(*  Title:   HOL/Ring_and_Field.thy

     ID:      $Id$

     Author:  Gertrud Bauer and Markus Wenzel, TU Muenchen

              Lawrence C Paulson, University of Cambridge

              Revised and splitted into Ring_and_Field.thy and Group.thy 

              by Steven Obua, TU Muenchen, in May 2004

     License: GPL (GNU GENERAL PUBLIC LICENSE)

 *)

 header {* (Ordered) Rings and Fields *}

 theory Ring_and_Field = OrderedGroup:

 text {*

   The theory of partially ordered rings is taken from the books:

   \begin{itemize}

   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 

   \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963

   \end{itemize}

   Most of the used notions can also be looked up in 

   \begin{itemize}

   \item \emph{www.mathworld.com} by Eric Weisstein et. al.

   \item \emph{Algebra I} by van der Waerden, Springer.

   \end{itemize}

 *}

 axclass semiring \<subseteq> ab_semigroup_add, semigroup_mult

   left_distrib: "(a + b) * c = a * c + b * c"

   right_distrib: "a * (b + c) = a * b + a * c"

 axclass semiring_0 \<subseteq> semiring, comm_monoid_add

 axclass comm_semiring \<subseteq> ab_semigroup_add, ab_semigroup_mult  

   mult_commute: "a * b = b * a"

   distrib: "(a + b) * c = a * c + b * c"

 instance comm_semiring \<subseteq> semiring

 proof

   fix a b c :: 'a

   show "(a + b) * c = a * c + b * c" by (simp add: distrib)

   have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)

   also have "... = b * a + c * a" by (simp only: distrib)

   also have "... = a * b + a * c" by (simp add: mult_ac)

   finally show "a * (b + c) = a * b + a * c" by blast

 qed

 axclass comm_semiring_0 \<subseteq> comm_semiring, comm_monoid_add

 instance comm_semiring_0 \<subseteq> semiring_0 ..

 axclass axclass_0_neq_1 \<subseteq> zero, one

   zero_neq_one [simp]: "0 \<noteq> 1"

 axclass semiring_1 \<subseteq> axclass_0_neq_1, semiring_0, monoid_mult

 axclass comm_semiring_1 \<subseteq> axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *)

 instance comm_semiring_1 \<subseteq> semiring_1 ..

 axclass axclass_no_zero_divisors \<subseteq> zero, times

   no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"

 axclass comm_semiring_1_cancel \<subseteq> comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)

 axclass ring \<subseteq> semiring, ab_group_add

 instance ring \<subseteq> semiring_0 ..

 axclass comm_ring \<subseteq> comm_semiring_0, ab_group_add

 instance comm_ring \<subseteq> ring ..

 instance comm_ring \<subseteq> comm_semiring_0 ..

 axclass ring_1 \<subseteq> ring, semiring_1

 axclass comm_ring_1 \<subseteq> comm_ring, comm_semiring_1 (* previously ring *)

 instance comm_ring_1 \<subseteq> ring_1 ..

 instance comm_ring_1 \<subseteq> comm_semiring_1_cancel ..

 axclass idom \<subseteq> comm_ring_1, axclass_no_zero_divisors

 axclass field \<subseteq> comm_ring_1, inverse

   left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"

   divide_inverse:      "a / b = a * inverse b"

 lemma mult_zero_left [simp]: "0 * a = (0::'a::{semiring_0, cancel_semigroup_add})"

 proof -

   have "0*a + 0*a = 0*a + 0"

     by (simp add: left_distrib [symmetric])

   thus ?thesis 

     by (simp only: add_left_cancel)

 qed

 lemma mult_zero_right [simp]: "a * 0 = (0::'a::{semiring_0, cancel_semigroup_add})"

 proof -

   have "a*0 + a*0 = a*0 + 0"

     by (simp add: right_distrib [symmetric])

   thus ?thesis 

     by (simp only: add_left_cancel)

 qed

 lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"

 proof cases

   assume "a=0" thus ?thesis by simp

 next

   assume anz [simp]: "a\<noteq>0"

   { assume "a * b = 0"

     hence "inverse a * (a * b) = 0" by simp

     hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}

   thus ?thesis by force

 qed

 instance field \<subseteq> idom

 by (intro_classes, simp)

 axclass division_by_zero \<subseteq> zero, inverse

   inverse_zero [simp]: "inverse 0 = 0"

 subsection {* Distribution rules *}

 theorems ring_distrib = right_distrib left_distrib

 text{*For the @{text combine_numerals} simproc*}

 lemma combine_common_factor:

      "a*e + (b*e + c) = (a+b)*e + (c::'a::semiring)"

 by (simp add: left_distrib add_ac)

 lemma minus_mult_left: "- (a * b) = (-a) * (b::'a::ring)"

 apply (rule equals_zero_I)

 apply (simp add: left_distrib [symmetric]) 

 done

 lemma minus_mult_right: "- (a * b) = a * -(b::'a::ring)"

 apply (rule equals_zero_I)

 apply (simp add: right_distrib [symmetric]) 

 done

 lemma minus_mult_minus [simp]: "(- a) * (- b) = a * (b::'a::ring)"

   by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])

 lemma minus_mult_commute: "(- a) * b = a * (- b::'a::ring)"

   by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])

 lemma right_diff_distrib: "a * (b - c) = a * b - a * (c::'a::ring)"

 by (simp add: right_distrib diff_minus 

               minus_mult_left [symmetric] minus_mult_right [symmetric]) 

 lemma left_diff_distrib: "(a - b) * c = a * c - b * (c::'a::ring)"

 by (simp add: left_distrib diff_minus 

               minus_mult_left [symmetric] minus_mult_right [symmetric]) 

 axclass pordered_semiring \<subseteq> semiring_0, pordered_ab_semigroup_add 

   mult_left_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"

   mult_right_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> a * c <= b * c"

 axclass pordered_cancel_semiring \<subseteq> pordered_semiring, cancel_ab_semigroup_add

 axclass ordered_semiring_strict \<subseteq> semiring_0, ordered_cancel_ab_semigroup_add

   mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

   mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"

 instance ordered_semiring_strict \<subseteq> pordered_cancel_semiring

 apply intro_classes

 apply (case_tac "a < b & 0 < c")

 apply (auto simp add: mult_strict_left_mono order_less_le)

 apply (auto simp add: mult_strict_left_mono order_le_less)

 apply (simp add: mult_strict_right_mono)

 done

 axclass pordered_comm_semiring \<subseteq> comm_semiring_0, pordered_ab_semigroup_add

   mult_mono: "a <= b \<Longrightarrow> 0 <= c \<Longrightarrow> c * a <= c * b"

 axclass pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring, cancel_ab_semigroup_add

 instance pordered_cancel_comm_semiring \<subseteq> pordered_comm_semiring ..

 axclass ordered_comm_semiring_strict \<subseteq> comm_semiring_0, ordered_cancel_ab_semigroup_add

   mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

 instance pordered_comm_semiring \<subseteq> pordered_semiring

 by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)

 instance pordered_cancel_comm_semiring \<subseteq> pordered_cancel_semiring ..

 instance ordered_comm_semiring_strict \<subseteq> ordered_semiring_strict

 by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)

 instance ordered_comm_semiring_strict \<subseteq> pordered_cancel_comm_semiring

 apply (intro_classes)

 apply (case_tac "a < b & 0 < c")

 apply (auto simp add: mult_strict_left_mono order_less_le)

 apply (auto simp add: mult_strict_left_mono order_le_less)

 done

 axclass pordered_ring \<subseteq> ring, pordered_semiring 

 instance pordered_ring \<subseteq> pordered_ab_group_add ..

 instance pordered_ring \<subseteq> pordered_cancel_semiring ..

