src/HOL/Ring_and_Field.thy
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(*  Title:   HOL/Ring_and_Field.thy
ID:      $Id$
Author:  Gertrud Bauer, Steven Obua, Tobias Nipkow, 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}
*}

class semiring = ab_semigroup_add + semigroup_mult +
assumes left_distrib: "(a + b) * c = a * c + b * c"
assumes right_distrib: "a * (b + c) = a * b + a * c"
begin

text{*For the @{text combine_numerals} simproc*}
lemma combine_common_factor:
"a * e + (b * e + c) = (a + b) * e + c"
by (simp add: left_distrib add_ac)

end

class mult_zero = times + zero +
assumes mult_zero_left [simp]: "0 * a = 0"
assumes mult_zero_right [simp]: "a * 0 = 0"

class semiring_0 = semiring + comm_monoid_add + mult_zero

class semiring_0_cancel = semiring + comm_monoid_add + cancel_ab_semigroup_add
begin

subclass semiring_0
proof unfold_locales
fix a :: 'a
have "0 * a + 0 * a = 0 * a + 0"
by (simp add: left_distrib [symmetric])
thus "0 * a = 0"
by (simp only: add_left_cancel)
next
fix a :: 'a
have "a * 0 + a * 0 = a * 0 + 0"
by (simp add: right_distrib [symmetric])
thus "a * 0 = 0"
by (simp only: add_left_cancel)
qed

end

class comm_semiring = ab_semigroup_add + ab_semigroup_mult +
assumes distrib: "(a + b) * c = a * c + b * c"
begin

subclass semiring
proof unfold_locales
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

end

class comm_semiring_0 = comm_semiring + comm_monoid_add + mult_zero
begin

subclass semiring_0 by unfold_locales

end

class comm_semiring_0_cancel = comm_semiring + comm_monoid_add + cancel_ab_semigroup_add
begin

subclass semiring_0_cancel by unfold_locales

end

class zero_neq_one = zero + one +
assumes zero_neq_one [simp]: "0 \<noteq> 1"

class semiring_1 = zero_neq_one + semiring_0 + monoid_mult

class comm_semiring_1 = zero_neq_one + comm_semiring_0 + comm_monoid_mult
(*previously almost_semiring*)
begin

subclass semiring_1 by unfold_locales

end

class no_zero_divisors = zero + times +
assumes no_zero_divisors: "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"

class semiring_1_cancel = semiring + comm_monoid_add + zero_neq_one
+ cancel_ab_semigroup_add + monoid_mult

subclass (in semiring_1_cancel) semiring_0_cancel by unfold_locales

subclass (in semiring_1_cancel) semiring_1 by unfold_locales

class comm_semiring_1_cancel = comm_semiring + comm_monoid_add + comm_monoid_mult
+ zero_neq_one + cancel_ab_semigroup_add

subclass (in comm_semiring_1_cancel) semiring_1_cancel by unfold_locales
subclass (in comm_semiring_1_cancel) comm_semiring_0_cancel by unfold_locales
subclass (in comm_semiring_1_cancel) comm_semiring_1 by unfold_locales

class ring = semiring + ab_group_add

subclass (in ring) semiring_0_cancel by unfold_locales

context ring
begin

text {* Distribution rules *}

lemma minus_mult_left: "- (a * b) = - a * b"
by (rule equals_zero_I) (simp add: left_distrib [symmetric])

lemma minus_mult_right: "- (a * b) = a * - b"
by (rule equals_zero_I) (simp add: right_distrib [symmetric])

lemma minus_mult_minus [simp]: "- a * - b = a * b"
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])

lemma minus_mult_commute: "- a * b = a * - b"
by (simp add: minus_mult_left [symmetric] minus_mult_right [symmetric])

lemma right_diff_distrib: "a * (b - c) = a * b - a * c"
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"
by (simp add: left_distrib diff_minus
minus_mult_left [symmetric] minus_mult_right [symmetric])

lemmas ring_distribs =
right_distrib left_distrib left_diff_distrib right_diff_distrib

lemmas ring_simps =
add_ac
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
ring_distribs

lemma eq_add_iff1:
"a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
by (simp add: ring_simps)

lemma eq_add_iff2:
"a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
by (simp add: ring_simps)

end

lemmas ring_distribs =
right_distrib left_distrib left_diff_distrib right_diff_distrib

class comm_ring = comm_semiring + ab_group_add

subclass (in comm_ring) ring by unfold_locales
subclass (in comm_ring) comm_semiring_0 by unfold_locales

