# Theory Fields

```(*  Title:      HOL/Fields.thy
Author:     Gertrud Bauer
Author:     Steven Obua
Author:     Tobias Nipkow
Author:     Lawrence C Paulson
Author:     Markus Wenzel
Author:     Jeremy Avigad
*)

section ‹Fields›

theory Fields
imports Nat
begin

subsection ‹Division rings›

text ‹
A division ring is like a field, but without the commutativity requirement.
›

class inverse = divide +
fixes inverse :: "'a ⇒ 'a"
begin

abbreviation inverse_divide :: "'a ⇒ 'a ⇒ 'a"  (infixl "'/" 70)
where
"inverse_divide ≡ divide"

end

text ‹Setup for linear arithmetic prover›

ML_file ‹~~/src/Provers/Arith/fast_lin_arith.ML›
ML_file ‹Tools/lin_arith.ML›
setup ‹Lin_Arith.global_setup›
declaration ‹K (
Lin_Arith.init_arith_data
#> Lin_Arith.add_discrete_type \<^type_name>‹nat›
#> Lin_Arith.add_lessD @{thm Suc_leI}
#> Lin_Arith.add_simps @{thms simp_thms ring_distribs if_True if_False
minus_diff_eq
add_0_left add_0_right order_less_irrefl
zero_neq_one zero_less_one zero_le_one
zero_neq_one [THEN not_sym] not_one_le_zero not_one_less_zero
add_Suc add_Suc_right nat.inject
Suc_le_mono Suc_less_eq Zero_not_Suc
Suc_not_Zero le_0_eq One_nat_def}
#> Lin_Arith.add_simprocs [\<^simproc>‹group_cancel_add›, \<^simproc>‹group_cancel_diff›,
\<^simproc>‹group_cancel_eq›, \<^simproc>‹group_cancel_le›,
\<^simproc>‹group_cancel_less›,
\<^simproc>‹nateq_cancel_sums›,\<^simproc>‹natless_cancel_sums›,
\<^simproc>‹natle_cancel_sums›])›

simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) ≤ n" | "(m::nat) = n") =
‹K Lin_Arith.simproc› ― ‹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
\<^text>‹fast_nat_arith_simproc› anyway. However, it seems cheaper to activate the
solver all the time rather than add the additional check.›

lemmas [linarith_split] = nat_diff_split split_min split_max abs_split

text‹Lemmas ‹divide_simps› move division to the outside and eliminates them on (in)equalities.›

named_theorems divide_simps "rewrite rules to eliminate divisions"

class division_ring = ring_1 + inverse +
assumes left_inverse [simp]:  "a ≠ 0 ⟹ inverse a * a = 1"
assumes right_inverse [simp]: "a ≠ 0 ⟹ 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 ≠ 0" and b: "b ≠ 0"
show "a * b ≠ 0"
proof
assume ab: "a * b = 0"
hence "0 = inverse a * (a * b) * inverse b" by simp
also have "… = (inverse a * a) * (b * inverse b)"
by (simp only: mult.assoc)
also have "… = 1" using a b by simp
finally show False by simp
qed
qed

lemma nonzero_imp_inverse_nonzero:
"a ≠ 0 ⟹ inverse a ≠ 0"
proof
assume ianz: "inverse a = 0"
assume "a ≠ 0"
hence "1 = a * inverse a" by simp
also have "... = 0" by (simp add: ianz)
finally have "1 = 0" .