 axclass lordered_ring \<subseteq> pordered_ring, lordered_ab_group_abs

 axclass axclass_abs_if \<subseteq> minus, ord, zero

   abs_if: "abs a = (if (a < 0) then (-a) else a)"

 axclass ordered_ring_strict \<subseteq> ring, ordered_semiring_strict, axclass_abs_if

 instance ordered_ring_strict \<subseteq> lordered_ab_group ..

 instance ordered_ring_strict \<subseteq> lordered_ring

 by (intro_classes, simp add: abs_if join_eq_if)

 axclass pordered_comm_ring \<subseteq> comm_ring, pordered_comm_semiring

 axclass ordered_semidom \<subseteq> comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *)

   zero_less_one [simp]: "0 < 1"

 axclass ordered_idom \<subseteq> comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *)

 instance ordered_idom \<subseteq> ordered_ring_strict ..

 axclass ordered_field \<subseteq> field, ordered_idom

 lemma eq_add_iff1:

      "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"

 apply (simp add: diff_minus left_distrib)

 apply (simp add: diff_minus left_distrib add_ac)

 apply (simp add: compare_rls minus_mult_left [symmetric])

 done

 lemma eq_add_iff2:

      "(a*e + c = b*e + d) = (c = (b-a)*e + (d::'a::ring))"

 apply (simp add: diff_minus left_distrib add_ac)

 apply (simp add: compare_rls minus_mult_left [symmetric]) 

 done

 lemma less_add_iff1:

      "(a*e + c < b*e + d) = ((a-b)*e + c < (d::'a::pordered_ring))"

 apply (simp add: diff_minus left_distrib add_ac)

 apply (simp add: compare_rls minus_mult_left [symmetric]) 

 done

 lemma less_add_iff2:

      "(a*e + c < b*e + d) = (c < (b-a)*e + (d::'a::pordered_ring))"

 apply (simp add: diff_minus left_distrib add_ac)

 apply (simp add: compare_rls minus_mult_left [symmetric]) 

 done

 lemma le_add_iff1:

      "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::pordered_ring))"

 apply (simp add: diff_minus left_distrib add_ac)

 apply (simp add: compare_rls minus_mult_left [symmetric]) 

 done

 lemma le_add_iff2:

      "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::pordered_ring))"

 apply (simp add: diff_minus left_distrib add_ac)

 apply (simp add: compare_rls minus_mult_left [symmetric]) 

 done

 subsection {* Ordering Rules for Multiplication *}

 lemma mult_left_le_imp_le:

      "[|c*a \<le> c*b; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"

   by (force simp add: mult_strict_left_mono linorder_not_less [symmetric])

 lemma mult_right_le_imp_le:

      "[|a*c \<le> b*c; 0 < c|] ==> a \<le> (b::'a::ordered_semiring_strict)"

   by (force simp add: mult_strict_right_mono linorder_not_less [symmetric])

 lemma mult_left_less_imp_less:

      "[|c*a < c*b; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"

   by (force simp add: mult_left_mono linorder_not_le [symmetric])

 lemma mult_right_less_imp_less:

      "[|a*c < b*c; 0 \<le> c|] ==> a < (b::'a::ordered_semiring_strict)"

   by (force simp add: mult_right_mono linorder_not_le [symmetric])

 lemma mult_strict_left_mono_neg:

      "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring_strict)"

 apply (drule mult_strict_left_mono [of _ _ "-c"])

 apply (simp_all add: minus_mult_left [symmetric]) 

 done

 lemma mult_left_mono_neg:

      "[|b \<le> a; c \<le> 0|] ==> c * a \<le>  c * (b::'a::pordered_ring)"

 apply (drule mult_left_mono [of _ _ "-c"])

 apply (simp_all add: minus_mult_left [symmetric]) 

 done

 lemma mult_strict_right_mono_neg:

      "[|b < a; c < 0|] ==> a * c < b * (c::'a::ordered_ring_strict)"

 apply (drule mult_strict_right_mono [of _ _ "-c"])

 apply (simp_all add: minus_mult_right [symmetric]) 

 done

 lemma mult_right_mono_neg:

      "[|b \<le> a; c \<le> 0|] ==> a * c \<le>  (b::'a::pordered_ring) * c"

 apply (drule mult_right_mono [of _ _ "-c"])

 apply (simp)

 apply (simp_all add: minus_mult_right [symmetric]) 

 done

 subsection{* Products of Signs *}

 lemma mult_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"

 by (drule mult_strict_left_mono [of 0 b], auto)

 lemma mult_pos_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; 0 \<le> b |] ==> 0 \<le> a*b"

 by (drule mult_left_mono [of 0 b], auto)

 lemma mult_pos_neg: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> a*b < 0"

 by (drule mult_strict_left_mono [of b 0], auto)

 lemma mult_pos_neg_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> a*b \<le> 0"

 by (drule mult_left_mono [of b 0], auto)

 lemma mult_pos_neg2: "[| (0::'a::ordered_semiring_strict) < a; b < 0 |] ==> b*a < 0" 

 by (drule mult_strict_right_mono[of b 0], auto)

 lemma mult_pos_neg2_le: "[| (0::'a::pordered_cancel_semiring) \<le> a; b \<le> 0 |] ==> b*a \<le> 0" 

 by (drule mult_right_mono[of b 0], auto)

 lemma mult_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"

 by (drule mult_strict_right_mono_neg, auto)

 lemma mult_neg_le: "[| a \<le> (0::'a::pordered_ring); b \<le> 0 |] ==> 0 \<le> a*b"

 by (drule mult_right_mono_neg[of a 0 b ], auto)

 lemma zero_less_mult_pos:

      "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"

 apply (case_tac "b\<le>0") 

  apply (auto simp add: order_le_less linorder_not_less)

 apply (drule_tac mult_pos_neg [of a b]) 

  apply (auto dest: order_less_not_sym)

 done

 lemma zero_less_mult_pos2:

      "[| 0 < b*a; 0 < a|] ==> 0 < (b::'a::ordered_semiring_strict)"

 apply (case_tac "b\<le>0") 

  apply (auto simp add: order_le_less linorder_not_less)

 apply (drule_tac mult_pos_neg2 [of a b]) 

  apply (auto dest: order_less_not_sym)

 done

 lemma zero_less_mult_iff:

      "((0::'a::ordered_ring_strict) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"

 apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)

 apply (blast dest: zero_less_mult_pos) 

 apply (blast dest: zero_less_mult_pos2)

 done

 text{*A field has no "zero divisors", and this theorem holds without the

       assumption of an ordering.  See @{text field_mult_eq_0_iff} below.*}

 lemma mult_eq_0_iff [simp]: "(a*b = (0::'a::ordered_ring_strict)) = (a = 0 | b = 0)"

 apply (case_tac "a < 0")

 apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)

 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+

 done

 lemma zero_le_mult_iff:

      "((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"

 by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less

                    zero_less_mult_iff)

 lemma mult_less_0_iff:

      "(a*b < (0::'a::ordered_ring_strict)) = (0 < a & b < 0 | a < 0 & 0 < b)"

 apply (insert zero_less_mult_iff [of "-a" b]) 

 apply (force simp add: minus_mult_left[symmetric]) 

 done

 lemma mult_le_0_iff:

      "(a*b \<le> (0::'a::ordered_ring_strict)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"

 apply (insert zero_le_mult_iff [of "-a" b]) 

 apply (force simp add: minus_mult_left[symmetric]) 

 done

 lemma split_mult_pos_le: "(0 \<le> a & 0 \<le> b) | (a \<le> 0 & b \<le> 0) \<Longrightarrow> 0 \<le> a * (b::_::pordered_ring)"

 by (auto simp add: mult_pos_le mult_neg_le)

