src/HOL/Dense_Linear_Order.thy

 setup Ferrante_Rackoff_Data.setup
 
 context linorder
 begin
 
-lemma less_not_permute: "\<not> (x \<sqsubset> y \<and> y \<sqsubset> x)" by (simp add: not_less linear)
+lemma less_not_permute: "\<not> (x \<^loc>< y \<and> y \<^loc>< x)" by (simp add: not_less linear)
 
 lemma gather_simps: 
   shows 
-  "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> x \<sqsubset> u \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> (insert u U). x \<sqsubset> y) \<and> P x)"
-  and "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> l \<sqsubset> x \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> P x)"
-  "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> x \<sqsubset> u) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> (insert u U). x \<sqsubset> y))"
-  and "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y) \<and> l \<sqsubset> x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y))"  by auto
+  "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> x \<^loc>< u \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> (insert u U). x \<^loc>< y) \<and> P x)"
+  and "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> l \<^loc>< x \<and> P x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> P x)"
+  "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> x \<^loc>< u) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> (insert u U). x \<^loc>< y))"
+  and "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y) \<and> l \<^loc>< x) \<longleftrightarrow> (\<exists>x. (\<forall>y \<in> (insert l L). y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y))"  by auto
 
 lemma 
-  gather_start: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y \<^loc>< x) \<and> (\<forall>y\<in> {}. x \<sqsubset> y) \<and> P x)" 
+  gather_start: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y \<^loc>< x) \<and> (\<forall>y\<in> {}. x \<^loc>< y) \<and> P x)" 
   by simp
 
-text{* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
-lemma minf_lt:  "\<exists>z . \<forall>x. x \<sqsubset> z \<longrightarrow> (x \<sqsubset> t \<longleftrightarrow> True)" by auto
-lemma minf_gt: "\<exists>z . \<forall>x. x \<sqsubset> z \<longrightarrow>  (t \<sqsubset> x \<longleftrightarrow>  False)"
+text{* Theorems for @{text "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
+lemma minf_lt:  "\<exists>z . \<forall>x. x \<^loc>< z \<longrightarrow> (x \<^loc>< t \<longleftrightarrow> True)" by auto
+lemma minf_gt: "\<exists>z . \<forall>x. x \<^loc>< z \<longrightarrow>  (t \<^loc>< x \<longleftrightarrow>  False)"
   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
 
-lemma minf_le: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (x \<sqsubseteq> t \<longleftrightarrow> True)" by (auto simp add: less_le)
-lemma minf_ge: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (t \<sqsubseteq> x \<longleftrightarrow> False)"
+lemma minf_le: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (x \<^loc>\<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
+lemma minf_ge: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (t \<^loc>\<le> x \<longleftrightarrow> False)"
   by (auto simp add: less_le not_less not_le)
-lemma minf_eq: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
-lemma minf_neq: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
-lemma minf_P: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
-
-text{* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
-lemma pinf_gt:  "\<exists>z . \<forall>x. z \<sqsubset> x \<longrightarrow> (t \<sqsubset> x \<longleftrightarrow> True)" by auto
-lemma pinf_lt: "\<exists>z . \<forall>x. z \<sqsubset> x \<longrightarrow>  (x \<sqsubset> t \<longleftrightarrow>  False)"
+lemma minf_eq: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
+lemma minf_neq: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
+lemma minf_P: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
+
+text{* Theorems for @{text "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
+lemma pinf_gt:  "\<exists>z . \<forall>x. z \<^loc>< x \<longrightarrow> (t \<^loc>< x \<longleftrightarrow> True)" by auto
+lemma pinf_lt: "\<exists>z . \<forall>x. z \<^loc>< x \<longrightarrow>  (x \<^loc>< t \<longleftrightarrow>  False)"
   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
 
