src/HOL/Computational_Algebra/Field_as_Ring.thy

+(*  Title:      HOL/Library/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 \<Rightarrow> 'a" 
+  where [simp]: "normalize_field x = (if x = 0 then 0 else 1)"
+definition unit_factor_field :: "'a \<Rightarrow> 'a" 
+  where [simp]: "unit_factor_field x = x"
+definition euclidean_size_field :: "'a \<Rightarrow> nat" 
+  where [simp]: "euclidean_size_field x = (if x = 0 then 0 else 1)"
+definition mod_field :: "'a \<Rightarrow> 'a \<Rightarrow> '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 \<Rightarrow> _)"
+definition [simp]: "unit_factor_real = (unit_factor_field :: real \<Rightarrow> _)"
+definition [simp]: "euclidean_size_real = (euclidean_size_field :: real \<Rightarrow> _)"
+definition [simp]: "uniqueness_constraint_real = (top :: real \<Rightarrow> real \<Rightarrow> bool)"
+definition [simp]: "modulo_real = (mod_field :: real \<Rightarrow> _)"
+
+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 \<Rightarrow> real \<Rightarrow> real" where
+  "gcd_real = Euclidean_Algorithm.gcd"
+definition lcm_real :: "real \<Rightarrow> real \<Rightarrow> real" where
+  "lcm_real = Euclidean_Algorithm.lcm"
+definition Gcd_real :: "real set \<Rightarrow> real" where
+ "Gcd_real = Euclidean_Algorithm.Gcd"
+definition Lcm_real :: "real set \<Rightarrow> 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 \<Rightarrow> _)"
+definition [simp]: "unit_factor_rat = (unit_factor_field :: rat \<Rightarrow> _)"
+definition [simp]: "euclidean_size_rat = (euclidean_size_field :: rat \<Rightarrow> _)"
+definition [simp]: "uniqueness_constraint_rat = (top :: rat \<Rightarrow> rat \<Rightarrow> bool)"
+definition [simp]: "modulo_rat = (mod_field :: rat \<Rightarrow> _)"
+
+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 \<Rightarrow> rat \<Rightarrow> rat" where
+  "gcd_rat = Euclidean_Algorithm.gcd"
+definition lcm_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat" where
+  "lcm_rat = Euclidean_Algorithm.lcm"
+definition Gcd_rat :: "rat set \<Rightarrow> rat" where
+ "Gcd_rat = Euclidean_Algorithm.Gcd"
+definition Lcm_rat :: "rat set \<Rightarrow> 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 \<Rightarrow> _)"
+definition [simp]: "unit_factor_complex = (unit_factor_field :: complex \<Rightarrow> _)"
+definition [simp]: "euclidean_size_complex = (euclidean_size_field :: complex \<Rightarrow> _)"
+definition [simp]: "uniqueness_constraint_complex = (top :: complex \<Rightarrow> complex \<Rightarrow> bool)"
+definition [simp]: "modulo_complex = (mod_field :: complex \<Rightarrow> _)"
+
+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 \<Rightarrow> complex \<Rightarrow> complex" where
+  "gcd_complex = Euclidean_Algorithm.gcd"
+definition lcm_complex :: "complex \<Rightarrow> complex \<Rightarrow> complex" where
+  "lcm_complex = Euclidean_Algorithm.lcm"
+definition Gcd_complex :: "complex set \<Rightarrow> complex" where
+ "Gcd_complex = Euclidean_Algorithm.Gcd"
+definition Lcm_complex :: "complex set \<Rightarrow> 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