 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)" 

 by (auto simp add: mult_pos_neg_le mult_pos_neg2_le)

 lemma zero_le_square: "(0::'a::ordered_ring_strict) \<le> a*a"

 by (simp add: zero_le_mult_iff linorder_linear) 

 text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}

       theorems available to members of @{term ordered_idom} *}

 instance ordered_idom \<subseteq> ordered_semidom

 proof

   have "(0::'a) \<le> 1*1" by (rule zero_le_square)

   thus "(0::'a) < 1" by (simp add: order_le_less) 

 qed

 instance ordered_ring_strict \<subseteq> axclass_no_zero_divisors 

 by (intro_classes, simp)

 instance ordered_idom \<subseteq> idom ..

 text{*All three types of comparision involving 0 and 1 are covered.*}

 declare zero_neq_one [THEN not_sym, simp]

 lemma zero_le_one [simp]: "(0::'a::ordered_semidom) \<le> 1"

   by (rule zero_less_one [THEN order_less_imp_le]) 

 lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) \<le> 0"

 by (simp add: linorder_not_le) 

 lemma not_one_less_zero [simp]: "~ (1::'a::ordered_semidom) < 0"

 by (simp add: linorder_not_less) 

 subsection{*More Monotonicity*}

 lemma mult_left_mono_neg:

      "[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::pordered_ring)"

 apply (drule mult_left_mono [of _ _ "-c"]) 

 apply (simp_all add: minus_mult_left [symmetric]) 

 done

 lemma mult_right_mono_neg:

      "[|b \<le> a; c \<le> 0|] ==> a * c \<le> b * (c::'a::pordered_ring)"

 apply (drule mult_right_mono [of _ _ "-c"]) 

 apply (simp_all add: minus_mult_right [symmetric]) 

 done  

 text{*Strict monotonicity in both arguments*}

 lemma mult_strict_mono:

      "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"

 apply (case_tac "c=0")

  apply (simp add: mult_pos) 

 apply (erule mult_strict_right_mono [THEN order_less_trans])

  apply (force simp add: order_le_less) 

 apply (erule mult_strict_left_mono, assumption)

 done

 text{*This weaker variant has more natural premises*}

 lemma mult_strict_mono':

      "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"

 apply (rule mult_strict_mono)

 apply (blast intro: order_le_less_trans)+

 done

 lemma mult_mono:

      "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 

       ==> a * c  \<le>  b * (d::'a::pordered_semiring)"

 apply (erule mult_right_mono [THEN order_trans], assumption)

 apply (erule mult_left_mono, assumption)

 done

 lemma less_1_mult: "[| 1 < m; 1 < n |] ==> 1 < m*(n::'a::ordered_semidom)"

 apply (insert mult_strict_mono [of 1 m 1 n]) 

 apply (simp add:  order_less_trans [OF zero_less_one]) 

 done

 subsection{*Cancellation Laws for Relationships With a Common Factor*}

 text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},

    also with the relations @{text "\<le>"} and equality.*}

 lemma mult_less_cancel_right:

     "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"

 apply (case_tac "c = 0")

 apply (auto simp add: linorder_neq_iff mult_strict_right_mono 

                       mult_strict_right_mono_neg)

 apply (auto simp add: linorder_not_less 

                       linorder_not_le [symmetric, of "a*c"]

                       linorder_not_le [symmetric, of a])

 apply (erule_tac [!] notE)

 apply (auto simp add: order_less_imp_le mult_right_mono 

                       mult_right_mono_neg)

 done

 lemma mult_less_cancel_left:

     "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring_strict)))"

 apply (case_tac "c = 0")

 apply (auto simp add: linorder_neq_iff mult_strict_left_mono 

                       mult_strict_left_mono_neg)

 apply (auto simp add: linorder_not_less 

                       linorder_not_le [symmetric, of "c*a"]

                       linorder_not_le [symmetric, of a])

 apply (erule_tac [!] notE)

 apply (auto simp add: order_less_imp_le mult_left_mono 

                       mult_left_mono_neg)

 done

 lemma mult_le_cancel_right:

      "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"

 by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)

 lemma mult_le_cancel_left:

      "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring_strict)))"

 by (simp add: linorder_not_less [symmetric] mult_less_cancel_left)

 lemma mult_less_imp_less_left:

       assumes less: "c*a < c*b" and nonneg: "0 \<le> c"

       shows "a < (b::'a::ordered_semiring_strict)"

 proof (rule ccontr)

   assume "~ a < b"

   hence "b \<le> a" by (simp add: linorder_not_less)

   hence "c*b \<le> c*a" by (rule mult_left_mono)

   with this and less show False 

     by (simp add: linorder_not_less [symmetric])

 qed

 lemma mult_less_imp_less_right:

   assumes less: "a*c < b*c" and nonneg: "0 <= c"

   shows "a < (b::'a::ordered_semiring_strict)"

 proof (rule ccontr)

   assume "~ a < b"

   hence "b \<le> a" by (simp add: linorder_not_less)

   hence "b*c \<le> a*c" by (rule mult_right_mono)

   with this and less show False 

     by (simp add: linorder_not_less [symmetric])

 qed  

 text{*Cancellation of equalities with a common factor*}

 lemma mult_cancel_right [simp]:

      "(a*c = b*c) = (c = (0::'a::ordered_ring_strict) | a=b)"

 apply (cut_tac linorder_less_linear [of 0 c])

 apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono

              simp add: linorder_neq_iff)

 done

 text{*These cancellation theorems require an ordering. Versions are proved

       below that work for fields without an ordering.*}

 lemma mult_cancel_left [simp]:

      "(c*a = c*b) = (c = (0::'a::ordered_ring_strict) | a=b)"

 apply (cut_tac linorder_less_linear [of 0 c])

 apply (force dest: mult_strict_left_mono_neg mult_strict_left_mono

              simp add: linorder_neq_iff)

 done

 text{*This list of rewrites decides ring equalities by ordered rewriting.*}

 lemmas ring_eq_simps =

   mult_ac

   left_distrib right_distrib left_diff_distrib right_diff_distrib

   add_ac

   add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2

   diff_eq_eq eq_diff_eq

 thm ring_eq_simps

 subsection {* Fields *}

 lemma right_inverse [simp]:

       assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"

 proof -

   have "a * inverse a = inverse a * a" by (simp add: mult_ac)

   also have "... = 1" using not0 by simp

   finally show ?thesis .

 qed

 lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"

 proof

   assume neq: "b \<noteq> 0"

   {

     hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)

     also assume "a / b = 1"

     finally show "a = b" by simp

   next

     assume "a = b"

     with neq show "a / b = 1" by (simp add: divide_inverse)

   }

 qed

 lemma nonzero_inverse_eq_divide: "a \<noteq> 0 ==> inverse (a::'a::field) = 1/a"

 by (simp add: divide_inverse)

 lemma divide_self [simp]: "a \<noteq> 0 ==> a / (a::'a::field) = 1"

   by (simp add: divide_inverse)

 lemma divide_zero [simp]: "a / 0 = (0::'a::{field,division_by_zero})"

 by (simp add: divide_inverse)

 lemma divide_zero_left [simp]: "0/a = (0::'a::field)"

 by (simp add: divide_inverse)

 lemma inverse_eq_divide: "inverse (a::'a::field) = 1/a"

 by (simp add: divide_inverse)

 lemma add_divide_distrib: "(a+b)/(c::'a::field) = a/c + b/c"

 by (simp add: divide_inverse left_distrib) 

 text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement

       of an ordering.*}

 lemma field_mult_eq_0_iff [simp]: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"

 proof cases

   assume "a=0" thus ?thesis by simp

 next

   assume anz [simp]: "a\<noteq>0"

   { assume "a * b = 0"

     hence "inverse a * (a * b) = 0" by simp

     hence "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])}

   thus ?thesis by force

 qed

 text{*Cancellation of equalities with a common factor*}

 lemma field_mult_cancel_right_lemma:

       assumes cnz: "c \<noteq> (0::'a::field)"

 	  and eq:  "a*c = b*c"

 	 shows "a=b"

 proof -

   have "(a * c) * inverse c = (b * c) * inverse c"

     by (simp add: eq)

   thus "a=b"

     by (simp add: mult_assoc cnz)

 qed

 lemma field_mult_cancel_right [simp]:

      "(a*c = b*c) = (c = (0::'a::field) | a=b)"

 proof cases

   assume "c=0" thus ?thesis by simp

 next

   assume "c\<noteq>0" 

   thus ?thesis by (force dest: field_mult_cancel_right_lemma)

 qed

 lemma field_mult_cancel_left [simp]:

      "(c*a = c*b) = (c = (0::'a::field) | a=b)"

   by (simp add: mult_commute [of c] field_mult_cancel_right) 

 lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"

 proof

   assume ianz: "inverse a = 0"

   assume "a \<noteq> 0"

   hence "1 = a * inverse a" by simp

   also have "... = 0" by (simp add: ianz)

   finally have "1 = (0::'a::field)" .

   thus False by (simp add: eq_commute)

 qed

 subsection{*Basic Properties of @{term inverse}*}

 lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"

 apply (rule ccontr) 

 apply (blast dest: nonzero_imp_inverse_nonzero) 

 done

 lemma inverse_nonzero_imp_nonzero:

    "inverse a = 0 ==> a = (0::'a::field)"

 apply (rule ccontr) 

 apply (blast dest: nonzero_imp_inverse_nonzero) 

 done

 lemma inverse_nonzero_iff_nonzero [simp]:

    "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"

 by (force dest: inverse_nonzero_imp_nonzero) 

 lemma nonzero_inverse_minus_eq:

       assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"

 proof -

   have "-a * inverse (- a) = -a * - inverse a"

     by simp

   thus ?thesis 

     by (simp only: field_mult_cancel_left, simp)

 qed

 lemma inverse_minus_eq [simp]:

    "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";

 proof cases

   assume "a=0" thus ?thesis by (simp add: inverse_zero)

 next

   assume "a\<noteq>0" 

   thus ?thesis by (simp add: nonzero_inverse_minus_eq)

 qed

 lemma nonzero_inverse_eq_imp_eq:

       assumes inveq: "inverse a = inverse b"

 	  and anz:  "a \<noteq> 0"

 	  and bnz:  "b \<noteq> 0"

 	 shows "a = (b::'a::field)"

 proof -

   have "a * inverse b = a * inverse a"

     by (simp add: inveq)

   hence "(a * inverse b) * b = (a * inverse a) * b"

     by simp

   thus "a = b"

     by (simp add: mult_assoc anz bnz)

 qed

 lemma inverse_eq_imp_eq:

      "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"

 apply (case_tac "a=0 | b=0") 

  apply (force dest!: inverse_zero_imp_zero

               simp add: eq_commute [of "0::'a"])

 apply (force dest!: nonzero_inverse_eq_imp_eq) 

 done

 lemma inverse_eq_iff_eq [simp]:

      "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"

 by (force dest!: inverse_eq_imp_eq) 

 lemma nonzero_inverse_inverse_eq:

       assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"

   proof -

   have "(inverse (inverse a) * inverse a) * a = a" 

     by (simp add: nonzero_imp_inverse_nonzero)

   thus ?thesis

     by (simp add: mult_assoc)

   qed

 lemma inverse_inverse_eq [simp]:

      "inverse(inverse (a::'a::{field,division_by_zero})) = a"

   proof cases

     assume "a=0" thus ?thesis by simp

   next

     assume "a\<noteq>0" 

     thus ?thesis by (simp add: nonzero_inverse_inverse_eq)

   qed

 lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"

   proof -

   have "inverse 1 * 1 = (1::'a::field)" 

     by (rule left_inverse [OF zero_neq_one [symmetric]])

   thus ?thesis  by simp

   qed

 lemma nonzero_inverse_mult_distrib: 

       assumes anz: "a \<noteq> 0"

           and bnz: "b \<noteq> 0"

       shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"

   proof -

   have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 

     by (simp add: field_mult_eq_0_iff anz bnz)

   hence "inverse(a*b) * a = inverse(b)" 

     by (simp add: mult_assoc bnz)

   hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 

     by simp

   thus ?thesis

     by (simp add: mult_assoc anz)

   qed

 text{*This version builds in division by zero while also re-orienting

       the right-hand side.*}

 lemma inverse_mult_distrib [simp]:

      "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"

   proof cases

     assume "a \<noteq> 0 & b \<noteq> 0" 

     thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)

   next

     assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

     thus ?thesis  by force

   qed

 text{*There is no slick version using division by zero.*}

 lemma inverse_add:

      "[|a \<noteq> 0;  b \<noteq> 0|]

       ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"

 apply (simp add: left_distrib mult_assoc)

 apply (simp add: mult_commute [of "inverse a"]) 

 apply (simp add: mult_assoc [symmetric] add_commute)

 done

 lemma inverse_divide [simp]:

       "inverse (a/b) = b / (a::'a::{field,division_by_zero})"

   by (simp add: divide_inverse mult_commute)

 lemma nonzero_mult_divide_cancel_left:

   assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 

     shows "(c*a)/(c*b) = a/(b::'a::field)"

 proof -

   have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"

     by (simp add: field_mult_eq_0_iff divide_inverse 

                   nonzero_inverse_mult_distrib)

   also have "... =  a * inverse b * (inverse c * c)"

     by (simp only: mult_ac)

   also have "... =  a * inverse b"

     by simp

     finally show ?thesis 

     by (simp add: divide_inverse)

 qed

 lemma mult_divide_cancel_left:

      "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"

 apply (case_tac "b = 0")

 apply (simp_all add: nonzero_mult_divide_cancel_left)

 done

 lemma nonzero_mult_divide_cancel_right:

      "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"

 by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 

 lemma mult_divide_cancel_right:

      "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"

 apply (case_tac "b = 0")

 apply (simp_all add: nonzero_mult_divide_cancel_right)

 done

 (*For ExtractCommonTerm*)

 lemma mult_divide_cancel_eq_if:

      "(c*a) / (c*b) = 

       (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"

   by (simp add: mult_divide_cancel_left)

 lemma divide_1 [simp]: "a/1 = (a::'a::field)"

   by (simp add: divide_inverse)

 lemma times_divide_eq_right [simp]: "a * (b/c) = (a*b) / (c::'a::field)"

 by (simp add: divide_inverse mult_assoc)

 lemma times_divide_eq_left: "(b/c) * a = (b*a) / (c::'a::field)"

 by (simp add: divide_inverse mult_ac)

 lemma divide_divide_eq_right [simp]:

      "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"

 by (simp add: divide_inverse mult_ac)

 lemma divide_divide_eq_left [simp]:

      "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"

 by (simp add: divide_inverse mult_assoc)

 subsection {* Division and Unary Minus *}

 lemma nonzero_minus_divide_left: "b \<noteq> 0 ==> - (a/b) = (-a) / (b::'a::field)"

 by (simp add: divide_inverse minus_mult_left)

 lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"

 by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)

 lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"

 by (simp add: divide_inverse nonzero_inverse_minus_eq)

 lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"

 by (simp add: divide_inverse minus_mult_left [symmetric])

 lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"

 by (simp add: divide_inverse minus_mult_right [symmetric])

 text{*The effect is to extract signs from divisions*}

 declare minus_divide_left  [symmetric, simp]

 declare minus_divide_right [symmetric, simp]

 text{*Also, extract signs from products*}

 declare minus_mult_left [symmetric, simp]

 declare minus_mult_right [symmetric, simp]

 lemma minus_divide_divide [simp]:

      "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"

 apply (case_tac "b=0", simp) 

 apply (simp add: nonzero_minus_divide_divide) 

 done

 lemma diff_divide_distrib: "(a-b)/(c::'a::field) = a/c - b/c"

 by (simp add: diff_minus add_divide_distrib) 

 subsection {* Ordered Fields *}

 lemma positive_imp_inverse_positive: 

       assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"

   proof -

   have "0 < a * inverse a" 

     by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)

   thus "0 < inverse a" 

     by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)

   qed

 lemma negative_imp_inverse_negative:

      "a < 0 ==> inverse a < (0::'a::ordered_field)"

   by (insert positive_imp_inverse_positive [of "-a"], 

       simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 

 lemma inverse_le_imp_le:

       assumes invle: "inverse a \<le> inverse b"

 	  and apos:  "0 < a"

 	 shows "b \<le> (a::'a::ordered_field)"

   proof (rule classical)

   assume "~ b \<le> a"

   hence "a < b"

     by (simp add: linorder_not_le)

   hence bpos: "0 < b"

     by (blast intro: apos order_less_trans)

   hence "a * inverse a \<le> a * inverse b"

     by (simp add: apos invle order_less_imp_le mult_left_mono)

   hence "(a * inverse a) * b \<le> (a * inverse b) * b"

     by (simp add: bpos order_less_imp_le mult_right_mono)

   thus "b \<le> a"

     by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)

   qed

 lemma inverse_positive_imp_positive:

       assumes inv_gt_0: "0 < inverse a"

           and [simp]:   "a \<noteq> 0"

         shows "0 < (a::'a::ordered_field)"

   proof -

   have "0 < inverse (inverse a)"

     by (rule positive_imp_inverse_positive)

   thus "0 < a"

     by (simp add: nonzero_inverse_inverse_eq)

   qed

 lemma inverse_positive_iff_positive [simp]:

       "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"

 apply (case_tac "a = 0", simp)

 apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)

 done

 lemma inverse_negative_imp_negative:

       assumes inv_less_0: "inverse a < 0"

           and [simp]:   "a \<noteq> 0"

         shows "a < (0::'a::ordered_field)"

   proof -

   have "inverse (inverse a) < 0"

     by (rule negative_imp_inverse_negative)

   thus "a < 0"

     by (simp add: nonzero_inverse_inverse_eq)

   qed

 lemma inverse_negative_iff_negative [simp]:

       "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"

 apply (case_tac "a = 0", simp)

 apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)

 done

 lemma inverse_nonnegative_iff_nonnegative [simp]:

       "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"

 by (simp add: linorder_not_less [symmetric])

 lemma inverse_nonpositive_iff_nonpositive [simp]:

       "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"

 by (simp add: linorder_not_less [symmetric])

 subsection{*Anti-Monotonicity of @{term inverse}*}

 lemma less_imp_inverse_less:

       assumes less: "a < b"

 	  and apos:  "0 < a"

 	shows "inverse b < inverse (a::'a::ordered_field)"

   proof (rule ccontr)

   assume "~ inverse b < inverse a"

   hence "inverse a \<le> inverse b"

     by (simp add: linorder_not_less)

   hence "~ (a < b)"

     by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])

   thus False

     by (rule notE [OF _ less])

   qed

 lemma inverse_less_imp_less:

    "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"

 apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])

 apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 

 done

 text{*Both premises are essential. Consider -1 and 1.*}

 lemma inverse_less_iff_less [simp]:

      "[|0 < a; 0 < b|] 

       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"

 by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 

 lemma le_imp_inverse_le:

    "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"

   by (force simp add: order_le_less less_imp_inverse_less)

 lemma inverse_le_iff_le [simp]:

      "[|0 < a; 0 < b|] 

       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"

 by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 

 text{*These results refer to both operands being negative.  The opposite-sign

 case is trivial, since inverse preserves signs.*}

 lemma inverse_le_imp_le_neg:

    "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"

   apply (rule classical) 

   apply (subgoal_tac "a < 0") 

    prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 

   apply (insert inverse_le_imp_le [of "-b" "-a"])

   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

   done

 lemma less_imp_inverse_less_neg:

    "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"

   apply (subgoal_tac "a < 0") 

    prefer 2 apply (blast intro: order_less_trans) 

   apply (insert less_imp_inverse_less [of "-b" "-a"])

   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

   done

 lemma inverse_less_imp_less_neg:

    "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"

   apply (rule classical) 

   apply (subgoal_tac "a < 0") 

    prefer 2

    apply (force simp add: linorder_not_less intro: order_le_less_trans) 

   apply (insert inverse_less_imp_less [of "-b" "-a"])

   apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

   done

 lemma inverse_less_iff_less_neg [simp]:

      "[|a < 0; b < 0|] 

       ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"

   apply (insert inverse_less_iff_less [of "-b" "-a"])

   apply (simp del: inverse_less_iff_less 

 	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

   done

 lemma le_imp_inverse_le_neg:

    "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"

   by (force simp add: order_le_less less_imp_inverse_less_neg)

 lemma inverse_le_iff_le_neg [simp]:

      "[|a < 0; b < 0|] 

       ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"

 by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 

 subsection{*Inverses and the Number One*}

 lemma one_less_inverse_iff:

     "(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"proof cases

   assume "0 < x"

     with inverse_less_iff_less [OF zero_less_one, of x]

     show ?thesis by simp

 next

   assume notless: "~ (0 < x)"

   have "~ (1 < inverse x)"

   proof

     assume "1 < inverse x"

     also with notless have "... \<le> 0" by (simp add: linorder_not_less)

     also have "... < 1" by (rule zero_less_one) 

     finally show False by auto

   qed

   with notless show ?thesis by simp

 qed

 lemma inverse_eq_1_iff [simp]:

     "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"

 by (insert inverse_eq_iff_eq [of x 1], simp) 

 lemma one_le_inverse_iff:

    "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"

 by (force simp add: order_le_less one_less_inverse_iff zero_less_one 

                     eq_commute [of 1]) 

 lemma inverse_less_1_iff:

    "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"

 by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 

 lemma inverse_le_1_iff:

    "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"

 by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 

 subsection{*Division and Signs*}

 lemma zero_less_divide_iff:

      "((0::'a::{ordered_field,division_by_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"

 by (simp add: divide_inverse zero_less_mult_iff)

 lemma divide_less_0_iff:

      "(a/b < (0::'a::{ordered_field,division_by_zero})) = 

       (0 < a & b < 0 | a < 0 & 0 < b)"

 by (simp add: divide_inverse mult_less_0_iff)

 lemma zero_le_divide_iff:

      "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =

       (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"

 by (simp add: divide_inverse zero_le_mult_iff)

 lemma divide_le_0_iff:

      "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =

       (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"

 by (simp add: divide_inverse mult_le_0_iff)

 lemma divide_eq_0_iff [simp]:

      "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"

 by (simp add: divide_inverse field_mult_eq_0_iff)

 subsection{*Simplification of Inequalities Involving Literal Divisors*}

 lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"

 proof -

   assume less: "0<c"

   hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"

     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

   also have "... = (a*c \<le> b)"

     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"

 proof -

   assume less: "c<0"

   hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"

     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

   also have "... = (b \<le> a*c)"

     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma le_divide_eq:

   "(a \<le> b/c) = 

    (if 0 < c then a*c \<le> b

              else if c < 0 then b \<le> a*c

              else  a \<le> (0::'a::{ordered_field,division_by_zero}))"

 apply (case_tac "c=0", simp) 

 apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 

 done

 lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"

 proof -

   assume less: "0<c"

   hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"

     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

   also have "... = (b \<le> a*c)"

     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"

 proof -

   assume less: "c<0"

   hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"

     by (simp add: mult_le_cancel_right order_less_not_sym [OF less])

   also have "... = (a*c \<le> b)"

     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma divide_le_eq:

   "(b/c \<le> a) = 

    (if 0 < c then b \<le> a*c

              else if c < 0 then a*c \<le> b

              else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"

 apply (case_tac "c=0", simp) 

 apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 

 done

 lemma pos_less_divide_eq:

      "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"

 proof -

   assume less: "0<c"

   hence "(a < b/c) = (a*c < (b/c)*c)"

     by (simp add: mult_less_cancel_right order_less_not_sym [OF less])

   also have "... = (a*c < b)"

