Theory Ring_and_Field (Isabelle2005: October 2005)

# Theory Ring_and_Field

Up to index of Isabelle/HOL

theory Ring_and_Field
imports OrderedGroup
begin

    (*  Title:   HOL/Ring_and_Field.thy
        ID:      $Id: Ring_and_Field.thy,v 1.52 2005/08/16 16:53:12 paulson Exp $
        Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel,
                 with contributions by Jeremy Avigad
    *)

    header {* (Ordered) Rings and Fields *}

    theory Ring_and_Field
    imports OrderedGroup
    begin

    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 \url{http://www.mathworld.com} by Eric Weisstein et. al.
      \item \emph{Algebra I} by van der Waerden, Springer.
      \end{itemize}
    *}

    axclass semiring ⊆ 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 ⊆ semiring, comm_monoid_add

    axclass semiring_0_cancel ⊆ semiring_0, cancel_ab_semigroup_add

    axclass comm_semiring ⊆ ab_semigroup_add, ab_semigroup_mult
      distrib: "(a + b) * c = a * c + b * c"

    instance comm_semiring ⊆ 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 ⊆ comm_semiring, comm_monoid_add

    instance comm_semiring_0 ⊆ semiring_0 ..

    axclass comm_semiring_0_cancel ⊆ comm_semiring_0, cancel_ab_semigroup_add

    instance comm_semiring_0_cancel ⊆ semiring_0_cancel ..

    axclass axclass_0_neq_1 ⊆ zero, one
      zero_neq_one [simp]: "0 ≠ 1"

    axclass semiring_1 ⊆ axclass_0_neq_1, semiring_0, monoid_mult

    axclass comm_semiring_1 ⊆ axclass_0_neq_1, comm_semiring_0, comm_monoid_mult (* previously almost_semiring *)

    instance comm_semiring_1 ⊆ semiring_1 ..

    axclass axclass_no_zero_divisors ⊆ zero, times
      no_zero_divisors: "a ≠ 0 ==> b ≠ 0 ==> a * b ≠ 0"

    axclass semiring_1_cancel ⊆ semiring_1, cancel_ab_semigroup_add

    instance semiring_1_cancel ⊆ semiring_0_cancel ..

    axclass comm_semiring_1_cancel ⊆ comm_semiring_1, cancel_ab_semigroup_add (* previously semiring *)

    instance comm_semiring_1_cancel ⊆ semiring_1_cancel ..

    instance comm_semiring_1_cancel ⊆ comm_semiring_0_cancel ..

    axclass ring ⊆ semiring, ab_group_add

    instance ring ⊆ semiring_0_cancel ..

    axclass comm_ring ⊆ comm_semiring_0, ab_group_add

    instance comm_ring ⊆ ring ..

    instance comm_ring ⊆ comm_semiring_0_cancel ..

    axclass ring_1 ⊆ ring, semiring_1

    instance ring_1 ⊆ semiring_1_cancel ..

    axclass comm_ring_1 ⊆ comm_ring, comm_semiring_1 (* previously ring *)

    instance comm_ring_1 ⊆ ring_1 ..

    instance comm_ring_1 ⊆ comm_semiring_1_cancel ..

    axclass idom ⊆ comm_ring_1, axclass_no_zero_divisors

    axclass field ⊆ comm_ring_1, inverse
      left_inverse [simp]: "a ≠ 0 ==> inverse a * a = 1"
      divide_inverse:      "a / b = a * inverse b"

    lemma mult_zero_left [simp]: "0 * a = (0::'a::semiring_0_cancel)"
    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)"
    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≠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 ⊆ idom
    by (intro_classes, simp)

    axclass division_by_zero ⊆ 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 ⊆ semiring_0, pordered_ab_semigroup_add
      mult_left_mono: "a <= b ==> 0 <= c ==> c * a <= c * b"
      mult_right_mono: "a <= b ==> 0 <= c ==> a * c <= b * c"

    axclass pordered_cancel_semiring ⊆ pordered_semiring, cancel_ab_semigroup_add

    instance pordered_cancel_semiring ⊆ semiring_0_cancel ..

    axclass ordered_semiring_strict ⊆ semiring_0, ordered_cancel_ab_semigroup_add
      mult_strict_left_mono: "a < b ==> 0 < c ==> c * a < c * b"
      mult_strict_right_mono: "a < b ==> 0 < c ==> a * c < b * c"

    instance ordered_semiring_strict ⊆ semiring_0_cancel ..

    instance ordered_semiring_strict ⊆ 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 ⊆ comm_semiring_0, pordered_ab_semigroup_add
      mult_mono: "a <= b ==> 0 <= c ==> c * a <= c * b"

    axclass pordered_cancel_comm_semiring ⊆ pordered_comm_semiring, cancel_ab_semigroup_add

    instance pordered_cancel_comm_semiring ⊆ pordered_comm_semiring ..

    axclass ordered_comm_semiring_strict ⊆ comm_semiring_0, ordered_cancel_ab_semigroup_add
      mult_strict_mono: "a < b ==> 0 < c ==> c * a < c * b"

    instance pordered_comm_semiring ⊆ pordered_semiring
    by (intro_classes, insert mult_mono, simp_all add: mult_commute, blast+)

    instance pordered_cancel_comm_semiring ⊆ pordered_cancel_semiring ..

    instance ordered_comm_semiring_strict ⊆ ordered_semiring_strict
    by (intro_classes, insert mult_strict_mono, simp_all add: mult_commute, blast+)

    instance ordered_comm_semiring_strict ⊆ 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 ⊆ ring, pordered_semiring

    instance pordered_ring ⊆ pordered_ab_group_add ..

    instance pordered_ring ⊆ pordered_cancel_semiring ..

    axclass lordered_ring ⊆ pordered_ring, lordered_ab_group_abs

    instance lordered_ring ⊆ lordered_ab_group_meet ..

    instance lordered_ring ⊆ lordered_ab_group_join ..

    axclass axclass_abs_if ⊆ minus, ord, zero
      abs_if: "abs a = (if (a < 0) then (-a) else a)"

    axclass ordered_ring_strict ⊆ ring, ordered_semiring_strict, axclass_abs_if

    instance ordered_ring_strict ⊆ lordered_ab_group ..

    instance ordered_ring_strict ⊆ lordered_ring
    by (intro_classes, simp add: abs_if join_eq_if)

    axclass pordered_comm_ring ⊆ comm_ring, pordered_comm_semiring

    axclass ordered_semidom ⊆ comm_semiring_1_cancel, ordered_comm_semiring_strict (* previously ordered_semiring *)
      zero_less_one [simp]: "0 < 1"

    axclass ordered_idom ⊆ comm_ring_1, ordered_comm_semiring_strict, axclass_abs_if (* previously ordered_ring *)

    instance ordered_idom ⊆ ordered_ring_strict ..

    axclass ordered_field ⊆ field, ordered_idom

    lemmas linorder_neqE_ordered_idom =
     linorder_neqE[where 'a = "?'b::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 ≤ 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 le_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

    subsection {* Ordering Rules for Multiplication *}

    lemma mult_left_le_imp_le:
         "[|c*a ≤ c*b; 0 < c|] ==> a ≤ (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 ≤ b*c; 0 < c|] ==> a ≤ (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 ≤ 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 ≤ 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 ≤ a; c ≤ 0|] ==> c * a ≤  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 ≤ a; c ≤ 0|] ==> a * c ≤  (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_pos: "[| (0::'a::ordered_semiring_strict) < a; 0 < b |] ==> 0 < a*b"
    by (drule mult_strict_left_mono [of 0 b], auto)

    lemma mult_nonneg_nonneg: "[| (0::'a::pordered_cancel_semiring) ≤ a; 0 ≤ b |] ==> 0 ≤ 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_nonneg_nonpos: "[| (0::'a::pordered_cancel_semiring) ≤ a; b ≤ 0 |] ==> a*b ≤ 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_nonneg_nonpos2: "[| (0::'a::pordered_cancel_semiring) ≤ a; b ≤ 0 |] ==> b*a ≤ 0"
    by (drule mult_right_mono[of b 0], auto)

    lemma mult_neg_neg: "[| a < (0::'a::ordered_ring_strict); b < 0 |] ==> 0 < a*b"
    by (drule mult_strict_right_mono_neg, auto)

    lemma mult_nonpos_nonpos: "[| a ≤ (0::'a::pordered_ring); b ≤ 0 |] ==> 0 ≤ 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≤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≤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_pos
      mult_neg_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) ≤ a*b) = (0 ≤ a & 0 ≤ b | a ≤ 0 & b ≤ 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 ≤ (0::'a::ordered_ring_strict)) = (0 ≤ a & b ≤ 0 | a ≤ 0 & 0 ≤ 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 ≤ a & 0 ≤ b) | (a ≤ 0 & b ≤ 0) ==> 0 ≤ a * (b::_::pordered_ring)"
    by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)

    lemma split_mult_neg_le: "(0 ≤ a & b ≤ 0) | (a ≤ 0 & 0 ≤ b) ==> a * b ≤ (0::_::pordered_cancel_semiring)"
    by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

    lemma zero_le_square: "(0::'a::ordered_ring_strict) ≤ 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 ⊆ ordered_semidom
    proof
      have "(0::'a) ≤ 1*1" by (rule zero_le_square)
      thus "(0::'a) < 1" by (simp add: order_le_less)
    qed

    instance ordered_ring_strict ⊆ axclass_no_zero_divisors
    by (intro_classes, simp)

    instance ordered_idom ⊆ idom ..

