src/HOL/Fields.thy
 author haftmann Fri, 14 Jun 2019 08:34:27 +0000 changeset 70344 050104f01bf9 parent 70343 e54caaa38ad9 child 70356 4a327c061870 permissions -rw-r--r--
moved comment to approproiate place
```
(*  Title:      HOL/Fields.thy
Author:     Gertrud Bauer
Author:     Steven Obua
Author:     Tobias Nipkow
Author:     Lawrence C Paulson
Author:     Markus Wenzel
*)

section \<open>Fields\<close>

theory Fields
imports Nat
begin

context idom
begin

lemma inj_mult_left [simp]: \<open>inj ((*) a) \<longleftrightarrow> a \<noteq> 0\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
proof
assume ?P
show ?Q
proof
assume \<open>a = 0\<close>
with \<open>?P\<close> have "inj ((*) 0)"
by simp
moreover have "0 * 0 = 0 * 1"
by simp
ultimately have "0 = 1"
by (rule injD)
then show False
by simp
qed
next
assume ?Q then show ?P
by (auto intro: injI)
qed

end

subsection \<open>Division rings\<close>

text \<open>
A division ring is like a field, but without the commutativity requirement.
\<close>

class inverse = divide +
fixes inverse :: "'a \<Rightarrow> 'a"
begin

abbreviation inverse_divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "'/" 70)
where
"inverse_divide \<equiv> divide"

end

text \<open>Setup for linear arithmetic prover\<close>

ML_file \<open>~~/src/Provers/Arith/fast_lin_arith.ML\<close>
ML_file \<open>Tools/lin_arith.ML\<close>
setup \<open>Lin_Arith.global_setup\<close>
declaration \<open>K Lin_Arith.setup\<close>

simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) \<le> n" | "(m::nat) = n") =
\<open>K Lin_Arith.simproc\<close>
(* Because of this simproc, the arithmetic solver is really only
useful to detect inconsistencies among the premises for subgoals which are
*not* themselves (in)equalities, because the latter activate
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the

lemmas [arith_split] = nat_diff_split split_min split_max

context linordered_nonzero_semiring
begin
lemma of_nat_max: "of_nat (max x y) = max (of_nat x) (of_nat y)"
by (auto simp: max_def)

lemma of_nat_min: "of_nat (min x y) = min (of_nat x) (of_nat y)"
by (auto simp: min_def)
end

text\<open>Lemmas \<open>divide_simps\<close> move division to the outside and eliminates them on (in)equalities.\<close>

named_theorems divide_simps "rewrite rules to eliminate divisions"

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"
assumes divide_inverse: "a / b = a * inverse b"
assumes inverse_zero [simp]: "inverse 0 = 0"
begin

subclass ring_1_no_zero_divisors
proof
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

lemma nonzero_imp_inverse_nonzero:
"a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
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" .
thus False by (simp add: eq_commute)
qed

lemma inverse_zero_imp_zero:
"inverse a = 0 \<Longrightarrow> a = 0"
apply (rule classical)
apply (drule nonzero_imp_inverse_nonzero)
apply auto
done

lemma inverse_unique:
assumes ab: "a * b = 1"
shows "inverse a = b"
proof -
have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
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_minus_eq:
"a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
by (rule inverse_unique) simp

lemma nonzero_inverse_inverse_eq:
"a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
by (rule inverse_unique) simp

lemma nonzero_inverse_eq_imp_eq:
assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
shows "a = b"
proof -
from \<open>inverse a = inverse b\<close>
have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> show "a = b"
qed

lemma inverse_1 [simp]: "inverse 1 = 1"
by (rule inverse_unique) simp

lemma nonzero_inverse_mult_distrib:
assumes "a \<noteq> 0" and "b \<noteq> 0"
shows "inverse (a * b) = inverse b * inverse a"
proof -
have "a * (b * inverse b) * inverse a = 1" using assms by simp
hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult.assoc)
thus ?thesis by (rule inverse_unique)
qed

"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"

lemma division_ring_inverse_diff:
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"

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.assoc)
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"

lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"

lemma inverse_eq_divide [field_simps, divide_simps]: "inverse a = 1 / a"

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

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

