src/HOL/Ring_and_Field.thy

new theorems

(*  Title:   HOL/Ring_and_Field.thy
    ID:      $Id$
    Author:  Gertrud Bauer and Markus Wenzel, TU Muenchen
             Lawrence C Paulson, University of Cambridge
    License: GPL (GNU GENERAL PUBLIC LICENSE)
*)

header {*
  \title{Ring and field structures}
  \author{Gertrud Bauer and Markus Wenzel}
*}

theory Ring_and_Field = Inductive:

text{*Lemmas and extension to semirings by L. C. Paulson*}

subsection {* Abstract algebraic structures *}

axclass semiring \<subseteq> zero, one, plus, times
  add_assoc: "(a + b) + c = a + (b + c)"
  add_commute: "a + b = b + a"
  add_0 [simp]: "0 + a = a"

  mult_assoc: "(a * b) * c = a * (b * c)"
  mult_commute: "a * b = b * a"
  mult_1 [simp]: "1 * a = a"

  left_distrib: "(a + b) * c = a * c + b * c"
  zero_neq_one [simp]: "0 \<noteq> 1"

axclass ring \<subseteq> semiring, minus
  left_minus [simp]: "- a + a = 0"
  diff_minus: "a - b = a + (-b)"

axclass ordered_semiring \<subseteq> semiring, linorder
  add_left_mono: "a \<le> b ==> c + a \<le> c + b"
  mult_strict_left_mono: "a < b ==> 0 < c ==> c * a < c * b"

axclass ordered_ring \<subseteq> ordered_semiring, ring
  abs_if: "\<bar>a\<bar> = (if a < 0 then -a else a)"

axclass field \<subseteq> ring, inverse
  left_inverse [simp]: "a \<noteq> 0 ==> inverse a * a = 1"
  divide_inverse:      "b \<noteq> 0 ==> a / b = a * inverse b"

axclass ordered_field \<subseteq> ordered_ring, field

axclass division_by_zero \<subseteq> zero, inverse
  inverse_zero [simp]: "inverse 0 = 0"
  divide_zero [simp]: "a / 0 = 0"


subsection {* Derived Rules for Addition *}

lemma add_0_right [simp]: "a + 0 = (a::'a::semiring)"
proof -
  have "a + 0 = 0 + a" by (simp only: add_commute)
  also have "... = a" by simp
  finally show ?thesis .
qed

lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::semiring))"
  by (rule mk_left_commute [of "op +", OF add_assoc add_commute])

theorems add_ac = add_assoc add_commute add_left_commute

lemma right_minus [simp]: "a + -(a::'a::ring) = 0"
proof -
  have "a + -a = -a + a" by (simp add: add_ac)
  also have "... = 0" by simp
  finally show ?thesis .
qed

lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ring))"
proof
  have "a = a - b + b" by (simp add: diff_minus add_ac)
  also assume "a - b = 0"
  finally show "a = b" by simp
next
  assume "a = b"
  thus "a - b = 0" by (simp add: diff_minus)
qed

lemma add_left_cancel [simp]:
     "(a + b = a + c) = (b = (c::'a::ring))"
proof
  assume eq: "a + b = a + c"
  hence "(-a + a) + b = (-a + a) + c"
    by (simp only: eq add_assoc)
  thus "b = c" by simp
next
  assume eq: "b = c"
  thus "a + b = a + c" by simp
qed

lemma add_right_cancel [simp]:
     "(b + a = c + a) = (b = (c::'a::ring))"
  by (simp add: add_commute)

lemma minus_minus [simp]: "- (- (a::'a::ring)) = a"
  proof (rule add_left_cancel [of "-a", THEN iffD1])
    show "(-a + -(-a) = -a + a)"
    by simp
  qed

lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ring)"
apply (rule right_minus_eq [THEN iffD1, symmetric])
apply (simp add: diff_minus add_commute) 
done

lemma minus_zero [simp]: "- 0 = (0::'a::ring)"
by (simp add: equals_zero_I)

lemma diff_self [simp]: "a - (a::'a::ring) = 0"
  by (simp add: diff_minus)

lemma diff_0 [simp]: "(0::'a::ring) - a = -a"
by (simp add: diff_minus)

lemma diff_0_right [simp]: "a - (0::'a::ring) = a" 
by (simp add: diff_minus)

lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::ring)"
by (simp add: diff_minus)

lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ring))" 
  proof 
    assume "- a = - b"
    hence "- (- a) = - (- b)"
      by simp
    thus "a=b" by simp
  next
    assume "a=b"
    thus "-a = -b" by simp
  qed

lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ring))"
by (subst neg_equal_iff_equal [symmetric], simp)

lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ring))"
by (subst neg_equal_iff_equal [symmetric], simp)

text{*The next two equations can make the simplifier loop!*}

lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ring))"
  proof -
  have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal)
  thus ?thesis by (simp add: eq_commute)
  qed

lemma minus_equation_iff: "(- a = b) = (- (b::'a::ring) = a)"
  proof -
  have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal)
  thus ?thesis by (simp add: eq_commute)
  qed


subsection {* Derived rules for multiplication *}

lemma mult_1_right [simp]: "a * (1::'a::semiring) = a"
proof -
  have "a * 1 = 1 * a" by (simp add: mult_commute)
  also have "... = a" by simp
  finally show ?thesis .
qed

lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::semiring))"
  by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute])

theorems mult_ac = mult_assoc mult_commute mult_left_commute

lemma mult_left_zero [simp]: "0 * a = (0::'a::ring)"
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_right_zero [simp]: "a * 0 = (0::'a::ring)"
  by (simp add: mult_commute)


subsection {* Distribution rules *}

lemma right_distrib: "a * (b + c) = a * b + a * (c::'a::semiring)"
proof -
  have "a * (b + c) = (b + c) * a" by (simp add: mult_ac)
  also have "... = b * a + c * a" by (simp only: left_distrib)
  also have "... = a * b + a * c" by (simp add: mult_ac)
  finally show ?thesis .
qed

