  thus ?thesis by force
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   710
qed
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   711
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   712
text{*Cancellation of equalities with a common factor*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   713
lemma field_mult_cancel_right_lemma:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   714
      assumes cnz: "c \<noteq> (0::'a::field)"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   715
	  and eq:  "a*c = b*c"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   716
	 shows "a=b"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   717
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   718
  have "(a * c) * inverse c = (b * c) * inverse c"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   719
    by (simp add: eq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   720
  thus "a=b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   721
    by (simp add: mult_assoc cnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   722
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   723
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   724
lemma field_mult_cancel_right [simp]:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   725
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   726
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   727
  assume "c=0" thus ?thesis by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   728
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   729
  assume "c\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   730
  thus ?thesis by (force dest: field_mult_cancel_right_lemma)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   731
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   732
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   733
lemma field_mult_cancel_left [simp]:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   734
     "(c*a = c*b) = (c = (0::'a::field) | a=b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   735
  by (simp add: mult_commute [of c] field_mult_cancel_right) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   736
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   737
lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   738
proof
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   739
  assume ianz: "inverse a = 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   740
  assume "a \<noteq> 0"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   741
  hence "1 = a * inverse a" by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   742
  also have "... = 0" by (simp add: ianz)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   743
  finally have "1 = (0::'a::field)" .
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   744
  thus False by (simp add: eq_commute)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   745
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   746
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   747
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   748
subsection{*Basic Properties of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   749
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   750
lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   751
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   752
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   753
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   754
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   755
lemma inverse_nonzero_imp_nonzero:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   756
   "inverse a = 0 ==> a = (0::'a::field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   757
apply (rule ccontr) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   758
apply (blast dest: nonzero_imp_inverse_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   759
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   760
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   761
lemma inverse_nonzero_iff_nonzero [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   762
   "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   763
by (force dest: inverse_nonzero_imp_nonzero) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   764
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   765
lemma nonzero_inverse_minus_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   766
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   767
proof -
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   768
  have "-a * inverse (- a) = -a * - inverse a"
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   769
    by simp
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   770
  thus ?thesis 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   771
    by (simp only: field_mult_cancel_left, simp)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   772
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   773
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   774
lemma inverse_minus_eq [simp]:
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   775
   "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   776
proof cases
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   777
  assume "a=0" thus ?thesis by (simp add: inverse_zero)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   778
next
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   779
  assume "a\<noteq>0" 
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   780
  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   781
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   782
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   783
lemma nonzero_inverse_eq_imp_eq:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   784
      assumes inveq: "inverse a = inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   785
	  and anz:  "a \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   786
	  and bnz:  "b \<noteq> 0"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
   787
	 shows "a = (b::'a::field)"
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   788
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   789
  have "a * inverse b = a * inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   790
    by (simp add: inveq)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   791
  hence "(a * inverse b) * b = (a * inverse a) * b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   792
    by simp
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   793
  thus "a = b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   794
    by (simp add: mult_assoc anz bnz)
14377
f454b3004f8f tidying up, especially the Complex numbers
paulson
parents: 14370
diff changeset
   795
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   796
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   797
lemma inverse_eq_imp_eq:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   798
     "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   799
apply (case_tac "a=0 | b=0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   800
 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
   801
              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
   802
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
   803
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   804
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   805
lemma inverse_eq_iff_eq [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   806
     "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   807
by (force dest!: inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   808
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   809
lemma nonzero_inverse_inverse_eq:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   810
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   811
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   812
  have "(inverse (inverse a) * inverse a) * a = a" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   813
    by (simp add: nonzero_imp_inverse_nonzero)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   814
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   815
    by (simp add: mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   816
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   817
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   818
lemma inverse_inverse_eq [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   819
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   820
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   821
    assume "a=0" thus ?thesis by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   822
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   823
    assume "a\<noteq>0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   824
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   825
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   826
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   827
lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   828
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   829
  have "inverse 1 * 1 = (1::'a::field)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   830
    by (rule left_inverse [OF zero_neq_one [symmetric]])
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   831
  thus ?thesis  by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   832
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   833
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   834
lemma inverse_unique: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   835
  assumes ab: "a*b = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   836
  shows "inverse a = (b::'a::field)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   837
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   838
  have "a \<noteq> 0" using ab by auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   839
  moreover have "inverse a * (a * b) = inverse a" by (simp add: ab) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   840
  ultimately show ?thesis by (simp add: mult_assoc [symmetric]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   841
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15010
diff changeset
   842
14270
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   843
lemma nonzero_inverse_mult_distrib: 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   844
      assumes anz: "a \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   845
          and bnz: "b \<noteq> 0"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   846
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   847
  proof -
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   848
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   849