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

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

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

lemma divide_minus_left [simp]: "(-a) / b = - (a / b)"

lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
using add_divide_distrib [of a "- b" c] by simp

lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
proof -
assume [simp]: "c \<noteq> 0"
have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
proof -
assume [simp]: "c \<noteq> 0"
have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma nonzero_neg_divide_eq_eq [field_simps]: "b \<noteq> 0 \<Longrightarrow> - (a / b) = c \<longleftrightarrow> - a = c * b"
using nonzero_divide_eq_eq[of b "-a" c] by simp

lemma nonzero_neg_divide_eq_eq2 [field_simps]: "b \<noteq> 0 \<Longrightarrow> c = - (a / b) \<longleftrightarrow> c * b = - a"
using nonzero_neg_divide_eq_eq[of b a c] by auto

lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"

lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
by (drule sym) (simp add: divide_inverse mult.assoc)

"z \<noteq> 0 \<Longrightarrow> x + y / z = (x * z + y) / z"

"z \<noteq> 0 \<Longrightarrow> x / z + y = (x + y * z) / z"

lemma diff_divide_eq_iff [field_simps]:
"z \<noteq> 0 \<Longrightarrow> x - y / z = (x * z - y) / z"
by (simp add: diff_divide_distrib nonzero_eq_divide_eq eq_diff_eq)

"z \<noteq> 0 \<Longrightarrow> - (x / z) + y = (- x + y * z) / z"

lemma divide_diff_eq_iff [field_simps]:
"z \<noteq> 0 \<Longrightarrow> x / z - y = (x - y * z) / z"

lemma minus_divide_diff_eq_iff [field_simps]:
"z \<noteq> 0 \<Longrightarrow> - (x / z) - y = (- x - y * z) / z"

lemma division_ring_divide_zero [simp]:
"a / 0 = 0"

lemma divide_self_if [simp]:
"a / a = (if a = 0 then 0 else 1)"
by simp

lemma inverse_nonzero_iff_nonzero [simp]:
"inverse a = 0 \<longleftrightarrow> a = 0"
by rule (fact inverse_zero_imp_zero, simp)

lemma inverse_minus_eq [simp]:
"inverse (- a) = - inverse a"
proof cases
assume "a=0" thus ?thesis by simp
next
assume "a\<noteq>0"
thus ?thesis by (simp add: nonzero_inverse_minus_eq)
qed

lemma inverse_inverse_eq [simp]:
"inverse (inverse a) = 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_eq_imp_eq:
"inverse a = inverse b \<Longrightarrow> a = b"
by (drule arg_cong [where f="inverse"], simp)

lemma inverse_eq_iff_eq [simp]:
"inverse a = inverse b \<longleftrightarrow> a = b"
by (force dest!: inverse_eq_imp_eq)

lemma mult_commute_imp_mult_inverse_commute:
assumes "y * x = x * y"
shows   "inverse y * x = x * inverse y"
proof (cases "y=0")
case False
hence "x * inverse y = inverse y * y * x * inverse y"
by simp
also have "\<dots> = inverse y * (x * y * inverse y)"
finally show ?thesis by (simp add: mult.assoc False)
qed simp

lemmas mult_inverse_of_nat_commute =
mult_commute_imp_mult_inverse_commute[OF mult_of_nat_commute]

lemma divide_divide_eq_left':
"(a / b) / c = a / (c * b)"
by (cases "b = 0 \<or> c = 0")
(auto simp: divide_inverse mult.assoc nonzero_inverse_mult_distrib)

"a + b / z = (if z = 0 then a else (a * z + b) / z)"
"a / z + b = (if z = 0 then b else (a + b * z) / z)"
"- (a / z) + b = (if z = 0 then b else (-a + b * z) / z)"
"a - b / z = (if z = 0 then a else (a * z - b) / z)"
"a / z - b = (if z = 0 then -b else (a - b * z) / z)"
"- (a / z) - b = (if z = 0 then -b else (- a - b * z) / z)"
minus_divide_diff_eq_iff)

lemma [divide_simps]:
shows divide_eq_eq: "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
and eq_divide_eq: "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
and minus_divide_eq_eq: "- (b / c) = a \<longleftrightarrow> (if c \<noteq> 0 then - b = a * c else a = 0)"
and eq_minus_divide_eq: "a = - (b / c) \<longleftrightarrow> (if c \<noteq> 0 then a * c = - b else a = 0)"

end

subsection \<open>Fields\<close>

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

subclass division_ring
proof
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)
next
fix a b :: 'a
show "a / b = a * inverse b" by (rule field_divide_inverse)
next
show "inverse 0 = 0"
by (fact field_inverse_zero)
qed

subclass idom_divide
proof
fix b a
assume "b \<noteq> 0"
then show "a * b / b = a"
next
fix a
show "a / 0 = 0"
qed

text\<open>There is no slick version using division by zero.\<close>
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = (a + b) * inverse a * inverse b"

lemma nonzero_mult_divide_mult_cancel_left [simp]:
assumes [simp]: "c \<noteq> 0"
shows "(c * a) / (c * b) = a / b"
proof (cases "b = 0")
case True then show ?thesis by simp
next
case False
then have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
also have "... =  a * inverse b * (inverse c * c)"
by (simp only: ac_simps)
also have "... =  a * inverse b" by simp
finally show ?thesis by (simp add: divide_inverse)
qed

lemma nonzero_mult_divide_mult_cancel_right [simp]:
"c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)

lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"

lemma divide_inverse_commute: "a / b = inverse b * a"

assumes "y \<noteq> 0" and "z \<noteq> 0"
shows "x / y + w / z = (x * z + w * y) / (y * z)"
proof -
have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
using assms by simp
also have "\<dots> = (x * z + y * w) / (y * z)"
finally show ?thesis
by (simp only: mult.commute)
qed

text\<open>Special Cancellation Simprules for Division\<close>

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

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

lemma nonzero_mult_divide_mult_cancel_left2 [simp]:
"c \<noteq> 0 \<Longrightarrow> (c * a) / (b * c) = a / b"
using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)

lemma nonzero_mult_divide_mult_cancel_right2 [simp]:
"c \<noteq> 0 \<Longrightarrow> (a * c) / (c * b) = a / b"
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: ac_simps)

lemma diff_frac_eq:
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"

lemma frac_eq_eq:
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"

lemma divide_minus1 [simp]: "x / - 1 = - x"
using nonzero_minus_divide_right [of "1" x] by simp

text\<open>This version builds in division by zero while also re-orienting
the right-hand side.\<close>
lemma inverse_mult_distrib [simp]:
"inverse (a * b) = inverse a * inverse b"
proof cases
assume "a \<noteq> 0 \<and> b \<noteq> 0"
thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps)
next
assume "\<not> (a \<noteq> 0 \<and> b \<noteq> 0)"
thus ?thesis by force
qed

lemma inverse_divide [simp]:
"inverse (a / b) = b / a"

text \<open>Calculations with fractions\<close>

text\<open>There is a whole bunch of simp-rules just for class \<open>field\<close> but none for class \<open>field\<close> and \<open>nonzero_divides\<close>
because the latter are covered by a simproc.\<close>

lemmas mult_divide_mult_cancel_left = nonzero_mult_divide_mult_cancel_left

lemmas mult_divide_mult_cancel_right = nonzero_mult_divide_mult_cancel_right

lemma divide_divide_eq_right [simp]:
"a / (b / c) = (a * c) / b"

lemma divide_divide_eq_left [simp]:
"(a / b) / c = a / (b * c)"

lemma divide_divide_times_eq:
"(x / y) / (z / w) = (x * w) / (y * z)"
by simp

text \<open>Special Cancellation Simprules for Division\<close>

lemma mult_divide_mult_cancel_left_if [simp]:
shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
by simp

text \<open>Division and Unary Minus\<close>

lemma minus_divide_right:
"- (a / b) = a / - b"

lemma divide_minus_right [simp]:
"a / - b = - (a / b)"

lemma minus_divide_divide:
"(- a) / (- b) = a / b"
by (cases "b=0") (simp_all add: nonzero_minus_divide_divide)

lemma inverse_eq_1_iff [simp]:
"inverse x = 1 \<longleftrightarrow> x = 1"
by (insert inverse_eq_iff_eq [of x 1], simp)

lemma divide_eq_0_iff [simp]:
"a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"

lemma divide_cancel_right [simp]:
"a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
by (cases "c=0") (simp_all add: divide_inverse)

lemma divide_cancel_left [simp]:
"c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b"
by (cases "c=0") (simp_all add: divide_inverse)

lemma divide_eq_1_iff [simp]:
"a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
by (cases "b=0") (simp_all add: right_inverse_eq)

lemma one_eq_divide_iff [simp]:
"1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
by (simp add: eq_commute [of 1])

lemma divide_eq_minus_1_iff:
"(a / b = - 1) \<longleftrightarrow> b \<noteq> 0 \<and> a = - b"
using divide_eq_1_iff by fastforce

lemma times_divide_times_eq:
"(x / y) * (z / w) = (x * z) / (y * w)"
by simp

"y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"

"y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"

lemma dvd_field_iff:
"a dvd b \<longleftrightarrow> (a = 0 \<longrightarrow> b = 0)"
proof (cases "a = 0")
case False
then have "b = a * (b / a)"
then have "a dvd b" ..