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_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ring)"
apply (rule equals_zero_I)
apply (simp add: add_ac) 
done

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 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: mult_commute [of _ c] right_diff_distrib) 

lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ring)"
by (simp add: diff_minus add_commute) 


subsection {* Ordering Rules for Addition *}

lemma add_right_mono: "a \<le> (b::'a::ordered_semiring) ==> a + c \<le> b + c"
by (simp add: add_commute [of _ c] add_left_mono)

text {* non-strict, in both arguments *}
lemma add_mono: "[|a \<le> b;  c \<le> d|] ==> a + c \<le> b + (d::'a::ordered_semiring)"
  apply (erule add_right_mono [THEN order_trans])
  apply (simp add: add_commute add_left_mono)
  done

lemma add_strict_left_mono:
     "a < b ==> c + a < c + (b::'a::ordered_ring)"
 by (simp add: order_less_le add_left_mono) 

lemma add_strict_right_mono:
     "a < b ==> a + c < b + (c::'a::ordered_ring)"
 by (simp add: add_commute [of _ c] add_strict_left_mono)

text{*Strict monotonicity in both arguments*}
lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::ordered_ring)"
apply (erule add_strict_right_mono [THEN order_less_trans])
apply (erule add_strict_left_mono)
done

lemma add_less_imp_less_left:
      assumes less: "c + a < c + b"  shows "a < (b::'a::ordered_ring)"
  proof -
  have "-c + (c + a) < -c + (c + b)"
    by (rule add_strict_left_mono [OF less])
  thus "a < b" by (simp add: add_assoc [symmetric])
  qed

lemma add_less_imp_less_right:
      "a + c < b + c ==> a < (b::'a::ordered_ring)"
apply (rule add_less_imp_less_left [of c])
apply (simp add: add_commute)  
done

lemma add_less_cancel_left [simp]:
    "(c+a < c+b) = (a < (b::'a::ordered_ring))"
by (blast intro: add_less_imp_less_left add_strict_left_mono) 

lemma add_less_cancel_right [simp]:
    "(a+c < b+c) = (a < (b::'a::ordered_ring))"
by (blast intro: add_less_imp_less_right add_strict_right_mono)

lemma add_le_cancel_left [simp]:
    "(c+a \<le> c+b) = (a \<le> (b::'a::ordered_ring))"
by (simp add: linorder_not_less [symmetric]) 

lemma add_le_cancel_right [simp]:
    "(a+c \<le> b+c) = (a \<le> (b::'a::ordered_ring))"
by (simp add: linorder_not_less [symmetric]) 

lemma add_le_imp_le_left:
      "c + a \<le> c + b ==> a \<le> (b::'a::ordered_ring)"
by simp

lemma add_le_imp_le_right:
      "a + c \<le> b + c ==> a \<le> (b::'a::ordered_ring)"
by simp


subsection {* Ordering Rules for Unary Minus *}

lemma le_imp_neg_le:
      assumes "a \<le> (b::'a::ordered_ring)" shows "-b \<le> -a"
  proof -
  have "-a+a \<le> -a+b"
    by (rule add_left_mono) 
  hence "0 \<le> -a+b"
    by simp
  hence "0 + (-b) \<le> (-a + b) + (-b)"
    by (rule add_right_mono) 
  thus ?thesis
    by (simp add: add_assoc)
  qed

lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::ordered_ring))"
  proof 
    assume "- b \<le> - a"
    hence "- (- a) \<le> - (- b)"
      by (rule le_imp_neg_le)
    thus "a\<le>b" by simp
  next
    assume "a\<le>b"
    thus "-b \<le> -a" by (rule le_imp_neg_le)
  qed

lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::ordered_ring))"
by (subst neg_le_iff_le [symmetric], simp)

lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::ordered_ring))"
by (subst neg_le_iff_le [symmetric], simp)

lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::ordered_ring))"
by (force simp add: order_less_le) 

lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::ordered_ring))"
by (subst neg_less_iff_less [symmetric], simp)

lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::ordered_ring))"
by (subst neg_less_iff_less [symmetric], simp)

text{*The next several equations can make the simplifier loop!*}

lemma less_minus_iff: "(a < - b) = (b < - (a::'a::ordered_ring))"
  proof -
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
  thus ?thesis by simp
  qed

lemma minus_less_iff: "(- a < b) = (- b < (a::'a::ordered_ring))"
  proof -
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
  thus ?thesis by simp
  qed

lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::ordered_ring))"
apply (simp add: linorder_not_less [symmetric])
apply (rule minus_less_iff) 
done

lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::ordered_ring))"
apply (simp add: linorder_not_less [symmetric])
apply (rule less_minus_iff) 
done


subsection{*Subtraction Laws*}

lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ring)"
by (simp add: diff_minus add_ac)

lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ring)"
by (simp add: diff_minus add_ac)

lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ring))"
by (auto simp add: diff_minus add_assoc)

lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ring) = c)"
by (auto simp add: diff_minus add_assoc)

lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ring))"
by (simp add: diff_minus add_ac)

lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ring)"
by (simp add: diff_minus add_ac)

text{*Further subtraction laws for ordered rings*}

lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::ordered_ring))"
proof -
  have  "(a < b) = (a + (- b) < b + (-b))"  
    by (simp only: add_less_cancel_right)
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
  finally show ?thesis .
qed

lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::ordered_ring))"
apply (subst less_iff_diff_less_0)
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
apply (simp add: diff_minus add_ac)
done

lemma less_diff_eq: "(a < c-b) = (a + (b::'a::ordered_ring) < c)"
apply (subst less_iff_diff_less_0)
apply (rule less_iff_diff_less_0 [of _ "c-b", THEN ssubst])
apply (simp add: diff_minus add_ac)
done

lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::ordered_ring))"
by (simp add: linorder_not_less [symmetric] less_diff_eq)

lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::ordered_ring) \<le> c)"
by (simp add: linorder_not_less [symmetric] diff_less_eq)

text{*This list of rewrites simplifies (in)equalities by bringing subtractions
  to the top and then moving negative terms to the other side.
  Use with @{text add_ac}*}
lemmas compare_rls =
       diff_minus [symmetric]
       add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
       diff_less_eq less_diff_eq diff_le_eq le_diff_eq
       diff_eq_eq eq_diff_eq


subsection{*Lemmas for the @{text cancel_numerals} simproc*}

lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ring))"
by (simp add: compare_rls)

lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::ordered_ring))"
by (simp add: compare_rls)

lemma eq_add_iff1:
     "(a*e + c = b*e + d) = ((a-b)*e + c = (d::'a::ring))"
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::ordered_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::ordered_ring))"
apply (simp add: diff_minus left_distrib add_ac)
apply (simp add: compare_rls minus_mult_left [symmetric]) 
done

lemma le_add_iff1:
     "(a*e + c \<le> b*e + d) = ((a-b)*e + c \<le> (d::'a::ordered_ring))"
apply (simp add: diff_minus left_distrib add_ac)
apply (simp add: compare_rls minus_mult_left [symmetric]) 
done

lemma le_add_iff2:
     "(a*e + c \<le> b*e + d) = (c \<le> (b-a)*e + (d::'a::ordered_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_strict_right_mono:
     "[|a < b; 0 < c|] ==> a * c < b * (c::'a::ordered_semiring)"
by (simp add: mult_commute [of _ c] mult_strict_left_mono)

lemma mult_left_mono:
     "[|a \<le> b; 0 \<le> c|] ==> c * a \<le> c * (b::'a::ordered_ring)"
  apply (case_tac "c=0", simp)
  apply (force simp add: mult_strict_left_mono order_le_less) 
  done

lemma mult_right_mono:
     "[|a \<le> b; 0 \<le> c|] ==> a*c \<le> b * (c::'a::ordered_ring)"
  by (simp add: mult_left_mono mult_commute [of _ c]) 

lemma mult_strict_left_mono_neg:
     "[|b < a; c < 0|] ==> c * a < c * (b::'a::ordered_ring)"
apply (drule mult_strict_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)"
apply (drule mult_strict_right_mono [of _ _ "-c"])
apply (simp_all add: minus_mult_right [symmetric]) 
done


subsection{* Products of Signs *}

lemma mult_pos: "[| (0::'a::ordered_ring) < a; 0 < b |] ==> 0 < a*b"
by (drule mult_strict_left_mono [of 0 b], auto)

lemma mult_pos_neg: "[| (0::'a::ordered_ring) < a; b < 0 |] ==> a*b < 0"
by (drule mult_strict_left_mono [of b 0], auto)

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

lemma zero_less_mult_pos: "[| 0 < a*b; 0 < a|] ==> 0 < (b::'a::ordered_ring)"
apply (case_tac "b\<le>0") 
 apply (auto simp add: order_le_less linorder_not_less)
apply (drule_tac mult_pos_neg [of a b]) 
 apply (auto dest: order_less_not_sym)
done

lemma zero_less_mult_iff:
     "((0::'a::ordered_ring) < a*b) = (0 < a & 0 < b | a < 0 & b < 0)"
apply (auto simp add: order_le_less linorder_not_less mult_pos mult_neg)
apply (blast dest: zero_less_mult_pos) 
apply (simp add: mult_commute [of a b]) 
apply (blast dest: zero_less_mult_pos) 
done

text{*A field has no "zero divisors", so this theorem should hold 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)) = (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) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
                   zero_less_mult_iff)

lemma mult_less_0_iff:
     "(a*b < (0::'a::ordered_ring)) = (0 < a & b < 0 | a < 0 & 0 < b)"
apply (insert zero_less_mult_iff [of "-a" b]) 
apply (force simp add: minus_mult_left[symmetric]) 
done

lemma mult_le_0_iff:
     "(a*b \<le> (0::'a::ordered_ring)) = (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
apply (insert zero_le_mult_iff [of "-a" b]) 
apply (force simp add: minus_mult_left[symmetric]) 
done

lemma zero_le_square: "(0::'a::ordered_ring) \<le> a*a"
by (simp add: zero_le_mult_iff linorder_linear) 

lemma zero_less_one: "(0::'a::ordered_ring) < 1"
apply (insert zero_le_square [of 1]) 
apply (simp add: order_less_le) 
done

lemma zero_le_one: "(0::'a::ordered_ring) \<le> 1"
  by (rule zero_less_one [THEN order_less_imp_le]) 


subsection{*More Monotonicity*}

lemma mult_left_mono_neg:
     "[|b \<le> a; c \<le> 0|] ==> c * a \<le> c * (b::'a::ordered_ring)"
apply (drule mult_left_mono [of _ _ "-c"]) 
apply (simp_all add: minus_mult_left [symmetric]) 
done

lemma mult_right_mono_neg:
     "[|b \<le> a; c \<le> 0|] ==> a * c \<le> b * (c::'a::ordered_ring)"
  by (simp add: mult_left_mono_neg mult_commute [of _ c]) 

text{*Strict monotonicity in both arguments*}
lemma mult_strict_mono:
     "[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_ring)"
apply (case_tac "c=0")
 apply (simp add: mult_pos) 
apply (erule mult_strict_right_mono [THEN order_less_trans])
 apply (force simp add: order_le_less) 
apply (erule mult_strict_left_mono, assumption)
done

text{*This weaker variant has more natural premises*}
lemma mult_strict_mono':
     "[| a<b; c<d; 0 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_ring)"
apply (rule mult_strict_mono)
apply (blast intro: order_le_less_trans)+
done

lemma mult_mono:
     "[|a \<le> b; c \<le> d; 0 \<le> b; 0 \<le> c|] 
      ==> a * c  \<le>  b * (d::'a::ordered_ring)"
apply (erule mult_right_mono [THEN order_trans], assumption)
apply (erule mult_left_mono, assumption)
done