thus False by (simp add: eq_commute)
qed

lemma inverse_zero_imp_zero:
assumes "inverse a = 0" shows "a = 0"
proof (rule ccontr)
assume "a ≠ 0"
then have "inverse a ≠ 0"
by (simp add: nonzero_imp_inverse_nonzero)
with assms show False
by auto
qed

lemma inverse_unique:
assumes ab: "a * b = 1"
shows "inverse a = b"
proof -
have "a ≠ 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 ≠ 0 ⟹ inverse (- a) = - inverse a"
by (rule inverse_unique) simp

lemma nonzero_inverse_inverse_eq:
"a ≠ 0 ⟹ inverse (inverse a) = a"
by (rule inverse_unique) simp

lemma nonzero_inverse_eq_imp_eq:
assumes "inverse a = inverse b" and "a ≠ 0" and "b ≠ 0"
shows "a = b"
proof -
from ‹inverse a = inverse b›
have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
with ‹a ≠ 0› and ‹b ≠ 0› show "a = b"
by (simp add: nonzero_inverse_inverse_eq)
qed

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

lemma nonzero_inverse_mult_distrib:
assumes "a ≠ 0" and "b ≠ 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

lemma division_ring_inverse_add:
"a ≠ 0 ⟹ b ≠ 0 ⟹ inverse a + inverse b = inverse a * (a + b) * inverse b"
by (simp add: algebra_simps)

lemma division_ring_inverse_diff:
"a ≠ 0 ⟹ b ≠ 0 ⟹ inverse a - inverse b = inverse a * (b - a) * inverse b"
by (simp add: algebra_simps)

lemma right_inverse_eq: "b ≠ 0 ⟹ a / b = 1 ⟷ a = b"
proof
assume neq: "b ≠ 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 ≠ 0 ⟹ inverse a = 1 / a"
by (simp add: divide_inverse)

lemma divide_self [simp]: "a ≠ 0 ⟹ a / a = 1"
by (simp add: divide_inverse)

lemma inverse_eq_divide [field_simps, field_split_simps, divide_simps]: "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 algebra_simps)

lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
by (simp add: divide_inverse mult.assoc)

lemma minus_divide_left: "- (a / b) = (-a) / b"
by (simp add: divide_inverse)

lemma nonzero_minus_divide_right: "b ≠ 0 ⟹ - (a / b) = a / (- b)"
by (simp add: divide_inverse nonzero_inverse_minus_eq)

lemma nonzero_minus_divide_divide: "b ≠ 0 ⟹ (-a) / (-b) = a / b"
by (simp add: divide_inverse nonzero_inverse_minus_eq)

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

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 ≠ 0 ⟹ a = b / c ⟷ a * c = b"
proof -
assume [simp]: "c ≠ 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 [field_simps]: "c ≠ 0 ⟹ b / c = a ⟷ b = a * c"
proof -
assume [simp]: "c ≠ 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 nonzero_neg_divide_eq_eq [field_simps]: "b ≠ 0 ⟹ - (a / b) = c ⟷ - a = c * b"
using nonzero_divide_eq_eq[of b "-a" c] by simp

lemma nonzero_neg_divide_eq_eq2 [field_simps]: "b ≠ 0 ⟹ c = - (a / b) ⟷ c * b = - a"
using nonzero_neg_divide_eq_eq[of b a c] by auto

lemma divide_eq_imp: "c ≠ 0 ⟹ b = a * c ⟹ b / c = a"
by (simp add: divide_inverse mult.assoc)

lemma eq_divide_imp: "c ≠ 0 ⟹ a * c = b ⟹ a = b / c"
by (drule sym) (simp add: divide_inverse mult.assoc)

lemma add_divide_eq_iff [field_simps]:
"z ≠ 0 ⟹ x + y / z = (x * z + y) / z"
by (simp add: add_divide_distrib nonzero_eq_divide_eq)

lemma divide_add_eq_iff [field_simps]:
"z ≠ 0 ⟹ x / z + y = (x + y * z) / z"
by (simp add: add_divide_distrib nonzero_eq_divide_eq)

lemma diff_divide_eq_iff [field_simps]:
"z ≠ 0 ⟹ x - y / z = (x * z - y) / z"
by (simp add: diff_divide_distrib nonzero_eq_divide_eq eq_diff_eq)

lemma minus_divide_add_eq_iff [field_simps]:
"z ≠ 0 ⟹ - (x / z) + y = (- x + y * z) / z"
by (simp add: add_divide_distrib diff_divide_eq_iff)

lemma divide_diff_eq_iff [field_simps]:
"z ≠ 0 ⟹ x / z - y = (x - y * z) / z"
by (simp add: field_simps)

lemma minus_divide_diff_eq_iff [field_simps]:
"z ≠ 0 ⟹ - (x / z) - y = (- x - y * z) / z"
by (simp add: divide_diff_eq_iff[symmetric])

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

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 ⟷ a = 0"
by (rule iffI) (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≠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≠0"
thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
qed

lemma inverse_eq_imp_eq:
"inverse a = inverse b ⟹ a = b"
by (drule arg_cong [where f="inverse"], simp)

lemma inverse_eq_iff_eq [simp]:
"inverse a = inverse b ⟷ 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 "… = inverse y * (x * y * inverse y)"
by (simp add: mult.assoc assms)
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 ∨ c = 0")
(auto simp: divide_inverse mult.assoc nonzero_inverse_mult_distrib)

