src/HOL/Presburger.thy

   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
-  "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (d dvd x + s) = (d dvd x + s)"
-  "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})<z. (d dvd x + s) = (d dvd x + s)"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
   "\<exists>z.\<forall>x<z. F = F"
   by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
 
 lemma pinf:
   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 

   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
-  "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (d dvd x + s) = (d dvd x + s)"
-  "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})>z. (d dvd x + s) = (d dvd x + s)"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
   "\<exists>z.\<forall>x>z. F = F"
   by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
 
 lemma inf_period:
   "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
     \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
   "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
     \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
-  "(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
-  "(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
+  "(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
+  "(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
   "\<forall>x k. F = F"
 apply (auto elim!: dvdE simp add: algebra_simps)
 unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric]
 unfolding dvd_def mult_commute [of d] 
 by auto

 next
   case (step i)
   {fix x
     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
-      by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
+      by (simp add: algebra_simps)
     ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
   thus ?case ..
 qed
 
 lemma  minusinfinity:

 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
 apply(fastsimp)
 done
 
-theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Ring_and_Field.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
+theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Rings.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
   apply (rule eq_reflection [symmetric])
   apply (rule iffI)
   defer
   apply (erule exE)
   apply (rule_tac x = "l * x" in exI)