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

text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
   also with the relations @{text "\<le>"} and equality.*}

lemma mult_less_cancel_right:
    "(a*c < b*c) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring)))"
apply (case_tac "c = 0")
apply (auto simp add: linorder_neq_iff mult_strict_right_mono 
                      mult_strict_right_mono_neg)
apply (auto simp add: linorder_not_less 
                      linorder_not_le [symmetric, of "a*c"]
                      linorder_not_le [symmetric, of a])
apply (erule_tac [!] notE)
apply (auto simp add: order_less_imp_le mult_right_mono 
                      mult_right_mono_neg)
done

lemma mult_less_cancel_left:
    "(c*a < c*b) = ((0 < c & a < b) | (c < 0 & b < (a::'a::ordered_ring)))"
by (simp add: mult_commute [of c] mult_less_cancel_right)

lemma mult_le_cancel_right:
     "(a*c \<le> b*c) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring)))"
by (simp add: linorder_not_less [symmetric] mult_less_cancel_right)

lemma mult_le_cancel_left:
     "(c*a \<le> c*b) = ((0<c --> a\<le>b) & (c<0 --> b \<le> (a::'a::ordered_ring)))"
by (simp add: mult_commute [of c] mult_le_cancel_right)

lemma mult_less_imp_less_left:
    "[|c*a < c*b; 0 < c|] ==> a < (b::'a::ordered_ring)"
  by (force elim: order_less_asym simp add: mult_less_cancel_left)

lemma mult_less_imp_less_right:
    "[|a*c < b*c; 0 < c|] ==> a < (b::'a::ordered_ring)"
  by (force elim: order_less_asym simp add: mult_less_cancel_right)

text{*Cancellation of equalities with a common factor*}
lemma mult_cancel_right [simp]:
     "(a*c = b*c) = (c = (0::'a::ordered_ring) | 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) | a=b)"
by (simp add: mult_commute [of c] mult_cancel_right)


subsection {* Fields *}

lemma right_inverse [simp]:
      assumes not0: "a \<noteq> 0" shows "a * inverse (a::'a::field) = 1"
proof -
  have "a * inverse a = inverse a * a" by (simp add: mult_ac)
  also have "... = 1" using not0 by simp
  finally show ?thesis .
qed

lemma right_inverse_eq: "b \<noteq> 0 ==> (a / b = 1) = (a = (b::'a::field))"
proof
  assume neq: "b \<noteq> 0"
  {
    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
    also assume "a / b = 1"
    finally show "a = b" by simp
  next
    assume "a = b"
    with neq show "a / b = 1" by (simp add: divide_inverse)
  }
qed

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

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

lemma divide_inverse_zero: "a/b = a * inverse(b::'a::{field,division_by_zero})"
apply (case_tac "b = 0")
apply (simp_all add: divide_inverse)
done

lemma divide_zero_left [simp]: "0/a = (0::'a::{field,division_by_zero})"
by (simp add: divide_inverse_zero)

lemma inverse_eq_divide: "inverse (a::'a::{field,division_by_zero}) = 1/a"
by (simp add: divide_inverse_zero)

lemma nonzero_add_divide_distrib: "c \<noteq> 0 ==> (a+b)/(c::'a::field) = a/c + b/c"
by (simp add: divide_inverse left_distrib) 

lemma add_divide_distrib: "(a+b)/(c::'a::{field,division_by_zero}) = a/c + b/c"
apply (case_tac "c=0", simp) 
apply (simp add: nonzero_add_divide_distrib) 
done

text{*Compared with @{text mult_eq_0_iff}, this version removes the requirement
      of an ordering.*}
lemma field_mult_eq_0_iff: "(a*b = (0::'a::field)) = (a = 0 | b = 0)"
  proof cases
    assume "a=0" thus ?thesis by simp
  next
    assume anz [simp]: "a\<noteq>0"
    thus ?thesis
    proof auto
      assume "a * b = 0"
      hence "inverse a * (a * b) = 0" by simp
      thus "b = 0"  by (simp (no_asm_use) add: mult_assoc [symmetric])
    qed
  qed

text{*Cancellation of equalities with a common factor*}
lemma field_mult_cancel_right_lemma:
      assumes cnz: "c \<noteq> (0::'a::field)"
	  and eq:  "a*c = b*c"
	 shows "a=b"
  proof -
  have "(a * c) * inverse c = (b * c) * inverse c"
    by (simp add: eq)
  thus "a=b"
    by (simp add: mult_assoc cnz)
  qed

lemma field_mult_cancel_right:
     "(a*c = b*c) = (c = (0::'a::field) | a=b)"
  proof cases
    assume "c=0" thus ?thesis by simp
  next
    assume "c\<noteq>0" 
    thus ?thesis by (force dest: field_mult_cancel_right_lemma)
  qed

lemma field_mult_cancel_left:
     "(c*a = c*b) = (c = (0::'a::field) | a=b)"
  by (simp add: mult_commute [of c] field_mult_cancel_right) 

lemma nonzero_imp_inverse_nonzero: "a \<noteq> 0 ==> inverse a \<noteq> (0::'a::field)"
  proof
  assume ianz: "inverse a = 0"
  assume "a \<noteq> 0"
  hence "1 = a * inverse a" by simp
  also have "... = 0" by (simp add: ianz)
  finally have "1 = (0::'a::field)" .
  thus False by (simp add: eq_commute)
  qed