    by (simp add: field_mult_eq_0_iff anz bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   850
  hence "inverse(a*b) * a = inverse(b)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   851
    by (simp add: mult_assoc bnz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   852
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   853
    by simp
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   854
  thus ?thesis
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   855
    by (simp add: mult_assoc anz)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   856
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   857
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   858
text{*This version builds in division by zero while also re-orienting
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   859
      the right-hand side.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   860
lemma inverse_mult_distrib [simp]:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   861
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   862
  proof cases
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   863
    assume "a \<noteq> 0 & b \<noteq> 0" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   864
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   865
  next
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   866
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   867
    thus ?thesis  by force
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   868
  qed
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   869
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   870
text{*There is no slick version using division by zero.*}
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   871
lemma inverse_add:
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   872
     "[|a \<noteq> 0;  b \<noteq> 0|]
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   873
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   874
apply (simp add: left_distrib mult_assoc)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   875
apply (simp add: mult_commute [of "inverse a"]) 
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   876
apply (simp add: mult_assoc [symmetric] add_commute)
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   877
done
342451d763f9 More re-organising of numerical theorems
paulson
parents: 14269
diff changeset
   878
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   879
lemma inverse_divide [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   880
      "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   881
  by (simp add: divide_inverse mult_commute)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   882
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   883
lemma nonzero_mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   884
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   885
    shows "(c*a)/(c*b) = a/(b::'a::field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   886
proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   887
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   888
    by (simp add: field_mult_eq_0_iff divide_inverse 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   889
                  nonzero_inverse_mult_distrib)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   890
  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
   891
    by (simp only: mult_ac)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   892
  also have "... =  a * inverse b"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   893
    by simp
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   894
    finally show ?thesis 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   895
    by (simp add: divide_inverse)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   896
qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   897
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   898
lemma mult_divide_cancel_left:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   899
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   900
apply (case_tac "b = 0")
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   901
apply (simp_all add: nonzero_mult_divide_cancel_left)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   902
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   903
14321
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   904
lemma nonzero_mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   905
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   906
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   907
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   908
lemma mult_divide_cancel_right:
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   909
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   910
apply (case_tac "b = 0")
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   911
apply (simp_all add: nonzero_mult_divide_cancel_right)
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   912
done
55c688d2eefa new theorems
paulson
parents: 14305
diff changeset
   913
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   914
(*For ExtractCommonTerm*)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   915
lemma mult_divide_cancel_eq_if:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   916
     "(c*a) / (c*b) = 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   917
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   918
  by (simp add: mult_divide_cancel_left)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   919
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   920
lemma divide_1 [simp]: "a/1 = (a::'a::field)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   921
  by (simp add: divide_inverse)
14284
f1abe67c448a re-organisation of Real/RealArith0.ML; more `Isar scripts
paulson
parents: 14277
diff changeset
   922
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   923
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
   924
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   925
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   926
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
   927
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   928
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   929
lemma divide_divide_eq_right [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   930
     "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
   931
by (simp add: divide_inverse mult_ac)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   932
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   933
lemma divide_divide_eq_left [simp]:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   934
     "(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
   935
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   936
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   937
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   938
subsubsection{*Special Cancellation Simprules for Division*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   939
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   940
lemma mult_divide_cancel_left_if [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   941
  fixes c :: "'a :: {field,division_by_zero}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   942
  shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   943
by (simp add: mult_divide_cancel_left)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   944
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   945
lemma mult_divide_cancel_right_if [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   946
  fixes c :: "'a :: {field,division_by_zero}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   947
  shows "(a*c) / (b*c) = (if c=0 then 0 else a/b)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   948
by (simp add: mult_divide_cancel_right)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   949
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   950
lemma mult_divide_cancel_left_if1 [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   951
  fixes c :: "'a :: {field,division_by_zero}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   952
  shows "c / (c*b) = (if c=0 then 0 else 1/b)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   953
apply (insert mult_divide_cancel_left_if [of c 1 b]) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   954
apply (simp del: mult_divide_cancel_left_if)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   955
done
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   956
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   957
lemma mult_divide_cancel_left_if2 [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   958
  fixes c :: "'a :: {field,division_by_zero}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   959
  shows "(c*a) / c = (if c=0 then 0 else a)" 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   960
apply (insert mult_divide_cancel_left_if [of c a 1]) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   961
apply (simp del: mult_divide_cancel_left_if)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   962
done
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   963
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   964
lemma mult_divide_cancel_right_if1 [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   965
  fixes c :: "'a :: {field,division_by_zero}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   966
  shows "c / (b*c) = (if c=0 then 0 else 1/b)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   967
apply (insert mult_divide_cancel_right_if [of 1 c b]) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   968
apply (simp del: mult_divide_cancel_right_if)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   969
done
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   970
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   971
lemma mult_divide_cancel_right_if2 [simp]:
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   972
  fixes c :: "'a :: {field,division_by_zero}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   973
  shows "(a*c) / c = (if c=0 then 0 else a)" 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   974
apply (insert mult_divide_cancel_right_if [of a c 1]) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   975
apply (simp del: mult_divide_cancel_right_if)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   976
done
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   977
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   978
text{*Two lemmas for cancelling the denominator*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   979
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   980
lemma times_divide_self_right [simp]: 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   981