-lemma pinf_ge: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (t \<sqsubseteq> x \<longleftrightarrow> True)" by (auto simp add: less_le)
-lemma pinf_le: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (x \<sqsubseteq> t \<longleftrightarrow> False)"
+lemma pinf_ge: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (t \<^loc>\<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
+lemma pinf_le: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (x \<^loc>\<le> t \<longleftrightarrow> False)"
   by (auto simp add: less_le not_less not_le)
-lemma pinf_eq: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
-lemma pinf_neq: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
-lemma pinf_P: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
-
-lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<sqsubset> t \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
-lemma nmi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t \<sqsubset> x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)"
+lemma pinf_eq: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
+lemma pinf_neq: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
+lemma pinf_P: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
+
+lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<^loc>< t \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
+lemma nmi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t \<^loc>< x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)"
   by (auto simp add: le_less)
-lemma  nmi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<sqsubseteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
-lemma  nmi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<sqsubseteq> x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
-lemma  nmi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
-lemma  nmi_neq: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
-lemma  nmi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
-lemma  nmi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x) ;
-  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)\<rbrakk> \<Longrightarrow>
-  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
-lemma  nmi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x) ;
-  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)\<rbrakk> \<Longrightarrow>
-  \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<sqsubseteq> x)" by auto
-
-lemma  npi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<sqsubset> t \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by (auto simp add: le_less)
-lemma  npi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<sqsubset> x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
-lemma  npi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<sqsubseteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
-lemma  npi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<sqsubseteq> x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
-lemma  npi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
-lemma  npi_neq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u )" by auto
-lemma  npi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
-lemma  npi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u) ;  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)\<rbrakk>
-  \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
-lemma  npi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)\<rbrakk>
-  \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<sqsubseteq> u)" by auto
-
-lemma lin_dense_lt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> x \<sqsubset> t \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> y \<sqsubset> t)"
+lemma  nmi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<^loc>\<le> t \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
+lemma  nmi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<^loc>\<le> x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
+lemma  nmi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
+lemma  nmi_neq: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
+lemma  nmi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
+lemma  nmi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x) ;
+  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)\<rbrakk> \<Longrightarrow>
+  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
+lemma  nmi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x) ;
+  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)\<rbrakk> \<Longrightarrow>
+  \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<^loc>\<le> x)" by auto
+
+lemma  npi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<^loc>< t \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by (auto simp add: le_less)
+lemma  npi_gt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<^loc>< x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
+lemma  npi_le: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<^loc>\<le> t \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
+lemma  npi_ge: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<^loc>\<le> x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
+lemma  npi_eq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
+lemma  npi_neq: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u )" by auto
+lemma  npi_P: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
+lemma  npi_conj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u) ;  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)\<rbrakk>
+  \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
+lemma  npi_disj: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)\<rbrakk>
+  \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<^loc>\<le> u)" by auto
+
+lemma lin_dense_lt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> x \<^loc>< t \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> y \<^loc>< t)"
 proof(clarsimp)
-  fix x l u y  assume tU: "t \<in> U" and noU: "\<forall>t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U" and lx: "l \<sqsubset> x"