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

lemma inverse_zero_imp_zero: "inverse a = 0 ==> a = (0::'a::field)"
apply (rule ccontr) 
apply (blast dest: nonzero_imp_inverse_nonzero) 
done

lemma inverse_nonzero_imp_nonzero:
   "inverse a = 0 ==> a = (0::'a::field)"
apply (rule ccontr) 
apply (blast dest: nonzero_imp_inverse_nonzero) 
done

lemma inverse_nonzero_iff_nonzero [simp]:
   "(inverse a = 0) = (a = (0::'a::{field,division_by_zero}))"
by (force dest: inverse_nonzero_imp_nonzero) 

lemma nonzero_inverse_minus_eq:
      assumes [simp]: "a\<noteq>0"  shows "inverse(-a) = -inverse(a::'a::field)"
  proof -
    have "-a * inverse (- a) = -a * - inverse a"
      by simp
    thus ?thesis 
      by (simp only: field_mult_cancel_left, simp)
  qed

lemma inverse_minus_eq [simp]:
     "inverse(-a) = -inverse(a::'a::{field,division_by_zero})";
  proof cases
    assume "a=0" thus ?thesis by (simp add: inverse_zero)
  next
    assume "a\<noteq>0" 
    thus ?thesis by (simp add: nonzero_inverse_minus_eq)
  qed

lemma nonzero_inverse_eq_imp_eq:
      assumes inveq: "inverse a = inverse b"
	  and anz:  "a \<noteq> 0"
	  and bnz:  "b \<noteq> 0"
	 shows "a = (b::'a::field)"
  proof -
  have "a * inverse b = a * inverse a"
    by (simp add: inveq)
  hence "(a * inverse b) * b = (a * inverse a) * b"
    by simp
  thus "a = b"
    by (simp add: mult_assoc anz bnz)
  qed

lemma inverse_eq_imp_eq:
     "inverse a = inverse b ==> a = (b::'a::{field,division_by_zero})"
apply (case_tac "a=0 | b=0") 
 apply (force dest!: inverse_zero_imp_zero
              simp add: eq_commute [of "0::'a"])
apply (force dest!: nonzero_inverse_eq_imp_eq) 
done

lemma inverse_eq_iff_eq [simp]:
     "(inverse a = inverse b) = (a = (b::'a::{field,division_by_zero}))"
by (force dest!: inverse_eq_imp_eq) 

lemma nonzero_inverse_inverse_eq:
      assumes [simp]: "a \<noteq> 0"  shows "inverse(inverse (a::'a::field)) = a"
  proof -
  have "(inverse (inverse a) * inverse a) * a = a" 
    by (simp add: nonzero_imp_inverse_nonzero)
  thus ?thesis
    by (simp add: mult_assoc)
  qed

lemma inverse_inverse_eq [simp]:
     "inverse(inverse (a::'a::{field,division_by_zero})) = a"
  proof cases
    assume "a=0" thus ?thesis by simp
  next
    assume "a\<noteq>0" 
    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
  qed

lemma inverse_1 [simp]: "inverse 1 = (1::'a::field)"
  proof -
  have "inverse 1 * 1 = (1::'a::field)" 
    by (rule left_inverse [OF zero_neq_one [symmetric]])
  thus ?thesis  by simp
  qed

lemma nonzero_inverse_mult_distrib: 
      assumes anz: "a \<noteq> 0"
          and bnz: "b \<noteq> 0"
      shows "inverse(a*b) = inverse(b) * inverse(a::'a::field)"
  proof -
  have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)" 
    by (simp add: field_mult_eq_0_iff anz bnz)
  hence "inverse(a*b) * a = inverse(b)" 
    by (simp add: mult_assoc bnz)
  hence "inverse(a*b) * a * inverse(a) = inverse(b) * inverse(a)" 
    by simp
  thus ?thesis
    by (simp add: mult_assoc anz)
  qed

text{*This version builds in division by zero while also re-orienting
      the right-hand side.*}
lemma inverse_mult_distrib [simp]:
     "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
  proof cases
    assume "a \<noteq> 0 & b \<noteq> 0" 
    thus ?thesis  by (simp add: nonzero_inverse_mult_distrib mult_commute)
  next
    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
    thus ?thesis  by force
  qed

text{*There is no slick version using division by zero.*}
lemma inverse_add:
     "[|a \<noteq> 0;  b \<noteq> 0|]
      ==> inverse a + inverse b = (a+b) * inverse a * inverse (b::'a::field)"
apply (simp add: left_distrib mult_assoc)
apply (simp add: mult_commute [of "inverse a"]) 
apply (simp add: mult_assoc [symmetric] add_commute)
done

lemma nonzero_mult_divide_cancel_left:
  assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" 
    shows "(c*a)/(c*b) = a/(b::'a::field)"
proof -
  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
    by (simp add: field_mult_eq_0_iff divide_inverse 
                  nonzero_inverse_mult_distrib)
  also have "... =  a * inverse b * (inverse c * c)"
    by (simp only: mult_ac)
  also have "... =  a * inverse b"
    by simp
    finally show ?thesis 
    by (simp add: divide_inverse)
qed

lemma mult_divide_cancel_left:
     "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
apply (case_tac "b = 0")
apply (simp_all add: nonzero_mult_divide_cancel_left)
done

lemma nonzero_mult_divide_cancel_right:
     "[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (b*c) = a/(b::'a::field)"
by (simp add: mult_commute [of _ c] nonzero_mult_divide_cancel_left) 

lemma mult_divide_cancel_right:
     "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
apply (case_tac "b = 0")
apply (simp_all add: nonzero_mult_divide_cancel_right)
done