     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma neg_less_divide_eq:

  "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"

 proof -

   assume less: "c<0"

   hence "(a < b/c) = ((b/c)*c < a*c)"

     by (simp add: mult_less_cancel_right order_less_not_sym [OF less])

   also have "... = (b < a*c)"

     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma less_divide_eq:

   "(a < b/c) = 

    (if 0 < c then a*c < b

              else if c < 0 then b < a*c

              else  a < (0::'a::{ordered_field,division_by_zero}))"

 apply (case_tac "c=0", simp) 

 apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 

 done

 lemma pos_divide_less_eq:

      "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"

 proof -

   assume less: "0<c"

   hence "(b/c < a) = ((b/c)*c < a*c)"

     by (simp add: mult_less_cancel_right order_less_not_sym [OF less])

   also have "... = (b < a*c)"

     by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma neg_divide_less_eq:

  "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"

 proof -

   assume less: "c<0"

   hence "(b/c < a) = (a*c < (b/c)*c)"

     by (simp add: mult_less_cancel_right order_less_not_sym [OF less])

   also have "... = (a*c < b)"

     by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma divide_less_eq:

   "(b/c < a) = 

    (if 0 < c then b < a*c

              else if c < 0 then a*c < b

              else 0 < (a::'a::{ordered_field,division_by_zero}))"

 apply (case_tac "c=0", simp) 

 apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 

 done

 lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"

 proof -

   assume [simp]: "c\<noteq>0"

   have "(a = b/c) = (a*c = (b/c)*c)"

     by (simp add: field_mult_cancel_right)

   also have "... = (a*c = b)"

     by (simp add: divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma eq_divide_eq:

   "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"

 by (simp add: nonzero_eq_divide_eq) 

 lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"

 proof -

   assume [simp]: "c\<noteq>0"

   have "(b/c = a) = ((b/c)*c = a*c)"

     by (simp add: field_mult_cancel_right)

   also have "... = (b = a*c)"

     by (simp add: divide_inverse mult_assoc) 

   finally show ?thesis .

 qed

 lemma divide_eq_eq:

   "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"

 by (force simp add: nonzero_divide_eq_eq) 

 subsection{*Cancellation Laws for Division*}

 lemma divide_cancel_right [simp]:

      "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"

 apply (case_tac "c=0", simp) 

 apply (simp add: divide_inverse field_mult_cancel_right) 

 done

 lemma divide_cancel_left [simp]:

      "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 

 apply (case_tac "c=0", simp) 

 apply (simp add: divide_inverse field_mult_cancel_left) 

 done

 subsection {* Division and the Number One *}

 text{*Simplify expressions equated with 1*}

 lemma divide_eq_1_iff [simp]:

      "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

 apply (case_tac "b=0", simp) 

 apply (simp add: right_inverse_eq) 

 done

 lemma one_eq_divide_iff [simp]:

      "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"

 by (simp add: eq_commute [of 1])  

 lemma zero_eq_1_divide_iff [simp]:

      "((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"

 apply (case_tac "a=0", simp) 

 apply (auto simp add: nonzero_eq_divide_eq) 

 done

 lemma one_divide_eq_0_iff [simp]:

      "(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"

 apply (case_tac "a=0", simp) 

 apply (insert zero_neq_one [THEN not_sym]) 

 apply (auto simp add: nonzero_divide_eq_eq) 

 done

 text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}

 declare zero_less_divide_iff [of "1", simp]

 declare divide_less_0_iff [of "1", simp]

 declare zero_le_divide_iff [of "1", simp]

 declare divide_le_0_iff [of "1", simp]

 subsection {* Ordering Rules for Division *}

 lemma divide_strict_right_mono:

      "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"

 by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 

               positive_imp_inverse_positive) 

 lemma divide_right_mono:

      "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"

   by (force simp add: divide_strict_right_mono order_le_less) 

 text{*The last premise ensures that @{term a} and @{term b} 

       have the same sign*}

 lemma divide_strict_left_mono:

        "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"

 by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 

       order_less_imp_not_eq order_less_imp_not_eq2  

       less_imp_inverse_less less_imp_inverse_less_neg) 

 lemma divide_left_mono:

      "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"

   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 

    prefer 2 

    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 

   apply (case_tac "c=0", simp add: divide_inverse)

   apply (force simp add: divide_strict_left_mono order_le_less) 

   done

 lemma divide_strict_left_mono_neg:

      "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"

   apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 

    prefer 2 

    apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 

   apply (drule divide_strict_left_mono [of _ _ "-c"]) 

    apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 

   done

 lemma divide_strict_right_mono_neg:

      "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"

 apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 

 apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 

 done

 subsection {* Ordered Fields are Dense *}

 lemma less_add_one: "a < (a+1::'a::ordered_semidom)"

 proof -

   have "a+0 < (a+1::'a::ordered_semidom)"

     by (blast intro: zero_less_one add_strict_left_mono) 

   thus ?thesis by simp

 qed

 lemma zero_less_two: "0 < (1+1::'a::ordered_semidom)"

   by (blast intro: order_less_trans zero_less_one less_add_one) 

 lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"

 by (simp add: zero_less_two pos_less_divide_eq right_distrib) 

 lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"

 by (simp add: zero_less_two pos_divide_less_eq right_distrib) 

 lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"

 by (blast intro!: less_half_sum gt_half_sum)

 subsection {* Absolute Value *}

 lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"

   by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) 

 lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))" 

 proof -

   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"

   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"

   have a: "(abs a) * (abs b) = ?x"

     by (simp only: abs_prts[of a] abs_prts[of b] ring_eq_simps)

   {

     fix u v :: 'a

     have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow> u * v = ?y"

       apply (subst prts[of u], subst prts[of v])

       apply (simp add: left_distrib right_distrib add_ac) 

       done

   }

   note b = this[OF refl[of a] refl[of b]]

   note addm = add_mono[of "0::'a" _ "0::'a", simplified]

   note addm2 = add_mono[of _ "0::'a" _ "0::'a", simplified]

   have xy: "- ?x <= ?y"

     apply (simp add: compare_rls)

     apply (rule add_le_imp_le_left[of "-(pprt a * nprt b + nprt a * pprt b)"])

     apply (simp add: add_ac)

     proof -

       let ?r = "nprt a * nprt b +(nprt a * nprt b + (nprt a * pprt b + (pprt a * nprt b + (pprt a * pprt b + (pprt a * pprt b +

 	(- (nprt a * pprt b) + - (pprt a * nprt b)))))))"

       let ?rr = "nprt a * nprt b + nprt a * nprt b + ((nprt a * pprt b) + (- (nprt a * pprt b))) + ((pprt a * nprt b) + - (pprt a * nprt b))

 	+ pprt a * pprt b + pprt a * pprt b"

       have a:"?r = ?rr" by (simp only: add_ac)      

       have "0 <= ?rr"

 	apply (simp)

 	apply (rule addm)+

 	apply (simp_all add: mult_neg_le mult_pos_le)

 	done

       with a show "0 <= ?r" by simp

     qed

   have yx: "?y <= ?x"

     apply (simp add: add_ac)

     apply (simp add: compare_rls)

     apply (rule add_le_imp_le_right[of _ "-(pprt a * pprt b)"])

     apply (simp add: add_ac)

     apply (rule addm2, (simp add: mult_pos_neg_le mult_pos_neg2_le)+)+

     done

   have i1: "a*b <= abs a * abs b" by (simp only: a b yx)

   have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)

   show ?thesis

     apply (rule abs_leI)

     apply (simp add: i1)

     apply (simp add: i2[simplified minus_le_iff])

     done

 qed

 lemma abs_eq_mult: 

   assumes "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"

   shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"

 proof -

   have s: "(0 <= a*b) | (a*b <= 0)"

     apply (auto)    

     apply (rule_tac split_mult_pos_le)

     apply (rule_tac contrapos_np[of "a*b <= 0"])

     apply (simp)

     apply (rule_tac split_mult_neg_le)

     apply (insert prems)

     apply (blast)

     done

   have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"

     by (simp add: prts[symmetric])

   show ?thesis

   proof cases

     assume "0 <= a * b"

     then show ?thesis

       apply (simp_all add: mulprts abs_prts)

       apply (insert prems)

       apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]

