Theory Field_as_Ring

theory Field_as_Ring
imports Complex_Main Euclidean_Algorithm
```(*  Title:      HOL/Computational_Algebra/Field_as_Ring.thy
Author:     Manuel Eberl
*)

theory Field_as_Ring
imports
Complex_Main
Euclidean_Algorithm
begin

context field
begin

subclass idom_divide ..

definition normalize_field :: "'a ⇒ 'a"
where [simp]: "normalize_field x = (if x = 0 then 0 else 1)"
definition unit_factor_field :: "'a ⇒ 'a"
where [simp]: "unit_factor_field x = x"
definition euclidean_size_field :: "'a ⇒ nat"
where [simp]: "euclidean_size_field x = (if x = 0 then 0 else 1)"
definition mod_field :: "'a ⇒ 'a ⇒ 'a"
where [simp]: "mod_field x y = (if y = 0 then x else 0)"

end

instantiation real :: unique_euclidean_ring
begin

definition [simp]: "normalize_real = (normalize_field :: real ⇒ _)"
definition [simp]: "unit_factor_real = (unit_factor_field :: real ⇒ _)"
definition [simp]: "euclidean_size_real = (euclidean_size_field :: real ⇒ _)"
definition [simp]: "uniqueness_constraint_real = (top :: real ⇒ real ⇒ bool)"
definition [simp]: "modulo_real = (mod_field :: real ⇒ _)"

instance
by standard
(simp_all add: dvd_field_iff divide_simps split: if_splits)

end

instantiation real :: euclidean_ring_gcd
begin

definition gcd_real :: "real ⇒ real ⇒ real" where
"gcd_real = Euclidean_Algorithm.gcd"
definition lcm_real :: "real ⇒ real ⇒ real" where
"lcm_real = Euclidean_Algorithm.lcm"
definition Gcd_real :: "real set ⇒ real" where
"Gcd_real = Euclidean_Algorithm.Gcd"
definition Lcm_real :: "real set ⇒ real" where
"Lcm_real = Euclidean_Algorithm.Lcm"

instance by standard (simp_all add: gcd_real_def lcm_real_def Gcd_real_def Lcm_real_def)

end

instantiation rat :: unique_euclidean_ring
begin

definition [simp]: "normalize_rat = (normalize_field :: rat ⇒ _)"
definition [simp]: "unit_factor_rat = (unit_factor_field :: rat ⇒ _)"
definition [simp]: "euclidean_size_rat = (euclidean_size_field :: rat ⇒ _)"
definition [simp]: "uniqueness_constraint_rat = (top :: rat ⇒ rat ⇒ bool)"
definition [simp]: "modulo_rat = (mod_field :: rat ⇒ _)"

instance
by standard
(simp_all add: dvd_field_iff divide_simps split: if_splits)

end

instantiation rat :: euclidean_ring_gcd
begin

definition gcd_rat :: "rat ⇒ rat ⇒ rat" where
"gcd_rat = Euclidean_Algorithm.gcd"
definition lcm_rat :: "rat ⇒ rat ⇒ rat" where
"lcm_rat = Euclidean_Algorithm.lcm"
definition Gcd_rat :: "rat set ⇒ rat" where
"Gcd_rat = Euclidean_Algorithm.Gcd"
definition Lcm_rat :: "rat set ⇒ rat" where
"Lcm_rat = Euclidean_Algorithm.Lcm"

instance by standard (simp_all add: gcd_rat_def lcm_rat_def Gcd_rat_def Lcm_rat_def)

end

instantiation complex :: unique_euclidean_ring
begin

definition [simp]: "normalize_complex = (normalize_field :: complex ⇒ _)"
definition [simp]: "unit_factor_complex = (unit_factor_field :: complex ⇒ _)"
definition [simp]: "euclidean_size_complex = (euclidean_size_field :: complex ⇒ _)"
definition [simp]: "uniqueness_constraint_complex = (top :: complex ⇒ complex ⇒ bool)"
definition [simp]: "modulo_complex = (mod_field :: complex ⇒ _)"

instance
by standard
(simp_all add: dvd_field_iff divide_simps split: if_splits)

end

instantiation complex :: euclidean_ring_gcd
begin

definition gcd_complex :: "complex ⇒ complex ⇒ complex" where
"gcd_complex = Euclidean_Algorithm.gcd"
definition lcm_complex :: "complex ⇒ complex ⇒ complex" where
"lcm_complex = Euclidean_Algorithm.lcm"
definition Gcd_complex :: "complex set ⇒ complex" where
"Gcd_complex = Euclidean_Algorithm.Gcd"
definition Lcm_complex :: "complex set ⇒ complex" where
"Lcm_complex = Euclidean_Algorithm.Lcm"

instance by standard (simp_all add: gcd_complex_def lcm_complex_def Gcd_complex_def Lcm_complex_def)

end

end
```