-    and xu: "x\<sqsubset>u"  and px: "x \<sqsubset> t" and ly: "l\<sqsubset>y" and yu:"y \<sqsubset> u"
+  fix x l u y  assume tU: "t \<in> U" and noU: "\<forall>t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U" and lx: "l \<^loc>< x"
+    and xu: "x\<^loc><u"  and px: "x \<^loc>< t" and ly: "l\<^loc><y" and yu:"y \<^loc>< u"
   from tU noU ly yu have tny: "t\<noteq>y" by auto
-  {assume H: "t \<sqsubset> y"
+  {assume H: "t \<^loc>< y"
     from less_trans[OF lx px] less_trans[OF H yu]
-    have "l \<sqsubset> t \<and> t \<sqsubset> u"  by simp
+    have "l \<^loc>< t \<and> t \<^loc>< u"  by simp
     with tU noU have "False" by auto}
-  hence "\<not> t \<sqsubset> y"  by auto hence "y \<sqsubseteq> t" by (simp add: not_less)
-  thus "y \<sqsubset> t" using tny by (simp add: less_le)
-qed
-
-lemma lin_dense_gt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l \<sqsubset> x \<and> x \<sqsubset> u \<and> t \<sqsubset> x \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> t \<sqsubset> y)"
+  hence "\<not> t \<^loc>< y"  by auto hence "y \<^loc>\<le> t" by (simp add: not_less)
+  thus "y \<^loc>< t" using tny by (simp add: less_le)
+qed
+
+lemma lin_dense_gt: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l \<^loc>< x \<and> x \<^loc>< u \<and> t \<^loc>< x \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> t \<^loc>< y)"
 proof(clarsimp)
   fix x l u y
-  assume tU: "t \<in> U" and noU: "\<forall>t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U" and lx: "l \<sqsubset> x" and xu: "x\<sqsubset>u"
-  and px: "t \<sqsubset> x" and ly: "l\<sqsubset>y" and yu:"y \<sqsubset> u"
+  assume tU: "t \<in> U" and noU: "\<forall>t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U" and lx: "l \<^loc>< x" and xu: "x\<^loc><u"
+  and px: "t \<^loc>< x" and ly: "l\<^loc><y" and yu:"y \<^loc>< u"
   from tU noU ly yu have tny: "t\<noteq>y" by auto
-  {assume H: "y\<sqsubset> t"
-    from less_trans[OF ly H] less_trans[OF px xu] have "l \<sqsubset> t \<and> t \<sqsubset> u" by simp
+  {assume H: "y\<^loc>< t"
+    from less_trans[OF ly H] less_trans[OF px xu] have "l \<^loc>< t \<and> t \<^loc>< u" by simp
     with tU noU have "False" by auto}
-  hence "\<not> y\<sqsubset>t"  by auto hence "t \<sqsubseteq> y" by (auto simp add: not_less)
-  thus "t \<sqsubset> y" using tny by (simp add:less_le)
-qed
-
-lemma lin_dense_le: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> x \<sqsubseteq> t \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> y\<sqsubseteq> t)"
+  hence "\<not> y\<^loc><t"  by auto hence "t \<^loc>\<le> y" by (auto simp add: not_less)
+  thus "t \<^loc>< y" using tny by (simp add:less_le)
+qed
+
+lemma lin_dense_le: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> x \<^loc>\<le> t \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> y\<^loc>\<le> t)"
 proof(clarsimp)
   fix x l u y
-  assume tU: "t \<in> U" and noU: "\<forall>t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U" and lx: "l \<sqsubset> x" and xu: "x\<sqsubset>u"
-  and px: "x \<sqsubseteq> t" and ly: "l\<sqsubset>y" and yu:"y \<sqsubset> u"
+  assume tU: "t \<in> U" and noU: "\<forall>t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U" and lx: "l \<^loc>< x" and xu: "x\<^loc><u"
+  and px: "x \<^loc>\<le> t" and ly: "l\<^loc><y" and yu:"y \<^loc>< u"
   from tU noU ly yu have tny: "t\<noteq>y" by auto
-  {assume H: "t \<sqsubset> y"
+  {assume H: "t \<^loc>< y"
     from less_le_trans[OF lx px] less_trans[OF H yu]
-    have "l \<sqsubset> t \<and> t \<sqsubset> u" by simp
+    have "l \<^loc>< t \<and> t \<^loc>< u" by simp
     with tU noU have "False" by auto}
-  hence "\<not> t \<sqsubset> y"  by auto thus "y \<sqsubseteq> t" by (simp add: not_less)
-qed
-
-lemma lin_dense_ge: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> t \<sqsubseteq> x \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> t \<sqsubseteq> y)"
+  hence "\<not> t \<^loc>< y"  by auto thus "y \<^loc>\<le> t" by (simp add: not_less)
+qed
+
+lemma lin_dense_ge: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> t \<^loc>\<le> x \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> t \<^loc>\<le> y)"
 proof(clarsimp)
   fix x l u y
-  assume tU: "t \<in> U" and noU: "\<forall>t. l \<sqsubset> t \<and> t \<sqsubset> u \<longrightarrow> t \<notin> U" and lx: "l \<sqsubset> x" and xu: "x\<sqsubset>u"
-  and px: "t \<sqsubseteq> x" and ly: "l\<sqsubset>y" and yu:"y \<sqsubset> u"
+  assume tU: "t \<in> U" and noU: "\<forall>t. l \<^loc>< t \<and> t \<^loc>< u \<longrightarrow> t \<notin> U" and lx: "l \<^loc>< x" and xu: "x\<^loc><u"
+  and px: "t \<^loc>\<le> x" and ly: "l\<^loc><y" and yu:"y \<^loc>< u"
   from tU noU ly yu have tny: "t\<noteq>y" by auto
-  {assume H: "y\<sqsubset> t"
+  {assume H: "y\<^loc>< t"
     from less_trans[OF ly H] le_less_trans[OF px xu]
-    have "l \<sqsubset> t \<and> t \<sqsubset> u" by simp
+    have "l \<^loc>< t \<and> t \<^loc>< u" by simp
     with tU noU have "False" by auto}
-  hence "\<not> y\<sqsubset>t"  by auto thus "t \<sqsubseteq> y" by (simp add: not_less)
-qed
-lemma lin_dense_eq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> x = t   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> y= t)"  by auto
-lemma lin_dense_neq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> x \<noteq> t   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> y\<noteq> t)"  by auto
-lemma lin_dense_P: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P)"  by auto
+  hence "\<not> y\<^loc><t"  by auto thus "t \<^loc>\<le> y" by (simp add: not_less)
+qed
+lemma lin_dense_eq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> x = t   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> y= t)"  by auto
+lemma lin_dense_neq: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> x \<noteq> t   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> y\<noteq> t)"  by auto
+lemma lin_dense_P: "\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P)"  by auto
 