class ring_1 = ring + zero_neq_one + monoid_mult

subclass (in ring_1) semiring_1_cancel by unfold_locales

class comm_ring_1 = comm_ring + zero_neq_one + comm_monoid_mult
(*previously ring*)

subclass (in comm_ring_1) ring_1 by unfold_locales
subclass (in comm_ring_1) comm_semiring_1_cancel by unfold_locales

class ring_no_zero_divisors = ring + no_zero_divisors
begin

lemma mult_eq_0_iff [simp]:
shows "a * b = 0 \<longleftrightarrow> (a = 0 \<or> b = 0)"
proof (cases "a = 0 \<or> b = 0")
case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
then show ?thesis using no_zero_divisors by simp
next
case True then show ?thesis by auto
qed

end

class ring_1_no_zero_divisors = ring_1 + ring_no_zero_divisors

class idom = comm_ring_1 + no_zero_divisors
begin

subclass ring_1_no_zero_divisors by unfold_locales

end

class division_ring = ring_1 + inverse +
assumes left_inverse [simp]:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
begin

subclass ring_1_no_zero_divisors
proof unfold_locales
fix a b :: 'a
assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
show "a * b \<noteq> 0"
proof
assume ab: "a * b = 0"
hence "0 = inverse a * (a * b) * inverse b"
by simp
also have "\<dots> = (inverse a * a) * (b * inverse b)"
by (simp only: mult_assoc)
also have "\<dots> = 1"
using a b by simp
finally show False
by simp
qed
qed

end

class field = comm_ring_1 + inverse +
assumes field_inverse:  "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
assumes divide_inverse: "a / b = a * inverse b"

subclass (in field) division_ring
proof unfold_locales
fix a :: 'a
assume "a \<noteq> 0"
thus "inverse a * a = 1" by (rule field_inverse)
thus "a * inverse a = 1" by (simp only: mult_commute)
qed

subclass (in field) idom by unfold_locales

context field
begin

lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
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 \<Longrightarrow> inverse a = 1 / a"
by (simp add: divide_inverse)

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

lemma divide_zero_left [simp]: "0 / a = 0"
by (simp add: divide_inverse)

lemma inverse_eq_divide: "inverse a = 1 / a"
by (simp add: divide_inverse)

lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
by (simp add: divide_inverse ring_distribs)

end

class division_by_zero = zero + inverse +
assumes inverse_zero [simp]: "inverse 0 = 0"

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)

class mult_mono = times + zero + ord +
assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"

class pordered_semiring = mult_mono + semiring_0 + pordered_ab_semigroup_add
begin

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

lemma mult_mono':
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c
\<Longrightarrow> a * c \<le> b * d"
apply (rule mult_mono)
apply (fast intro: order_trans)+
done

end

class pordered_cancel_semiring = mult_mono + pordered_ab_semigroup_add
+ semiring + comm_monoid_add + cancel_ab_semigroup_add

subclass (in pordered_cancel_semiring) semiring_0_cancel by unfold_locales
subclass (in pordered_cancel_semiring) pordered_semiring by unfold_locales

context pordered_cancel_semiring
begin

lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
by (drule mult_left_mono [of zero b], auto)

lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
by (drule mult_left_mono [of b zero], auto)

lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
by (drule mult_right_mono [of b zero], auto)

lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> (0::_::pordered_cancel_semiring)"
by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)

end

class ordered_semiring = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add + mult_mono

subclass (in ordered_semiring) pordered_cancel_semiring by unfold_locales

context ordered_semiring
begin

lemma mult_left_less_imp_less:
"c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
by (force simp add: mult_left_mono not_le [symmetric])

lemma mult_right_less_imp_less:
"a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
by (force simp add: mult_right_mono not_le [symmetric])

end

class ordered_semiring_strict = semiring + comm_monoid_add + ordered_cancel_ab_semigroup_add +
assumes mult_strict_left_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
assumes mult_strict_right_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> a * c < b * c"

subclass (in ordered_semiring_strict) semiring_0_cancel by unfold_locales

subclass (in ordered_semiring_strict) ordered_semiring
proof unfold_locales
fix a b c :: 'a
assume A: "a \<le> b" "0 \<le> c"
from A show "c * a \<le> c * b"
unfolding le_less
using mult_strict_left_mono by (cases "c = 0") auto
from A show "a * c \<le> b * c"
unfolding le_less
using mult_strict_right_mono by (cases "c = 0") auto
qed