lemma add_divide_eq_if_simps [field_split_simps, divide_simps]:
"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)"
by (simp_all add: add_divide_eq_iff divide_add_eq_iff diff_divide_eq_iff divide_diff_eq_iff
minus_divide_diff_eq_iff)

lemma [field_split_simps, divide_simps]:
shows divide_eq_eq: "b / c = a ⟷ (if c ≠ 0 then b = a * c else a = 0)"
and eq_divide_eq: "a = b / c ⟷ (if c ≠ 0 then a * c = b else a = 0)"
and minus_divide_eq_eq: "- (b / c) = a ⟷ (if c ≠ 0 then - b = a * c else a = 0)"
and eq_minus_divide_eq: "a = - (b / c) ⟷ (if c ≠ 0 then a * c = - b else a = 0)"
by (auto simp add:  field_simps)

end

subsection ‹Fields›

class field = comm_ring_1 + inverse +
assumes field_inverse: "a ≠ 0 ⟹ 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 ≠ 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 ≠ 0"
then show "a * b / b = a"
by (simp add: divide_inverse ac_simps)
next
fix a
show "a / 0 = 0"
by (simp add: divide_inverse)
qed

text‹There is no slick version using division by zero.›
lemma inverse_add:
"a ≠ 0 ⟹ b ≠ 0 ⟹ inverse a + inverse b = (a + b) * inverse a * inverse b"
by (simp add: division_ring_inverse_add ac_simps)

lemma nonzero_mult_divide_mult_cancel_left [simp]:
assumes [simp]: "c ≠ 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)"
by (simp add: divide_inverse nonzero_inverse_mult_distrib)
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 ≠ 0 ⟹ (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"
by (simp add: divide_inverse ac_simps)

lemma divide_inverse_commute: "a / b = inverse b * a"
by (simp add: divide_inverse mult.commute)

lemma add_frac_eq:
assumes "y ≠ 0" and "z ≠ 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 "… = (x * z + y * w) / (y * z)"
by (simp only: add_divide_distrib)
finally show ?thesis
by (simp only: mult.commute)
qed

text‹Special Cancellation Simprules for Division›

lemma nonzero_divide_mult_cancel_right [simp]:
"b ≠ 0 ⟹ 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 ≠ 0 ⟹ 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 ≠ 0 ⟹ (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 ≠ 0 ⟹ (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 ≠ 0 ⟹ z ≠ 0 ⟹ x / y - w / z = (x * z - w * y) / (y * z)"
by (simp add: field_simps)

lemma frac_eq_eq:
"y ≠ 0 ⟹ z ≠ 0 ⟹ (x / y = w / z) = (x * z = w * y)"
by (simp add: field_simps)

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

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"
proof cases
assume "a ≠ 0 ∧ b ≠ 0"
thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps)
next
assume "¬ (a ≠ 0 ∧ b ≠ 0)"
thus ?thesis by force
qed

lemma inverse_divide [simp]:
"inverse (a / b) = b / a"
by (simp add: divide_inverse mult.commute)

text ‹Calculations with fractions›

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

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"
by (simp add: divide_inverse ac_simps)

lemma divide_divide_eq_left [simp]:
"(a / b) / c = a / (b * c)"
by (simp add: divide_inverse mult.assoc)

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

text ‹Special Cancellation Simprules for Division›

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

text ‹Division and Unary Minus›

lemma minus_divide_right:
"- (a / b) = a / - b"
by (simp add: divide_inverse)

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

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 ⟷ x = 1"
using inverse_eq_iff_eq [of x 1] by simp

lemma divide_eq_0_iff [simp]:
"a / b = 0 ⟷ a = 0 ∨ b = 0"
by (simp add: divide_inverse)

lemma divide_cancel_right [simp]:
"a / c = b / c ⟷ c = 0 ∨ a = b"
by (cases "c=0") (simp_all add: divide_inverse)

lemma divide_cancel_left [simp]:
"c / a = c / b ⟷ c = 0 ∨ a = b"
by (cases "c=0") (simp_all add: divide_inverse)

lemma divide_eq_1_iff [simp]:
"a / b = 1 ⟷ b ≠ 0 ∧ a = b"
by (cases "b=0") (simp_all add: right_inverse_eq)

lemma one_eq_divide_iff [simp]:
"1 = a / b ⟷ b ≠ 0 ∧ a = b"
by (simp add: eq_commute [of 1])

lemma divide_eq_minus_1_iff:
"(a / b = - 1) ⟷ b ≠ 0 ∧ a = - b"
using divide_eq_1_iff by fastforce

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

lemma add_frac_num:
"y ≠ 0 ⟹ x / y + z = (x + z * y) / y"
by (simp add: add_divide_distrib)

lemma add_num_frac:
"y ≠ 0 ⟹ z + x / y = (x + z * y) / y"
by (simp add: add_divide_distrib add.commute)

lemma dvd_field_iff:
"a dvd b ⟷ (a = 0 ⟶ b = 0)"
proof (cases "a = 0")
case False
then have "b = a * (b / a)"
by (simp add: field_simps)
then have "a dvd b" ..