  fixes a :: "'a :: {field,division_by_zero}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   982
  shows "a * (b/a) = (if a=0 then 0 else b)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   983
by (simp add: times_divide_eq_right)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   984
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   985
lemma times_divide_self_left [simp]: 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   986
  fixes a :: "'a :: {field,division_by_zero}"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   987
  shows "(b/a) * a = (if a=0 then 0 else b)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   988
by (simp add: times_divide_eq_left)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   989
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   990
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   991
subsection {* Division and Unary Minus *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   992
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   993
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
   994
by (simp add: divide_inverse minus_mult_left)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   995
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   996
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
   997
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
   998
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   999
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
  1000
by (simp add: divide_inverse nonzero_inverse_minus_eq)
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1001
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1002
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
  1003
by (simp add: divide_inverse minus_mult_left [symmetric])
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1004
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1005
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
  1006
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
  1007
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1008
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1009
text{*The effect is to extract signs from divisions*}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1010
declare minus_divide_left  [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1011
declare minus_divide_right [symmetric, simp]
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1012
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1013
text{*Also, extract signs from products*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1014
declare minus_mult_left [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1015
declare minus_mult_right [symmetric, simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1016
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1017
lemma minus_divide_divide [simp]:
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1018
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1019
apply (case_tac "b=0", simp) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1020
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1021
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1022
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1023
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
  1024
by (simp add: diff_minus add_divide_distrib) 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14377
diff changeset
  1025
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1026
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1027
subsection {* Ordered Fields *}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1028
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1029
lemma positive_imp_inverse_positive: 
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1030
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1031
  proof -
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1032
  have "0 < a * inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1033
    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
  1034
  thus "0 < inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1035
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1036
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1037
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1038
lemma negative_imp_inverse_negative:
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1039
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1040
  by (insert positive_imp_inverse_positive [of "-a"], 
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1041
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1042
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1043
lemma inverse_le_imp_le:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1044
      assumes invle: "inverse a \<le> inverse b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1045
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1046
	 shows "b \<le> (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1047
  proof (rule classical)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1048
  assume "~ b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1049
  hence "a < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1050
    by (simp add: linorder_not_le)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1051
  hence bpos: "0 < b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1052
    by (blast intro: apos order_less_trans)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1053
  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
  1054
    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
  1055
  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
  1056
    by (simp add: bpos order_less_imp_le mult_right_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1057
  thus "b \<le> a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1058
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1059
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1060
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1061
lemma inverse_positive_imp_positive:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1062
      assumes inv_gt_0: "0 < inverse a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1063
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1064
        shows "0 < (a::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1065
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1066
  have "0 < inverse (inverse a)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1067
    by (rule positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1068
  thus "0 < a"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1069
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1070
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1071
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1072
lemma inverse_positive_iff_positive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1073
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1074
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1075
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1076
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1077
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1078
lemma inverse_negative_imp_negative:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1079
      assumes inv_less_0: "inverse a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1080
          and [simp]:   "a \<noteq> 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1081
        shows "a < (0::'a::ordered_field)"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1082
  proof -
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1083
  have "inverse (inverse a) < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1084
    by (rule negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1085
  thus "a < 0"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1086
    by (simp add: nonzero_inverse_inverse_eq)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1087
  qed
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1088
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1089
lemma inverse_negative_iff_negative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1090
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1091
apply (case_tac "a = 0", simp)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1092
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1093
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1094
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1095
lemma inverse_nonnegative_iff_nonnegative [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1096
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1097
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1098
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1099
lemma inverse_nonpositive_iff_nonpositive [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1100
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1101
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1102
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1103
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1104
subsection{*Anti-Monotonicity of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1105
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1106
lemma less_imp_inverse_less:
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1107
      assumes less: "a < b"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1108
	  and apos:  "0 < a"
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 14268
diff changeset
  1109
	shows "inverse b < inverse (a::'a::ordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1110
  proof (rule ccontr)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1111
  assume "~ inverse b < inverse a"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1112
  hence "inverse a \<le> inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1113
    by (simp add: linorder_not_less)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1114
  hence "~ (a < b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1115
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1116
  thus False
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1117
    by (rule notE [OF _ less])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1118
  qed
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1119
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1120
lemma inverse_less_imp_less:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1121
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1122
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1123
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1124
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1125
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1126
text{*Both premises are essential. Consider -1 and 1.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1127