 lemma lin_dense_conj:
-  "\<lbrakk>\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P1 x
-  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P1 y) ;
-  \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P2 x
-  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
-  \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> (P1 x \<and> P2 x)
-  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> (P1 y \<and> P2 y))"
+  "\<lbrakk>\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P1 x
+  \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P1 y) ;
+  \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P2 x
+  \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
+  \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> (P1 x \<and> P2 x)
+  \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> (P1 y \<and> P2 y))"
   by blast
 lemma lin_dense_disj:
-  "\<lbrakk>\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P1 x
-  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P1 y) ;
-  \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P2 x
-  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
-  \<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> (P1 x \<or> P2 x)
-  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> (P1 y \<or> P2 y))"
+  "\<lbrakk>\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P1 x
+  \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P1 y) ;
+  \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P2 x
+  \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
+  \<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> (P1 x \<or> P2 x)
+  \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> (P1 y \<or> P2 y))"
   by blast
 
-lemma npmibnd: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)\<rbrakk>
-  \<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')"
+lemma npmibnd: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<^loc>\<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<^loc>\<le> u)\<rbrakk>
+  \<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<^loc>\<le> x \<and> x \<^loc>\<le> u')"
 by auto
 
 lemma finite_set_intervals:
-  assumes px: "P x" and lx: "l \<sqsubseteq> x" and xu: "x \<sqsubseteq> u" and linS: "l\<in> S"
-  and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<sqsubseteq> x" and Su: "\<forall> x\<in> S. x \<sqsubseteq> u"
-  shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a \<sqsubset> y \<and> y \<sqsubset> b \<longrightarrow> y \<notin> S) \<and> a \<sqsubseteq> x \<and> x \<sqsubseteq> b \<and> P x"
-proof-
-  let ?Mx = "{y. y\<in> S \<and> y \<sqsubseteq> x}"
-  let ?xM = "{y. y\<in> S \<and> x \<sqsubseteq> y}"
+  assumes px: "P x" and lx: "l \<^loc>\<le> x" and xu: "x \<^loc>\<le> u" and linS: "l\<in> S"
+  and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<^loc>\<le> x" and Su: "\<forall> x\<in> S. x \<^loc>\<le> u"
+  shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a \<^loc>< y \<and> y \<^loc>< b \<longrightarrow> y \<notin> S) \<and> a \<^loc>\<le> x \<and> x \<^loc>\<le> b \<and> P x"
+proof-
+  let ?Mx = "{y. y\<in> S \<and> y \<^loc>\<le> x}"
+  let ?xM = "{y. y\<in> S \<and> x \<^loc>\<le> y}"
   let ?a = "Max ?Mx"
   let ?b = "Min ?xM"
   have MxS: "?Mx \<subseteq> S" by blast
   hence fMx: "finite ?Mx" using fS finite_subset by auto
   from lx linS have linMx: "l \<in> ?Mx" by blast
   hence Mxne: "?Mx \<noteq> {}" by blast
   have xMS: "?xM \<subseteq> S" by blast
   hence fxM: "finite ?xM" using fS finite_subset by auto
   from xu uinS have linxM: "u \<in> ?xM" by blast
   hence xMne: "?xM \<noteq> {}" by blast
-  have ax:"?a \<sqsubseteq> x" using Mxne fMx by auto
-  have xb:"x \<sqsubseteq> ?b" using xMne fxM by auto
+  have ax:"?a \<^loc>\<le> x" using Mxne fMx by auto
+  have xb:"x \<^loc>\<le> ?b" using xMne fxM by auto
   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
-  have noy:"\<forall> y. ?a \<sqsubset> y \<and> y \<sqsubset> ?b \<longrightarrow> y \<notin> S"
+  have noy:"\<forall> y. ?a \<^loc>< y \<and> y \<^loc>< ?b \<longrightarrow> y \<notin> S"
   proof(clarsimp)
-    fix y   assume ay: "?a \<sqsubset> y" and yb: "y \<sqsubset> ?b" and yS: "y \<in> S"
+    fix y   assume ay: "?a \<^loc>< y" and yb: "y \<^loc>< ?b" and yS: "y \<in> S"
     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear)
-    moreover {assume "y \<in> ?Mx" hence "y \<sqsubseteq> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
-    moreover {assume "y \<in> ?xM" hence "?b \<sqsubseteq> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
+    moreover {assume "y \<in> ?Mx" hence "y \<^loc>\<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
+    moreover {assume "y \<in> ?xM" hence "?b \<^loc>\<le> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
     ultimately show "False" by blast
   qed
   from ainS binS noy ax xb px show ?thesis by blast
 qed
 