 	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id] order_antisym mult_pos_neg_le[of a b] mult_pos_neg2_le[of b a])

       done

   next

     assume "~(0 <= a*b)"

     with s have "a*b <= 0" by simp

     then show ?thesis

       apply (simp_all add: mulprts abs_prts)

       apply (insert prems)

       apply (auto simp add: ring_eq_simps iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_zero_pprt]

 	iff2imp[OF le_zero_iff_pprt_id] iff2imp[OF zero_le_iff_nprt_id] order_antisym mult_pos_le[of a b] mult_neg_le[of a b])

       done

   qed

 qed

 lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)" 

 by (simp add: abs_eq_mult linorder_linear)

 lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"

 by (simp add: abs_if) 

 lemma nonzero_abs_inverse:

      "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"

 apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 

                       negative_imp_inverse_negative)

 apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 

 done

 lemma abs_inverse [simp]:

      "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 

       inverse (abs a)"

 apply (case_tac "a=0", simp) 

 apply (simp add: nonzero_abs_inverse) 

 done

 lemma nonzero_abs_divide:

      "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"

 by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 

 lemma abs_divide:

      "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"

 apply (case_tac "b=0", simp) 

 apply (simp add: nonzero_abs_divide) 

 done

 lemma abs_mult_less:

      "[| abs a < c; abs b < d |] ==> abs a * abs b < c*(d::'a::ordered_idom)"

 proof -

   assume ac: "abs a < c"

   hence cpos: "0<c" by (blast intro: order_le_less_trans abs_ge_zero)

   assume "abs b < d"

   thus ?thesis by (simp add: ac cpos mult_strict_mono) 

 qed

 lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_idom))"

 by (force simp add: order_eq_iff le_minus_self_iff minus_le_self_iff)

 lemma less_minus_self_iff: "(a < -a) = (a < (0::'a::ordered_idom))"

 by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)

 lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_idom))" 

 apply (simp add: order_less_le abs_le_iff)  

 apply (auto simp add: abs_if minus_le_self_iff eq_minus_self_iff)

 apply (simp add: le_minus_self_iff linorder_neq_iff) 

 done

 text{*Moving this up spoils many proofs using @{text mult_le_cancel_right}*}

 declare times_divide_eq_left [simp]

 ML {*

 val left_distrib = thm "left_distrib";

 val right_distrib = thm "right_distrib";

 val mult_commute = thm "mult_commute";

 val distrib = thm "distrib";

 val zero_neq_one = thm "zero_neq_one";

 val no_zero_divisors = thm "no_zero_divisors";

 val left_inverse = thm "left_inverse";

 val divide_inverse = thm "divide_inverse";

 val mult_zero_left = thm "mult_zero_left";

 val mult_zero_right = thm "mult_zero_right";

 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";

 val inverse_zero = thm "inverse_zero";

 val ring_distrib = thms "ring_distrib";

 val combine_common_factor = thm "combine_common_factor";

 val minus_mult_left = thm "minus_mult_left";

 val minus_mult_right = thm "minus_mult_right";

 val minus_mult_minus = thm "minus_mult_minus";

 val minus_mult_commute = thm "minus_mult_commute";

 val right_diff_distrib = thm "right_diff_distrib";

 val left_diff_distrib = thm "left_diff_distrib";

 val mult_left_mono = thm "mult_left_mono";

 val mult_right_mono = thm "mult_right_mono";

 val mult_strict_left_mono = thm "mult_strict_left_mono";

 val mult_strict_right_mono = thm "mult_strict_right_mono";

 val mult_mono = thm "mult_mono";

 val mult_strict_mono = thm "mult_strict_mono";

 val abs_if = thm "abs_if";

 val zero_less_one = thm "zero_less_one";

 val eq_add_iff1 = thm "eq_add_iff1";

 val eq_add_iff2 = thm "eq_add_iff2";

 val less_add_iff1 = thm "less_add_iff1";

 val less_add_iff2 = thm "less_add_iff2";

 val le_add_iff1 = thm "le_add_iff1";

 val le_add_iff2 = thm "le_add_iff2";

 val mult_left_le_imp_le = thm "mult_left_le_imp_le";

 val mult_right_le_imp_le = thm "mult_right_le_imp_le";

 val mult_left_less_imp_less = thm "mult_left_less_imp_less";

 val mult_right_less_imp_less = thm "mult_right_less_imp_less";

 val mult_strict_left_mono_neg = thm "mult_strict_left_mono_neg";

 val mult_left_mono_neg = thm "mult_left_mono_neg";

 val mult_strict_right_mono_neg = thm "mult_strict_right_mono_neg";

 val mult_right_mono_neg = thm "mult_right_mono_neg";

 val mult_pos = thm "mult_pos";

 val mult_pos_le = thm "mult_pos_le";

 val mult_pos_neg = thm "mult_pos_neg";

 val mult_pos_neg_le = thm "mult_pos_neg_le";

 val mult_pos_neg2 = thm "mult_pos_neg2";

 val mult_pos_neg2_le = thm "mult_pos_neg2_le";

 val mult_neg = thm "mult_neg";

 val mult_neg_le = thm "mult_neg_le";

 val zero_less_mult_pos = thm "zero_less_mult_pos";

 val zero_less_mult_pos2 = thm "zero_less_mult_pos2";

 val zero_less_mult_iff = thm "zero_less_mult_iff";

 val mult_eq_0_iff = thm "mult_eq_0_iff";

 val zero_le_mult_iff = thm "zero_le_mult_iff";

 val mult_less_0_iff = thm "mult_less_0_iff";

 val mult_le_0_iff = thm "mult_le_0_iff";

 val split_mult_pos_le = thm "split_mult_pos_le";

 val split_mult_neg_le = thm "split_mult_neg_le";

 val zero_le_square = thm "zero_le_square";

 val zero_le_one = thm "zero_le_one";

 val not_one_le_zero = thm "not_one_le_zero";

 val not_one_less_zero = thm "not_one_less_zero";

 val mult_left_mono_neg = thm "mult_left_mono_neg";

 val mult_right_mono_neg = thm "mult_right_mono_neg";

 val mult_strict_mono = thm "mult_strict_mono";

 val mult_strict_mono' = thm "mult_strict_mono'";

 val mult_mono = thm "mult_mono";

 val less_1_mult = thm "less_1_mult";

 val mult_less_cancel_right = thm "mult_less_cancel_right";

 val mult_less_cancel_left = thm "mult_less_cancel_left";

 val mult_le_cancel_right = thm "mult_le_cancel_right";

 val mult_le_cancel_left = thm "mult_le_cancel_left";

 val mult_less_imp_less_left = thm "mult_less_imp_less_left";

 val mult_less_imp_less_right = thm "mult_less_imp_less_right";

 val mult_cancel_right = thm "mult_cancel_right";

 val mult_cancel_left = thm "mult_cancel_left";

 val ring_eq_simps = thms "ring_eq_simps";

 val right_inverse = thm "right_inverse";

 val right_inverse_eq = thm "right_inverse_eq";

 val nonzero_inverse_eq_divide = thm "nonzero_inverse_eq_divide";

 val divide_self = thm "divide_self";

 val divide_zero = thm "divide_zero";

 val divide_zero_left = thm "divide_zero_left";

 val inverse_eq_divide = thm "inverse_eq_divide";

 val add_divide_distrib = thm "add_divide_distrib";

 val field_mult_eq_0_iff = thm "field_mult_eq_0_iff";