(*For ExtractCommonTerm*)
lemma mult_divide_cancel_eq_if:
     "(c*a) / (c*b) = 
      (if c=0 then 0 else a / (b::'a::{field,division_by_zero}))"
  by (simp add: mult_divide_cancel_left)

lemma divide_1 [simp]: "a/1 = (a::'a::field)"
  by (simp add: divide_inverse [OF not_sym])

lemma times_divide_eq_right [simp]:
     "a * (b/c) = (a*b) / (c::'a::{field,division_by_zero})"
by (simp add: divide_inverse_zero mult_assoc)

lemma times_divide_eq_left [simp]:
     "(b/c) * a = (b*a) / (c::'a::{field,division_by_zero})"
by (simp add: divide_inverse_zero mult_ac)

lemma divide_divide_eq_right [simp]:
     "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
by (simp add: divide_inverse_zero mult_ac)

lemma divide_divide_eq_left [simp]:
     "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
by (simp add: divide_inverse_zero mult_assoc)


subsection {* Division and Unary Minus *}

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

lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a/b) = a / -(b::'a::field)"
by (simp add: divide_inverse nonzero_inverse_minus_eq minus_mult_right)

lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a)/(-b) = a / (b::'a::field)"
by (simp add: divide_inverse nonzero_inverse_minus_eq)

lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::{field,division_by_zero})"
apply (case_tac "b=0", simp) 
apply (simp add: nonzero_minus_divide_left) 
done

lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
apply (case_tac "b=0", simp) 
by (rule nonzero_minus_divide_right) 

text{*The effect is to extract signs from divisions*}
declare minus_divide_left  [symmetric, simp]
declare minus_divide_right [symmetric, simp]

lemma minus_divide_divide [simp]:
     "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
apply (case_tac "b=0", simp) 
apply (simp add: nonzero_minus_divide_divide) 
done


subsection {* Ordered Fields *}

lemma positive_imp_inverse_positive: 
      assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::ordered_field)"
  proof -
  have "0 < a * inverse a" 
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one)
  thus "0 < inverse a" 
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
  qed

lemma negative_imp_inverse_negative:
     "a < 0 ==> inverse a < (0::'a::ordered_field)"
  by (insert positive_imp_inverse_positive [of "-a"], 
      simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 

lemma inverse_le_imp_le:
      assumes invle: "inverse a \<le> inverse b"
	  and apos:  "0 < a"
	 shows "b \<le> (a::'a::ordered_field)"
  proof (rule classical)
  assume "~ b \<le> a"
  hence "a < b"
    by (simp add: linorder_not_le)
  hence bpos: "0 < b"
    by (blast intro: apos order_less_trans)
  hence "a * inverse a \<le> a * inverse b"
    by (simp add: apos invle order_less_imp_le mult_left_mono)
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
    by (simp add: bpos order_less_imp_le mult_right_mono)
  thus "b \<le> a"
    by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
  qed

lemma inverse_positive_imp_positive:
      assumes inv_gt_0: "0 < inverse a"
          and [simp]:   "a \<noteq> 0"
        shows "0 < (a::'a::ordered_field)"
  proof -
  have "0 < inverse (inverse a)"
    by (rule positive_imp_inverse_positive)
  thus "0 < a"
    by (simp add: nonzero_inverse_inverse_eq)
  qed

lemma inverse_positive_iff_positive [simp]:
      "(0 < inverse a) = (0 < (a::'a::{ordered_field,division_by_zero}))"
apply (case_tac "a = 0", simp)
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
done

lemma inverse_negative_imp_negative:
      assumes inv_less_0: "inverse a < 0"
          and [simp]:   "a \<noteq> 0"
        shows "a < (0::'a::ordered_field)"
  proof -
  have "inverse (inverse a) < 0"
    by (rule negative_imp_inverse_negative)
  thus "a < 0"
    by (simp add: nonzero_inverse_inverse_eq)
  qed

lemma inverse_negative_iff_negative [simp]:
      "(inverse a < 0) = (a < (0::'a::{ordered_field,division_by_zero}))"
apply (case_tac "a = 0", simp)
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
done

lemma inverse_nonnegative_iff_nonnegative [simp]:
      "(0 \<le> inverse a) = (0 \<le> (a::'a::{ordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric])

lemma inverse_nonpositive_iff_nonpositive [simp]:
      "(inverse a \<le> 0) = (a \<le> (0::'a::{ordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric])


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

lemma less_imp_inverse_less:
      assumes less: "a < b"
	  and apos:  "0 < a"
	shows "inverse b < inverse (a::'a::ordered_field)"
  proof (rule ccontr)
  assume "~ inverse b < inverse a"
  hence "inverse a \<le> inverse b"
    by (simp add: linorder_not_less)
  hence "~ (a < b)"
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
  thus False
    by (rule notE [OF _ less])
  qed

lemma inverse_less_imp_less:
   "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::ordered_field)"
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
done

text{*Both premises are essential. Consider -1 and 1.*}
lemma inverse_less_iff_less [simp]:
     "[|0 < a; 0 < b|] 
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 

lemma le_imp_inverse_le:
   "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
  by (force simp add: order_le_less less_imp_inverse_less)

lemma inverse_le_iff_le [simp]:
     "[|0 < a; 0 < b|] 
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 


text{*These results refer to both operands being negative.  The opposite-sign
case is trivial, since inverse preserves signs.*}
lemma inverse_le_imp_le_neg:
   "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::ordered_field)"
  apply (rule classical) 
  apply (subgoal_tac "a < 0") 
   prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
  apply (insert inverse_le_imp_le [of "-b" "-a"])
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  done

lemma less_imp_inverse_less_neg:
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::ordered_field)"
  apply (subgoal_tac "a < 0") 
   prefer 2 apply (blast intro: order_less_trans) 
  apply (insert less_imp_inverse_less [of "-b" "-a"])
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  done

lemma inverse_less_imp_less_neg:
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::ordered_field)"
  apply (rule classical) 
  apply (subgoal_tac "a < 0") 
   prefer 2
   apply (force simp add: linorder_not_less intro: order_le_less_trans) 
  apply (insert inverse_less_imp_less [of "-b" "-a"])
  apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  done