context ordered_semiring_strict
begin

lemma mult_left_le_imp_le:
"c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
by (force simp add: mult_strict_left_mono _not_less [symmetric])

lemma mult_right_le_imp_le:
"a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
by (force simp add: mult_strict_right_mono not_less [symmetric])

lemma mult_pos_pos:
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
by (drule mult_strict_left_mono [of zero b], auto)

lemma mult_pos_neg:
"0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
by (drule mult_strict_left_mono [of b zero], auto)

lemma mult_pos_neg2:
"0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
by (drule mult_strict_right_mono [of b zero], auto)

lemma zero_less_mult_pos:
"0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
apply (cases "b\<le>0")
apply (auto simp add: le_less not_less)
apply (drule_tac mult_pos_neg [of a b])
apply (auto dest: less_not_sym)
done

lemma zero_less_mult_pos2:
"0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
apply (cases "b\<le>0")
apply (auto simp add: le_less not_less)
apply (drule_tac mult_pos_neg2 [of a b])
apply (auto dest: less_not_sym)
done

end

class mult_mono1 = times + zero + ord +
assumes mult_mono1: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"

class pordered_comm_semiring = comm_semiring_0
+ pordered_ab_semigroup_add + mult_mono1

class pordered_cancel_comm_semiring = comm_semiring_0_cancel
+ pordered_ab_semigroup_add + mult_mono1
begin

subclass pordered_comm_semiring by unfold_locales

end

class ordered_comm_semiring_strict = comm_semiring_0 + ordered_cancel_ab_semigroup_add +
assumes mult_strict_mono: "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

subclass (in pordered_comm_semiring) pordered_semiring
proof unfold_locales
fix a b c :: 'a
assume "a \<le> b" "0 \<le> c"
thus "c * a \<le> c * b" by (rule mult_mono1)
thus "a * c \<le> b * c" by (simp only: mult_commute)
qed

subclass (in pordered_cancel_comm_semiring) pordered_cancel_semiring
by unfold_locales

subclass (in ordered_comm_semiring_strict) ordered_semiring_strict
proof unfold_locales
fix a b c :: 'a
assume "a < b" "0 < c"
thus "c * a < c * b" by (rule mult_strict_mono)
thus "a * c < b * c" by (simp only: mult_commute)
qed

subclass (in ordered_comm_semiring_strict) pordered_cancel_comm_semiring
proof unfold_locales
fix a b c :: 'a
assume "a \<le> b" "0 \<le> c"
thus "c * a \<le> c * b"
unfolding le_less
using mult_strict_mono by (cases "c = 0") auto
qed

class pordered_ring = ring + pordered_cancel_semiring

subclass (in pordered_ring) pordered_ab_group_add by unfold_locales

context pordered_ring
begin

lemmas ring_simps = ring_simps group_simps

lemma less_add_iff1:
"a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
by (simp add: ring_simps)

lemma less_add_iff2:
"a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
by (simp add: ring_simps)

lemma le_add_iff1:
"a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
by (simp add: ring_simps)

lemma le_add_iff2:
"a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
by (simp add: ring_simps)

lemma mult_left_mono_neg:
"b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
apply (drule mult_left_mono [of _ _ "uminus c"])
apply (simp_all add: minus_mult_left [symmetric])
done

lemma mult_right_mono_neg:
"b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
apply (drule mult_right_mono [of _ _ "uminus c"])
apply (simp_all add: minus_mult_right [symmetric])
done

lemma mult_nonpos_nonpos:
"a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
by (drule mult_right_mono_neg [of a zero b]) auto

lemma split_mult_pos_le:
"(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)

end

class lordered_ring = pordered_ring + lordered_ab_group_abs

subclass (in lordered_ring) lordered_ab_group_meet by unfold_locales
subclass (in lordered_ring) lordered_ab_group_join by unfold_locales

class abs_if = minus + ord + zero + abs +
assumes abs_if: "\<bar>a\<bar> = (if a < 0 then (- a) else a)"

class sgn_if = sgn + zero + one + minus + ord +
assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"

class ordered_ring = ring + ordered_semiring
+ lordered_ab_group + abs_if
-- {*FIXME: should inherit from @{text ordered_ab_group_add} rather than
@{text lordered_ab_group}*}

instance ordered_ring \<subseteq> lordered_ring
proof
fix x :: 'a
show "\<bar>x\<bar> = sup x (- x)"
by (simp only: abs_if sup_eq_if)
qed