 lemma finite_set_intervals2:
-  assumes px: "P x" and lx: "l \<sqsubseteq> x" and xu: "x \<sqsubseteq> u" and linS: "l\<in> S"
-  and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<sqsubseteq> x" and Su: "\<forall> x\<in> S. x \<sqsubseteq> u"
-  shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a \<sqsubset> y \<and> y \<sqsubset> b \<longrightarrow> y \<notin> S) \<and> a \<sqsubset> x \<and> x \<sqsubset> b \<and> P x)"
+  assumes px: "P x" and lx: "l \<^loc>\<le> x" and xu: "x \<^loc>\<le> u" and linS: "l\<in> S"
+  and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<^loc>\<le> x" and Su: "\<forall> x\<in> S. x \<^loc>\<le> u"
+  shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a \<^loc>< y \<and> y \<^loc>< b \<longrightarrow> y \<notin> S) \<and> a \<^loc>< x \<and> x \<^loc>< b \<and> P x)"
 proof-
   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
   obtain a and b where
-    as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a \<sqsubset> y \<and> y \<sqsubset> b \<longrightarrow> y \<notin> S"
-    and axb: "a \<sqsubseteq> x \<and> x \<sqsubseteq> b \<and> P x"  by auto
-  from axb have "x= a \<or> x= b \<or> (a \<sqsubset> x \<and> x \<sqsubset> b)" by (auto simp add: le_less)
+    as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a \<^loc>< y \<and> y \<^loc>< b \<longrightarrow> y \<notin> S"
+    and axb: "a \<^loc>\<le> x \<and> x \<^loc>\<le> b \<and> P x"  by auto
+  from axb have "x= a \<or> x= b \<or> (a \<^loc>< x \<and> x \<^loc>< b)" by (auto simp add: le_less)
   thus ?thesis using px as bs noS by blast
 qed
 
 end
 

 context dense_linear_order
 begin
 
 lemma dlo_qe_bnds: 
   assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
-  shows "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y)) \<equiv> (\<forall> l \<in> L. \<forall>u \<in> U. l \<sqsubset> u)"
+  shows "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y)) \<equiv> (\<forall> l \<in> L. \<forall>u \<in> U. l \<^loc>< u)"
 proof (simp only: atomize_eq, rule iffI)
   assume H: "\<exists>x. (\<forall>y\<in>L. y \<^loc>< x) \<and> (\<forall>y\<in>U. x \<^loc>< y)"
   then obtain x where xL: "\<forall>y\<in>L. y \<^loc>< x" and xU: "\<forall>y\<in>U. x \<^loc>< y" by blast
   {fix l u assume l: "l \<in> L" and u: "u \<in> U"
     from less_trans[OF xL[rule_format, OF l] xU[rule_format, OF u]]
-    have "l \<sqsubset> u" .}
+    have "l \<^loc>< u" .}
   thus "\<forall>l\<in>L. \<forall>u\<in>U. l \<^loc>< u" by blast
 next
   assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l \<^loc>< u"
   let ?ML = "Max L"
   let ?MU = "Min U"  
-  from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<sqsubseteq> ?ML" by auto
-  from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<sqsubseteq> u" by auto
-  from th1 th2 H have "?ML \<sqsubset> ?MU" by auto
-  with dense obtain w where th3: "?ML \<sqsubset> w" and th4: "w \<sqsubset> ?MU" by blast
-  from th3 th1' have "\<forall>l \<in> L. l \<sqsubset> w" by auto
-  moreover from th4 th2' have "\<forall>u \<in> U. w \<sqsubset> u" by auto
+  from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<^loc>\<le> ?ML" by auto
+  from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<^loc>\<le> u" by auto
+  from th1 th2 H have "?ML \<^loc>< ?MU" by auto
+  with dense obtain w where th3: "?ML \<^loc>< w" and th4: "w \<^loc>< ?MU" by blast
+  from th3 th1' have "\<forall>l \<in> L. l \<^loc>< w" by auto
+  moreover from th4 th2' have "\<forall>u \<in> U. w \<^loc>< u" by auto
   ultimately show "\<exists>x. (\<forall>y\<in>L. y \<^loc>< x) \<and> (\<forall>y\<in>U. x \<^loc>< y)" by auto
 qed
 
