# Theory Fraction_Field

```(*  Title:      HOL/Computational_Algebra/Fraction_Field.thy
Author:     Amine Chaieb, University of Cambridge
*)

section‹The fraction field of any integral domain›

theory Fraction_Field
imports Main
begin

subsection ‹General fractions construction›

subsubsection ‹Construction of the type of fractions›

context idom begin

definition fractrel :: "'a × 'a ⇒ 'a * 'a ⇒ bool" where
"fractrel = (λx y. snd x ≠ 0 ∧ snd y ≠ 0 ∧ fst x * snd y = fst y * snd x)"

lemma fractrel_iff [simp]:
"fractrel x y ⟷ snd x ≠ 0 ∧ snd y ≠ 0 ∧ fst x * snd y = fst y * snd x"

lemma symp_fractrel: "symp fractrel"

lemma transp_fractrel: "transp fractrel"
proof (rule transpI, unfold split_paired_all)
fix a b a' b' a'' b'' :: 'a
assume A: "fractrel (a, b) (a', b')"
assume B: "fractrel (a', b') (a'', b'')"
have "b' * (a * b'') = b'' * (a * b')" by (simp add: ac_simps)
also from A have "a * b' = a' * b" by auto
also have "b'' * (a' * b) = b * (a' * b'')" by (simp add: ac_simps)
also from B have "a' * b'' = a'' * b'" by auto
also have "b * (a'' * b') = b' * (a'' * b)" by (simp add: ac_simps)
finally have "b' * (a * b'') = b' * (a'' * b)" .
moreover from B have "b' ≠ 0" by auto
ultimately have "a * b'' = a'' * b" by simp
with A B show "fractrel (a, b) (a'', b'')" by auto
qed

lemma part_equivp_fractrel: "part_equivp fractrel"
using _ symp_fractrel transp_fractrel
by(rule part_equivpI)(rule exI[where x="(0, 1)"]; simp)

end

quotient_type (overloaded) 'a fract = "'a :: idom × 'a" / partial: "fractrel"
by(rule part_equivp_fractrel)

subsubsection ‹Representation and basic operations›

lift_definition Fract :: "'a :: idom ⇒ 'a ⇒ 'a fract"
is "λa b. if b = 0 then (0, 1) else (a, b)"
by simp

lemma Fract_cases [cases type: fract]:
obtains (Fract) a b where "q = Fract a b" "b ≠ 0"
by transfer simp

lemma Fract_induct [case_names Fract, induct type: fract]:
"(⋀a b. b ≠ 0 ⟹ P (Fract a b)) ⟹ P q"
by (cases q) simp

lemma eq_fract:
shows "⋀a b c d. b ≠ 0 ⟹ d ≠ 0 ⟹ Fract a b = Fract c d ⟷ a * d = c * b"
and "⋀a. Fract a 0 = Fract 0 1"
and "⋀a c. Fract 0 a = Fract 0 c"
by(transfer; simp)+

instantiation fract :: (idom) comm_ring_1
begin

lift_definition zero_fract :: "'a fract" is "(0, 1)" by simp

lemma Zero_fract_def: "0 = Fract 0 1"
by transfer simp

lift_definition one_fract :: "'a fract" is "(1, 1)" by simp

lemma One_fract_def: "1 = Fract 1 1"
by transfer simp

lift_definition plus_fract :: "'a fract ⇒ 'a fract ⇒ 'a fract"
is "λq r. (fst q * snd r + fst r * snd q, snd q * snd r)"

"⟦ b ≠ 0; d ≠ 0 ⟧ ⟹ Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
by transfer simp

lift_definition uminus_fract :: "'a fract ⇒ 'a fract"
is "λx. (- fst x, snd x)"
by simp

lemma minus_fract [simp]:
fixes a b :: "'a::idom"
shows "- Fract a b = Fract (- a) b"
by transfer simp

lemma minus_fract_cancel [simp]: "Fract (- a) (- b) = Fract a b"
by (cases "b = 0") (simp_all add: eq_fract)

definition diff_fract_def: "q - r = q + - (r::'a fract)"

lemma diff_fract [simp]:
"⟦ b ≠ 0; d ≠ 0 ⟧ ⟹ Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"

lift_definition times_fract :: "'a fract ⇒ 'a fract ⇒ 'a fract"
is "λq r. (fst q * fst r, snd q * snd r)"

lemma mult_fract [simp]: "Fract (a::'a::idom) b * Fract c d = Fract (a * c) (b * d)"