(* The "strict" suffix can be seen as describing the combination of ordered_ring and no_zero_divisors.
Basically, ordered_ring + no_zero_divisors = ordered_ring_strict.
*)
class ordered_ring_strict = ring + ordered_semiring_strict
+ lordered_ab_group + abs_if
-- {*FIXME: should inherit from @{text ordered_ab_group_add} rather than
@{text lordered_ab_group}*}

instance ordered_ring_strict \<subseteq> ordered_ring by intro_classes

context ordered_ring_strict
begin

lemma mult_strict_left_mono_neg:
"b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
apply (drule mult_strict_left_mono [of _ _ "uminus c"])
apply (simp_all add: minus_mult_left [symmetric])
done

lemma mult_strict_right_mono_neg:
"b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
apply (drule mult_strict_right_mono [of _ _ "uminus c"])
apply (simp_all add: minus_mult_right [symmetric])
done

lemma mult_neg_neg:
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
by (drule mult_strict_right_mono_neg, auto)

end

lemma zero_less_mult_iff:
fixes a :: "'a::ordered_ring_strict"
shows "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
apply (auto simp add: le_less not_less mult_pos_pos mult_neg_neg)
apply (blast dest: zero_less_mult_pos)
apply (blast dest: zero_less_mult_pos2)
done

instance ordered_ring_strict \<subseteq> ring_no_zero_divisors
apply intro_classes
apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
apply (force dest: mult_strict_right_mono_neg mult_strict_right_mono)+
done

lemma zero_le_mult_iff:
"((0::'a::ordered_ring_strict) \<le> a*b) = (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
by (auto simp add: eq_commute [of 0] order_le_less linorder_not_less
zero_less_mult_iff)

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

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

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

lemma not_square_less_zero[simp]: "\<not> (a * a < (0::'a::ordered_ring_strict))"
by (simp add: not_less)

text{*This list of rewrites simplifies ring terms by multiplying
everything out and bringing sums and products into a canonical form
(by ordered rewriting). As a result it decides ring equalities but
also helps with inequalities. *}
lemmas ring_simps = group_simps ring_distribs

class pordered_comm_ring = comm_ring + pordered_comm_semiring

subclass (in pordered_comm_ring) pordered_ring by unfold_locales

subclass (in pordered_comm_ring) pordered_cancel_comm_semiring by unfold_locales

class ordered_semidom = comm_semiring_1_cancel + ordered_comm_semiring_strict +
(*previously ordered_semiring*)
assumes zero_less_one [simp]: "0 < 1"
begin

lemma pos_add_strict:
shows "0 < a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
using add_strict_mono [of zero a b c] by simp

end

class ordered_idom =
comm_ring_1 +
ordered_comm_semiring_strict +
lordered_ab_group +
abs_if + sgn_if
(*previously ordered_ring*)

instance ordered_idom \<subseteq> ordered_ring_strict ..

instance ordered_idom \<subseteq> pordered_comm_ring ..

class ordered_field = field + ordered_idom

lemma linorder_neqE_ordered_idom:
fixes x y :: "'a :: ordered_idom"
assumes "x \<noteq> y" obtains "x < y" | "y < x"
using assms by (rule linorder_neqE)

-- {* FIXME continue localization here *}

text{*Proving axiom @{text zero_less_one} makes all @{text ordered_semidom}
theorems available to members of @{term ordered_idom} *}

instance ordered_idom \<subseteq> ordered_semidom
proof
have "(0::'a) \<le> 1*1" by (rule zero_le_square)
thus "(0::'a) < 1" by (simp add: order_le_less)
qed

instance ordered_idom \<subseteq> 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) \<le> 1"
by (rule zero_less_one [THEN order_less_imp_le])

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

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

subsection{*More Monotonicity*}

text{*Strict monotonicity in both arguments*}
lemma mult_strict_mono:
"[|a<b; c<d; 0<b; 0\<le>c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
apply (cases "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 \<le> a; 0 \<le> c|] ==> a * c < b * (d::'a::ordered_semiring_strict)"
apply (rule mult_strict_mono)
apply (blast intro: order_le_less_trans)+
done

lemma 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 "\<le>"} 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 (cases "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 (cases "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 \<le> c --> a < b) & (c \<le> 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 \<le> c --> a < b) & (c \<le> 0 --> b < a))"
by (insert mult_less_cancel_left_disj [of c a b], auto)

