Theory Fraction_Field

theory Fraction_Field
imports Main
(*  Title:      HOL/Library/Fraction_Field.thy
Author: Amine Chaieb, University of Cambridge
*)


header{* A formalization of the fraction field of any integral domain;
generalization of theory Rat from int to any integral domain *}


theory Fraction_Field
imports Main
begin

subsection {* General fractions construction *}

subsubsection {* Construction of the type of fractions *}

definition fractrel :: "(('a::idom * 'a ) * ('a * 'a)) set" where
"fractrel = {(x, y). snd x ≠ 0 ∧ snd y ≠ 0 ∧ fst x * snd y = fst y * snd x}"

lemma fractrel_iff [simp]:
"(x, y) ∈ fractrel <-> snd x ≠ 0 ∧ snd y ≠ 0 ∧ fst x * snd y = fst y * snd x"
by (simp add: fractrel_def)

lemma refl_fractrel: "refl_on {x. snd x ≠ 0} fractrel"
by (auto simp add: refl_on_def fractrel_def)

lemma sym_fractrel: "sym fractrel"
by (simp add: fractrel_def sym_def)

lemma trans_fractrel: "trans fractrel"
proof (rule transI, unfold split_paired_all)
fix a b a' b' a'' b'' :: 'a
assume A: "((a, b), (a', b')) ∈ fractrel"
assume B: "((a', b'), (a'', b'')) ∈ fractrel"
have "b' * (a * b'') = b'' * (a * b')" by (simp add: mult_ac)
also from A have "a * b' = a' * b" by auto
also have "b'' * (a' * b) = b * (a' * b'')" by (simp add: mult_ac)
also from B have "a' * b'' = a'' * b'" by auto
also have "b * (a'' * b') = b' * (a'' * b)" by (simp add: mult_ac)
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 "((a, b), (a'', b'')) ∈ fractrel" by auto
qed

lemma equiv_fractrel: "equiv {x. snd x ≠ 0} fractrel"
by (rule equivI [OF refl_fractrel sym_fractrel trans_fractrel])

lemmas UN_fractrel = UN_equiv_class [OF equiv_fractrel]
lemmas UN_fractrel2 = UN_equiv_class2 [OF equiv_fractrel equiv_fractrel]

lemma equiv_fractrel_iff [iff]:
assumes "snd x ≠ 0" and "snd y ≠ 0"
shows "fractrel `` {x} = fractrel `` {y} <-> (x, y) ∈ fractrel"
by (rule eq_equiv_class_iff, rule equiv_fractrel) (auto simp add: assms)

definition "fract = {(x::'a×'a). snd x ≠ (0::'a::idom)} // fractrel"

typedef 'a fract = "fract :: ('a * 'a::idom) set set"
unfolding fract_def
proof
have "(0::'a, 1::'a) ∈ {x. snd x ≠ 0}" by simp
then show "fractrel `` {(0::'a, 1)} ∈ {x. snd x ≠ 0} // fractrel" by (rule quotientI)
qed

lemma fractrel_in_fract [simp]: "snd x ≠ 0 ==> fractrel `` {x} ∈ fract"
by (simp add: fract_def quotientI)

declare Abs_fract_inject [simp] Abs_fract_inverse [simp]


subsubsection {* Representation and basic operations *}

definition Fract :: "'a::idom => 'a => 'a fract" where
"Fract a b = Abs_fract (fractrel `` {if b = 0 then (0, 1) else (a, b)})"

code_datatype Fract

lemma Fract_cases [cases type: fract]:
obtains (Fract) a b where "q = Fract a b" "b ≠ 0"
by (cases q) (clarsimp simp add: Fract_def fract_def quotient_def)

lemma Fract_induct [case_names Fract, induct type: fract]:
shows "(!!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 (simp_all add: Fract_def)

instantiation fract :: (idom) "{comm_ring_1,power}"
begin

definition Zero_fract_def [code_unfold]: "0 = Fract 0 1"

definition One_fract_def [code_unfold]: "1 = Fract 1 1"

definition add_fract_def:
"q + r = Abs_fract (\<Union>x ∈ Rep_fract q. \<Union>y ∈ Rep_fract r.
fractrel `` {(fst x * snd y + fst y * snd x, snd x * snd y)})"


lemma add_fract [simp]:
assumes "b ≠ (0::'a::idom)"
and "d ≠ 0"
shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
proof -
have "(λx y. fractrel``{(fst x * snd y + fst y * snd x, snd x * snd y :: 'a)})
respects2 fractrel"

apply (rule equiv_fractrel [THEN congruent2_commuteI])
apply (auto simp add: algebra_simps)
unfolding mult_assoc[symmetric]
done
with assms show ?thesis by (simp add: Fract_def add_fract_def UN_fractrel2)
qed

definition minus_fract_def:
"- q = Abs_fract (\<Union>x ∈ Rep_fract q. fractrel `` {(- fst x, snd x)})"

lemma minus_fract [simp, code]: "- Fract a b = Fract (- a) (b::'a::idom)"
proof -
have "(λx. fractrel `` {(- fst x, snd x :: 'a)}) respects fractrel"
by (simp add: congruent_def split_paired_all)
then show ?thesis by (simp add: Fract_def minus_fract_def UN_fractrel)
qed