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

    lemmas one_neq_zero = zero_neq_one [THEN not_sym]
    declare one_neq_zero [simp]

    lemma zero_le_one [simp]: "(0::'a::ordered_semidom) ≤ 1"
      by (rule zero_less_one [THEN order_less_imp_le])

    lemma not_one_le_zero [simp]: "~ (1::'a::ordered_semidom) ≤ 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*}

    text{*Strict monotonicity in both arguments*}
    lemma mult_strict_mono:
         "[|a<b; c<d; 0<b; 0≤c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
    apply (case_tac "c=0")
     apply (simp add: mult_pos_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 ≤ a; 0 ≤ 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 ≤ b; c ≤ d; 0 ≤ b; 0 ≤ c|]
          ==> a * c  ≤  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

    lemma mult_less_le_imp_less: "(a::'a::ordered_semiring_strict) < b ==>
        c <= d ==> 0 <= a ==> 0 < c ==> a * c < b * d"
      apply (subgoal_tac "a * c < b * c")
      apply (erule order_less_le_trans)
      apply (erule mult_left_mono)
      apply simp
      apply (erule mult_strict_right_mono)
      apply assumption
    done

    lemma mult_le_less_imp_less: "(a::'a::ordered_semiring_strict) <= b ==>
        c < d ==> 0 < a ==> 0 <= c ==> a * c < b * d"
      apply (subgoal_tac "a * c <= b * c")
      apply (erule order_le_less_trans)
      apply (erule mult_strict_left_mono)
      apply simp
      apply (erule mult_right_mono)
      apply simp
    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 "≤"} and equality.*}

    text{*These ``disjunction'' versions produce two cases when the comparison is
     an assumption, but effectively four when the comparison is a goal.*}

    lemma mult_less_cancel_right_disj:
        "(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_disj:
        "(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

    text{*The ``conjunction of implication'' lemmas produce two cases when the
    comparison is a goal, but give four when the comparison is an assumption.*}

    lemma mult_less_cancel_right:
      fixes c :: "'a :: ordered_ring_strict"
      shows      "(a*c < b*c) = ((0 ≤ c --> a < b) & (c ≤ 0 --> b < a))"
    by (insert mult_less_cancel_right_disj [of a c b], auto)

    lemma mult_less_cancel_left:
      fixes c :: "'a :: ordered_ring_strict"
      shows      "(c*a < c*b) = ((0 ≤ c --> a < b) & (c ≤ 0 --> b < a))"
    by (insert mult_less_cancel_left_disj [of c a b], auto)

    lemma mult_le_cancel_right:
         "(a*c ≤ b*c) = ((0<c --> a≤b) & (c<0 --> b ≤ (a::'a::ordered_ring_strict)))"
    by (simp add: linorder_not_less [symmetric] mult_less_cancel_right_disj)

    lemma mult_le_cancel_left:
         "(c*a ≤ c*b) = ((0<c --> a≤b) & (c<0 --> b ≤ (a::'a::ordered_ring_strict)))"
    by (simp add: linorder_not_less [symmetric] mult_less_cancel_left_disj)

    lemma mult_less_imp_less_left:
          assumes less: "c*a < c*b" and nonneg: "0 ≤ c"
          shows "a < (b::'a::ordered_semiring_strict)"
    proof (rule ccontr)
      assume "~ a < b"
      hence "b ≤ a" by (simp add: linorder_not_less)
      hence "c*b ≤ 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 ≤ a" by (simp add: linorder_not_less)
      hence "b*c ≤ 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

    subsubsection{*Special Cancellation Simprules for Multiplication*}

    text{*These also produce two cases when the comparison is a goal.*}

    lemma mult_le_cancel_right1:
      fixes c :: "'a :: ordered_idom"
      shows "(c ≤ b*c) = ((0<c --> 1≤b) & (c<0 --> b ≤ 1))"
    by (insert mult_le_cancel_right [of 1 c b], simp)

    lemma mult_le_cancel_right2:
      fixes c :: "'a :: ordered_idom"
      shows "(a*c ≤ c) = ((0<c --> a≤1) & (c<0 --> 1 ≤ a))"
    by (insert mult_le_cancel_right [of a c 1], simp)

    lemma mult_le_cancel_left1:
      fixes c :: "'a :: ordered_idom"
      shows "(c ≤ c*b) = ((0<c --> 1≤b) & (c<0 --> b ≤ 1))"
    by (insert mult_le_cancel_left [of c 1 b], simp)

    lemma mult_le_cancel_left2:
      fixes c :: "'a :: ordered_idom"
      shows "(c*a ≤ c) = ((0<c --> a≤1) & (c<0 --> 1 ≤ a))"
    by (insert mult_le_cancel_left [of c a 1], simp)

    lemma mult_less_cancel_right1:
      fixes c :: "'a :: ordered_idom"
      shows "(c < b*c) = ((0 ≤ c --> 1<b) & (c ≤ 0 --> b < 1))"
    by (insert mult_less_cancel_right [of 1 c b], simp)

    lemma mult_less_cancel_right2:
      fixes c :: "'a :: ordered_idom"
      shows "(a*c < c) = ((0 ≤ c --> a<1) & (c ≤ 0 --> 1 < a))"
    by (insert mult_less_cancel_right [of a c 1], simp)

    lemma mult_less_cancel_left1:
      fixes c :: "'a :: ordered_idom"
      shows "(c < c*b) = ((0 ≤ c --> 1<b) & (c ≤ 0 --> b < 1))"
    by (insert mult_less_cancel_left [of c 1 b], simp)

    lemma mult_less_cancel_left2:
      fixes c :: "'a :: ordered_idom"
      shows "(c*a < c) = ((0 ≤ c --> a<1) & (c ≤ 0 --> 1 < a))"
    by (insert mult_less_cancel_left [of c a 1], simp)

    lemma mult_cancel_right1 [simp]:
    fixes c :: "'a :: ordered_idom"
      shows "(c = b*c) = (c = 0 | b=1)"
    by (insert mult_cancel_right [of 1 c b], force)

    lemma mult_cancel_right2 [simp]:
    fixes c :: "'a :: ordered_idom"
      shows "(a*c = c) = (c = 0 | a=1)"
    by (insert mult_cancel_right [of a c 1], simp)

    lemma mult_cancel_left1 [simp]:
    fixes c :: "'a :: ordered_idom"
      shows "(c = c*b) = (c = 0 | b=1)"
    by (insert mult_cancel_left [of c 1 b], force)

    lemma mult_cancel_left2 [simp]:
    fixes c :: "'a :: ordered_idom"
      shows "(c*a = c) = (c = 0 | a=1)"
    by (insert mult_cancel_left [of c a 1], simp)

    text{*Simprules for comparisons where common factors can be cancelled.*}
    lemmas mult_compare_simps =
        mult_le_cancel_right mult_le_cancel_left
        mult_le_cancel_right1 mult_le_cancel_right2
        mult_le_cancel_left1 mult_le_cancel_left2
        mult_less_cancel_right mult_less_cancel_left
        mult_less_cancel_right1 mult_less_cancel_right2
        mult_less_cancel_left1 mult_less_cancel_left2
        mult_cancel_right mult_cancel_left
        mult_cancel_right1 mult_cancel_right2
        mult_cancel_left1 mult_cancel_left2

    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
      group_eq_simps
    (*  add_ac
      add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
      diff_eq_eq eq_diff_eq *)

    subsection {* Fields *}

    lemma right_inverse [simp]:
          assumes not0: "a ≠ 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 ≠ 0 ==> (a / b = 1) = (a = (b::'a::field))"
    proof
      assume neq: "b ≠ 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 ≠ 0 ==> inverse (a::'a::field) = 1/a"
    by (simp add: divide_inverse)

    lemma divide_self: "a ≠ 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_self_if [simp]:
         "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
      by (simp add: divide_self)

    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≠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 ≠ (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≠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 ≠ 0 ==> inverse a ≠ (0::'a::field)"
    proof
      assume ianz: "inverse a = 0"
      assume "a ≠ 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≠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≠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 ≠ 0"
              and bnz:  "b ≠ 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 ≠ 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≠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 inverse_unique:
      assumes ab: "a*b = 1"
      shows "inverse a = (b::'a::field)"
    proof -
      have "a ≠ 0" using ab by auto
      moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
      ultimately show ?thesis by (simp add: mult_assoc [symmetric])
    qed

    lemma nonzero_inverse_mult_distrib:
          assumes anz: "a ≠ 0"
              and bnz: "b ≠ 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 ≠ 0 & b ≠ 0"
        thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
      next
        assume "~ (a ≠ 0 & b ≠ 0)"
        thus ?thesis  by force
      qed

    text{*There is no slick version using division by zero.*}
    lemma inverse_add:
         "[|a ≠ 0;  b ≠ 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)

    subsection {* Calculations with fractions *}

    lemma nonzero_mult_divide_cancel_left:
      assumes [simp]: "b≠0" and [simp]: "c≠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≠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≠0; c≠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≠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: "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)

    lemma add_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
        x / y + w / z = (x * z + w * y) / (y * z)"
      apply (subgoal_tac "x / y = (x * z) / (y * z)")
      apply (erule ssubst)
      apply (subgoal_tac "w / z = (w * y) / (y * z)")
      apply (erule ssubst)
      apply (rule add_divide_distrib [THEN sym])
      apply (subst mult_commute)
      apply (erule nonzero_mult_divide_cancel_left [THEN sym])
      apply assumption
      apply (erule nonzero_mult_divide_cancel_right [THEN sym])
      apply assumption
    done