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

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

lemma mult_less_imp_less_left:
assumes less: "c*a < c*b" and nonneg: "0 \<le> c"
shows "a < (b::'a::ordered_semiring_strict)"
proof (rule ccontr)
assume "~ a < b"
hence "b \<le> a" by (simp add: linorder_not_less)
hence "c*b \<le> c*a" using nonneg by (rule mult_left_mono)
with this and less show False
by (simp add: linorder_not_less [symmetric])
qed

lemma mult_less_imp_less_right:
assumes less: "a*c < b*c" and nonneg: "0 <= c"
shows "a < (b::'a::ordered_semiring_strict)"
proof (rule ccontr)
assume "~ a < b"
hence "b \<le> a" by (simp add: linorder_not_less)
hence "b*c \<le> a*c" using nonneg 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,noatp]:
fixes a b c :: "'a::ring_no_zero_divisors"
shows "(a * c = b * c) = (c = 0 \<or> a = b)"
proof -
have "(a * c = b * c) = ((a - b) * c = 0)"
by (simp add: ring_distribs)
thus ?thesis
by (simp add: disj_commute)
qed

lemma mult_cancel_left [simp,noatp]:
fixes a b c :: "'a::ring_no_zero_divisors"
shows "(c * a = c * b) = (c = 0 \<or> a = b)"
proof -
have "(c * a = c * b) = (c * (a - b) = 0)"
by (simp add: ring_distribs)
thus ?thesis
by simp
qed

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 \<le> b*c) = ((0<c --> 1\<le>b) & (c<0 --> b \<le> 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 \<le> c) = ((0<c --> a\<le>1) & (c<0 --> 1 \<le> a))"
by (insert mult_le_cancel_right [of a c 1], simp)

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

lemma mult_cancel_right1 [simp]:
fixes c :: "'a :: ring_1_no_zero_divisors"
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 :: ring_1_no_zero_divisors"
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 :: ring_1_no_zero_divisors"
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 :: ring_1_no_zero_divisors"
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

(* what ordering?? this is a straight instance of mult_eq_0_iff
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::division_ring)) = (a = 0 | b = 0)"
by simp
*)
(* subsumed by mult_cancel lemmas on ring_no_zero_divisors
text{*Cancellation of equalities with a common factor*}
lemma field_mult_cancel_right_lemma:
assumes cnz: "c \<noteq> (0::'a::division_ring)"
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::division_ring) | a=b)"
by simp

lemma field_mult_cancel_left [simp]:
"(c*a = c*b) = (c = (0::'a::division_ring) | a=b)"
by simp
*)
lemma nonzero_imp_inverse_nonzero:
"a \<noteq> 0 ==> inverse a \<noteq> (0::'a::division_ring)"
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::division_ring)" .
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::division_ring)"
apply (rule ccontr)
apply (blast dest: nonzero_imp_inverse_nonzero)
done

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

lemma inverse_nonzero_iff_nonzero [simp]:
"(inverse a = 0) = (a = (0::'a::{division_ring,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::division_ring)"
proof -
have "-a * inverse (- a) = -a * - inverse a"
by simp
thus ?thesis
by (simp only: mult_cancel_left, simp)
qed

lemma inverse_minus_eq [simp]:
"inverse(-a) = -inverse(a::'a::{division_ring,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::division_ring)"
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::{division_ring,division_by_zero})"
apply (cases "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::{division_ring,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::division_ring)) = 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::{division_ring,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::division_ring)"
proof -
have "inverse 1 * 1 = (1::'a::division_ring)"
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::division_ring)"
proof -
have "a \<noteq> 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 \<noteq> 0"
and bnz: "b \<noteq> 0"
shows "inverse(a*b) = inverse(b) * inverse(a::'a::division_ring)"
proof -
have "inverse(a*b) * (a * b) * inverse(b) = inverse(b)"
by (simp add: 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

lemma division_ring_inverse_add:
"[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
==> inverse a + inverse b = inverse a * (a+b) * inverse b"
by (simp add: ring_simps)

lemma division_ring_inverse_diff:
"[|(a::'a::division_ring) \<noteq> 0; b \<noteq> 0|]
==> inverse a - inverse b = inverse a * (b-a) * inverse b"
by (simp add: ring_simps)

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)"
by (simp add: division_ring_inverse_add mult_ac)

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 *}

text{* There is a whole bunch of simp-rules just for class @{text
field} but none for class @{text field} and @{text nonzero_divides}
because the latter are covered by a simproc. *}

lemma nonzero_mult_divide_mult_cancel_left[simp,noatp]:
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: 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_mult_cancel_left:
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
apply (cases "b = 0")
apply (simp_all add: nonzero_mult_divide_mult_cancel_left)
done