lemma inverse_less_iff_less [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1128
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1129
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1130
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1131
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1132
lemma le_imp_inverse_le:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1133
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1134
  by (force simp add: order_le_less less_imp_inverse_less)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1135
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1136
lemma inverse_le_iff_le [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1137
     "[|0 < a; 0 < b|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1138
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1139
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1140
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1141
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1142
text{*These results refer to both operands being negative.  The opposite-sign
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1143
case is trivial, since inverse preserves signs.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1144
lemma inverse_le_imp_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1145
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1146
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1147
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1148
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1149
  apply (insert inverse_le_imp_le [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1150
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1151
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1152
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1153
lemma less_imp_inverse_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1154
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1155
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1156
   prefer 2 apply (blast intro: order_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1157
  apply (insert less_imp_inverse_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1158
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1159
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1160
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1161
lemma inverse_less_imp_less_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1162
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1163
  apply (rule classical) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1164
  apply (subgoal_tac "a < 0") 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1165
   prefer 2
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1166
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1167
  apply (insert inverse_less_imp_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1168
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1169
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1170
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1171
lemma inverse_less_iff_less_neg [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1172
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1173
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1174
  apply (insert inverse_less_iff_less [of "-b" "-a"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1175
  apply (simp del: inverse_less_iff_less 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1176
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1177
  done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1178
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1179
lemma le_imp_inverse_le_neg:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1180
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1181
  by (force simp add: order_le_less less_imp_inverse_less_neg)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1182
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1183
lemma inverse_le_iff_le_neg [simp]:
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1184
     "[|a < 0; b < 0|] 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1185
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
  1186
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  1187
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1188
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1189
subsection{*Inverses and the Number One*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1190
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1191
lemma one_less_inverse_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1192
    "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1193
  assume "0 < x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1194
    with inverse_less_iff_less [OF zero_less_one, of x]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1195
    show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1196
next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1197
  assume notless: "~ (0 < x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1198
  have "~ (1 < inverse x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1199
  proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1200
    assume "1 < inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1201
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1202
    also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1203
    finally show False by auto
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1204
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1205
  with notless show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1206
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1207
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1208
lemma inverse_eq_1_iff [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1209
    "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1210
by (insert inverse_eq_iff_eq [of x 1], simp) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1211
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1212
lemma one_le_inverse_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1213
   "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1214
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1215
                    eq_commute [of 1]) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1216
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1217
lemma inverse_less_1_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1218
   "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1219
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1220
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1221
lemma inverse_le_1_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1222
   "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1223
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1224
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
  1225
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1226
subsection{*Division and Signs*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1227
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1228
lemma zero_less_divide_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1229
     "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1230
by (simp add: divide_inverse zero_less_mult_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1231
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1232
lemma divide_less_0_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1233
     "(a/b < (0::'a::{ordered_field,division_by_zero})) = 
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1234
      (0 < a & b < 0 | a < 0 & 0 < b)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1235
by (simp add: divide_inverse mult_less_0_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1236
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1237
lemma zero_le_divide_iff:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1238
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1239
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1240
by (simp add: divide_inverse zero_le_mult_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1241
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1242
lemma divide_le_0_iff:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1243
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1244
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1245
by (simp add: divide_inverse mult_le_0_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1246
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1247
lemma divide_eq_0_iff [simp]:
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1248
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
  1249
by (simp add: divide_inverse field_mult_eq_0_iff)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
  1250
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1251
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1252
subsection{*Simplification of Inequalities Involving Literal Divisors*}
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1253
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1254
lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1255
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1256
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1257
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1258
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1259
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1260
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1261
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1262
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1263
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1264
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1265
lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1266
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1267
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1268
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1269
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1270
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1271
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1272
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1273
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1274
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1275
lemma le_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1276
  "(a \<le> b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1277
   (if 0 < c then a*c \<le> b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1278
             else if c < 0 then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1279
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1280
apply (case_tac "c=0", simp) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
  1281
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)