    subsubsection{*Special Cancellation Simprules for Division*}

    lemma mult_divide_cancel_left_if [simp]:
      fixes c :: "'a :: {field,division_by_zero}"
      shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
    by (simp add: mult_divide_cancel_left)

    lemma mult_divide_cancel_right_if [simp]:
      fixes c :: "'a :: {field,division_by_zero}"
      shows "(a*c) / (b*c) = (if c=0 then 0 else a/b)"
    by (simp add: mult_divide_cancel_right)

    lemma mult_divide_cancel_left_if1 [simp]:
      fixes c :: "'a :: {field,division_by_zero}"
      shows "c / (c*b) = (if c=0 then 0 else 1/b)"
    apply (insert mult_divide_cancel_left_if [of c 1 b])
    apply (simp del: mult_divide_cancel_left_if)
    done

    lemma mult_divide_cancel_left_if2 [simp]:
      fixes c :: "'a :: {field,division_by_zero}"
      shows "(c*a) / c = (if c=0 then 0 else a)"
    apply (insert mult_divide_cancel_left_if [of c a 1])
    apply (simp del: mult_divide_cancel_left_if)
    done

    lemma mult_divide_cancel_right_if1 [simp]:
      fixes c :: "'a :: {field,division_by_zero}"
      shows "c / (b*c) = (if c=0 then 0 else 1/b)"
    apply (insert mult_divide_cancel_right_if [of 1 c b])
    apply (simp del: mult_divide_cancel_right_if)
    done

    lemma mult_divide_cancel_right_if2 [simp]:
      fixes c :: "'a :: {field,division_by_zero}"
      shows "(a*c) / c = (if c=0 then 0 else a)"
    apply (insert mult_divide_cancel_right_if [of a c 1])
    apply (simp del: mult_divide_cancel_right_if)
    done

    text{*Two lemmas for cancelling the denominator*}

    lemma times_divide_self_right [simp]:
      fixes a :: "'a :: {field,division_by_zero}"
      shows "a * (b/a) = (if a=0 then 0 else b)"
    by (simp add: times_divide_eq_right)

    lemma times_divide_self_left [simp]:
      fixes a :: "'a :: {field,division_by_zero}"
      shows "(b/a) * a = (if a=0 then 0 else b)"
    by (simp add: times_divide_eq_left)

    subsection {* Division and Unary Minus *}

    lemma nonzero_minus_divide_left: "b ≠ 0 ==> - (a/b) = (-a) / (b::'a::field)"
    by (simp add: divide_inverse minus_mult_left)

    lemma nonzero_minus_divide_right: "b ≠ 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 ≠ 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*}
    lemmas divide_minus_left = minus_divide_left [symmetric]
    lemmas divide_minus_right = minus_divide_right [symmetric]
    declare divide_minus_left [simp]   divide_minus_right [simp]

    text{*Also, extract signs from products*}
    lemmas mult_minus_left = minus_mult_left [symmetric]
    lemmas mult_minus_right = minus_mult_right [symmetric]
    declare mult_minus_left [simp]   mult_minus_right [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)

    lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
        x / y - w / z = (x * z - w * y) / (y * z)"
      apply (subst diff_def)+
      apply (subst minus_divide_left)
      apply (subst add_frac_eq)
      apply simp_all
    done

    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 ≤ inverse b"
              and apos:  "0 < a"
             shows "b ≤ (a::'a::ordered_field)"
      proof (rule classical)
      assume "~ b ≤ 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 ≤ a * inverse b"
        by (simp add: apos invle order_less_imp_le mult_left_mono)
      hence "(a * inverse a) * b ≤ (a * inverse b) * b"
        by (simp add: bpos order_less_imp_le mult_right_mono)
      thus "b ≤ 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 ≠ 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 ≠ 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 ≤ inverse a) = (0 ≤ (a::'a::{ordered_field,division_by_zero}))"
    by (simp add: linorder_not_less [symmetric])

    lemma inverse_nonpositive_iff_nonpositive [simp]:
          "(inverse a ≤ 0) = (a ≤ (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 ≤ 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 ≤ b; 0 < a|] ==> inverse b ≤ 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 ≤ inverse b) = (b ≤ (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 ≤ 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_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 ≤ b; b < 0|] ==> inverse b ≤ 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 ≤ inverse b) = (b ≤ (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 "... ≤ 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 ≤ inverse x) = (0 < x & x ≤ (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 ≤ 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 ≤ 1) = (x ≤ 0 | 1 ≤ (x::'a::{ordered_field,division_by_zero}))"
    by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)

    subsection{*Simplification of Inequalities Involving Literal Divisors*}

    lemma pos_le_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_le_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_le_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_le_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 le_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_le_divide_eq neg_le_divide_eq linorder_neq_iff)
    done

    lemma pos_divide_le_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_le_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_le_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_le_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_le_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_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_disj 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_disj 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_disj 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_disj 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≠0 ==> ((a::'a::field) = b/c) = (a*c = b)"
    proof -
      assume [simp]: "c≠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≠0 then a*c = b else a=0)"
    by (simp add: nonzero_eq_divide_eq)

    lemma nonzero_divide_eq_eq: "c≠0 ==> (b/c = (a::'a::field)) = (b = a*c)"
    proof -
      assume [simp]: "c≠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≠0 then b = a*c else a=0)"
    by (force simp add: nonzero_divide_eq_eq)

    lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
        b = a * c ==> b / c = a"
      by (subst divide_eq_eq, simp)

    lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
        a * c = b ==> a = b / c"
      by (subst eq_divide_eq, simp)

    lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
        (x / y = w / z) = (x * z = w * y)"
      apply (subst nonzero_eq_divide_eq)
      apply assumption
      apply (subst times_divide_eq_left)
      apply (erule nonzero_divide_eq_eq)
    done

    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}) ≤ a/b) =
          (0 ≤ a & 0 ≤ b | a ≤ 0 & b ≤ 0)"
    by (simp add: divide_inverse zero_le_mult_iff)

    lemma divide_le_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_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)

    lemma divide_pos_pos: "0 < (x::'a::ordered_field) ==>
        0 < y ==> 0 < x / y"
      apply (subst pos_less_divide_eq)
      apply assumption
      apply simp
    done

    lemma divide_nonneg_pos: "0 <= (x::'a::ordered_field) ==> 0 < y ==>
        0 <= x / y"
      apply (subst pos_le_divide_eq)
      apply assumption
      apply simp
    done

    lemma divide_neg_pos: "(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
      apply (subst pos_divide_less_eq)
      apply assumption
      apply simp
    done

    lemma divide_nonpos_pos: "(x::'a::ordered_field) <= 0 ==>
        0 < y ==> x / y <= 0"
      apply (subst pos_divide_le_eq)
      apply assumption
      apply simp
    done

    lemma divide_pos_neg: "0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
      apply (subst neg_divide_less_eq)
      apply assumption
      apply simp
    done

    lemma divide_nonneg_neg: "0 <= (x::'a::ordered_field) ==>
        y < 0 ==> x / y <= 0"
      apply (subst neg_divide_le_eq)
      apply assumption
      apply simp
    done

    lemma divide_neg_neg: "(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
      apply (subst neg_less_divide_eq)
      apply assumption
      apply simp
    done

    lemma divide_nonpos_neg: "(x::'a::ordered_field) <= 0 ==> y < 0 ==>
        0 <= x / y"
      apply (subst neg_le_divide_eq)
      apply assumption
      apply simp
    done

    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 ≠ 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 ≠ 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"}*}
    lemmas zero_less_divide_1_iff = zero_less_divide_iff [of "1"]
    lemmas divide_less_0_1_iff = divide_less_0_iff [of "1"]
    lemmas zero_le_divide_1_iff = zero_le_divide_iff [of "1"]
    lemmas divide_le_0_1_iff = divide_le_0_iff [of "1"]

    declare zero_less_divide_1_iff [simp]
    declare divide_less_0_1_iff [simp]
    declare zero_le_divide_1_iff [simp]
    declare divide_le_0_1_iff [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 ≤ b; 0 ≤ c|] ==> a/c ≤ b/(c::'a::{ordered_field,division_by_zero})"
      by (force simp add: divide_strict_right_mono order_le_less)

    lemma divide_right_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b
        ==> c <= 0 ==> b / c <= a / c"
      apply (drule divide_right_mono [of _ _ "- c"])
      apply auto
    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

