src/HOL/Library/Abstract_Rat.thy

 lemma INum_int [simp]: "INum i\<^sub>N = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
   by (simp_all add: INum_def)
 
 lemma isnormNum_unique[simp]: 
   assumes na: "isnormNum x" and nb: "isnormNum y" 
-  shows "((INum x ::'a::{ring_char_0,field, division_ring_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
+  shows "((INum x ::'a::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
 proof
   have "\<exists> a b a' b'. x = (a,b) \<and> y = (a',b')" by auto
   then obtain a b a' b' where xy[simp]: "x = (a,b)" "y=(a',b')" by blast
   assume H: ?lhs 
   {assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"

 next
   assume ?rhs thus ?lhs by simp
 qed
 
 
-lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = (0::'a::{ring_char_0, field,division_ring_inverse_zero})) = (x = 0\<^sub>N)"
+lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"
   unfolding INum_int(2)[symmetric]
   by (rule isnormNum_unique, simp_all)
 
-lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::{field, ring_char_0}) / (of_int d) = 
+lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) = 
     of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
 proof -
   assume "d ~= 0"
   hence dz: "of_int d \<noteq> (0::'a)" by (simp add: of_int_eq_0_iff)
   let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"

     using cong[OF refl[of ?f] eq] by simp
   then show ?thesis by (simp add: add_divide_distrib algebra_simps prems)
 qed
 
 lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
-    (of_int(n div d)::'a::{field, ring_char_0}) = of_int n / of_int d"
+    (of_int(n div d)::'a::field_char_0) = of_int n / of_int d"
   apply (frule of_int_div_aux [of d n, where ?'a = 'a])
   apply simp
   apply (simp add: dvd_eq_mod_eq_0)
 done
 
 
-lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{ring_char_0,field, division_ring_inverse_zero})"
+lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
 proof-
   have "\<exists> a b. x = (a,b)" by auto
   then obtain a b where x[simp]: "x = (a,b)" by blast
   {assume "a=0 \<or> b = 0" hence ?thesis
       by (simp add: INum_def normNum_def split_def Let_def)}

     from of_int_div[OF g, where ?'a = 'a]
     have ?thesis by (auto simp add: INum_def normNum_def split_def Let_def)}
   ultimately show ?thesis by blast
 qed
 
-lemma INum_normNum_iff: "(INum x ::'a::{field, division_ring_inverse_zero, ring_char_0}) = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")
+lemma INum_normNum_iff: "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")
 proof -
   have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
     by (simp del: normNum)
   also have "\<dots> = ?lhs" by simp
   finally show ?thesis by simp
 qed
 
-lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {ring_char_0,division_ring_inverse_zero,field})"
+lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})"
 proof-
 let ?z = "0:: 'a"
   have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
   then obtain a b a' b' where x[simp]: "x = (a,b)" 
     and y[simp]: "y = (a',b')" by blast

     ultimately have ?thesis using aa' bb' 
       by (simp add: Nadd_def INum_def normNum_def x y Let_def) }
   ultimately show ?thesis by blast
 qed
 
-lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {ring_char_0,division_ring_inverse_zero,field}) "
+lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero}) "
 proof-
   let ?z = "0::'a"
   have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
   then obtain a b a' b' where x: "x = (a,b)" and y: "y = (a',b')" by blast
   {assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" hence ?thesis 

 qed
 
 lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
   by (simp add: Nneg_def split_def INum_def)
 
-lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {ring_char_0,division_ring_inverse_zero,field})"
+lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
 by (simp add: Nsub_def split_def)
 
-lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: {division_ring_inverse_zero,field}) / (INum x)"
+lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)"
   by (simp add: Ninv_def INum_def split_def)
 
-lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {ring_char_0, division_ring_inverse_zero,field})" by (simp add: Ndiv_def)
+lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})" by (simp add: Ndiv_def)
 
 lemma Nlt0_iff[simp]: assumes nx: "isnormNum x" 
-  shows "((INum x :: 'a :: {ring_char_0,division_ring_inverse_zero,linordered_field})< 0) = 0>\<^sub>N x "
+  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x "
 proof-
   have " \<exists> a b. x = (a,b)" by simp
   then obtain a b where x[simp]:"x = (a,b)" by blast
   {assume "a = 0" hence ?thesis by (simp add: Nlt0_def INum_def) }
   moreover

     have ?thesis by (simp add: Nlt0_def INum_def)}
   ultimately show ?thesis by blast
 qed
 
 lemma Nle0_iff[simp]:assumes nx: "isnormNum x" 
-  shows "((INum x :: 'a :: {ring_char_0,division_ring_inverse_zero,linordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
+  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"
 proof-
   have " \<exists> a b. x = (a,b)" by simp
   then obtain a b where x[simp]:"x = (a,b)" by blast
   {assume "a = 0" hence ?thesis by (simp add: Nle0_def INum_def) }
   moreover

     from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
     have ?thesis by (simp add: Nle0_def INum_def)}
   ultimately show ?thesis by blast
 qed
 
-lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {ring_char_0,division_ring_inverse_zero,linordered_field})> 0) = 0<\<^sub>N x"
+lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
 proof-
   have " \<exists> a b. x = (a,b)" by simp
   then obtain a b where x[simp]:"x = (a,b)" by blast
   {assume "a = 0" hence ?thesis by (simp add: Ngt0_def INum_def) }
   moreover

     from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
     have ?thesis by (simp add: Ngt0_def INum_def)}
   ultimately show ?thesis by blast
 qed
 lemma Nge0_iff[simp]:assumes nx: "isnormNum x" 
-  shows "((INum x :: 'a :: {ring_char_0,division_ring_inverse_zero,linordered_field}) \<ge> 0) = 0\<le>\<^sub>N x"
+  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
 proof-
   have " \<exists> a b. x = (a,b)" by simp
   then obtain a b where x[simp]:"x = (a,b)" by blast
   {assume "a = 0" hence ?thesis by (simp add: Nge0_def INum_def) }
   moreover

     have ?thesis by (simp add: Nge0_def INum_def)}
   ultimately show ?thesis by blast
 qed
 
 lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
-  shows "((INum x :: 'a :: {ring_char_0,division_ring_inverse_zero,linordered_field}) < INum y) = (x <\<^sub>N y)"
+  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)"
 proof-
   let ?z = "0::'a"
   have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" using nx ny by simp
   also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
   finally show ?thesis by (simp add: Nlt_def)
 qed
 
 lemma Nle_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
-  shows "((INum x :: 'a :: {ring_char_0,division_ring_inverse_zero,linordered_field})\<le> INum y) = (x \<le>\<^sub>N y)"
+  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)"
 proof-
   have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" using nx ny by simp
   also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" using Nle0_iff[OF Nsub_normN[OF ny]] by simp
   finally show ?thesis by (simp add: Nle_def)
 qed
 
 lemma Nadd_commute:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "x +\<^sub>N y = y +\<^sub>N x"
 proof-
   have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
   have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp
   with isnormNum_unique[OF n] show ?thesis by simp
 qed
 
 lemma [simp]:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "(0, b) +\<^sub>N y = normNum y"
     and "(a, 0) +\<^sub>N y = normNum y" 
     and "x +\<^sub>N (0, b) = normNum x"
     and "x +\<^sub>N (a, 0) = normNum x"
   apply (simp add: Nadd_def split_def)

   apply (subst Nadd_commute, simp add: Nadd_def split_def)
   apply (subst Nadd_commute, simp add: Nadd_def split_def)
   done
 
 lemma normNum_nilpotent_aux[simp]:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   assumes nx: "isnormNum x" 
   shows "normNum x = x"
 proof-
   let ?a = "normNum x"
   have n: "isnormNum ?a" by simp

   with isnormNum_unique[OF n nx]  
   show ?thesis by simp
 qed
 
 lemma normNum_nilpotent[simp]:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "normNum (normNum x) = normNum x"
   by simp
 
 lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"
   by (simp_all add: normNum_def)
 
 lemma normNum_Nadd:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
 
 lemma Nadd_normNum1[simp]:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "normNum x +\<^sub>N y = x +\<^sub>N y"
 proof-
   have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
   have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp
   also have "\<dots> = INum (x +\<^sub>N y)" by simp
   finally show ?thesis using isnormNum_unique[OF n] by simp
 qed
 
 lemma Nadd_normNum2[simp]:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "x +\<^sub>N normNum y = x +\<^sub>N y"
 proof-
   have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
   have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp
   also have "\<dots> = INum (x +\<^sub>N y)" by simp
   finally show ?thesis using isnormNum_unique[OF n] by simp
 qed
 
 lemma Nadd_assoc:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
 proof-
   have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
   with isnormNum_unique[OF n] show ?thesis by simp

 
 lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
   by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute)
 
 lemma Nmul_assoc:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   assumes nx: "isnormNum x" and ny:"isnormNum y" and nz:"isnormNum z"
   shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
 proof-
   from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))" 
     by simp_all
   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
   with isnormNum_unique[OF n] show ?thesis by simp
 qed
 
 lemma Nsub0:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
 proof-
   { fix h :: 'a
     from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"] 
     have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp

 
 lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
   by (simp_all add: Nmul_def Let_def split_def)
 
 lemma Nmul_eq0[simp]:
-  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_ring_inverse_zero,field})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   assumes nx:"isnormNum x" and ny: "isnormNum y"
   shows "(x*\<^sub>N y = 0\<^sub>N) = (x = 0\<^sub>N \<or> y = 0\<^sub>N)"
 proof-
   { fix h :: 'a
     have " \<exists> a b a' b'. x = (a,b) \<and> y= (a',b')" by auto