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

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

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)

lemmas times_divide_eq = times_divide_eq_right times_divide_eq_left

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

lemma divide_divide_eq_left [simp,noatp]:
"(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_mult_cancel_left [THEN sym])
apply assumption
apply (erule nonzero_mult_divide_mult_cancel_right [THEN sym])
apply assumption
done

subsubsection{*Special Cancellation Simprules for Division*}

lemma mult_divide_mult_cancel_left_if[simp,noatp]:
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_mult_cancel_left)

lemma nonzero_mult_divide_cancel_right[simp,noatp]:
"b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
using nonzero_mult_divide_mult_cancel_right[of 1 b a] by simp

lemma nonzero_mult_divide_cancel_left[simp,noatp]:
"a \<noteq> 0 \<Longrightarrow> a * b / a = (b::'a::field)"
using nonzero_mult_divide_mult_cancel_left[of 1 a b] by simp

lemma nonzero_divide_mult_cancel_right[simp,noatp]:
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> b / (a * b) = 1/(a::'a::field)"
using nonzero_mult_divide_mult_cancel_right[of a b 1] by simp

lemma nonzero_divide_mult_cancel_left[simp,noatp]:
"\<lbrakk> a\<noteq>0; b\<noteq>0 \<rbrakk> \<Longrightarrow> a / (a * b) = 1/(b::'a::field)"
using nonzero_mult_divide_mult_cancel_left[of b a 1] by simp

lemma nonzero_mult_divide_mult_cancel_left2[simp,noatp]:
"[|b\<noteq>0; c\<noteq>0|] ==> (c*a) / (b*c) = a/(b::'a::field)"
using nonzero_mult_divide_mult_cancel_left[of b c a] by(simp add:mult_ac)

lemma nonzero_mult_divide_mult_cancel_right2[simp,noatp]:
"[|b\<noteq>0; c\<noteq>0|] ==> (a*c) / (c*b) = a/(b::'a::field)"
using nonzero_mult_divide_mult_cancel_right[of b c a] by(simp add:mult_ac)

subsection {* Division and Unary Minus *}

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

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

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

lemma minus_divide_left: "- (a/b) = (-a) / (b::'a::field)"
by (simp add: divide_inverse minus_mult_left [symmetric])

lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
by (simp add: divide_inverse minus_mult_right [symmetric])

text{*The effect is to extract signs from divisions*}
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 (cases "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 add_divide_eq_iff:
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x + y/z = (z*x + y)/z"
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)

lemma divide_add_eq_iff:
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z + y = (x + z*y)/z"
by(simp add:add_divide_distrib nonzero_mult_divide_cancel_left)

lemma diff_divide_eq_iff:
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x - y/z = (z*x - y)/z"
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)

lemma divide_diff_eq_iff:
"(z::'a::field) \<noteq> 0 \<Longrightarrow> x/z - y = (x - z*y)/z"
by(simp add:diff_divide_distrib nonzero_mult_divide_cancel_left)

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
also have "... = (a*c = b)" by (simp add: divide_inverse mult_assoc)
finally show ?thesis .
qed

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

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)

lemmas field_eq_simps = ring_simps
(* pull / out*)
add_divide_eq_iff divide_add_eq_iff
diff_divide_eq_iff divide_diff_eq_iff
(* multiply eqn *)
nonzero_eq_divide_eq nonzero_divide_eq_eq
(* is added later:
times_divide_eq_left times_divide_eq_right
*)

text{*An example:*}
lemma fixes a b c d e f :: "'a::field"
shows "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f \<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
apply(simp add:field_eq_simps)
apply(simp)
done

lemma diff_frac_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
x / y - w / z = (x * z - w * y) / (y * z)"
by (simp add:field_eq_simps times_divide_eq)

lemma frac_eq_eq: "(y::'a::field) ~= 0 ==> z ~= 0 ==>
(x / y = w / z) = (x * z = w * y)"
by (simp add:field_eq_simps times_divide_eq)