with False show ?thesis
by simp
qed simp

lemma inj_divide_right [simp]:
"inj (λb. b / a) ⟷ a ≠ 0"
proof -
have "(λb. b / a) = (*) (inverse a)"
by (simp add: field_simps fun_eq_iff)
then have "inj (λy. y / a) ⟷ inj ((*) (inverse a))"
by simp
also have "… ⟷ inverse a ≠ 0"
by simp
also have "… ⟷ a ≠ 0"
by simp
finally show ?thesis
by simp
qed

end

class field_char_0 = field + ring_char_0

subsection ‹Ordered fields›

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"
by (simp add: sgn_mult)
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)"
by (simp add: ac_simps)
with False show ?thesis
by (simp add: sgn_eq_0_iff)
qed

lemma abs_inverse [simp]:
"¦inverse a¦ = inverse ¦a¦"
proof -
from sgn_mult_abs [of "inverse a"] sgn_mult_abs [of a]
have "inverse (sgn a) * ¦inverse a¦ = inverse (sgn a * ¦a¦)"
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]:
"¦a / b¦ = ¦a¦ / ¦b¦"
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 ⟹ inverse a < 0"
using positive_imp_inverse_positive [of "-a"]
by (simp add: nonzero_inverse_minus_eq less_imp_not_eq)

lemma inverse_le_imp_le:
assumes invle: "inverse a ≤ inverse b" and apos: "0 < a"
shows "b ≤ a"
proof (rule classical)
assume "¬ b ≤ a"
hence "a < b"  by (simp add: linorder_not_le)
hence bpos: "0 < b"  by (blast intro: apos less_trans)
hence "a * inverse a ≤ a * inverse b"
by (simp add: apos invle less_imp_le mult_left_mono)
hence "(a * inverse a) * b ≤ (a * inverse b) * b"
by (simp add: bpos less_imp_le mult_right_mono)
thus "b ≤ 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 ≠ 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 ≠ 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:
"∀x. ∃y. y < x"
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 "∃y. y < x" by blast
qed

lemma linordered_field_no_ub:
"∀ x. ∃y. y > x"
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 "∃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 "¬ inverse b < inverse a"
hence "inverse a ≤ inverse b" by simp
hence "¬ (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:
assumes "inverse a < inverse b" "0 < a"
shows "b < a"
proof -
have "a ≠ b"
using assms by (simp add: less_le)
moreover have "b ≤ a"
using assms by (force simp: less_le dest: inverse_le_imp_le)
ultimately show ?thesis
by (simp add: less_le)
qed

text‹Both premises are essential. Consider -1 and 1.›
lemma inverse_less_iff_less [simp]:
"0 < a ⟹ 0 < b ⟹ inverse a < inverse b ⟷ b < a"
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)

lemma le_imp_inverse_le:
"a ≤ b ⟹ 0 < a ⟹ inverse b ≤ inverse a"
by (force simp add: le_less less_imp_inverse_less)

lemma inverse_le_iff_le [simp]:
"0 < a ⟹ 0 < b ⟹ inverse a ≤ inverse b ⟷ b ≤ a"
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:
assumes "inverse a ≤ inverse b" "b < 0"
shows "b ≤ a"
proof (rule classical)
assume "¬ b ≤ a"
with ‹b < 0› have "a < 0"
by force
with assms show "b ≤ a"
using inverse_le_imp_le [of "-b" "-a"] by (simp add: nonzero_inverse_minus_eq)
qed

lemma less_imp_inverse_less_neg:
assumes "a < b" "b < 0"
shows "inverse b < inverse a"
proof -
have "a < 0"
using assms by (blast intro: less_trans)
with less_imp_inverse_less [of "-b" "-a"] show ?thesis
by (simp add: nonzero_inverse_minus_eq assms)
qed

lemma inverse_less_imp_less_neg:
assumes "inverse a < inverse b" "b < 0"
shows "b < a"
proof (rule classical)
assume "¬ b < a"
with ‹b < 0› have "a < 0"
by force
with inverse_less_imp_less [of "-b" "-a"] show ?thesis
by (simp add: nonzero_inverse_minus_eq assms)
qed