with False show ?thesis
by simp
qed simp

lemma inj_divide_right [simp]:
"inj (\<lambda>b. b / a) \<longleftrightarrow> a \<noteq> 0"
proof -
have "(\<lambda>b. b / a) = (*) (inverse a)"
then have "inj (\<lambda>y. y / a) \<longleftrightarrow> inj ((*) (inverse a))"
by simp
also have "\<dots> \<longleftrightarrow> inverse a \<noteq> 0"
by simp
also have "\<dots> \<longleftrightarrow> a \<noteq> 0"
by simp
finally show ?thesis
by simp
qed

end

class field_char_0 = field + ring_char_0

subsection \<open>Ordered fields\<close>

class field_abs_sgn = field + idom_abs_sgn
begin

lemma sgn_inverse [simp]:
"sgn (inverse a) = inverse (sgn a)"
proof (cases "a = 0")
case True then show ?thesis by simp
next
case False
then have "a * inverse a = 1"
by simp
then have "sgn (a * inverse a) = sgn 1"
by simp
then have "sgn a * sgn (inverse a) = 1"
then have "inverse (sgn a) * (sgn a * sgn (inverse a)) = inverse (sgn a) * 1"
by simp
then have "(inverse (sgn a) * sgn a) * sgn (inverse a) = inverse (sgn a)"
with False show ?thesis
qed

lemma abs_inverse [simp]:
"\<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
proof -
from sgn_mult_abs [of "inverse a"] sgn_mult_abs [of a]
have "inverse (sgn a) * \<bar>inverse a\<bar> = inverse (sgn a * \<bar>a\<bar>)"
by simp
then show ?thesis by (auto simp add: sgn_eq_0_iff)
qed

lemma sgn_divide [simp]:
"sgn (a / b) = sgn a / sgn b"
unfolding divide_inverse sgn_mult by simp

lemma abs_divide [simp]:
"\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
unfolding divide_inverse abs_mult by simp

end

class linordered_field = field + linordered_idom
begin

lemma positive_imp_inverse_positive:
assumes a_gt_0: "0 < a"
shows "0 < inverse a"
proof -
have "0 < a * inverse a"
by (simp add: a_gt_0 [THEN less_imp_not_eq2])
thus "0 < inverse a"
by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
qed

lemma negative_imp_inverse_negative:
"a < 0 \<Longrightarrow> inverse a < 0"
by (insert positive_imp_inverse_positive [of "-a"],

lemma inverse_le_imp_le:
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
shows "b \<le> a"
proof (rule classical)
assume "\<not> b \<le> a"
hence "a < b"  by (simp add: linorder_not_le)
hence bpos: "0 < b"  by (blast intro: apos less_trans)
hence "a * inverse a \<le> a * inverse b"
by (simp add: apos invle less_imp_le mult_left_mono)
hence "(a * inverse a) * b \<le> (a * inverse b) * b"
by (simp add: bpos less_imp_le mult_right_mono)
thus "b \<le> a"  by (simp add: mult.assoc apos bpos 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"
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_negative_imp_negative:
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
shows "a < 0"
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 linordered_field_no_lb:
"\<forall>x. \<exists>y. y < x"
proof
fix x::'a
have m1: "- (1::'a) < 0" by simp
have "(- 1) + x < x" by simp
thus "\<exists>y. y < x" by blast
qed

lemma linordered_field_no_ub:
"\<forall> x. \<exists>y. y > x"
proof
fix x::'a
have m1: " (1::'a) > 0" by simp
have "1 + x > x" by simp
thus "\<exists>y. y > x" by blast
qed

lemma less_imp_inverse_less:
assumes less: "a < b" and apos:  "0 < a"
shows "inverse b < inverse a"
proof (rule ccontr)
assume "\<not> inverse b < inverse a"
hence "inverse a \<le> inverse b" by simp
hence "\<not> (a < b)"
by (simp add: 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 \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
apply (simp add: less_le [of "inverse a"] less_le [of "b"])
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
done

text\<open>Both premises are essential. Consider -1 and 1.\<close>
lemma inverse_less_iff_less [simp]:
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)

lemma le_imp_inverse_le:
"a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
by (force simp add: le_less less_imp_inverse_less)

lemma inverse_le_iff_le [simp]:
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)

text\<open>These results refer to both operands being negative.  The opposite-sign
case is trivial, since inverse preserves signs.\<close>
lemma inverse_le_imp_le_neg:
"inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
apply (rule classical)
apply (subgoal_tac "a < 0")
prefer 2 apply force
apply (insert inverse_le_imp_le [of "-b" "-a"])
done

lemma less_imp_inverse_less_neg:
"a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
apply (subgoal_tac "a < 0")
prefer 2 apply (blast intro: less_trans)
apply (insert less_imp_inverse_less [of "-b" "-a"])
done

lemma inverse_less_imp_less_neg:
"inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
apply (rule classical)
apply (subgoal_tac "a < 0")
prefer 2
apply force
apply (insert inverse_less_imp_less [of "-b" "-a"])
done

lemma inverse_less_iff_less_neg [simp]:
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
apply (insert inverse_less_iff_less [of "-b" "-a"])
apply (simp del: inverse_less_iff_less
done

lemma le_imp_inverse_le_neg:
"a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
by (force simp add: le_less less_imp_inverse_less_neg)

lemma inverse_le_iff_le_neg [simp]:
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)

lemma one_less_inverse:
"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
using less_imp_inverse_less [of a 1, unfolded inverse_1] .