lemma inverse_less_iff_less_neg [simp]:
     "[|a < 0; b < 0|] 
      ==> (inverse a < inverse b) = (b < (a::'a::ordered_field))"
  apply (insert inverse_less_iff_less [of "-b" "-a"])
  apply (simp del: inverse_less_iff_less 
	      add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
  done

lemma le_imp_inverse_le_neg:
   "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::ordered_field)"
  by (force simp add: order_le_less less_imp_inverse_less_neg)

lemma inverse_le_iff_le_neg [simp]:
     "[|a < 0; b < 0|] 
      ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::ordered_field))"
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 


subsection{*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 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_zero mult_less_0_iff)

lemma zero_le_divide_iff:
     "((0::'a::{ordered_field,division_by_zero}) \<le> a/b) =
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
by (simp add: divide_inverse_zero zero_le_mult_iff)

lemma divide_le_0_iff:
     "(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
by (simp add: divide_inverse_zero 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_zero field_mult_eq_0_iff)


subsection{*Simplification of Inequalities Involving Literal Divisors*}

lemma pos_le_divide_eq: "0 < (c::'a::ordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
proof -
  assume less: "0<c"
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  also have "... = (a*c \<le> b)"
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  finally show ?thesis .
qed


lemma neg_le_divide_eq: "c < (0::'a::ordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
proof -
  assume less: "c<0"
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  also have "... = (b \<le> a*c)"
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  finally show ?thesis .
qed

lemma le_divide_eq:
  "(a \<le> b/c) = 
   (if 0 < c then a*c \<le> b
             else if c < 0 then b \<le> a*c
             else  a \<le> (0::'a::{ordered_field,division_by_zero}))"
apply (case_tac "c=0", simp) 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
done

lemma pos_divide_le_eq: "0 < (c::'a::ordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
proof -
  assume less: "0<c"
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  also have "... = (b \<le> a*c)"
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  finally show ?thesis .
qed

lemma neg_divide_le_eq: "c < (0::'a::ordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
proof -
  assume less: "c<0"
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
  also have "... = (a*c \<le> b)"
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  finally show ?thesis .
qed

lemma divide_le_eq:
  "(b/c \<le> a) = 
   (if 0 < c then b \<le> a*c
             else if c < 0 then a*c \<le> b
             else 0 \<le> (a::'a::{ordered_field,division_by_zero}))"
apply (case_tac "c=0", simp) 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
done


lemma pos_less_divide_eq:
     "0 < (c::'a::ordered_field) ==> (a < b/c) = (a*c < b)"
proof -
  assume less: "0<c"
  hence "(a < b/c) = (a*c < (b/c)*c)"
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
  also have "... = (a*c < b)"
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  finally show ?thesis .
qed

lemma neg_less_divide_eq:
 "c < (0::'a::ordered_field) ==> (a < b/c) = (b < a*c)"
proof -
  assume less: "c<0"
  hence "(a < b/c) = ((b/c)*c < a*c)"
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
  also have "... = (b < a*c)"
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  finally show ?thesis .
qed

lemma less_divide_eq:
  "(a < b/c) = 
   (if 0 < c then a*c < b
             else if c < 0 then b < a*c
             else  a < (0::'a::{ordered_field,division_by_zero}))"
apply (case_tac "c=0", simp) 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
done

lemma pos_divide_less_eq:
     "0 < (c::'a::ordered_field) ==> (b/c < a) = (b < a*c)"
proof -
  assume less: "0<c"
  hence "(b/c < a) = ((b/c)*c < a*c)"
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
  also have "... = (b < a*c)"
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
  finally show ?thesis .
qed

lemma neg_divide_less_eq:
 "c < (0::'a::ordered_field) ==> (b/c < a) = (a*c < b)"
proof -
  assume less: "c<0"
  hence "(b/c < a) = (a*c < (b/c)*c)"
    by (simp add: mult_less_cancel_right order_less_not_sym [OF less])
  also have "... = (a*c < b)"
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
  finally show ?thesis .
qed

lemma divide_less_eq:
  "(b/c < a) = 
   (if 0 < c then b < a*c
             else if c < 0 then a*c < b
             else 0 < (a::'a::{ordered_field,division_by_zero}))"
apply (case_tac "c=0", simp) 
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
done

lemma nonzero_eq_divide_eq: "c\<noteq>0 ==> ((a::'a::field) = b/c) = (a*c = b)"
proof -
  assume [simp]: "c\<noteq>0"
  have "(a = b/c) = (a*c = (b/c)*c)"
    by (simp add: field_mult_cancel_right)
  also have "... = (a*c = b)"
    by (simp add: divide_inverse mult_assoc) 
  finally show ?thesis .
qed

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

lemma nonzero_divide_eq_eq: "c\<noteq>0 ==> (b/c = (a::'a::field)) = (b = a*c)"
proof -
  assume [simp]: "c\<noteq>0"
  have "(b/c = a) = ((b/c)*c = a*c)"
    by (simp add: field_mult_cancel_right)
  also have "... = (b = a*c)"
    by (simp add: divide_inverse mult_assoc) 
  finally show ?thesis .
qed

lemma divide_eq_eq:
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
by (force simp add: nonzero_divide_eq_eq) 

subsection{*Cancellation Laws for Division*}

lemma divide_cancel_right [simp]:
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
apply (case_tac "c=0", simp) 
apply (simp add: divide_inverse_zero 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_zero field_mult_cancel_left) 
done


subsection {* Ordering Rules for Division *}

lemma divide_strict_right_mono:
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::ordered_field)"
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
              positive_imp_inverse_positive) 

lemma divide_right_mono:
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{ordered_field,division_by_zero})"
  by (force simp add: divide_strict_right_mono order_le_less) 


text{*The last premise ensures that @{term a} and @{term b} 
      have the same sign*}
lemma divide_strict_left_mono:
       "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
by (force simp add: zero_less_mult_iff divide_inverse mult_strict_left_mono 
      order_less_imp_not_eq order_less_imp_not_eq2  
      less_imp_inverse_less less_imp_inverse_less_neg) 