lemma inverse_less_iff_less_neg [simp]:
"a < 0 ⟹ b < 0 ⟹ inverse a < inverse b ⟷ b < a"
using inverse_less_iff_less [of "-b" "-a"]
by (simp del: inverse_less_iff_less add: nonzero_inverse_minus_eq)

lemma le_imp_inverse_le_neg:
"a ≤ b ⟹ b < 0 ⟹ inverse b ≤ inverse a"
by (force simp add: le_less less_imp_inverse_less_neg)

lemma inverse_le_iff_le_neg [simp]:
"a < 0 ⟹ b < 0 ⟹ inverse a ≤ inverse b ⟷ b ≤ a"
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)

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

lemma one_le_inverse:
"0 < a ⟹ a ≤ 1 ⟹ 1 ≤ 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 ≤ b / c ⟷ a * c ≤ b"
proof -
from assms have "a ≤ b / c ⟷ a * c ≤ (b / c) * c"
using mult_le_cancel_right [of a c "b * inverse c"] by (auto simp add: field_simps)
also have "... ⟷ a * c ≤ 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 ⟷ a * c < b"
proof -
from assms have "a < b / c ⟷ 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 ⟷ b < a * c"
proof -
from assms have "a < b / c ⟷ (b / c) * c < a * c"
using mult_less_cancel_right [of "b / c" c a] by auto
also have "... ⟷ 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 ≤ b / c ⟷ b ≤ a * c"
proof -
from assms have "a ≤ b / c ⟷ (b / c) * c ≤ a * c"
using mult_le_cancel_right [of "b * inverse c" c a] by (auto simp add: field_simps)
also have "... ⟷ b ≤ 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 ≤ a ⟷ b ≤ a * c"
proof -
from assms have "b / c ≤ a ⟷ (b / c) * c ≤ a * c"
using mult_le_cancel_right [of "b / c" c a] by auto
also have "... ⟷ b ≤ 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 ⟷ b < a * c"
proof -
from assms have "b / c < a ⟷ (b / c) * c < a * c"
using mult_less_cancel_right [of "b / c" c a] by auto
also have "... ⟷ 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 ≤ a ⟷ a * c ≤ b"
proof -
from assms have "b / c ≤ a ⟷ a * c ≤ (b / c) * c"
using mult_le_cancel_right [of a c "b / c"] by auto
also have "... ⟷ a * c ≤ 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 ⟷ a * c < b"
proof -
from assms have "b / c < a ⟷ 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_eq [OF assms] divide_inverse mult.assoc)
finally show ?thesis .
qed

text‹The following ‹field_simps› rules are necessary, as minus is always moved atop of
division but we want to get rid of division.›

lemma pos_le_minus_divide_eq [field_simps]: "0 < c ⟹ a ≤ - (b / c) ⟷ a * c ≤ - b"
unfolding minus_divide_left by (rule pos_le_divide_eq)

lemma neg_le_minus_divide_eq [field_simps]: "c < 0 ⟹ a ≤ - (b / c) ⟷ - b ≤ a * c"
unfolding minus_divide_left by (rule neg_le_divide_eq)

lemma pos_less_minus_divide_eq [field_simps]: "0 < c ⟹ a < - (b / c) ⟷ a * c < - b"
unfolding minus_divide_left by (rule pos_less_divide_eq)

lemma neg_less_minus_divide_eq [field_simps]: "c < 0 ⟹ a < - (b / c) ⟷ - b < a * c"
unfolding minus_divide_left by (rule neg_less_divide_eq)

lemma pos_minus_divide_less_eq [field_simps]: "0 < c ⟹ - (b / c) < a ⟷ - b < a * c"
unfolding minus_divide_left by (rule pos_divide_less_eq)

lemma neg_minus_divide_less_eq [field_simps]: "c < 0 ⟹ - (b / c) < a ⟷ a * c < - b"
unfolding minus_divide_left by (rule neg_divide_less_eq)

lemma pos_minus_divide_le_eq [field_simps]: "0 < c ⟹ - (b / c) ≤ a ⟷ - b ≤ a * c"
unfolding minus_divide_left by (rule pos_divide_le_eq)

lemma neg_minus_divide_le_eq [field_simps]: "c < 0 ⟹ - (b / c) ≤ a ⟷ a * c ≤ - b"
unfolding minus_divide_left by (rule neg_divide_le_eq)

lemma frac_less_eq:
"y ≠ 0 ⟹ z ≠ 0 ⟹ x / y < w / z ⟷ (x * z - w * y) / (y * z) < 0"