lemma one_le_inverse:
"0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
using le_imp_inverse_le [of a 1, unfolded inverse_1] .

lemma pos_le_divide_eq [field_simps]:
assumes "0 < c"
shows "a \<le> b / c \<longleftrightarrow> a * c \<le> b"
proof -
from assms have "a \<le> b / c \<longleftrightarrow> a * c \<le> (b / c) * c"
using mult_le_cancel_right [of a c "b * inverse c"] by (auto simp add: field_simps)
also have "... \<longleftrightarrow> a * c \<le> b"
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma pos_less_divide_eq [field_simps]:
assumes "0 < c"
shows "a < b / c \<longleftrightarrow> a * c < b"
proof -
from assms have "a < b / c \<longleftrightarrow> a * c < (b / c) * c"
using mult_less_cancel_right [of a c "b / c"] by auto
also have "... = (a*c < b)"
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma neg_less_divide_eq [field_simps]:
assumes "c < 0"
shows "a < b / c \<longleftrightarrow> b < a * c"
proof -
from assms have "a < b / c \<longleftrightarrow> (b / c) * c < a * c"
using mult_less_cancel_right [of "b / c" c a] by auto
also have "... \<longleftrightarrow> b < a * c"
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma neg_le_divide_eq [field_simps]:
assumes "c < 0"
shows "a \<le> b / c \<longleftrightarrow> b \<le> a * c"
proof -
from assms have "a \<le> b / c \<longleftrightarrow> (b / c) * c \<le> a * c"
using mult_le_cancel_right [of "b * inverse c" c a] by (auto simp add: field_simps)
also have "... \<longleftrightarrow> b \<le> a * c"
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma pos_divide_le_eq [field_simps]:
assumes "0 < c"
shows "b / c \<le> a \<longleftrightarrow> b \<le> a * c"
proof -
from assms have "b / c \<le> a \<longleftrightarrow> (b / c) * c \<le> a * c"
using mult_le_cancel_right [of "b / c" c a] by auto
also have "... \<longleftrightarrow> b \<le> a * c"
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma pos_divide_less_eq [field_simps]:
assumes "0 < c"
shows "b / c < a \<longleftrightarrow> b < a * c"
proof -
from assms have "b / c < a \<longleftrightarrow> (b / c) * c < a * c"
using mult_less_cancel_right [of "b / c" c a] by auto
also have "... \<longleftrightarrow> b < a * c"
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma neg_divide_le_eq [field_simps]:
assumes "c < 0"
shows "b / c \<le> a \<longleftrightarrow> a * c \<le> b"
proof -
from assms have "b / c \<le> a \<longleftrightarrow> a * c \<le> (b / c) * c"
using mult_le_cancel_right [of a c "b / c"] by auto
also have "... \<longleftrightarrow> a * c \<le> b"
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

lemma neg_divide_less_eq [field_simps]:
assumes "c < 0"
shows "b / c < a \<longleftrightarrow> a * c < b"
proof -
from assms have "b / c < a \<longleftrightarrow> a * c < b / c * c"
using mult_less_cancel_right [of a c "b / c"] by auto
also have "... \<longleftrightarrow> a * c < b"
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

text\<open>The following \<open>field_simps\<close> rules are necessary, as minus is always moved atop of
division but we want to get rid of division.\<close>

lemma pos_le_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> a * c \<le> - b"
unfolding minus_divide_left by (rule pos_le_divide_eq)

lemma neg_le_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> - b \<le> a * c"
unfolding minus_divide_left by (rule neg_le_divide_eq)