 lemma dlo_qe_noub: 
   assumes ne: "L \<noteq> {}" and fL: "finite L"
-  shows "(\<exists>x. (\<forall>y \<in> L. y \<sqsubset> x) \<and> (\<forall>y \<in> {}. x \<sqsubset> y)) \<equiv> True"
+  shows "(\<exists>x. (\<forall>y \<in> L. y \<^loc>< x) \<and> (\<forall>y \<in> {}. x \<^loc>< y)) \<equiv> True"
 proof(simp add: atomize_eq)
-  from gt_ex[rule_format, of "Max L"] obtain M where M: "Max L \<sqsubset> M" by blast
-  from ne fL have "\<forall>x \<in> L. x \<sqsubseteq> Max L" by simp
-  with M have "\<forall>x\<in>L. x \<sqsubset> M" by (auto intro: le_less_trans)
+  from gt_ex[rule_format, of "Max L"] obtain M where M: "Max L \<^loc>< M" by blast
+  from ne fL have "\<forall>x \<in> L. x \<^loc>\<le> Max L" by simp
+  with M have "\<forall>x\<in>L. x \<^loc>< M" by (auto intro: le_less_trans)
   thus "\<exists>x. \<forall>y\<in>L. y \<^loc>< x" by blast
 qed
 
 lemma dlo_qe_nolb: 
   assumes ne: "U \<noteq> {}" and fU: "finite U"
-  shows "(\<exists>x. (\<forall>y \<in> {}. y \<sqsubset> x) \<and> (\<forall>y \<in> U. x \<sqsubset> y)) \<equiv> True"
+  shows "(\<exists>x. (\<forall>y \<in> {}. y \<^loc>< x) \<and> (\<forall>y \<in> U. x \<^loc>< y)) \<equiv> True"
 proof(simp add: atomize_eq)
-  from lt_ex[rule_format, of "Min U"] obtain M where M: "M \<sqsubset> Min U" by blast
-  from ne fU have "\<forall>x \<in> U. Min U \<sqsubseteq> x" by simp
-  with M have "\<forall>x\<in>U. M \<sqsubset> x" by (auto intro: less_le_trans)
+  from lt_ex[rule_format, of "Min U"] obtain M where M: "M \<^loc>< Min U" by blast
+  from ne fU have "\<forall>x \<in> U. Min U \<^loc>\<le> x" by simp
+  with M have "\<forall>x\<in>U. M \<^loc>< x" by (auto intro: less_le_trans)
   thus "\<exists>x. \<forall>y\<in>U. x \<^loc>< y" by blast
 qed
 
 lemma exists_neq: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" 
   using gt_ex[rule_format, of t] by auto
 
 lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq 
   le_less neq_iff linear less_not_permute
 
-lemma axiom: "dense_linear_order (op \<sqsubseteq>) (op \<sqsubset>)" .
-lemma atoms: includes meta_term_syntax
-  shows "TERM (op \<sqsubset> :: 'a \<Rightarrow> _)" and "TERM (op \<sqsubseteq>)" and "TERM (op = :: 'a \<Rightarrow> _)" .
+lemma axiom: "dense_linear_order (op \<^loc>\<le>) (op \<^loc><)" .
+lemma atoms:
+  includes meta_term_syntax
+  shows "TERM (less :: 'a \<Rightarrow> _)"
+    and "TERM (less_eq :: 'a \<Rightarrow> _)"
+    and "TERM (op = :: 'a \<Rightarrow> _)" .
 
 declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
 declare dlo_simps[langfordsimp]
 
 end

 section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *}
 
 text {* Linear order without upper bounds *}
 
 class linorder_no_ub = linorder +
-  assumes gt_ex: "\<exists>y. x \<sqsubset> y"
+  assumes gt_ex: "\<exists>y. x \<^loc>< y"
 begin
 
-lemma ge_ex: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
-
-text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
+lemma ge_ex: "\<exists>y. x \<^loc>\<le> y" using gt_ex by auto
+
+text {* Theorems for @{text "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
 lemma pinf_conj:
-  assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
-  and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
-  shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
-proof-
-  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
-     and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
-  from gt_ex obtain z where z:"max z1 z2 \<sqsubset> z" by blast
-  from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
-  {fix x assume H: "z \<sqsubset> x"
+  assumes ex1: "\<exists>z1. \<forall>x. z1 \<^loc>< x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+  and ex2: "\<exists>z2. \<forall>x. z2 \<^loc>< x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+  shows "\<exists>z. \<forall>x. z \<^loc><  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
+proof-
+  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<^loc>< x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+     and z2: "\<forall>x. z2 \<^loc>< x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+  from gt_ex obtain z where z:"max z1 z2 \<^loc>< z" by blast
+  from z have zz1: "z1 \<^loc>< z" and zz2: "z2 \<^loc>< z" by simp_all
+  {fix x assume H: "z \<^loc>< x"
     from less_trans[OF zz1 H] less_trans[OF zz2 H]
     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   }
   thus ?thesis by blast
 qed
 
