src/HOL/Decision_Procs/Rat_Pair.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::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
+  shows "((INum x ::'a::{field_char_0, field}) = INum y) = (x = y)" (is "?lhs = ?rhs")
 proof
   obtain a b where x: "x = (a, b)" by (cases x)
   obtain a' b' where y: "y = (a', b')" by (cases y)
   assume H: ?lhs
   { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"

   assume ?rhs thus ?lhs by simp
 qed
 
 
 lemma isnormNum0[simp]:
-    "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"
+    "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field})) = (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_char_0) / (of_int d) =
     of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"

 
 lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
     (of_int(n div d)::'a::field_char_0) = of_int n / of_int d"
   using of_int_div_aux [of d n, where ?'a = 'a] by simp
 
-lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
+lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field})"
 proof -
   obtain a b where x: "x = (a, b)" by (cases x)
   { assume "a = 0 \<or> b = 0"
     hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) }
   moreover

     have ?thesis by (auto simp add: x INum_def normNum_def split_def Let_def) }
   ultimately show ?thesis by blast
 qed
 
 lemma INum_normNum_iff:
-  "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y"
+  "(INum x ::'a::{field_char_0, field}) = 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 :: {field_char_0, field_inverse_zero})"
+lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field})"
 proof -
   let ?z = "0:: 'a"
   obtain a b where x: "x = (a, b)" by (cases x)
   obtain a' b' where y: "y = (a', b')" by (cases y)
   { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0"

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

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

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

   ultimately show ?thesis by blast
 qed
 
 lemma Ngt0_iff[simp]:
   assumes nx: "isnormNum x"
-  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
+  shows "((INum x :: 'a :: {field_char_0, linordered_field})> 0) = 0<\<^sub>N x"
 proof -
   obtain a b where x: "x = (a, b)" by (cases x)
   { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) }
   moreover
   { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx

   ultimately show ?thesis by blast
 qed
 
 lemma Nge0_iff[simp]:
   assumes nx: "isnormNum x"
-  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
+  shows "((INum x :: 'a :: {field_char_0, linordered_field}) \<ge> 0) = 0\<le>\<^sub>N x"
 proof -
   obtain a b where x: "x = (a, b)" by (cases x)
   { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) }
   moreover
   { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx

   ultimately show ?thesis by blast
 qed
 
 lemma Nlt_iff[simp]:
   assumes nx: "isnormNum x" and ny: "isnormNum y"
-  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)"
+  shows "((INum x :: 'a :: {field_char_0, linordered_field}) < 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))"

   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 :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)"
+  shows "((INum x :: 'a :: {field_char_0, linordered_field})\<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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   assumes nx: "isnormNum x"
   shows "normNum x = x"
 proof -
   let ?a = "normNum x"
   have n: "isnormNum ?a" by simp
   have th: "INum ?a = (INum x ::'a)" by simp
   with isnormNum_unique[OF n nx] show ?thesis by simp
 qed
 
 lemma normNum_nilpotent[simp]:
-  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
 
 lemma Nadd_normNum1[simp]:
-  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   assumes x: "isnormNum x" and y: "isnormNum y"
   shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
 proof -
   fix h :: 'a
   from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]

 
 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::{field_char_0, field_inverse_zero})"
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
   assumes nx: "isnormNum x" and ny: "isnormNum y"
   shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
 proof -
   fix h :: 'a
   obtain a b where x: "x = (a, b)" by (cases x)