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 nz: "a \<noteq> 0"
shows "0 < (a::'a::ordered_field)"
proof -
have "0 < inverse (inverse a)"
using inv_gt_0 by (rule positive_imp_inverse_positive)
thus "0 < a"
using nz 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 (cases "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 nz:  "a \<noteq> 0"
shows "a < (0::'a::ordered_field)"
proof -
have "inverse (inverse a) < 0"
using inv_less_0 by (rule negative_imp_inverse_negative)
thus "a < 0" using nz 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 (cases "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])

lemma ordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::ordered_field)"
proof
fix x::'a
have m1: "- (1::'a) < 0" by simp
from add_strict_right_mono[OF m1, where c=x]
have "(- 1) + x < x" by simp
thus "\<exists>y. y < x" by blast
qed

lemma ordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::ordered_field)"
proof
fix x::'a
have m1: " (1::'a) > 0" by simp
from add_strict_right_mono[OF m1, where c=x]
have "1 + x > x" by simp
thus "\<exists>y. y > x" by blast
qed

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

subsection{*Inverses and the Number One*}

lemma one_less_inverse_iff:
"(1 < inverse x) = (0 < x & x < (1::'a::{ordered_field,division_by_zero}))"
proof cases
assume "0 < x"
with inverse_less_iff_less [OF zero_less_one, of x]
show ?thesis by simp
next
assume notless: "~ (0 < x)"
have "~ (1 < inverse x)"
proof
assume "1 < inverse x"
also with notless have "... \<le> 0" by (simp add: linorder_not_less)
also have "... < 1" by (rule zero_less_one)
finally show False by auto
qed
with notless show ?thesis by simp
qed

lemma inverse_eq_1_iff [simp]:
"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
by (insert inverse_eq_iff_eq [of x 1], simp)

lemma one_le_inverse_iff:
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{ordered_field,division_by_zero}))"
by (force simp add: order_le_less one_less_inverse_iff zero_less_one
eq_commute [of 1])

lemma inverse_less_1_iff:
"(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{ordered_field,division_by_zero}))"
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff)

lemma inverse_le_1_iff:
"(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{ordered_field,division_by_zero}))"
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)

subsection{*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 (cases "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 (cases "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 (cases "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 (cases "c=0", simp)
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff)
done

subsection{*Field simplification*}

text{* Lemmas @{text field_simps} multiply with denominators in
in(equations) if they can be proved to be non-zero (for equations) or
positive/negative (for inequations). *}

lemmas field_simps = field_eq_simps
(* multiply ineqn *)
pos_divide_less_eq neg_divide_less_eq
pos_less_divide_eq neg_less_divide_eq
pos_divide_le_eq neg_divide_le_eq
pos_le_divide_eq neg_le_divide_eq

text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
of positivity/negativity needed for @{text field_simps}. Have not added @{text
sign_simps} to @{text field_simps} because the former can lead to case
explosions. *}

lemmas sign_simps = group_simps
zero_less_mult_iff  mult_less_0_iff

(* Only works once linear arithmetic is installed:
text{*An example:*}
lemma fixes a b c d e f :: "'a::ordered_field"
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
prefer 2 apply(simp add:sign_simps)
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
prefer 2 apply(simp add:sign_simps)
apply(simp add:field_simps)
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}) \<le> a/b) =
(0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
by (simp add: divide_inverse zero_le_mult_iff)

lemma divide_le_0_iff:
"(a/b \<le> (0::'a::{ordered_field,division_by_zero})) =
(0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
by (simp add: divide_inverse mult_le_0_iff)

lemma divide_eq_0_iff [simp,noatp]:
"(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
by (simp add: divide_inverse)

lemma divide_pos_pos:
"0 < (x::'a::ordered_field) ==> 0 < y ==> 0 < x / y"
by(simp add:field_simps)

lemma divide_nonneg_pos:
"0 <= (x::'a::ordered_field) ==> 0 < y ==> 0 <= x / y"
by(simp add:field_simps)

lemma divide_neg_pos:
"(x::'a::ordered_field) < 0 ==> 0 < y ==> x / y < 0"
by(simp add:field_simps)

lemma divide_nonpos_pos:
"(x::'a::ordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
by(simp add:field_simps)

lemma divide_pos_neg:
"0 < (x::'a::ordered_field) ==> y < 0 ==> x / y < 0"
by(simp add:field_simps)

lemma divide_nonneg_neg:
"0 <= (x::'a::ordered_field) ==> y < 0 ==> x / y <= 0"
by(simp add:field_simps)

lemma divide_neg_neg:
"(x::'a::ordered_field) < 0 ==> y < 0 ==> 0 < x / y"
by(simp add:field_simps)

lemma divide_nonpos_neg:
"(x::'a::ordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
by(simp add:field_simps)

subsection{*Cancellation Laws for Division*}

lemma divide_cancel_right [simp,noatp]:
"(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
apply (cases "c=0", simp)
apply (simp add: divide_inverse)
done