 lemma pinf_disj:
-  assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
-  and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
-  shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
-proof-
-  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
-     and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
-  from gt_ex obtain z where z:"max z1 z2 \<sqsubset> z" by blast
-  from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
-  {fix x assume H: "z \<sqsubset> x"
+  assumes ex1: "\<exists>z1. \<forall>x. z1 \<^loc>< x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+  and ex2: "\<exists>z2. \<forall>x. z2 \<^loc>< x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+  shows "\<exists>z. \<forall>x. z \<^loc><  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
+proof-
+  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<^loc>< x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+     and z2: "\<forall>x. z2 \<^loc>< x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+  from gt_ex obtain z where z:"max z1 z2 \<^loc>< z" by blast
+  from z have zz1: "z1 \<^loc>< z" and zz2: "z2 \<^loc>< z" by simp_all
+  {fix x assume H: "z \<^loc>< x"
     from less_trans[OF zz1 H] less_trans[OF zz2 H]
     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   }
   thus ?thesis by blast
 qed
 
-lemma pinf_ex: assumes ex:"\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
-proof-
-  from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
-  from gt_ex obtain x where x: "z \<sqsubset> x" by blast
+lemma pinf_ex: assumes ex:"\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
+proof-
+  from ex obtain z where z: "\<forall>x. z \<^loc>< x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
+  from gt_ex obtain x where x: "z \<^loc>< x" by blast
   from z x p1 show ?thesis by blast
 qed
 
 end
 
 text {* Linear order without upper bounds *}
 
 class linorder_no_lb = linorder +
-  assumes lt_ex: "\<exists>y. y \<sqsubset> x"
+  assumes lt_ex: "\<exists>y. y \<^loc>< x"
 begin
 
-lemma le_ex: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
-
-
-text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
+lemma le_ex: "\<exists>y. y \<^loc>\<le> x" using lt_ex by auto
+
+
+text {* Theorems for @{text "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
 lemma minf_conj:
-  assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
-  and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
-  shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
-proof-
-  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
-  from lt_ex obtain z where z:"z \<sqsubset> min z1 z2" by blast
-  from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
-  {fix x assume H: "x \<sqsubset> z"
+  assumes ex1: "\<exists>z1. \<forall>x. x \<^loc>< z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+  and ex2: "\<exists>z2. \<forall>x. x \<^loc>< z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+  shows "\<exists>z. \<forall>x. x \<^loc><  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
+proof-
+  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<^loc>< z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<^loc>< z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+  from lt_ex obtain z where z:"z \<^loc>< min z1 z2" by blast
+  from z have zz1: "z \<^loc>< z1" and zz2: "z \<^loc>< z2" by simp_all
+  {fix x assume H: "x \<^loc>< z"
     from less_trans[OF H zz1] less_trans[OF H zz2]
     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
   }
   thus ?thesis by blast
 qed
 
 lemma minf_disj:
-  assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
-  and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
-  shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
-proof-
-  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
-  from lt_ex obtain z where z:"z \<sqsubset> min z1 z2" by blast
-  from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
-  {fix x assume H: "x \<sqsubset> z"
+  assumes ex1: "\<exists>z1. \<forall>x. x \<^loc>< z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
+  and ex2: "\<exists>z2. \<forall>x. x \<^loc>< z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
+  shows "\<exists>z. \<forall>x. x \<^loc><  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
+proof-
+  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<^loc>< z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"and z2: "\<forall>x. x \<^loc>< z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
+  from lt_ex obtain z where z:"z \<^loc>< min z1 z2" by blast
+  from z have zz1: "z \<^loc>< z1" and zz2: "z \<^loc>< z2" by simp_all
+  {fix x assume H: "x \<^loc>< z"
     from less_trans[OF H zz1] less_trans[OF H zz2]
     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
   }
   thus ?thesis by blast
 qed
 