by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )

lemma frac_le_eq:
"y ≠ 0 ⟹ z ≠ 0 ⟹ x / y ≤ w / z ⟷ (x * z - w * y) / (y * z) ≤ 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"
by(simp add:field_simps)

lemma divide_nonneg_pos:
"0 ≤ x ⟹ 0 < y ⟹ 0 ≤ x / y"
by(simp add:field_simps)

lemma divide_neg_pos:
"x < 0 ⟹ 0 < y ⟹ x / y < 0"
by(simp add:field_simps)

lemma divide_nonpos_pos:
"x ≤ 0 ⟹ 0 < y ⟹ x / y ≤ 0"
by(simp add:field_simps)

lemma divide_pos_neg:
"0 < x ⟹ y < 0 ⟹ x / y < 0"
by(simp add:field_simps)

lemma divide_nonneg_neg:
"0 ≤ x ⟹ y < 0 ⟹ x / y ≤ 0"
by(simp add:field_simps)

lemma divide_neg_neg:
"x < 0 ⟹ y < 0 ⟹ 0 < x / y"
by(simp add:field_simps)

lemma divide_nonpos_neg:
"x ≤ 0 ⟹ y < 0 ⟹ 0 ≤ x / y"
by(simp add:field_simps)

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:
assumes "b < a" "c < 0" shows "a / c < b / c"
proof -
have "b / - c < a / - c"
by (rule divide_strict_right_mono) (use assms in auto)
then show ?thesis
by (simp add: less_imp_not_eq)
qed

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"
by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)

lemma divide_left_mono:
"⟦b ≤ a; 0 ≤ c; 0 < a*b⟧ ⟹ c / a ≤ 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"
by(simp add:field_simps)

lemma mult_imp_div_pos_less: "0 < y ⟹ x < z * y ⟹ x / y < z"
by(simp add:field_simps)

lemma mult_imp_less_div_pos: "0 < y ⟹ z * y < x ⟹ z < x / y"
by(simp add:field_simps)

lemma frac_le:
assumes "0 ≤ y" "x ≤ y" "0 < w" "w ≤ z"
shows "x / z ≤ y / w"
proof (rule mult_imp_div_pos_le)
show "z > 0"
using assms by simp
have "x ≤ y * z / w"
proof (rule mult_imp_le_div_pos [OF ‹0 < w›])
show "x * w ≤ y * z"
using assms by (auto intro: mult_mono)
qed
also have "... = y / w * z"
by simp
finally show "x ≤ y / w * z" .
qed

lemma frac_less:
assumes "0 ≤ x" "x < y" "0 < w" "w ≤ z"
shows "x / z < y / w"
proof (rule mult_imp_div_pos_less)
show "z > 0"
using assms by simp
have "x < y * z / w"
proof (rule mult_imp_less_div_pos [OF ‹0 < w›])
show "x * w < y * z"
using assms by (auto intro: mult_less_le_imp_less)
qed
also have "... = y / w * z"
by simp
finally show "x < y / w * z" .
qed

lemma frac_less2:
assumes "0 < x" "x ≤ y" "0 < w" "w < z"
shows "x / z < y / w"
proof (rule mult_imp_div_pos_less)
show "z > 0"
using assms by simp
show "x < y / w * z"
using assms by (force intro: mult_imp_less_div_pos mult_le_less_imp_less)
qed

lemma less_half_sum: "a < b ⟹ a < (a+b) / (1+1)"
by (simp add: field_simps zero_less_two)

lemma gt_half_sum: "a < b ⟹ (a+b)/(1+1) < b"
by (simp add: field_simps zero_less_two)

subclass unbounded_dense_linorder
proof
fix x y :: 'a
from less_add_one show "∃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 "∃y. y < x" ..
show "x < y ⟹ ∃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 ≠ 0 ⟹ ¦inverse a¦ = inverse ¦a¦"
by (rule abs_inverse)

lemma nonzero_abs_divide:
"b ≠ 0 ⟹ ¦a / b¦ = ¦a¦ / ¦b¦"
by (rule abs_divide)

lemma field_le_epsilon:
assumes e: "⋀e. 0 < e ⟹ x ≤ y + e"
shows "x ≤ 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 ≤ x + (y - t)" by (simp add: algebra_simps)
then have "0 ≤ y - t" by (simp only: add_le_cancel_left)
then show "t ≤ y" by (simp add: algebra_simps)
qed

lemma inverse_positive_iff_positive [simp]: "(0 < inverse a) = (0 < a)"
proof (cases "a = 0")
case False
then show ?thesis
by (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
qed auto

lemma inverse_negative_iff_negative [simp]: "(inverse a < 0) = (a < 0)"
proof (cases "a = 0")
case False
then show ?thesis