 val field_mult_cancel_right_lemma = thm "field_mult_cancel_right_lemma";

 val field_mult_cancel_right = thm "field_mult_cancel_right";

 val field_mult_cancel_left = thm "field_mult_cancel_left";

 val nonzero_imp_inverse_nonzero = thm "nonzero_imp_inverse_nonzero";

 val inverse_zero_imp_zero = thm "inverse_zero_imp_zero";

 val inverse_nonzero_imp_nonzero = thm "inverse_nonzero_imp_nonzero";

 val inverse_nonzero_iff_nonzero = thm "inverse_nonzero_iff_nonzero";

 val nonzero_inverse_minus_eq = thm "nonzero_inverse_minus_eq";

 val inverse_minus_eq = thm "inverse_minus_eq";

 val nonzero_inverse_eq_imp_eq = thm "nonzero_inverse_eq_imp_eq";

 val inverse_eq_imp_eq = thm "inverse_eq_imp_eq";

 val inverse_eq_iff_eq = thm "inverse_eq_iff_eq";

 val nonzero_inverse_inverse_eq = thm "nonzero_inverse_inverse_eq";

 val inverse_inverse_eq = thm "inverse_inverse_eq";

 val inverse_1 = thm "inverse_1";

 val nonzero_inverse_mult_distrib = thm "nonzero_inverse_mult_distrib";

 val inverse_mult_distrib = thm "inverse_mult_distrib";

 val inverse_add = thm "inverse_add";

 val inverse_divide = thm "inverse_divide";

 val nonzero_mult_divide_cancel_left = thm "nonzero_mult_divide_cancel_left";

 val mult_divide_cancel_left = thm "mult_divide_cancel_left";

 val nonzero_mult_divide_cancel_right = thm "nonzero_mult_divide_cancel_right";

 val mult_divide_cancel_right = thm "mult_divide_cancel_right";

 val mult_divide_cancel_eq_if = thm "mult_divide_cancel_eq_if";

 val divide_1 = thm "divide_1";

 val times_divide_eq_right = thm "times_divide_eq_right";

 val times_divide_eq_left = thm "times_divide_eq_left";

 val divide_divide_eq_right = thm "divide_divide_eq_right";

 val divide_divide_eq_left = thm "divide_divide_eq_left";

 val nonzero_minus_divide_left = thm "nonzero_minus_divide_left";

 val nonzero_minus_divide_right = thm "nonzero_minus_divide_right";

 val nonzero_minus_divide_divide = thm "nonzero_minus_divide_divide";

 val minus_divide_left = thm "minus_divide_left";

 val minus_divide_right = thm "minus_divide_right";

 val minus_divide_divide = thm "minus_divide_divide";

 val diff_divide_distrib = thm "diff_divide_distrib";

 val positive_imp_inverse_positive = thm "positive_imp_inverse_positive";

 val negative_imp_inverse_negative = thm "negative_imp_inverse_negative";

 val inverse_le_imp_le = thm "inverse_le_imp_le";

 val inverse_positive_imp_positive = thm "inverse_positive_imp_positive";

 val inverse_positive_iff_positive = thm "inverse_positive_iff_positive";

 val inverse_negative_imp_negative = thm "inverse_negative_imp_negative";

 val inverse_negative_iff_negative = thm "inverse_negative_iff_negative";

 val inverse_nonnegative_iff_nonnegative = thm "inverse_nonnegative_iff_nonnegative";

 val inverse_nonpositive_iff_nonpositive = thm "inverse_nonpositive_iff_nonpositive";

 val less_imp_inverse_less = thm "less_imp_inverse_less";

 val inverse_less_imp_less = thm "inverse_less_imp_less";

 val inverse_less_iff_less = thm "inverse_less_iff_less";

 val le_imp_inverse_le = thm "le_imp_inverse_le";

 val inverse_le_iff_le = thm "inverse_le_iff_le";

 val inverse_le_imp_le_neg = thm "inverse_le_imp_le_neg";

 val less_imp_inverse_less_neg = thm "less_imp_inverse_less_neg";

 val inverse_less_imp_less_neg = thm "inverse_less_imp_less_neg";

 val inverse_less_iff_less_neg = thm "inverse_less_iff_less_neg";

 val le_imp_inverse_le_neg = thm "le_imp_inverse_le_neg";

 val inverse_le_iff_le_neg = thm "inverse_le_iff_le_neg";

 val one_less_inverse_iff = thm "one_less_inverse_iff";

 val inverse_eq_1_iff = thm "inverse_eq_1_iff";

 val one_le_inverse_iff = thm "one_le_inverse_iff";

 val inverse_less_1_iff = thm "inverse_less_1_iff";

 val inverse_le_1_iff = thm "inverse_le_1_iff";

 val zero_less_divide_iff = thm "zero_less_divide_iff";

 val divide_less_0_iff = thm "divide_less_0_iff";

 val zero_le_divide_iff = thm "zero_le_divide_iff";

 val divide_le_0_iff = thm "divide_le_0_iff";

 val divide_eq_0_iff = thm "divide_eq_0_iff";

 val pos_le_divide_eq = thm "pos_le_divide_eq";

 val neg_le_divide_eq = thm "neg_le_divide_eq";

 val le_divide_eq = thm "le_divide_eq";

 val pos_divide_le_eq = thm "pos_divide_le_eq";

 val neg_divide_le_eq = thm "neg_divide_le_eq";

 val divide_le_eq = thm "divide_le_eq";

 val pos_less_divide_eq = thm "pos_less_divide_eq";

 val neg_less_divide_eq = thm "neg_less_divide_eq";

 val less_divide_eq = thm "less_divide_eq";

 val pos_divide_less_eq = thm "pos_divide_less_eq";

 val neg_divide_less_eq = thm "neg_divide_less_eq";

 val divide_less_eq = thm "divide_less_eq";

 val nonzero_eq_divide_eq = thm "nonzero_eq_divide_eq";

 val eq_divide_eq = thm "eq_divide_eq";

 val nonzero_divide_eq_eq = thm "nonzero_divide_eq_eq";

 val divide_eq_eq = thm "divide_eq_eq";

 val divide_cancel_right = thm "divide_cancel_right";

 val divide_cancel_left = thm "divide_cancel_left";

 val divide_eq_1_iff = thm "divide_eq_1_iff";

 val one_eq_divide_iff = thm "one_eq_divide_iff";

 val zero_eq_1_divide_iff = thm "zero_eq_1_divide_iff";

 val one_divide_eq_0_iff = thm "one_divide_eq_0_iff";

 val divide_strict_right_mono = thm "divide_strict_right_mono";

 val divide_right_mono = thm "divide_right_mono";

 val divide_strict_left_mono = thm "divide_strict_left_mono";

 val divide_left_mono = thm "divide_left_mono";

 val divide_strict_left_mono_neg = thm "divide_strict_left_mono_neg";

 val divide_strict_right_mono_neg = thm "divide_strict_right_mono_neg";

 val less_add_one = thm "less_add_one";

 val zero_less_two = thm "zero_less_two";

 val less_half_sum = thm "less_half_sum";

 val gt_half_sum = thm "gt_half_sum";

 val dense = thm "dense";

 val abs_one = thm "abs_one";

 val abs_le_mult = thm "abs_le_mult";

 val abs_eq_mult = thm "abs_eq_mult";

 val abs_mult = thm "abs_mult";

 val abs_mult_self = thm "abs_mult_self";

 val nonzero_abs_inverse = thm "nonzero_abs_inverse";

 val abs_inverse = thm "abs_inverse";

 val nonzero_abs_divide = thm "nonzero_abs_divide";

 val abs_divide = thm "abs_divide";

 val abs_mult_less = thm "abs_mult_less";

 val eq_minus_self_iff = thm "eq_minus_self_iff";

 val less_minus_self_iff = thm "less_minus_self_iff";

 val abs_less_iff = thm "abs_less_iff";

 *}

 end