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]:
assumes "b ≠ 0" and "d ≠ 0"
shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
using assms by (simp add: diff_fract_def diff_minus)

definition mult_fract_def:
"q * r = Abs_fract (\<Union>x ∈ Rep_fract q. \<Union>y ∈ Rep_fract r.
fractrel``{(fst x * fst y, snd x * snd y)})"


lemma mult_fract [simp]: "Fract (a::'a::idom) b * Fract c d = Fract (a * c) (b * d)"
proof -
have "(λx y. fractrel `` {(fst x * fst y, snd x * snd y :: 'a)}) respects2 fractrel"
apply (rule equiv_fractrel [THEN congruent2_commuteI])
apply (auto simp add: algebra_simps)
done
then show ?thesis by (simp add: Fract_def mult_fract_def UN_fractrel2)
qed

lemma mult_fract_cancel:
assumes "c ≠ (0::'a)"
shows "Fract (c * a) (c * b) = Fract a b"
proof -
from assms have "Fract c c = Fract 1 1" by (simp add: Fract_def)
then show ?thesis by (simp add: mult_fract [symmetric])
qed

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 [code_post]:
"Fract 0 k = 0"
"Fract 1 1 = 1"
"Fract k 0 = 0"
by (cases "k = 0")
(simp_all add: Zero_fract_def One_fract_def eq_fract Fract_def)

lemma fract_expand [code_unfold]:
"0 = Fract 0 1"
"1 = Fract 1 1"
by (simp_all add: fract_collapse)

lemma Fract_cases_nonzero:
obtains (Fract) a b where "q = Fract a b" "b ≠ 0" "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_inverse_zero
begin

definition inverse_fract_def:
"inverse q = Abs_fract (\<Union>x ∈ Rep_fract q.
fractrel `` {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"
proof -
have *: "!!x. (0::'a) = x <-> x = 0" by auto
have "(λx. fractrel `` {if fst x = 0 then (0, 1) else (snd x, fst x :: 'a)}) respects fractrel"
by (auto simp add: congruent_def * algebra_simps)
then show ?thesis by (simp add: Fract_def inverse_fract_def UN_fractrel)
qed

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

lemma divide_fract [simp]: "Fract a b / Fract c d = Fract (a * d) (b * c)"
by (simp add: divide_fract_def)

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

end


subsubsection {* The ordered field of fractions over an ordered idom *}

lemma le_congruent2:
"(λx y::'a × 'a::linordered_idom.
{(fst x * snd y)*(snd x * snd y) ≤ (fst y * snd x)*(snd x * snd y)})
respects2 fractrel"

proof (clarsimp simp add: congruent2_def)
fix a b a' b' c d c' d' :: 'a
assume neq: "b ≠ 0" "b' ≠ 0" "d ≠ 0" "d' ≠ 0"
assume eq1: "a * b' = a' * b"
assume eq2: "c * d' = c' * d"

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" by (auto simp add: zero_less_mult_iff)
then have "?le a b c d =
((a * d) * (b * d) * (x * x) ≤ (c * b) * (b * d) * (x * x))"

by (simp add: mult_le_cancel_right)
also have "... = ?le (a * x) (b * x) c d"
by (simp add: mult_ac)
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')"
by (simp add: mult_ac)
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'"
by (simp add: mult_ac)
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

instantiation fract :: (linordered_idom) linorder
begin

definition le_fract_def:
"q ≤ r <-> the_elem (\<Union>x ∈ Rep_fract q. \<Union>y ∈ Rep_fract r.
{(fst x * snd y) * (snd x * snd y) ≤ (fst y * snd x) * (snd x * snd y)})"


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

lemma le_fract [simp]:
assumes "b ≠ 0" and "d ≠ 0"
shows "Fract a b ≤ Fract c d <-> (a * d) * (b * d) ≤ (c * b) * (b * d)"
by (simp add: Fract_def le_fract_def le_congruent2 UN_fractrel2 assms)

lemma less_fract [simp]:
assumes "b ≠ 0" and "d ≠ 0"
shows "Fract a b < Fract c d <-> (a * d) * (b * d) < (c * b) * (b * d)"
by (simp add: less_fract_def less_le_not_le mult_ac assms)

instance
proof
fix q r s :: "'a fract"
assume "q ≤ r" and "r ≤ s" thus "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" and 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: mult_ac)
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: mult_ac)
with dd have "(a * f) * (b * f) ≤ (e * b) * (b * f)"
by (simp add: mult_le_cancel_right)
with neq show ?thesis by simp
qed
qed
next
fix q r :: "'a fract"
assume "q ≤ r" and "r ≤ q" thus "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" and 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
thus ?thesis by (simp only: mult_ac)
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)
(simp add: mult_commute, rule linorder_linear)
qed

end

instantiation fract :: (linordered_idom) "{distrib_lattice, abs_if, sgn_if}"
begin

definition abs_fract_def: "¦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¦"
by (auto simp add: abs_fract_def Zero_fract_def le_less
eq_fract zero_less_mult_iff mult_less_0_iff split: abs_split)

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 intro_classes
(auto simp add: abs_fract_def sgn_fract_def
min_max.sup_inf_distrib1 inf_fract_def sup_fract_def)

end

instance fract :: (linordered_idom) linordered_field_inverse_zero
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"
by (auto simp add: zero_less_mult_iff)
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"
by (simp add: mult_le_cancel_right)
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"
by (auto simp add: zero_less_mult_iff)
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"
by (simp add: mult_less_cancel_right)
with neq show ?thesis
by (simp add: mult_ac)
qed
qed
qed

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)
have step': "!!a b. b < 0 ==> P (Fract a b)"
proof -
fix a::'a and b::'a
assume b: "b < 0"
then have "0 < -b" by simp
then have "P (Fract (-a) (-b))" by (rule step)
thus "P (Fract a b)" by (simp add: order_less_imp_not_eq [OF b])
qed
case (Fract a b)
thus "P q" by (force simp add: linorder_neq_iff step 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