by (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
qed auto

lemma inverse_nonnegative_iff_nonnegative [simp]: "0 ≤ inverse a ⟷ 0 ≤ a"
by (simp add: not_less [symmetric])

lemma inverse_nonpositive_iff_nonpositive [simp]: "inverse a ≤ 0 ⟷ a ≤ 0"
by (simp add: not_less [symmetric])

lemma one_less_inverse_iff: "1 < inverse x ⟷ 0 < x ∧ 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 ≤ inverse x ⟷ 0 < x ∧ x ≤ 1"
proof (cases "x = 1")
case True then show ?thesis by simp
next
case False then have "inverse x ≠ 1" by simp
then have "1 ≠ inverse x" by blast
then have "1 ≤ inverse x ⟷ 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 ⟷ x ≤ 0 ∨ 1 < x"
by (simp add: not_le [symmetric] one_le_inverse_iff)

lemma inverse_le_1_iff: "inverse x ≤ 1 ⟷ x ≤ 0 ∨ 1 ≤ x"
by (simp add: not_less [symmetric] one_less_inverse_iff)

lemma [field_split_simps, divide_simps]:
shows le_divide_eq: "a ≤ b / c ⟷ (if 0 < c then a * c ≤ b else if c < 0 then b ≤ a * c else a ≤ 0)"
and divide_le_eq: "b / c ≤ a ⟷ (if 0 < c then b ≤ a * c else if c < 0 then a * c ≤ b else 0 ≤ a)"
and less_divide_eq: "a < b / c ⟷ (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 ⟷ (if 0 < c then b < a * c else if c < 0 then a * c < b else 0 < a)"
and le_minus_divide_eq: "a ≤ - (b / c) ⟷ (if 0 < c then a * c ≤ - b else if c < 0 then - b ≤ a * c else a ≤ 0)"
and minus_divide_le_eq: "- (b / c) ≤ a ⟷ (if 0 < c then - b ≤ a * c else if c < 0 then a * c ≤ - b else 0 ≤ a)"
and less_minus_divide_eq: "a < - (b / c) ⟷ (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 ⟷ (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: order.antisym)

text ‹Division and Signs›

lemma
shows zero_less_divide_iff: "0 < a / b ⟷ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0"
and divide_less_0_iff: "a / b < 0 ⟷ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b"
and zero_le_divide_iff: "0 ≤ a / b ⟷ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0"
and divide_le_0_iff: "a / b ≤ 0 ⟷ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b"
by (auto simp add: field_split_simps)

text ‹Division and the Number One›

text‹Simplify expressions equated with 1›

lemma zero_eq_1_divide_iff [simp]: "0 = 1 / a ⟷ a = 0"
by (cases "a = 0") (auto simp: field_simps)

lemma one_divide_eq_0_iff [simp]: "1 / a = 0 ⟷ a = 0"
using zero_eq_1_divide_iff[of a] by simp

text‹Simplify expressions such as ‹0 < 1/x› to ‹0 < x››

lemma zero_le_divide_1_iff [simp]:
"0 ≤ 1 / a ⟷ 0 ≤ a"
by (simp add: zero_le_divide_iff)

lemma zero_less_divide_1_iff [simp]:
"0 < 1 / a ⟷ 0 < a"
by (simp add: zero_less_divide_iff)

lemma divide_le_0_1_iff [simp]:
"1 / a ≤ 0 ⟷ a ≤ 0"
by (simp add: divide_le_0_iff)

lemma divide_less_0_1_iff [simp]:
"1 / a < 0 ⟷ a < 0"
by (simp add: divide_less_0_iff)

lemma divide_right_mono:
"⟦a ≤ b; 0 ≤ c⟧ ⟹ a/c ≤ 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"
by (auto dest: divide_right_mono [of _ _ "- c"])

lemma divide_left_mono_neg: "a ≤ b ⟹ c ≤ 0 ⟹ 0 < a * b ⟹ c / a ≤ c / b"
by (auto simp add: mult.commute dest: divide_left_mono [of _ _ "- c"])

lemma inverse_le_iff: "inverse a ≤ inverse b ⟷ (0 < a * b ⟶ b ≤ a) ∧ (a * b ≤ 0 ⟶ a ≤ 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 ⟷ (0 < a * b ⟶ b < a) ∧ (a * b ≤ 0 ⟶ a < b)"
by (subst less_le) (auto simp: inverse_le_iff)

lemma divide_le_cancel: "a / c ≤ b / c ⟷ (0 < c ⟶ a ≤ b) ∧ (c < 0 ⟶ b ≤ a)"
by (simp add: divide_inverse mult_le_cancel_right)

lemma divide_less_cancel: "a / c < b / c ⟷ (0 < c ⟶ a < b) ∧ (c < 0 ⟶ b < a) ∧ c ≠ 0"
by (auto simp add: divide_inverse mult_less_cancel_right)