by transfer simp

lemma mult_fract_cancel:
"c ≠ 0 ⟹ Fract (c * a) (c * b) = Fract a b"
by transfer simp

instance
proof
fix q r s :: "'a fract"
show "(q * r) * s = q * (r * s)"
by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
show "q * r = r * q"
by (cases q, cases r) (simp add: eq_fract algebra_simps)
show "1 * q = q"
by (cases q) (simp add: One_fract_def eq_fract)
show "(q + r) + s = q + (r + s)"
by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
show "q + r = r + q"
by (cases q, cases r) (simp add: eq_fract algebra_simps)
show "0 + q = q"
by (cases q) (simp add: Zero_fract_def eq_fract)
show "- q + q = 0"
by (cases q) (simp add: Zero_fract_def eq_fract)
show "q - r = q + - r"
by (cases q, cases r) (simp add: eq_fract)
show "(q + r) * s = q * s + r * s"
by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps)
show "(0::'a fract) ≠ 1"
by (simp add: Zero_fract_def One_fract_def eq_fract)
qed

end

lemma of_nat_fract: "of_nat k = Fract (of_nat k) 1"
by (induct k) (simp_all add: Zero_fract_def One_fract_def)

lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k"
by (rule of_nat_fract [symmetric])

lemma fract_collapse:
"Fract 0 k = 0"
"Fract 1 1 = 1"
"Fract k 0 = 0"
by(transfer; simp)+

lemma fract_expand:
"0 = Fract 0 1"
"1 = Fract 1 1"

lemma Fract_cases_nonzero:
obtains (Fract) a b where "q = Fract a b" and "b ≠ 0" and "a ≠ 0"
| (0) "q = 0"
proof (cases "q = 0")
case True
then show thesis using 0 by auto
next
case False
then obtain a b where "q = Fract a b" and "b ≠ 0" by (cases q) auto
with False have "0 ≠ Fract a b" by simp
with ‹b ≠ 0› have "a ≠ 0" by (simp add: Zero_fract_def eq_fract)
with Fract ‹q = Fract a b› ‹b ≠ 0› show thesis by auto
qed

subsubsection ‹The field of rational numbers›

context idom
begin

subclass ring_no_zero_divisors ..

end

instantiation fract :: (idom) field
begin

lift_definition inverse_fract :: "'a fract ⇒ 'a fract"
is "λx. if fst x = 0 then (0, 1) else (snd x, fst x)"

lemma inverse_fract [simp]: "inverse (Fract a b) = Fract (b::'a::idom) a"
by transfer simp

definition divide_fract_def: "q div r = q * inverse (r:: 'a fract)"

lemma divide_fract [simp]: "Fract a b div Fract c d = Fract (a * d) (b * c)"

instance
proof
fix q :: "'a fract"
assume "q ≠ 0"
then show "inverse q * q = 1"
by (cases q rule: Fract_cases_nonzero)
next
fix q r :: "'a fract"
show "q div r = q * inverse r" by (simp add: divide_fract_def)
next
show "inverse 0 = (0:: 'a fract)"
qed

end

subsubsection ‹The ordered field of fractions over an ordered idom›

instantiation fract :: (linordered_idom) linorder
begin

lemma less_eq_fract_respect:
fixes a b a' b' c d c' d' :: 'a
assumes neq: "b ≠ 0"  "b' ≠ 0"  "d ≠ 0"  "d' ≠ 0"
assumes eq1: "a * b' = a' * b"
assumes eq2: "c * d' = c' * d"
shows "((a * d) * (b * d) ≤ (c * b) * (b * d)) ⟷ ((a' * d') * (b' * d') ≤ (c' * b') * (b' * d'))"
proof -
let ?le = "λa b c d. ((a * d) * (b * d) ≤ (c * b) * (b * d))"
{
fix a b c d x :: 'a
assume x: "x ≠ 0"
have "?le a b c d = ?le (a * x) (b * x) c d"
proof -
from x have "0 < x * x"
then have "?le a b c d =
((a * d) * (b * d) * (x * x) ≤ (c * b) * (b * d) * (x * x))"
also have "... = ?le (a * x) (b * x) c d"
finally show ?thesis .