lemma divide_left_mono:
     "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
   prefer 2 
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
  apply (case_tac "c=0", simp add: divide_inverse)
  apply (force simp add: divide_strict_left_mono order_le_less) 
  done

lemma divide_strict_left_mono_neg:
     "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::ordered_field)"
  apply (subgoal_tac "a \<noteq> 0 & b \<noteq> 0") 
   prefer 2 
   apply (force simp add: zero_less_mult_iff order_less_imp_not_eq) 
  apply (drule divide_strict_left_mono [of _ _ "-c"]) 
   apply (simp_all add: mult_commute nonzero_minus_divide_left [symmetric]) 
  done

lemma divide_strict_right_mono_neg:
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::ordered_field)"
apply (drule divide_strict_right_mono [of _ _ "-c"], simp) 
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
done


subsection {* Ordered Fields are Dense *}

lemma zero_less_two: "0 < (1+1::'a::ordered_field)"
proof -
  have "0 + 0 <  (1+1::'a::ordered_field)"
    by (blast intro: zero_less_one add_strict_mono) 
  thus ?thesis by simp
qed

lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::ordered_field)"
by (simp add: zero_less_two pos_less_divide_eq right_distrib) 

lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::ordered_field) < b"
by (simp add: zero_less_two pos_divide_less_eq right_distrib) 

lemma dense: "a < b ==> \<exists>r::'a::ordered_field. a < r & r < b"
by (blast intro!: less_half_sum gt_half_sum)


subsection {* Absolute Value *}

lemma abs_zero [simp]: "abs 0 = (0::'a::ordered_ring)"
by (simp add: abs_if)

lemma abs_one [simp]: "abs 1 = (1::'a::ordered_ring)"
  by (simp add: abs_if zero_less_one [THEN order_less_not_sym]) 

lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_ring)" 
apply (case_tac "a=0 | b=0", force) 
apply (auto elim: order_less_asym
            simp add: abs_if mult_less_0_iff linorder_neq_iff
                  minus_mult_left [symmetric] minus_mult_right [symmetric])  
done

lemma abs_eq_0 [simp]: "(abs a = 0) = (a = (0::'a::ordered_ring))"
by (simp add: abs_if)

lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::ordered_ring))"
by (simp add: abs_if linorder_neq_iff)

lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::ordered_ring)"
by (simp add: abs_if  order_less_not_sym [of a 0])

lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::ordered_ring)) = (a = 0)" 
by (simp add: order_le_less) 

lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::ordered_ring)"
apply (auto simp add: abs_if linorder_not_less order_less_not_sym [of 0 a])  
apply (drule order_antisym, assumption, simp) 
done

lemma abs_ge_zero [simp]: "(0::'a::ordered_ring) \<le> abs a"
apply (simp add: abs_if order_less_imp_le)
apply (simp add: linorder_not_less) 
done

lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::ordered_ring)"
  by (force elim: order_less_asym simp add: abs_if)

lemma abs_zero_iff [simp]: "(abs a = 0) = (a = (0::'a::ordered_ring))"
by (simp add: abs_if)

lemma abs_ge_self: "a \<le> abs (a::'a::ordered_ring)"
apply (simp add: abs_if)
apply (simp add: order_less_imp_le order_trans [of _ 0])
done

lemma abs_ge_minus_self: "-a \<le> abs (a::'a::ordered_ring)"
by (insert abs_ge_self [of "-a"], simp)

lemma nonzero_abs_inverse:
     "a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
                      negative_imp_inverse_negative)
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
done

lemma abs_inverse [simp]:
     "abs (inverse (a::'a::{ordered_field,division_by_zero})) = 
      inverse (abs a)"
apply (case_tac "a=0", simp) 
apply (simp add: nonzero_abs_inverse) 
done


lemma nonzero_abs_divide:
     "b \<noteq> 0 ==> abs (a / (b::'a::ordered_field)) = abs a / abs b"
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 

lemma abs_divide:
     "abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
apply (case_tac "b=0", simp) 
apply (simp add: nonzero_abs_divide) 
done

lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::ordered_ring)"
by (simp add: abs_if)

lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::ordered_ring))"
proof 
  assume ale: "a \<le> -a"
  show "a\<le>0"
    apply (rule classical) 
    apply (simp add: linorder_not_le) 
    apply (blast intro: ale order_trans order_less_imp_le
                        neg_0_le_iff_le [THEN iffD1]) 
    done
next
  assume "a\<le>0"
  hence "0 \<le> -a" by (simp only: neg_0_le_iff_le)
  thus "a \<le> -a"  by (blast intro: prems order_trans) 
qed

lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::ordered_ring))"
by (insert le_minus_self_iff [of "-a"], simp)

lemma eq_minus_self_iff: "(a = -a) = (a = (0::'a::ordered_ring))"
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_ring))"
by (simp add: order_less_le le_minus_self_iff eq_minus_self_iff)

lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::ordered_ring)"
apply (simp add: abs_if split: split_if_asm)
apply (rule order_trans [of _ "-a"]) 
 apply (simp add: less_minus_self_iff order_less_imp_le, assumption)
done

lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::ordered_ring)"
by (insert abs_le_D1 [of "-a"], simp)

lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::ordered_ring))"
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)

lemma abs_less_iff: "(abs a < b) = (a < b & -a < (b::'a::ordered_ring))" 
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:  linorder_not_less [symmetric]) 
apply (simp add: le_minus_self_iff linorder_neq_iff) 
apply (simp add:  linorder_not_less [symmetric]) 
done

lemma abs_triangle_ineq: "abs (a+b) \<le> abs a + abs (b::'a::ordered_ring)"
by (force simp add: abs_le_iff abs_ge_self abs_ge_minus_self add_mono)

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


end