text‹Simplify quotients that are compared with the value 1.›

lemma le_divide_eq_1:
"(1 ≤ b / a) = ((0 < a ∧ a ≤ b) ∨ (a < 0 ∧ b ≤ a))"
by (auto simp add: le_divide_eq)

lemma divide_le_eq_1:
"(b / a ≤ 1) = ((0 < a ∧ b ≤ a) ∨ (a < 0 ∧ a ≤ b) ∨ a=0)"
by (auto simp add: divide_le_eq)

lemma less_divide_eq_1:
"(1 < b / a) = ((0 < a ∧ a < b) ∨ (a < 0 ∧ b < a))"
by (auto simp add: less_divide_eq)

lemma divide_less_eq_1:
"(b / a < 1) = ((0 < a ∧ b < a) ∨ (a < 0 ∧ a < b) ∨ a=0)"
by (auto simp add: divide_less_eq)

lemma divide_nonneg_nonneg [simp]:
"0 ≤ x ⟹ 0 ≤ y ⟹ 0 ≤ x / y"
by (auto simp add: field_split_simps)

lemma divide_nonpos_nonpos:
"x ≤ 0 ⟹ y ≤ 0 ⟹ 0 ≤ x / y"
by (auto simp add: field_split_simps)

lemma divide_nonneg_nonpos:
"0 ≤ x ⟹ y ≤ 0 ⟹ x / y ≤ 0"
by (auto simp add: field_split_simps)

lemma divide_nonpos_nonneg:
"x ≤ 0 ⟹ 0 ≤ y ⟹ x / y ≤ 0"
by (auto simp add: field_split_simps)

text ‹Conditional Simplification Rules: No Case Splits›

lemma le_divide_eq_1_pos [simp]:
"0 < a ⟹ (1 ≤ b/a) = (a ≤ b)"
by (auto simp add: le_divide_eq)

lemma le_divide_eq_1_neg [simp]:
"a < 0 ⟹ (1 ≤ b/a) = (b ≤ a)"
by (auto simp add: le_divide_eq)

lemma divide_le_eq_1_pos [simp]:
"0 < a ⟹ (b/a ≤ 1) = (b ≤ a)"
by (auto simp add: divide_le_eq)

lemma divide_le_eq_1_neg [simp]:
"a < 0 ⟹ (b/a ≤ 1) = (a ≤ b)"
by (auto simp add: divide_le_eq)

lemma less_divide_eq_1_pos [simp]:
"0 < a ⟹ (1 < b/a) = (a < b)"
by (auto simp add: less_divide_eq)

lemma less_divide_eq_1_neg [simp]:
"a < 0 ⟹ (1 < b/a) = (b < a)"
by (auto simp add: less_divide_eq)

lemma divide_less_eq_1_pos [simp]:
"0 < a ⟹ (b/a < 1) = (b < a)"
by (auto simp add: divide_less_eq)

lemma divide_less_eq_1_neg [simp]:
"a < 0 ⟹ b/a < 1 ⟷ a < b"
by (auto simp add: divide_less_eq)

lemma eq_divide_eq_1 [simp]:
"(1 = b/a) = ((a ≠ 0 ∧ a = b))"
by (auto simp add: eq_divide_eq)

lemma divide_eq_eq_1 [simp]:
"(b/a = 1) = ((a ≠ 0 ∧ a = b))"
by (auto simp add: divide_eq_eq)

lemma abs_div_pos: "0 < y ⟹ ¦x¦ / y = ¦x / y¦"
by (simp add: order_less_imp_le)

lemma zero_le_divide_abs_iff [simp]: "(0 ≤ a / ¦b¦) = (0 ≤ a ∨ b = 0)"
by (auto simp: zero_le_divide_iff)

lemma divide_le_0_abs_iff [simp]: "(a / ¦b¦ ≤ 0) = (a ≤ 0 ∨ b = 0)"
by (auto simp: divide_le_0_iff)

lemma field_le_mult_one_interval:
assumes *: "⋀z. ⟦ 0 < z ; z < 1 ⟧ ⟹ z * x ≤ y"
shows "x ≤ 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 ‹0 < x›] by simp
next
assume "¬0 < x" hence "x ≤ 0" by simp
obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1::'a"] by auto
hence "x ≤ s * x" using mult_le_cancel_right[of 1 x s] ‹x ≤ 0› by auto
also note *[OF s]
finally show ?thesis .
qed

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

end

text ‹Min/max Simplification Rules›

lemma min_mult_distrib_left:
fixes x::"'a::linordered_idom"
shows "p * min x y = (if 0 ≤ 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 ≤ 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 ≤ p then min (x/p) (y/p) else max (x/p) (y/p))"
by (simp add: min_mult_distrib_right divide_inverse)

lemma max_mult_distrib_left:
fixes x::"'a::linordered_idom"
shows "p * max x y = (if 0 ≤ 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 ≤ 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 ≤ p then max (x/p) (y/p) else min (x/p) (y/p))"
by (simp add: max_mult_distrib_right divide_inverse)

hide_fact (open) field_inverse field_divide_inverse field_inverse_zero

code_identifier
code_module Fields ⇀ (SML) Arith and (OCaml) Arith and (Haskell) Arith

end
```