    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 ≤ a; 0 ≤ c; 0 < a*b|] ==> c / a ≤ c / (b::'a::ordered_field)"
      apply (subgoal_tac "a ≠ 0 & b ≠ 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_left_mono_neg: "(a::'a::{division_by_zero,ordered_field}) <= b
        ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
      apply (drule divide_left_mono [of _ _ "- c"])
      apply (auto simp add: mult_commute)
    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 ≠ 0 & b ≠ 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

    text{*Simplify quotients that are compared with the value 1.*}

    lemma le_divide_eq_1:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "(1 ≤ b / a) = ((0 < a & a ≤ b) | (a < 0 & b ≤ a))"
    by (auto simp add: le_divide_eq)

    lemma divide_le_eq_1:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "(b / a ≤ 1) = ((0 < a & b ≤ a) | (a < 0 & a ≤ b) | a=0)"
    by (auto simp add: divide_le_eq)

    lemma less_divide_eq_1:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
    by (auto simp add: less_divide_eq)

    lemma divide_less_eq_1:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
    by (auto simp add: divide_less_eq)

    subsection{*Conditional Simplification Rules: No Case Splits*}

    lemma le_divide_eq_1_pos [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "0 < a ==> (1 ≤ b / a) = (a ≤ b)"
    by (auto simp add: le_divide_eq)

    lemma le_divide_eq_1_neg [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "a < 0 ==> (1 ≤ b / a) = (b ≤ a)"
    by (auto simp add: le_divide_eq)

    lemma divide_le_eq_1_pos [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "0 < a ==> (b / a ≤ 1) = (b ≤ a)"
    by (auto simp add: divide_le_eq)

    lemma divide_le_eq_1_neg [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "a < 0 ==> (b / a ≤ 1) = (a ≤ b)"
    by (auto simp add: divide_le_eq)

    lemma less_divide_eq_1_pos [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "0 < a ==> (1 < b / a) = (a < b)"
    by (auto simp add: less_divide_eq)

    lemma less_divide_eq_1_neg [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "a < 0 ==> (1 < b / a) = (b < a)"
    by (auto simp add: less_divide_eq)

    lemma divide_less_eq_1_pos [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "0 < a ==> (b / a < 1) = (b < a)"
    by (auto simp add: divide_less_eq)

    lemma eq_divide_eq_1 [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "(1 = b / a) = ((a ≠ 0 & a = b))"
    by (auto simp add: eq_divide_eq)

    lemma divide_eq_eq_1 [simp]:
      fixes a :: "'a :: {ordered_field,division_by_zero}"
      shows "(b / a = 1) = ((a ≠ 0 & a = b))"
    by (auto simp add: divide_eq_eq)

    subsection {* Reasoning about inequalities with division *}

    lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
        ==> x * y <= x"
      by (auto simp add: mult_compare_simps);

    lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
        ==> y * x <= x"
      by (auto simp add: mult_compare_simps);

    lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
        x / y <= z";
      by (subst pos_divide_le_eq, assumption+);

    lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
        z <= x / y";
      by (subst pos_le_divide_eq, assumption+)

    lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
        x / y < z"
      by (subst pos_divide_less_eq, assumption+)

    lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
        z < x / y"
      by (subst pos_less_divide_eq, assumption+)

    lemma frac_le: "(0::'a::ordered_field) <= x ==>
        x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
      apply (rule mult_imp_div_pos_le)
      apply simp;
      apply (subst times_divide_eq_left);
      apply (rule mult_imp_le_div_pos, assumption)
      apply (rule mult_mono)
      apply simp_all
    done

    lemma frac_less: "(0::'a::ordered_field) <= x ==>
        x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
      apply (rule mult_imp_div_pos_less)
      apply simp;
      apply (subst times_divide_eq_left);
      apply (rule mult_imp_less_div_pos, assumption)
      apply (erule mult_less_le_imp_less)
      apply simp_all
    done

    lemma frac_less2: "(0::'a::ordered_field) < x ==>
        x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
      apply (rule mult_imp_div_pos_less)
      apply simp_all
      apply (subst times_divide_eq_left);
      apply (rule mult_imp_less_div_pos, assumption)
      apply (erule mult_le_less_imp_less)
      apply simp_all
    done

    lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left

    text{*It's not obvious whether these should be simprules or not.
      Their effect is to gather terms into one big fraction, like
      a*b*c / x*y*z. The rationale for that is unclear, but many proofs
      seem to need them.*}

    declare times_divide_eq [simp]

    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 ==> ∃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) ≤ (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: "[|u = a; v = b|] ==>
                  u * v = pprt a * pprt b + pprt a * nprt b +
                          nprt a * pprt b + nprt a * nprt b"
          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)
        apply (rule_tac y="0::'a" in order_trans)
        apply (rule addm2)
        apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
        apply (rule addm)
        apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
        done
      have yx: "?y <= ?x"
        apply (simp add:diff_def)
        apply (rule_tac y=0 in order_trans)
        apply (rule addm2, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
        apply (rule addm, (simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)+)
        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 ≤ a ∨ a ≤ 0) ∧ (0 ≤ b ∨ b ≤ 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])
            apply(drule (1) mult_nonneg_nonpos[of a b], simp)
            apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
          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)
          apply(drule (1) mult_nonneg_nonneg[of a b],simp)
          apply(drule (1) mult_nonpos_nonpos[of a b],simp)
          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 ≠ 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 ≠ 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
    by (simp add: divide_inverse abs_mult nonzero_abs_inverse)

    lemma abs_divide [simp]:
         "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

    lemma abs_mult_pos: "(0::'a::ordered_idom) <= x ==>
        (abs y) * x = abs (y * x)";
      apply (subst abs_mult);
      apply simp;
    done;

    lemma abs_div_pos: "(0::'a::{division_by_zero,ordered_field}) < y ==>
        abs x / y = abs (x / y)";
      apply (subst abs_divide);
      apply (simp add: order_less_imp_le);
    done;

    subsection {* Miscellaneous *}

    lemma linprog_dual_estimate:
      assumes
      "A * x ≤ (b::'a::lordered_ring)"
      "0 ≤ y"
      "abs (A - A') ≤ δA"
      "b ≤ b'"
      "abs (c - c') ≤ δc"
      "abs x ≤ r"
      shows
      "c * x ≤ y * b' + (y * δA + abs (y * A' - c') + δc) * r"
    proof -
      from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
      from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)
      have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: ring_eq_simps)
      from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
      have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
        by (simp only: 4 estimate_by_abs)
      have 6: "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <= abs (y * (A - A') + (y * A' - c') + (c'-c)) * abs x"
        by (simp add: abs_le_mult)
      have 7: "(abs (y * (A - A') + (y * A' - c') + (c'-c))) * abs x <= (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x"
        by(rule abs_triangle_ineq [THEN mult_right_mono]) simp
      have 8: " (abs (y * (A-A') + (y*A'-c')) + abs(c'-c)) * abs x <=  (abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x"
        by (simp add: abs_triangle_ineq mult_right_mono)
      have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x"
        by (simp add: abs_le_mult mult_right_mono)
      have 10: "c'-c = -(c-c')" by (simp add: ring_eq_simps)
      have 11: "abs (c'-c) = abs (c-c')"
        by (subst 10, subst abs_minus_cancel, simp)
      have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + δc) * abs x"
        by (simp add: 11 prems mult_right_mono)
      have 13: "(abs y * abs (A-A') + abs (y*A'-c') + δc) * abs x <= (abs y * δA + abs (y*A'-c') + δc) * abs x"
        by (simp add: prems mult_right_mono mult_left_mono)
      have r: "(abs y * δA + abs (y*A'-c') + δc) * abs x <=  (abs y * δA + abs (y*A'-c') + δc) * r"
        apply (rule mult_left_mono)
        apply (simp add: prems)
        apply (rule_tac add_mono[of "0::'a" _ "0", simplified])+
        apply (rule mult_left_mono[of "0" "δA", simplified])
        apply (simp_all)
        apply (rule order_trans[where y="abs (A-A')"], simp_all add: prems)
        apply (rule order_trans[where y="abs (c-c')"], simp_all add: prems)
        done
      from 6 7 8 9 12 13 r have 14:" abs((y * (A - A') + (y * A' - c') + (c'-c)) * x) <=(abs y * δA + abs (y*A'-c') + δc) * r"
        by (simp)
      show ?thesis
        apply (rule_tac le_add_right_mono[of _ _ "abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"])
        apply (simp_all only: 5 14[simplified abs_of_nonneg[of y, simplified prems]])
        done
    qed

    lemma le_ge_imp_abs_diff_1:
      assumes
      "A1 <= (A::'a::lordered_ring)"
      "A <= A2"
      shows "abs (A-A1) <= A2-A1"
    proof -
      have "0 <= A - A1"
      proof -
        have 1: "A - A1 = A + (- A1)" by simp
        show ?thesis by (simp only: 1 add_right_mono[of A1 A "-A1", simplified, simplified prems])
      qed
      then have "abs (A-A1) = A-A1" by (rule abs_of_nonneg)
      with prems show "abs (A-A1) <= (A2-A1)" by simp
    qed

    lemma mult_le_prts:
      assumes
      "a1 <= (a::'a::lordered_ring)"
      "a <= a2"
      "b1 <= b"
      "b <= b2"
      shows
      "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
    proof -
      have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
        apply (subst prts[symmetric])+
        apply simp
        done
      then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
        by (simp add: ring_eq_simps)
      moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
        by (simp_all add: prems mult_mono)
      moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
      proof -
        have "pprt a * nprt b <= pprt a * nprt b2"
          by (simp add: mult_left_mono prems)
        moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
          by (simp add: mult_right_mono_neg prems)
        ultimately show ?thesis
          by simp
      qed
      moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
      proof -
        have "nprt a * pprt b <= nprt a2 * pprt b"
          by (simp add: mult_right_mono prems)
        moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
          by (simp add: mult_left_mono_neg prems)
        ultimately show ?thesis
          by simp
      qed
      moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
      proof -
        have "nprt a * nprt b <= nprt a * nprt b1"
          by (simp add: mult_left_mono_neg prems)
        moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
          by (simp add: mult_right_mono_neg prems)
        ultimately show ?thesis
          by simp
      qed
      ultimately show ?thesis
        by - (rule add_mono | simp)+
    qed

    lemma mult_le_dual_prts:
      assumes
      "A * x ≤ (b::'a::lordered_ring)"
      "0 ≤ y"
      "A1 ≤ A"
      "A ≤ A2"
      "c1 ≤ c"
      "c ≤ c2"
      "r1 ≤ x"
      "x ≤ r2"
      shows
      "c * x ≤ y * b + (let s1 = c1 - y * A2; s2 = c2 - y * A1 in pprt s2 * pprt r2 + pprt s1 * nprt r2 + nprt s2 * pprt r1 + nprt s1 * nprt r1)"
      (is "_ <= _ + ?C")
    proof -
      from prems have "y * (A * x) <= y * b" by (simp add: mult_left_mono)
      moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: ring_eq_simps)
      ultimately have "c * x + (y * A - c) * x <= y * b" by simp
      then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)
      then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: ring_eq_simps)
      have s2: "c - y * A <= c2 - y * A1"
        by (simp add: diff_def prems add_mono mult_left_mono)
      have s1: "c1 - y * A2 <= c - y * A"
        by (simp add: diff_def prems add_mono mult_left_mono)
      have prts: "(c - y * A) * x <= ?C"
        apply (simp add: Let_def)
        apply (rule mult_le_prts)
        apply (simp_all add: prems s1 s2)
        done
      then have "y * b + (c - y * A) * x <= y * b + ?C"
        by simp
      with cx show ?thesis
        by(simp only:)
    qed