qed
} note le_factor = this

let ?D = "b * d" and ?D' = "b' * d'"
from neq have D: "?D ≠ 0" by simp
from neq have "?D' ≠ 0" by simp
then have "?le a b c d = ?le (a * ?D') (b * ?D') c d"
by (rule le_factor)
also have "... = ((a * b') * ?D * ?D' * d * d' ≤ (c * d') * ?D * ?D' * b * b')"
also have "... = ((a' * b) * ?D * ?D' * d * d' ≤ (c' * d) * ?D * ?D' * b * b')"
by (simp only: eq1 eq2)
also have "... = ?le (a' * ?D) (b' * ?D) c' d'"
also from D have "... = ?le a' b' c' d'"
by (rule le_factor [symmetric])
finally show "?le a b c d = ?le a' b' c' d'" .
qed

lift_definition less_eq_fract :: "'a fract ⇒ 'a fract ⇒ bool"
is "λq r. (fst q * snd r) * (snd q * snd r) ≤ (fst r * snd q) * (snd q * snd r)"

definition less_fract_def: "z < (w::'a fract) ⟷ z ≤ w ∧ ¬ w ≤ z"

lemma le_fract [simp]:
"⟦ b ≠ 0; d ≠ 0 ⟧ ⟹ Fract a b ≤ Fract c d ⟷ (a * d) * (b * d) ≤ (c * b) * (b * d)"
by transfer simp

lemma less_fract [simp]:
"⟦ b ≠ 0; d ≠ 0 ⟧ ⟹ Fract a b < Fract c d ⟷ (a * d) * (b * d) < (c * b) * (b * d)"
by (simp add: less_fract_def less_le_not_le ac_simps)

instance
proof
fix q r s :: "'a fract"
assume "q ≤ r" and "r ≤ s"
then show "q ≤ s"
proof (induct q, induct r, induct s)
fix a b c d e f :: 'a
assume neq: "b ≠ 0" "d ≠ 0" "f ≠ 0"
assume 1: "Fract a b ≤ Fract c d"
assume 2: "Fract c d ≤ Fract e f"
show "Fract a b ≤ Fract e f"
proof -
from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f"
by (auto simp add: zero_less_mult_iff linorder_neq_iff)
have "(a * d) * (b * d) * (f * f) ≤ (c * b) * (b * d) * (f * f)"
proof -
from neq 1 have "(a * d) * (b * d) ≤ (c * b) * (b * d)"
by simp
with ff show ?thesis by (simp add: mult_le_cancel_right)
qed
also have "... = (c * f) * (d * f) * (b * b)"
by (simp only: ac_simps)
also have "... ≤ (e * d) * (d * f) * (b * b)"
proof -
from neq 2 have "(c * f) * (d * f) ≤ (e * d) * (d * f)"
by simp
with bb show ?thesis by (simp add: mult_le_cancel_right)
qed
finally have "(a * f) * (b * f) * (d * d) ≤ e * b * (b * f) * (d * d)"
by (simp only: ac_simps)
with dd have "(a * f) * (b * f) ≤ (e * b) * (b * f)"
with neq show ?thesis by simp
qed
qed
next
fix q r :: "'a fract"
assume "q ≤ r" and "r ≤ q"
then show "q = r"
proof (induct q, induct r)
fix a b c d :: 'a
assume neq: "b ≠ 0" "d ≠ 0"
assume 1: "Fract a b ≤ Fract c d"
assume 2: "Fract c d ≤ Fract a b"
show "Fract a b = Fract c d"
proof -
from neq 1 have "(a * d) * (b * d) ≤ (c * b) * (b * d)"
by simp
also have "... ≤ (a * d) * (b * d)"
proof -
from neq 2 have "(c * b) * (d * b) ≤ (a * d) * (d * b)"
by simp
then show ?thesis by (simp only: ac_simps)
qed
finally have "(a * d) * (b * d) = (c * b) * (b * d)" .