lemma pos_less_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a < - (b / c) \<longleftrightarrow> a * c < - b"
unfolding minus_divide_left by (rule pos_less_divide_eq)

lemma neg_less_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a < - (b / c) \<longleftrightarrow> - b < a * c"
unfolding minus_divide_left by (rule neg_less_divide_eq)

lemma pos_minus_divide_less_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) < a \<longleftrightarrow> - b < a * c"
unfolding minus_divide_left by (rule pos_divide_less_eq)

lemma neg_minus_divide_less_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) < a \<longleftrightarrow> a * c < - b"
unfolding minus_divide_left by (rule neg_divide_less_eq)

lemma pos_minus_divide_le_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> - b \<le> a * c"
unfolding minus_divide_left by (rule pos_divide_le_eq)

lemma neg_minus_divide_le_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> a * c \<le> - b"
unfolding minus_divide_left by (rule neg_divide_le_eq)

lemma frac_less_eq:
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )

lemma frac_le_eq:
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )

lemma divide_pos_pos[simp]:
"0 < x ==> 0 < y ==> 0 < x / y"

lemma divide_nonneg_pos:
"0 <= x ==> 0 < y ==> 0 <= x / y"

lemma divide_neg_pos:
"x < 0 ==> 0 < y ==> x / y < 0"

lemma divide_nonpos_pos:
"x <= 0 ==> 0 < y ==> x / y <= 0"

lemma divide_pos_neg:
"0 < x ==> y < 0 ==> x / y < 0"

lemma divide_nonneg_neg:
"0 <= x ==> y < 0 ==> x / y <= 0"

lemma divide_neg_neg:
"x < 0 ==> y < 0 ==> 0 < x / y"

lemma divide_nonpos_neg:
"x <= 0 ==> y < 0 ==> 0 <= x / y"

lemma divide_strict_right_mono:
"[|a < b; 0 < c|] ==> a / c < b / c"
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono
positive_imp_inverse_positive)

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

text\<open>The last premise ensures that \<^term>\<open>a\<close> and \<^term>\<open>b\<close>
have the same sign\<close>
lemma divide_strict_left_mono:
"[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
by (auto simp: field_simps 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"
by (auto simp: field_simps zero_less_mult_iff mult_right_mono)

lemma divide_strict_left_mono_neg:
"[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)

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

lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
z <= x / y"

lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
x / y < z"

lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
z < x / y"

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

lemma frac_less: "0 <= 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 < x ==>
x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
apply (rule mult_imp_div_pos_less)
apply simp_all
apply (rule mult_imp_less_div_pos, assumption)
apply (erule mult_le_less_imp_less)
apply simp_all
done

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

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

subclass unbounded_dense_linorder
proof
fix x y :: 'a
from less_add_one show "\<exists>y. x < y" ..
from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
then have "x - 1 < x + 1 - 1" by simp
then have "x - 1 < x" by (simp add: algebra_simps)
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

subclass field_abs_sgn ..