    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_disj = thm "mult_less_cancel_right_disj";
    val mult_less_cancel_left_disj = thm "mult_less_cancel_left_disj";
    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

lemma mult_zero_left:

      (0::'a) * a = (0::'a)

lemma mult_zero_right:

      a * (0::'a) = (0::'a)

lemma field_mult_eq_0_iff:

      (a * b = (0::'a)) = (a = (0::'a) ∨ b = (0::'a))

### Distribution rules

theorems ring_distrib:

      a * (b + c) = a * b + a * c

      (a + b) * c = a * c + b * c

theorems ring_distrib:

      a * (b + c) = a * b + a * c

      (a + b) * c = a * c + b * c

lemma combine_common_factor:

      a * e + (b * e + c) = (a + b) * e + c

lemma minus_mult_left:

      - (a * b) = - a * b

lemma minus_mult_right:

      - (a * b) = a * - b

lemma minus_mult_minus:

      - a * - b = a * b

lemma minus_mult_commute:

      - a * b = a * - b

lemma right_diff_distrib:

      a * (b - c) = a * b - a * c

lemma left_diff_distrib:

      (a - b) * c = a * c - b * c

lemmas linorder_neqE_ordered_idom:

      [| x ≠ y; x < y ==> R; y < x ==> R |] ==> R

lemmas linorder_neqE_ordered_idom:

      [| x ≠ y; x < y ==> R; y < x ==> R |] ==> R

lemma eq_add_iff1:

      (a * e + c = b * e + d) = ((a - b) * e + c = d)

lemma eq_add_iff2:

      (a * e + c = b * e + d) = (c = (b - a) * e + d)

lemma less_add_iff1:

      (a * e + c < b * e + d) = ((a - b) * e + c < d)

lemma less_add_iff2:

      (a * e + c < b * e + d) = (c < (b - a) * e + d)

lemma le_add_iff1:

      (a * e + c ≤ b * e + d) = ((a - b) * e + c ≤ d)

lemma le_add_iff2:

      (a * e + c ≤ b * e + d) = (c ≤ (b - a) * e + d)

### Ordering Rules for Multiplication

lemma mult_left_le_imp_le:

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

lemma mult_right_le_imp_le:

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

lemma mult_left_less_imp_less:

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

lemma mult_right_less_imp_less:

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

lemma mult_strict_left_mono_neg:

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

lemma mult_left_mono_neg:

      [| b ≤ a; c ≤ (0::'a) |] ==> c * a ≤ c * b

lemma mult_strict_right_mono_neg:

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

lemma mult_right_mono_neg:

      [| b ≤ a; c ≤ (0::'a) |] ==> a * c ≤ b * c

### Products of Signs

lemma mult_pos_pos:

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

lemma mult_nonneg_nonneg:

      [| (0::'a) ≤ a; (0::'a) ≤ b |] ==> (0::'a) ≤ a * b

lemma mult_pos_neg:

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

lemma mult_nonneg_nonpos:

      [| (0::'a) ≤ a; b ≤ (0::'a) |] ==> a * b ≤ (0::'a)

lemma mult_pos_neg2:

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

lemma mult_nonneg_nonpos2:

      [| (0::'a) ≤ a; b ≤ (0::'a) |] ==> b * a ≤ (0::'a)

lemma mult_neg_neg:

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

lemma mult_nonpos_nonpos:

      [| a ≤ (0::'a); b ≤ (0::'a) |] ==> (0::'a) ≤ a * b

lemma zero_less_mult_pos:

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

lemma zero_less_mult_pos2:

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

lemma zero_less_mult_iff:

      ((0::'a) < a * b) = ((0::'a) < a ∧ (0::'a) < b ∨ a < (0::'a) ∧ b < (0::'a))

lemma mult_eq_0_iff:

      (a * b = (0::'a)) = (a = (0::'a) ∨ b = (0::'a))

lemma zero_le_mult_iff:

      ((0::'a) ≤ a * b) = ((0::'a) ≤ a ∧ (0::'a) ≤ b ∨ a ≤ (0::'a) ∧ b ≤ (0::'a))

lemma mult_less_0_iff:

      (a * b < (0::'a)) = ((0::'a) < a ∧ b < (0::'a) ∨ a < (0::'a) ∧ (0::'a) < b)

lemma mult_le_0_iff:

      (a * b ≤ (0::'a)) = ((0::'a) ≤ a ∧ b ≤ (0::'a) ∨ a ≤ (0::'a) ∧ (0::'a) ≤ b)

lemma split_mult_pos_le:

      (0::'a) ≤ a ∧ (0::'a) ≤ b ∨ a ≤ (0::'a) ∧ b ≤ (0::'a) ==> (0::'a) ≤ a * b

lemma split_mult_neg_le:

      (0::'a) ≤ a ∧ b ≤ (0::'a) ∨ a ≤ (0::'a) ∧ (0::'a) ≤ b ==> a * b ≤ (0::'a)

lemma zero_le_square:

      (0::'a) ≤ a * a

lemmas one_neq_zero:

      (1::'a1) ≠ (0::'a1)

lemmas one_neq_zero:

      (1::'a1) ≠ (0::'a1)

lemma zero_le_one:

      (0::'a) ≤ (1::'a)

lemma not_one_le_zero:

      ¬ (1::'a) ≤ (0::'a)

lemma not_one_less_zero:

      ¬ (1::'a) < (0::'a)

### More Monotonicity

lemma mult_strict_mono:

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

lemma mult_strict_mono':

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

lemma mult_mono:

      [| a ≤ b; c ≤ d; (0::'a) ≤ b; (0::'a) ≤ c |] ==> a * c ≤ b * d

lemma less_1_mult:

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

lemma mult_less_le_imp_less:

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

lemma mult_le_less_imp_less:

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

### Cancellation Laws for Relationships With a Common Factor

lemma mult_less_cancel_right_disj:

      (a * c < b * c) = ((0::'a) < c ∧ a < b ∨ c < (0::'a) ∧ b < a)

lemma mult_less_cancel_left_disj:

      (c * a < c * b) = ((0::'a) < c ∧ a < b ∨ c < (0::'a) ∧ b < a)

lemma mult_less_cancel_right:

      (a * c < b * c) = (((0::'a) ≤ c --> a < b) ∧ (c ≤ (0::'a) --> b < a))

lemma mult_less_cancel_left:

      (c * a < c * b) = (((0::'a) ≤ c --> a < b) ∧ (c ≤ (0::'a) --> b < a))

lemma mult_le_cancel_right:

      (a * c ≤ b * c) = (((0::'a) < c --> a ≤ b) ∧ (c < (0::'a) --> b ≤ a))

lemma mult_le_cancel_left:

      (c * a ≤ c * b) = (((0::'a) < c --> a ≤ b) ∧ (c < (0::'a) --> b ≤ a))

lemma mult_less_imp_less_left:

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

lemma mult_less_imp_less_right:

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

lemma mult_cancel_right:

      (a * c = b * c) = (c = (0::'a) ∨ a = b)

lemma mult_cancel_left:

      (c * a = c * b) = (c = (0::'a) ∨ a = b)

#### Special Cancellation Simprules for Multiplication

lemma mult_le_cancel_right1:

      (c ≤ b * c) = (((0::'a) < c --> (1::'a) ≤ b) ∧ (c < (0::'a) --> b ≤ (1::'a)))

lemma mult_le_cancel_right2:

      (a * c ≤ c) = (((0::'a) < c --> a ≤ (1::'a)) ∧ (c < (0::'a) --> (1::'a) ≤ a))

lemma mult_le_cancel_left1:

      (c ≤ c * b) = (((0::'a) < c --> (1::'a) ≤ b) ∧ (c < (0::'a) --> b ≤ (1::'a)))

lemma mult_le_cancel_left2:

      (c * a ≤ c) = (((0::'a) < c --> a ≤ (1::'a)) ∧ (c < (0::'a) --> (1::'a) ≤ a))

lemma mult_less_cancel_right1:

      (c < b * c) = (((0::'a) ≤ c --> (1::'a) < b) ∧ (c ≤ (0::'a) --> b < (1::'a)))

lemma mult_less_cancel_right2:

      (a * c < c) = (((0::'a) ≤ c --> a < (1::'a)) ∧ (c ≤ (0::'a) --> (1::'a) < a))

lemma mult_less_cancel_left1:

      (c < c * b) = (((0::'a) ≤ c --> (1::'a) < b) ∧ (c ≤ (0::'a) --> b < (1::'a)))

lemma mult_less_cancel_left2:

      (c * a < c) = (((0::'a) ≤ c --> a < (1::'a)) ∧ (c ≤ (0::'a) --> (1::'a) < a))

lemma mult_cancel_right1:

      (c = b * c) = (c = (0::'a) ∨ b = (1::'a))

lemma mult_cancel_right2:

      (a * c = c) = (c = (0::'a) ∨ a = (1::'a))

lemma mult_cancel_left1:

      (c = c * b) = (c = (0::'a) ∨ b = (1::'a))

lemma mult_cancel_left2:

      (c * a = c) = (c = (0::'a) ∨ a = (1::'a))

lemmas mult_compare_simps:

      (a * c ≤ b * c) = (((0::'a) < c --> a ≤ b) ∧ (c < (0::'a) --> b ≤ a))

      (c * a ≤ c * b) = (((0::'a) < c --> a ≤ b) ∧ (c < (0::'a) --> b ≤ a))

      (c ≤ b * c) = (((0::'a) < c --> (1::'a) ≤ b) ∧ (c < (0::'a) --> b ≤ (1::'a)))

      (a * c ≤ c) = (((0::'a) < c --> a ≤ (1::'a)) ∧ (c < (0::'a) --> (1::'a) ≤ a))

      (c ≤ c * b) = (((0::'a) < c --> (1::'a) ≤ b) ∧ (c < (0::'a) --> b ≤ (1::'a)))

      (c * a ≤ c) = (((0::'a) < c --> a ≤ (1::'a)) ∧ (c < (0::'a) --> (1::'a) ≤ a))

      (a * c < b * c) = (((0::'a) ≤ c --> a < b) ∧ (c ≤ (0::'a) --> b < a))

      (c * a < c * b) = (((0::'a) ≤ c --> a < b) ∧ (c ≤ (0::'a) --> b < a))

      (c < b * c) = (((0::'a) ≤ c --> (1::'a) < b) ∧ (c ≤ (0::'a) --> b < (1::'a)))

      (a * c < c) = (((0::'a) ≤ c --> a < (1::'a)) ∧ (c ≤ (0::'a) --> (1::'a) < a))

      (c < c * b) = (((0::'a) ≤ c --> (1::'a) < b) ∧ (c ≤ (0::'a) --> b < (1::'a)))

      (c * a < c) = (((0::'a) ≤ c --> a < (1::'a)) ∧ (c ≤ (0::'a) --> (1::'a) < a))

      (a * c = b * c) = (c = (0::'a) ∨ a = b)

      (c * a = c * b) = (c = (0::'a) ∨ a = b)

      (c = b * c) = (c = (0::'a) ∨ b = (1::'a))

      (a * c = c) = (c = (0::'a) ∨ a = (1::'a))

      (c = c * b) = (c = (0::'a) ∨ b = (1::'a))

      (c * a = c) = (c = (0::'a) ∨ a = (1::'a))

lemmas mult_compare_simps:

      (a * c ≤ b * c) = (((0::'a) < c --> a ≤ b) ∧ (c < (0::'a) --> b ≤ a))

      (c * a ≤ c * b) = (((0::'a) < c --> a ≤ b) ∧ (c < (0::'a) --> b ≤ a))

      (c ≤ b * c) = (((0::'a) < c --> (1::'a) ≤ b) ∧ (c < (0::'a) --> b ≤ (1::'a)))

      (a * c ≤ c) = (((0::'a) < c --> a ≤ (1::'a)) ∧ (c < (0::'a) --> (1::'a) ≤ a))

      (c ≤ c * b) = (((0::'a) < c --> (1::'a) ≤ b) ∧ (c < (0::'a) --> b ≤ (1::'a)))

      (c * a ≤ c) = (((0::'a) < c --> a ≤ (1::'a)) ∧ (c < (0::'a) --> (1::'a) ≤ a))

      (a * c < b * c) = (((0::'a) ≤ c --> a < b) ∧ (c ≤ (0::'a) --> b < a))

      (c * a < c * b) = (((0::'a) ≤ c --> a < b) ∧ (c ≤ (0::'a) --> b < a))

      (c < b * c) = (((0::'a) ≤ c --> (1::'a) < b) ∧ (c ≤ (0::'a) --> b < (1::'a)))

      (a * c < c) = (((0::'a) ≤ c --> a < (1::'a)) ∧ (c ≤ (0::'a) --> (1::'a) < a))

      (c < c * b) = (((0::'a) ≤ c --> (1::'a) < b) ∧ (c ≤ (0::'a) --> b < (1::'a)))

      (c * a < c) = (((0::'a) ≤ c --> a < (1::'a)) ∧ (c ≤ (0::'a) --> (1::'a) < a))

      (a * c = b * c) = (c = (0::'a) ∨ a = b)

      (c * a = c * b) = (c = (0::'a) ∨ a = b)

      (c = b * c) = (c = (0::'a) ∨ b = (1::'a))

      (a * c = c) = (c = (0::'a) ∨ a = (1::'a))

      (c = c * b) = (c = (0::'a) ∨ b = (1::'a))

      (c * a = c) = (c = (0::'a) ∨ a = (1::'a))

lemmas ring_eq_simps:

      (a + b) * c = a * c + b * c

      a * (b + c) = a * b + a * c

      (a - b) * c = a * c - b * c

      a * (b - c) = a * b - a * c

      a * b * c = a * (b * c)

      a * b = b * a

      a * (b * c) = b * (a * c)

      a + b + c = a + (b + c)

      a + b = b + a

      a + (b + c) = b + (a + c)

      a + (b - c) = a + b - c

      a - b + c = a + c - b

      a - b - c = a - (b + c)

      a - (b - c) = a + c - b

      (a - b = c) = (a = c + b)

      (a = c - b) = (a + b = c)

lemmas ring_eq_simps:

      (a + b) * c = a * c + b * c

      a * (b + c) = a * b + a * c

      (a - b) * c = a * c - b * c

      a * (b - c) = a * b - a * c

      a * b * c = a * (b * c)

      a * b = b * a

      a * (b * c) = b * (a * c)

      a + b + c = a + (b + c)

      a + b = b + a

      a + (b + c) = b + (a + c)

      a + (b - c) = a + b - c

      a - b + c = a + c - b

      a - b - c = a - (b + c)

      a - (b - c) = a + c - b

      (a - b = c) = (a = c + b)

      (a = c - b) = (a + b = c)

### Fields

lemma right_inverse:

      a ≠ (0::'a) ==> a * inverse a = (1::'a)

lemma right_inverse_eq:

      b ≠ (0::'a) ==> (a / b = (1::'a)) = (a = b)

lemma nonzero_inverse_eq_divide:

      a ≠ (0::'a) ==> inverse a = (1::'a) / a

lemma divide_self:

      a ≠ (0::'a) ==> a / a = (1::'a)

lemma divide_zero:

      a / (0::'a) = (0::'a)

lemma divide_self_if:

      a / a = (if a = (0::'a) then 0::'a else 1::'a)

lemma divide_zero_left:

      (0::'a) / a = (0::'a)

lemma inverse_eq_divide:

      inverse a = (1::'a) / a

lemma add_divide_distrib:

      (a + b) / c = a / c + b / c

lemma field_mult_eq_0_iff:

      (a * b = (0::'a)) = (a = (0::'a) ∨ b = (0::'a))

lemma field_mult_cancel_right_lemma:

      [| c ≠ (0::'a); a * c = b * c |] ==> a = b

lemma field_mult_cancel_right:

      (a * c = b * c) = (c = (0::'a) ∨ a = b)

lemma field_mult_cancel_left:

      (c * a = c * b) = (c = (0::'a) ∨ a = b)

lemma nonzero_imp_inverse_nonzero:

      a ≠ (0::'a) ==> inverse a ≠ (0::'a)

### Basic Properties of @{term inverse}

lemma inverse_zero_imp_zero:

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

lemma inverse_nonzero_imp_nonzero:

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

lemma inverse_nonzero_iff_nonzero:

      (inverse a = (0::'a)) = (a = (0::'a))

lemma nonzero_inverse_minus_eq:

      a ≠ (0::'a) ==> inverse (- a) = - inverse a

lemma inverse_minus_eq:

      inverse (- a) = - inverse a

lemma nonzero_inverse_eq_imp_eq:

      [| inverse a = inverse b; a ≠ (0::'a); b ≠ (0::'a) |] ==> a = b

lemma inverse_eq_imp_eq:

      inverse a = inverse b ==> a = b

lemma inverse_eq_iff_eq:

      (inverse a = inverse b) = (a = b)

lemma nonzero_inverse_inverse_eq:

      a ≠ (0::'a) ==> inverse (inverse a) = a

lemma inverse_inverse_eq:

      inverse (inverse a) = a

lemma inverse_1:

      inverse (1::'a) = (1::'a)

lemma inverse_unique:

      a * b = (1::'a) ==> inverse a = b

lemma nonzero_inverse_mult_distrib:

      [| a ≠ (0::'a); b ≠ (0::'a) |] ==> inverse (a * b) = inverse b * inverse a

lemma inverse_mult_distrib:

      inverse (a * b) = inverse a * inverse b

lemma inverse_add:

      [| a ≠ (0::'a); b ≠ (0::'a) |]
      ==> inverse a + inverse b = (a + b) * inverse a * inverse b

lemma inverse_divide:

      inverse (a / b) = b / a

### Calculations with fractions

lemma nonzero_mult_divide_cancel_left:

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

lemma mult_divide_cancel_left:

      c ≠ (0::'a) ==> c * a / (c * b) = a / b

lemma nonzero_mult_divide_cancel_right:

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

lemma mult_divide_cancel_right:

      c ≠ (0::'a) ==> a * c / (b * c) = a / b

lemma mult_divide_cancel_eq_if:

      c * a / (c * b) = (if c = (0::'a) then 0::'a else a / b)

lemma divide_1:

      a / (1::'a) = a

lemma times_divide_eq_right:

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

lemma times_divide_eq_left:

      b / c * a = b * a / c

lemma divide_divide_eq_right:

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

lemma divide_divide_eq_left:

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

lemma add_frac_eq:

      [| y ≠ (0::'a); z ≠ (0::'a) |] ==> x / y + w / z = (x * z + w * y) / (y * z)

#### Special Cancellation Simprules for Division

lemma mult_divide_cancel_left_if:

      c * a / (c * b) = (if c = (0::'a) then 0::'a else a / b)

lemma mult_divide_cancel_right_if:

      a * c / (b * c) = (if c = (0::'a) then 0::'a else a / b)

lemma mult_divide_cancel_left_if1:

      c / (c * b) = (if c = (0::'a) then 0::'a else (1::'a) / b)

lemma mult_divide_cancel_left_if2:

      c * a / c = (if c = (0::'a) then 0::'a else a)

lemma mult_divide_cancel_right_if1:

      c / (b * c) = (if c = (0::'a) then 0::'a else (1::'a) / b)

lemma mult_divide_cancel_right_if2:

      a * c / c = (if c = (0::'a) then 0::'a else a)

lemma times_divide_self_right:

      a * (b / a) = (if a = (0::'a) then 0::'a else b)

lemma times_divide_self_left:

      b / a * a = (if a = (0::'a) then 0::'a else b)

### Division and Unary Minus

lemma nonzero_minus_divide_left:

      b ≠ (0::'a) ==> - (a / b) = - a / b

lemma nonzero_minus_divide_right:

      b ≠ (0::'a) ==> - (a / b) = a / - b

lemma nonzero_minus_divide_divide:

      b ≠ (0::'a) ==> - a / - b = a / b

lemma minus_divide_left:

      - (a / b) = - a / b

lemma minus_divide_right:

      - (a / b) = a / - b

lemmas divide_minus_left:

      - a1 / b1 = - (a1 / b1)

lemmas divide_minus_left:

      - a1 / b1 = - (a1 / b1)

lemmas divide_minus_right:

      a1 / - b1 = - (a1 / b1)

lemmas divide_minus_right:

      a1 / - b1 = - (a1 / b1)

lemmas mult_minus_left:

      - a1 * b1 = - (a1 * b1)

lemmas mult_minus_left:

      - a1 * b1 = - (a1 * b1)

lemmas mult_minus_right:

      a1 * - b1 = - (a1 * b1)

lemmas mult_minus_right:

      a1 * - b1 = - (a1 * b1)

lemma minus_divide_divide:

      - a / - b = a / b

lemma diff_divide_distrib:

      (a - b) / c = a / c - b / c

lemma diff_frac_eq:

      [| y ≠ (0::'a); z ≠ (0::'a) |] ==> x / y - w / z = (x * z - w * y) / (y * z)

### Ordered Fields

lemma positive_imp_inverse_positive:

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

lemma negative_imp_inverse_negative:

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

lemma inverse_le_imp_le:

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

lemma inverse_positive_imp_positive:

      [| (0::'a) < inverse a; a ≠ (0::'a) |] ==> (0::'a) < a

lemma inverse_positive_iff_positive:

      ((0::'a) < inverse a) = ((0::'a) < a)

lemma inverse_negative_imp_negative:

      [| inverse a < (0::'a); a ≠ (0::'a) |] ==> a < (0::'a)

lemma inverse_negative_iff_negative:

      (inverse a < (0::'a)) = (a < (0::'a))

lemma inverse_nonnegative_iff_nonnegative:

      ((0::'a) ≤ inverse a) = ((0::'a) ≤ a)

lemma inverse_nonpositive_iff_nonpositive:

      (inverse a ≤ (0::'a)) = (a ≤ (0::'a))

### Anti-Monotonicity of @{term inverse}

lemma less_imp_inverse_less:

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

lemma inverse_less_imp_less:

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

lemma inverse_less_iff_less:

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

lemma le_imp_inverse_le:

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

lemma inverse_le_iff_le:

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

lemma inverse_le_imp_le_neg:

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

lemma less_imp_inverse_less_neg:

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

lemma inverse_less_imp_less_neg:

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

lemma inverse_less_iff_less_neg:

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

lemma le_imp_inverse_le_neg:

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

lemma inverse_le_iff_le_neg:

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

### Inverses and the Number One

lemma one_less_inverse_iff:

      ((1::'a) < inverse x) = ((0::'a) < x ∧ x < (1::'a))

lemma inverse_eq_1_iff:

      (inverse x = (1::'a)) = (x = (1::'a))

lemma one_le_inverse_iff:

      ((1::'a) ≤ inverse x) = ((0::'a) < x ∧ x ≤ (1::'a))

lemma inverse_less_1_iff:

      (inverse x < (1::'a)) = (x ≤ (0::'a) ∨ (1::'a) < x)

lemma inverse_le_1_iff:

      (inverse x ≤ (1::'a)) = (x ≤ (0::'a) ∨ (1::'a) ≤ x)

### Simplification of Inequalities Involving Literal Divisors

lemma pos_le_divide_eq:

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

lemma neg_le_divide_eq:

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

lemma le_divide_eq:

      (a ≤ b / c) =
      (if (0::'a) < c then a * c ≤ b
       else if c < (0::'a) then b ≤ a * c else a ≤ (0::'a))

lemma pos_divide_le_eq:

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

lemma neg_divide_le_eq:

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

lemma divide_le_eq:

      (b / c ≤ a) =
      (if (0::'a) < c then b ≤ a * c
       else if c < (0::'a) then a * c ≤ b else (0::'a) ≤ a)

lemma pos_less_divide_eq:

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

lemma neg_less_divide_eq:

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

lemma less_divide_eq:

      (a < b / c) =
      (if (0::'a) < c then a * c < b
       else if c < (0::'a) then b < a * c else a < (0::'a))

lemma pos_divide_less_eq:

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

lemma neg_divide_less_eq:

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

lemma divide_less_eq:

      (b / c < a) =
      (if (0::'a) < c then b < a * c
       else if c < (0::'a) then a * c < b else (0::'a) < a)

lemma nonzero_eq_divide_eq:

      c ≠ (0::'a) ==> (a = b / c) = (a * c = b)

lemma eq_divide_eq:

      (a = b / c) = (if c ≠ (0::'a) then a * c = b else a = (0::'a))

lemma nonzero_divide_eq_eq:

      c ≠ (0::'a) ==> (b / c = a) = (b = a * c)

lemma divide_eq_eq:

      (b / c = a) = (if c ≠ (0::'a) then b = a * c else a = (0::'a))

lemma divide_eq_imp:

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

lemma eq_divide_imp:

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

lemma frac_eq_eq:

      [| y ≠ (0::'a); z ≠ (0::'a) |] ==> (x / y = w / z) = (x * z = w * y)

### Division and Signs

lemma zero_less_divide_iff:

      ((0::'a) < a / b) = ((0::'a) < a ∧ (0::'a) < b ∨ a < (0::'a) ∧ b < (0::'a))

lemma divide_less_0_iff:

      (a / b < (0::'a)) = ((0::'a) < a ∧ b < (0::'a) ∨ a < (0::'a) ∧ (0::'a) < b)

lemma zero_le_divide_iff:

      ((0::'a) ≤ a / b) = ((0::'a) ≤ a ∧ (0::'a) ≤ b ∨ a ≤ (0::'a) ∧ b ≤ (0::'a))

lemma divide_le_0_iff:

      (a / b ≤ (0::'a)) = ((0::'a) ≤ a ∧ b ≤ (0::'a) ∨ a ≤ (0::'a) ∧ (0::'a) ≤ b)

lemma divide_eq_0_iff:

      (a / b = (0::'a)) = (a = (0::'a) ∨ b = (0::'a))

lemma divide_pos_pos:

      [| (0::'a) < x; (0::'a) < y |] ==> (0::'a) < x / y

lemma divide_nonneg_pos:

      [| (0::'a) ≤ x; (0::'a) < y |] ==> (0::'a) ≤ x / y

lemma divide_neg_pos:

      [| x < (0::'a); (0::'a) < y |] ==> x / y < (0::'a)

lemma divide_nonpos_pos:

      [| x ≤ (0::'a); (0::'a) < y |] ==> x / y ≤ (0::'a)

lemma divide_pos_neg:

      [| (0::'a) < x; y < (0::'a) |] ==> x / y < (0::'a)

lemma divide_nonneg_neg:

      [| (0::'a) ≤ x; y < (0::'a) |] ==> x / y ≤ (0::'a)

lemma divide_neg_neg:

      [| x < (0::'a); y < (0::'a) |] ==> (0::'a) < x / y

lemma divide_nonpos_neg:

      [| x ≤ (0::'a); y < (0::'a) |] ==> (0::'a) ≤ x / y

### Cancellation Laws for Division

lemma divide_cancel_right:

      (a / c = b / c) = (c = (0::'a) ∨ a = b)

lemma divide_cancel_left:

      (c / a = c / b) = (c = (0::'a) ∨ a = b)

### Division and the Number One

lemma divide_eq_1_iff:

      (a / b = (1::'a)) = (b ≠ (0::'a) ∧ a = b)

lemma one_eq_divide_iff:

      ((1::'a) = a / b) = (b ≠ (0::'a) ∧ a = b)

lemma zero_eq_1_divide_iff:

      ((0::'a) = (1::'a) / a) = (a = (0::'a))

lemma one_divide_eq_0_iff:

      ((1::'a) / a = (0::'a)) = (a = (0::'a))

lemmas zero_less_divide_1_iff:

      ((0::'a) < (1::'a) / b) =
      ((0::'a) < (1::'a) ∧ (0::'a) < b ∨ (1::'a) < (0::'a) ∧ b < (0::'a))

lemmas zero_less_divide_1_iff:

      ((0::'a) < (1::'a) / b) =
      ((0::'a) < (1::'a) ∧ (0::'a) < b ∨ (1::'a) < (0::'a) ∧ b < (0::'a))

lemmas divide_less_0_1_iff:

      ((1::'a) / b < (0::'a)) =
      ((0::'a) < (1::'a) ∧ b < (0::'a) ∨ (1::'a) < (0::'a) ∧ (0::'a) < b)

lemmas divide_less_0_1_iff:

      ((1::'a) / b < (0::'a)) =
      ((0::'a) < (1::'a) ∧ b < (0::'a) ∨ (1::'a) < (0::'a) ∧ (0::'a) < b)

lemmas zero_le_divide_1_iff:

      ((0::'a) ≤ (1::'a) / b) =
      ((0::'a) ≤ (1::'a) ∧ (0::'a) ≤ b ∨ (1::'a) ≤ (0::'a) ∧ b ≤ (0::'a))

lemmas zero_le_divide_1_iff:

      ((0::'a) ≤ (1::'a) / b) =
      ((0::'a) ≤ (1::'a) ∧ (0::'a) ≤ b ∨ (1::'a) ≤ (0::'a) ∧ b ≤ (0::'a))

lemmas divide_le_0_1_iff:

      ((1::'a) / b ≤ (0::'a)) =
      ((0::'a) ≤ (1::'a) ∧ b ≤ (0::'a) ∨ (1::'a) ≤ (0::'a) ∧ (0::'a) ≤ b)

lemmas divide_le_0_1_iff:

      ((1::'a) / b ≤ (0::'a)) =
      ((0::'a) ≤ (1::'a) ∧ b ≤ (0::'a) ∨ (1::'a) ≤ (0::'a) ∧ (0::'a) ≤ b)

### Ordering Rules for Division

lemma divide_strict_right_mono:

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

lemma divide_right_mono:

      [| a ≤ b; (0::'a) ≤ c |] ==> a / c ≤ b / c

lemma divide_right_mono_neg:

      [| a ≤ b; c ≤ (0::'a) |] ==> b / c ≤ a / c

lemma divide_strict_right_mono_neg:

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

lemma divide_strict_left_mono:

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

lemma divide_left_mono:

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

lemma divide_left_mono_neg:

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

lemma divide_strict_left_mono_neg:

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

lemma le_divide_eq_1:

      ((1::'a) ≤ b / a) = ((0::'a) < a ∧ a ≤ b ∨ a < (0::'a) ∧ b ≤ a)

lemma divide_le_eq_1:

      (b / a ≤ (1::'a)) = ((0::'a) < a ∧ b ≤ a ∨ a < (0::'a) ∧ a ≤ b ∨ a = (0::'a))

lemma less_divide_eq_1:

      ((1::'a) < b / a) = ((0::'a) < a ∧ a < b ∨ a < (0::'a) ∧ b < a)

lemma divide_less_eq_1:

      (b / a < (1::'a)) = ((0::'a) < a ∧ b < a ∨ a < (0::'a) ∧ a < b ∨ a = (0::'a))

### Conditional Simplification Rules: No Case Splits

lemma le_divide_eq_1_pos:

      (0::'a) < a ==> ((1::'a) ≤ b / a) = (a ≤ b)

lemma le_divide_eq_1_neg:

      a < (0::'a) ==> ((1::'a) ≤ b / a) = (b ≤ a)

lemma divide_le_eq_1_pos:

      (0::'a) < a ==> (b / a ≤ (1::'a)) = (b ≤ a)

lemma divide_le_eq_1_neg:

      a < (0::'a) ==> (b / a ≤ (1::'a)) = (a ≤ b)

lemma less_divide_eq_1_pos:

      (0::'a) < a ==> ((1::'a) < b / a) = (a < b)

lemma less_divide_eq_1_neg:

      a < (0::'a) ==> ((1::'a) < b / a) = (b < a)

lemma divide_less_eq_1_pos:

      (0::'a) < a ==> (b / a < (1::'a)) = (b < a)

lemma eq_divide_eq_1:

      ((1::'a) = b / a) = (a ≠ (0::'a) ∧ a = b)

lemma divide_eq_eq_1:

      (b / a = (1::'a)) = (a ≠ (0::'a) ∧ a = b)

### Reasoning about inequalities with division

lemma mult_right_le_one_le:

      [| (0::'a) ≤ x; (0::'a) ≤ y; y ≤ (1::'a) |] ==> x * y ≤ x

lemma mult_left_le_one_le:

      [| (0::'a) ≤ x; (0::'a) ≤ y; y ≤ (1::'a) |] ==> y * x ≤ x

lemma mult_imp_div_pos_le:

      [| (0::'a) < y; x ≤ z * y |] ==> x / y ≤ z

lemma mult_imp_le_div_pos:

      [| (0::'a) < y; z * y ≤ x |] ==> z ≤ x / y

lemma mult_imp_div_pos_less:

      [| (0::'a) < y; x < z * y |] ==> x / y < z

lemma mult_imp_less_div_pos:

      [| (0::'a) < y; z * y < x |] ==> z < x / y

lemma frac_le:

      [| (0::'a) ≤ x; x ≤ y; (0::'a) < w; w ≤ z |] ==> x / z ≤ y / w

lemma frac_less:

      [| (0::'a) ≤ x; x < y; (0::'a) < w; w ≤ z |] ==> x / z < y / w

lemma frac_less2:

      [| (0::'a) < x; x ≤ y; (0::'a) < w; w < z |] ==> x / z < y / w

lemmas times_divide_eq:

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

      b / c * a = b * a / c

lemmas times_divide_eq:

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

      b / c * a = b * a / c

### Ordered Fields are Dense

lemma less_add_one:

      a < a + (1::'a)

lemma zero_less_two:

      (0::'a) < (1::'a) + (1::'a)

lemma less_half_sum:

      a < b ==> a < (a + b) / ((1::'a) + (1::'a))

lemma gt_half_sum:

      a < b ==> (a + b) / ((1::'a) + (1::'a)) < b

lemma dense:

      a < b ==> ∃r>a. r < b

### Absolute Value

lemma abs_one:

      ¦1::'a¦ = (1::'a)

lemma abs_le_mult:

      ¦a * b¦ ≤ ¦a¦ * ¦b¦

lemma abs_eq_mult:

      ((0::'a) ≤ a ∨ a ≤ (0::'a)) ∧ ((0::'a) ≤ b ∨ b ≤ (0::'a))
      ==> ¦a * b¦ = ¦a¦ * ¦b¦

lemma abs_mult:

      ¦a * b¦ = ¦a¦ * ¦b¦

lemma abs_mult_self:

      ¦a¦ * ¦a¦ = a * a

lemma nonzero_abs_inverse:

      a ≠ (0::'a) ==> ¦inverse a¦ = inverse ¦a¦

lemma abs_inverse:

      ¦inverse a¦ = inverse ¦a¦

lemma nonzero_abs_divide:

      b ≠ (0::'a) ==> ¦a / b¦ = ¦a¦ / ¦b¦

lemma abs_divide:

      ¦a / b¦ = ¦a¦ / ¦b¦

lemma abs_mult_less:

      [| ¦a¦ < c; ¦b¦ < d |] ==> ¦a¦ * ¦b¦ < c * d

lemma eq_minus_self_iff:

      (a = - a) = (a = (0::'a))

lemma less_minus_self_iff:

      (a < - a) = (a < (0::'a))

lemma abs_less_iff:

      (¦a¦ < b) = (a < b ∧ - a < b)

lemma abs_mult_pos:

      (0::'a) ≤ x ==> ¦y¦ * x = ¦y * x¦

lemma abs_div_pos:

      (0::'a) < y ==> ¦x¦ / y = ¦x / y¦

### Miscellaneous

lemma linprog_dual_estimate:

      [| A * x ≤ b; (0::'a) ≤ y; ¦A - A'¦ ≤ δA; b ≤ b'; ¦c - c'¦ ≤ δc; ¦x¦ ≤ r |]
      ==> c * x ≤ y * b' + (y * δA + ¦y * A' - c'¦ + δc) * r

lemma le_ge_imp_abs_diff_1:

      [| A1.0 ≤ A; A ≤ A2.0 |] ==> ¦A - A1.0¦ ≤ A2.0 - A1.0

lemma mult_le_prts:

      [| a1.0 ≤ a; a ≤ a2.0; b1.0 ≤ b; b ≤ b2.0 |]
      ==> a * b
          ≤ pprt a2.0 * pprt b2.0 + pprt a1.0 * nprt b2.0 + nprt a2.0 * pprt b1.0 +
            nprt a1.0 * nprt b1.0

lemma mult_le_dual_prts:

      [| A * x ≤ b; (0::'a) ≤ y; A1.0 ≤ A; A ≤ A2.0; c1.0 ≤ c; c ≤ c2.0; r1.0 ≤ x;
         x ≤ r2.0 |]
      ==> c * x
          ≤ y * b +
            (let s1 = c1.0 - y * A2.0; s2 = c2.0 - y * A1.0
             in pprt s2 * pprt r2.0 + pprt s1 * nprt r2.0 + nprt s2 * pprt r1.0 +
                nprt s1 * nprt r1.0)