moreover from neq have "b * d ≠ 0" by simp
ultimately have "a * d = c * b" by simp
with neq show ?thesis by (simp add: eq_fract)
qed
qed
next
fix q r :: "'a fract"
show "q ≤ q"
by (induct q) simp
show "(q < r) = (q ≤ r ∧ ¬ r ≤ q)"
by (simp only: less_fract_def)
show "q ≤ r ∨ r ≤ q"
by (induct q, induct r)
qed

end

instantiation fract :: (linordered_idom) linordered_field
begin

definition abs_fract_def2:
"¦q¦ = (if q < 0 then -q else (q::'a fract))"

definition sgn_fract_def:
"sgn (q::'a fract) = (if q = 0 then 0 else if 0 < q then 1 else - 1)"

theorem abs_fract [simp]: "¦Fract a b¦ = Fract ¦a¦ ¦b¦"
unfolding abs_fract_def2 not_le [symmetric]
by transfer (auto simp add: zero_less_mult_iff le_less)

instance proof
fix q r s :: "'a fract"
assume "q ≤ r"
then show "s + q ≤ s + r"
proof (induct q, induct r, induct s)
fix a b c d e f :: 'a
assume neq: "b ≠ 0" "d ≠ 0" "f ≠ 0"
assume le: "Fract a b ≤ Fract c d"
show "Fract e f + Fract a b ≤ Fract e f + Fract c d"
proof -
let ?F = "f * f" from neq have F: "0 < ?F"
from neq le have "(a * d) * (b * d) ≤ (c * b) * (b * d)"
by simp
with F have "(a * d) * (b * d) * ?F * ?F ≤ (c * b) * (b * d) * ?F * ?F"
with neq show ?thesis by (simp add: field_simps)
qed
qed
next
fix q r s :: "'a fract"
assume "q < r" and "0 < s"
then show "s * q < s * r"
proof (induct q, induct r, induct s)
fix a b c d e f :: 'a
assume neq: "b ≠ 0" "d ≠ 0" "f ≠ 0"
assume le: "Fract a b < Fract c d"
assume gt: "0 < Fract e f"
show "Fract e f * Fract a b < Fract e f * Fract c d"
proof -
let ?E = "e * f" and ?F = "f * f"
from neq gt have "0 < ?E"
by (auto simp add: Zero_fract_def order_less_le eq_fract)
moreover from neq have "0 < ?F"
moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)"
by simp
ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F"
with neq show ?thesis
qed
qed
qed (fact sgn_fract_def abs_fract_def2)+

end

instantiation fract :: (linordered_idom) distrib_lattice
begin

definition inf_fract_def:
"(inf :: 'a fract ⇒ 'a fract ⇒ 'a fract) = min"

definition sup_fract_def:
"(sup :: 'a fract ⇒ 'a fract ⇒ 'a fract) = max"

instance
by standard (simp_all add: inf_fract_def sup_fract_def max_min_distrib2)

end

lemma fract_induct_pos [case_names Fract]:
fixes P :: "'a::linordered_idom fract ⇒ bool"
assumes step: "⋀a b. 0 < b ⟹ P (Fract a b)"
shows "P q"
proof (cases q)
case (Fract a b)
{
fix a b :: 'a
assume b: "b < 0"
have "P (Fract a b)"
proof -
from b have "0 < - b" by simp
then have "P (Fract (- a) (- b))"
by (rule step)
then show "P (Fract a b)"
by (simp add: order_less_imp_not_eq [OF b])
qed
}
with Fract show "P q"
by (auto simp add: linorder_neq_iff step)
qed

lemma zero_less_Fract_iff: "0 < b ⟹ 0 < Fract a b ⟷ 0 < a"
by (auto simp add: Zero_fract_def zero_less_mult_iff)

lemma Fract_less_zero_iff: "0 < b ⟹ Fract a b < 0 ⟷ a < 0"
by (auto simp add: Zero_fract_def mult_less_0_iff)

lemma zero_le_Fract_iff: "0 < b ⟹ 0 ≤ Fract a b ⟷ 0 ≤ a"
by (auto simp add: Zero_fract_def zero_le_mult_iff)

lemma Fract_le_zero_iff: "0 < b ⟹ Fract a b ≤ 0 ⟷ a ≤ 0"
by (auto simp add: Zero_fract_def mult_le_0_iff)

lemma one_less_Fract_iff: "0 < b ⟹ 1 < Fract a b ⟷ b < a"
by (auto simp add: One_fract_def mult_less_cancel_right_disj)

lemma Fract_less_one_iff: "0 < b ⟹ Fract a b < 1 ⟷ a < b"
by (auto simp add: One_fract_def mult_less_cancel_right_disj)

lemma one_le_Fract_iff: "0 < b ⟹ 1 ≤ Fract a b ⟷ b ≤ a"
by (auto simp add: One_fract_def mult_le_cancel_right)

lemma Fract_le_one_iff: "0 < b ⟹ Fract a b ≤ 1 ⟷ a ≤ b"
by (auto simp add: One_fract_def mult_le_cancel_right)

end
```