lemma divide_cancel_left [simp,noatp]:
"(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
apply (cases "c=0", simp)
apply (simp add: divide_inverse)
done

subsection {* Division and the Number One *}

text{*Simplify expressions equated with 1*}
lemma divide_eq_1_iff [simp,noatp]:
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
apply (cases "b=0", simp)
apply (simp add: right_inverse_eq)
done

lemma one_eq_divide_iff [simp,noatp]:
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
by (simp add: eq_commute [of 1])

lemma zero_eq_1_divide_iff [simp,noatp]:
"((0::'a::{ordered_field,division_by_zero}) = 1/a) = (a = 0)"
apply (cases "a=0", simp)
apply (auto simp add: nonzero_eq_divide_eq)
done

lemma one_divide_eq_0_iff [simp,noatp]:
"(1/a = (0::'a::{ordered_field,division_by_zero})) = (a = 0)"
apply (cases "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, simplified]
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]

declare zero_less_divide_1_iff [simp]
declare divide_less_0_1_iff [simp,noatp]
declare zero_le_divide_1_iff [simp]
declare divide_le_0_1_iff [simp,noatp]

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)

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(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)

lemma divide_left_mono:
"[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::ordered_field)"
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)

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)"
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)

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

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

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

lemma less_divide_eq_1 [noatp]:
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 [noatp]:
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,noatp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
by (auto simp add: le_divide_eq)

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

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

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

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

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

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

lemma divide_less_eq_1_neg [simp,noatp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
by (auto simp add: divide_less_eq)

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

lemma divide_eq_eq_1 [simp,noatp]:
fixes a :: "'a :: {ordered_field,division_by_zero}"
shows "(b/a = 1) = ((a \<noteq> 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(simp add:field_simps)

lemma mult_imp_div_pos_less: "0 < (y::'a::ordered_field) ==> x < z * y ==>
x / y < z"
by(simp add:field_simps)

lemma mult_imp_less_div_pos: "0 < (y::'a::ordered_field) ==> z * y < x ==>
z < x / y"
by(simp add:field_simps)

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)
thm mult_mono
thm mult_mono'
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

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 *}

context ordered_semidom
begin

lemma less_add_one: "a < a + 1"
proof -
have "a + 0 < a + 1"
by (blast intro: zero_less_one add_strict_left_mono)
thus ?thesis by simp
qed

lemma zero_less_two: "0 < 1 + 1"
by (blast intro: less_trans zero_less_one less_add_one)

end

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

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

instance ordered_field < dense_linear_order
proof
fix x y :: 'a
have "x < x + 1" by simp
then show "\<exists>y. x < y" ..
have "x - 1 < x" by simp
then show "\<exists>y. y < x" ..
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
qed

subsection {* Absolute Value *}

lemma mult_sgn_abs: "sgn x * abs x = (x::'a::{ordered_idom,linorder})"
using less_linear[of x 0]
by(auto simp: sgn_if abs_if)

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

lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lordered_ring))"
proof -
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
have a: "(abs a) * (abs b) = ?x"
by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
{
fix u v :: 'a
have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
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: ring_simps)
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 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
shows "abs (a*b) = abs a * abs (b::'a::lordered_ring)"
proof -
have s: "(0 <= a*b) | (a*b <= 0)"
apply (auto)
apply (rule_tac split_mult_pos_le)
apply (rule_tac contrapos_np[of "a*b <= 0"])
apply (simp)
apply (rule_tac split_mult_neg_le)
apply (insert prems)
apply (blast)
done
have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
by (simp add: prts[symmetric])
show ?thesis
proof cases
assume "0 <= a * b"
then show ?thesis
apply (simp_all add: mulprts abs_prts)
apply (insert prems)
apply (auto simp add:
ring_simps
iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
iffD1[OF le_zero_iff_pprt_id] iffD1[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_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 \<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 (cases "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 [simp]:
"abs (a / (b::'a::{ordered_field,division_by_zero})) = abs a / abs b"
apply (cases "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 {* Bounds of products via negative and positive Part *}

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_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_ge_prts:
assumes
"a1 <= (a::'a::lordered_ring)"
"a <= a2"
"b1 <= b"
"b <= b2"
shows
"a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
proof -
from prems have a1:"- a2 <= -a" by auto
from prems have a2: "-a <= -a1" by auto
from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 prems(3) prems(4), simplified nprt_neg pprt_neg]
have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp
then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
by (simp only: minus_le_iff)
then show ?thesis by simp
qed

end