lemma inverse_sgn [simp]:
"inverse (sgn a) = sgn a"
by (cases a 0 rule: linorder_cases) simp_all

lemma divide_sgn [simp]:
"a / sgn b = a * sgn b"
by (cases b 0 rule: linorder_cases) simp_all

lemma nonzero_abs_inverse:
"a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
by (rule abs_inverse)

lemma nonzero_abs_divide:
"b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
by (rule abs_divide)

lemma field_le_epsilon:
assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
shows "x \<le> y"
proof (rule dense_le)
fix t assume "t < x"
hence "0 < x - t" by (simp add: less_diff_eq)
from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
then show "t \<le> y" by (simp add: algebra_simps)
qed

lemma inverse_positive_iff_positive [simp]:
"(0 < inverse a) = (0 < a)"
apply (cases "a = 0", simp)
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
done

lemma inverse_negative_iff_negative [simp]:
"(inverse a < 0) = (a < 0)"
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 \<longleftrightarrow> 0 \<le> a"

lemma inverse_nonpositive_iff_nonpositive [simp]:
"inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"

lemma one_less_inverse_iff: "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
using less_trans[of 1 x 0 for x]
by (cases x 0 rule: linorder_cases) (auto simp add: field_simps)

lemma one_le_inverse_iff: "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
proof (cases "x = 1")
case True then show ?thesis by simp
next
case False then have "inverse x \<noteq> 1" by simp
then have "1 \<noteq> inverse x" by blast
then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
with False show ?thesis by (auto simp add: one_less_inverse_iff)
qed

lemma inverse_less_1_iff: "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
by (simp add: not_le [symmetric] one_le_inverse_iff)

lemma inverse_le_1_iff: "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
by (simp add: not_less [symmetric] one_less_inverse_iff)

lemma [divide_simps]:
shows le_divide_eq: "a \<le> b / c \<longleftrightarrow> (if 0 < c then a * c \<le> b else if c < 0 then b \<le> a * c else a \<le> 0)"
and divide_le_eq: "b / c \<le> a \<longleftrightarrow> (if 0 < c then b \<le> a * c else if c < 0 then a * c \<le> b else 0 \<le> a)"
and less_divide_eq: "a < b / c \<longleftrightarrow> (if 0 < c then a * c < b else if c < 0 then b < a * c else a < 0)"
and divide_less_eq: "b / c < a \<longleftrightarrow> (if 0 < c then b < a * c else if c < 0 then a * c < b else 0 < a)"
and le_minus_divide_eq: "a \<le> - (b / c) \<longleftrightarrow> (if 0 < c then a * c \<le> - b else if c < 0 then - b \<le> a * c else a \<le> 0)"
and minus_divide_le_eq: "- (b / c) \<le> a \<longleftrightarrow> (if 0 < c then - b \<le> a * c else if c < 0 then a * c \<le> - b else 0 \<le> a)"
and less_minus_divide_eq: "a < - (b / c) \<longleftrightarrow> (if 0 < c then a * c < - b else if c < 0 then - b < a * c else  a < 0)"
and minus_divide_less_eq: "- (b / c) < a \<longleftrightarrow> (if 0 < c then - b < a * c else if c < 0 then a * c < - b else 0 < a)"
by (auto simp: field_simps not_less dest: antisym)

text \<open>Division and Signs\<close>

lemma
shows zero_less_divide_iff: "0 < a / b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
and divide_less_0_iff: "a / b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
and zero_le_divide_iff: "0 \<le> a / b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
and divide_le_0_iff: "a / b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"

text \<open>Division and the Number One\<close>

text\<open>Simplify expressions equated with 1\<close>

lemma zero_eq_1_divide_iff [simp]: "0 = 1 / a \<longleftrightarrow> a = 0"
by (cases "a = 0") (auto simp: field_simps)

lemma one_divide_eq_0_iff [simp]: "1 / a = 0 \<longleftrightarrow> a = 0"
using zero_eq_1_divide_iff[of a] by simp

text\<open>Simplify expressions such as \<open>0 < 1/x\<close> to \<open>0 < x\<close>\<close>

lemma zero_le_divide_1_iff [simp]:
"0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"

lemma zero_less_divide_1_iff [simp]:
"0 < 1 / a \<longleftrightarrow> 0 < a"

lemma divide_le_0_1_iff [simp]:
"1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"

lemma divide_less_0_1_iff [simp]:
"1 / a < 0 \<longleftrightarrow> a < 0"

lemma divide_right_mono:
"[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
by (force simp add: divide_strict_right_mono le_less)

lemma divide_right_mono_neg: "a <= b
==> c <= 0 ==> b / c <= a / c"
apply (drule divide_right_mono [of _ _ "- c"])
apply auto
done

lemma divide_left_mono_neg: "a <= b
==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
apply (drule divide_left_mono [of _ _ "- c"])
done

lemma inverse_le_iff: "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
(auto simp add: field_simps zero_less_mult_iff mult_le_0_iff)

lemma inverse_less_iff: "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
by (subst less_le) (auto simp: inverse_le_iff)

lemma divide_le_cancel: "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"

lemma divide_less_cancel: "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
by (auto simp add: divide_inverse mult_less_cancel_right)

text\<open>Simplify quotients that are compared with the value 1.\<close>

lemma le_divide_eq_1:
"(1 \<le> b / a) = ((0 < a \<and> a \<le> b) \<or> (a < 0 \<and> b \<le> a))"

lemma divide_le_eq_1:
"(b / a \<le> 1) = ((0 < a \<and> b \<le> a) \<or> (a < 0 \<and> a \<le> b) \<or> a=0)"

lemma less_divide_eq_1:
"(1 < b / a) = ((0 < a \<and> a < b) \<or> (a < 0 \<and> b < a))"

lemma divide_less_eq_1:
"(b / a < 1) = ((0 < a \<and> b < a) \<or> (a < 0 \<and> a < b) \<or> a=0)"

lemma divide_nonneg_nonneg [simp]:
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x / y"

lemma divide_nonpos_nonpos:
"x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> 0 \<le> x / y"

lemma divide_nonneg_nonpos:
"0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> x / y \<le> 0"

lemma divide_nonpos_nonneg:
"x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x / y \<le> 0"

text \<open>Conditional Simplification Rules: No Case Splits\<close>

lemma le_divide_eq_1_pos [simp]:
"0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"

lemma le_divide_eq_1_neg [simp]:
"a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"

lemma divide_le_eq_1_pos [simp]:
"0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"

lemma divide_le_eq_1_neg [simp]:
"a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"

lemma less_divide_eq_1_pos [simp]:
"0 < a \<Longrightarrow> (1 < b/a) = (a < b)"

lemma less_divide_eq_1_neg [simp]:
"a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"

lemma divide_less_eq_1_pos [simp]:
"0 < a \<Longrightarrow> (b/a < 1) = (b < a)"

lemma divide_less_eq_1_neg [simp]:
"a < 0 \<Longrightarrow> b/a < 1 \<longleftrightarrow> a < b"

lemma eq_divide_eq_1 [simp]:
"(1 = b/a) = ((a \<noteq> 0 \<and> a = b))"

lemma divide_eq_eq_1 [simp]:
"(b/a = 1) = ((a \<noteq> 0 \<and> a = b))"

lemma abs_div_pos: "0 < y ==>
\<bar>x\<bar> / y = \<bar>x / y\<bar>"
apply (subst abs_divide)
done

lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / \<bar>b\<bar>) = (0 \<le> a \<or> b = 0)"
by (auto simp: zero_le_divide_iff)

lemma divide_le_0_abs_iff [simp]: "(a / \<bar>b\<bar> \<le> 0) = (a \<le> 0 \<or> b = 0)"
by (auto simp: divide_le_0_iff)

lemma field_le_mult_one_interval:
assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
shows "x \<le> y"
proof (cases "0 < x")
assume "0 < x"
thus ?thesis
using dense_le_bounded[of 0 1 "y/x"] *
unfolding le_divide_eq if_P[OF \<open>0 < x\<close>] by simp
next
assume "\<not>0 < x" hence "x \<le> 0" by simp
obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1::'a"] by auto
hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] \<open>x \<le> 0\<close> by auto
also note *[OF s]
finally show ?thesis .
qed

text\<open>For creating values between \<^term>\<open>u\<close> and \<^term>\<open>v\<close>.\<close>
lemma scaling_mono:
assumes "u \<le> v" "0 \<le> r" "r \<le> s"
shows "u + r * (v - u) / s \<le> v"
proof -
have "r/s \<le> 1" using assms
using divide_le_eq_1 by fastforce
then have "(r/s) * (v - u) \<le> 1 * (v - u)"
apply (rule mult_right_mono)
using assms by simp
then show ?thesis
qed

end

text \<open>Min/max Simplification Rules\<close>

lemma min_mult_distrib_left:
fixes x::"'a::linordered_idom"
shows "p * min x y = (if 0 \<le> p then min (p*x) (p*y) else max (p*x) (p*y))"
by (auto simp add: min_def max_def mult_le_cancel_left)

lemma min_mult_distrib_right:
fixes x::"'a::linordered_idom"
shows "min x y * p = (if 0 \<le> p then min (x*p) (y*p) else max (x*p) (y*p))"
by (auto simp add: min_def max_def mult_le_cancel_right)

lemma min_divide_distrib_right:
fixes x::"'a::linordered_field"
shows "min x y / p = (if 0 \<le> p then min (x/p) (y/p) else max (x/p) (y/p))"

lemma max_mult_distrib_left:
fixes x::"'a::linordered_idom"
shows "p * max x y = (if 0 \<le> p then max (p*x) (p*y) else min (p*x) (p*y))"
by (auto simp add: min_def max_def mult_le_cancel_left)

lemma max_mult_distrib_right:
fixes x::"'a::linordered_idom"
shows "max x y * p = (if 0 \<le> p then max (x*p) (y*p) else min (x*p) (y*p))"
by (auto simp add: min_def max_def mult_le_cancel_right)

lemma max_divide_distrib_right:
fixes x::"'a::linordered_field"
shows "max x y / p = (if 0 \<le> p then max (x/p) (y/p) else min (x/p) (y/p))"