-lemma minf_ex: assumes ex:"\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
-proof-
-  from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
-  from lt_ex obtain x where x: "x \<sqsubset> z" by blast
+lemma minf_ex: assumes ex:"\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
+proof-
+  from ex obtain z where z: "\<forall>x. x \<^loc>< z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
+  from lt_ex obtain x where x: "x \<^loc>< z" by blast
   from z x p1 show ?thesis by blast
 qed
 
 end
 
 
 class constr_dense_linear_order = linorder_no_lb + linorder_no_ub +
   fixes between
-  assumes between_less: "x \<sqsubset> y \<Longrightarrow> x \<sqsubset> between x y \<and> between x y \<sqsubset> y"
+  assumes between_less: "x \<^loc>< y \<Longrightarrow> x \<^loc>< between x y \<and> between x y \<^loc>< y"
      and  between_same: "between x x = x"
 
 instance advanced constr_dense_linear_order < dense_linear_order
   apply unfold_locales
   using gt_ex lt_ex between_less

 context constr_dense_linear_order
 begin
 
 lemma rinf_U:
   assumes fU: "finite U"
-  and lin_dense: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x
-  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
-  and nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')"
+  and lin_dense: "\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P x
+  \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P y )"
+  and nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<^loc>\<le> x \<and> x \<^loc>\<le> u')"
   and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
   shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
 proof-
   from ex obtain x where px: "P x" by blast
-  from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
-  then obtain u and u' where uU:"u\<in> U" and uU': "u' \<in> U" and ux:"u \<sqsubseteq> x" and xu':"x \<sqsubseteq> u'" by auto
+  from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<^loc>\<le> x \<and> x \<^loc>\<le> u'" by auto
+  then obtain u and u' where uU:"u\<in> U" and uU': "u' \<in> U" and ux:"u \<^loc>\<le> x" and xu':"x \<^loc>\<le> u'" by auto
   from uU have Une: "U \<noteq> {}" by auto
   let ?l = "Min U"
   let ?u = "Max U"
   have linM: "?l \<in> U" using fU Une by simp
   have uinM: "?u \<in> U" using fU Une by simp
-  have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
-  have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
-  have th:"?l \<sqsubseteq> u" using uU Une lM by auto
-  from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
-  have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
-  from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
+  have lM: "\<forall> t\<in> U. ?l \<^loc>\<le> t" using Une fU by auto
+  have Mu: "\<forall> t\<in> U. t \<^loc>\<le> ?u" using Une fU by auto
+  have th:"?l \<^loc>\<le> u" using uU Une lM by auto
+  from order_trans[OF th ux] have lx: "?l \<^loc>\<le> x" .
+  have th: "u' \<^loc>\<le> ?u" using uU' Une Mu by simp
+  from order_trans[OF xu' th] have xu: "x \<^loc>\<le> ?u" .
   from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
   have "(\<exists> s\<in> U. P s) \<or>
-      (\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x)" .
+      (\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<^loc>< y \<and> y \<^loc>< t2 \<longrightarrow> y \<notin> U) \<and> t1 \<^loc>< x \<and> x \<^loc>< t2 \<and> P x)" .
   moreover { fix u assume um: "u\<in>U" and pu: "P u"
     have "between u u = u" by (simp add: between_same)
     with um pu have "P (between u u)" by simp
     with um have ?thesis by blast}
   moreover{
-    assume "\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x"
+    assume "\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<^loc>< y \<and> y \<^loc>< t2 \<longrightarrow> y \<notin> U) \<and> t1 \<^loc>< x \<and> x \<^loc>< t2 \<and> P x"
       then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
-        and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x"
+        and noM: "\<forall> y. t1 \<^loc>< y \<and> y \<^loc>< t2 \<longrightarrow> y \<notin> U" and t1x: "t1 \<^loc>< x" and xt2: "x \<^loc>< t2" and px: "P x"
         by blast
-      from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
+      from less_trans[OF t1x xt2] have t1t2: "t1 \<^loc>< t2" .
       let ?u = "between t1 t2"
-      from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
+      from between_less t1t2 have t1lu: "t1 \<^loc>< ?u" and ut2: "?u \<^loc>< t2" by auto
       from lin_dense[rule_format, OF] noM t1x xt2 px t1lu ut2 have "P ?u" by blast
       with t1M t2M have ?thesis by blast}
     ultimately show ?thesis by blast
   qed
 
 theorem fr_eq:
   assumes fU: "finite U"
-  and lin_dense: "\<forall>x l u. (\<forall> t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x
-   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
-  and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
-  and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
-  and mi: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x = MP)"  and pi: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x = PP)"
+  and lin_dense: "\<forall>x l u. (\<forall> t. l \<^loc>< t \<and> t\<^loc>< u \<longrightarrow> t \<notin> U) \<and> l\<^loc>< x \<and> x \<^loc>< u \<and> P x
+   \<longrightarrow> (\<forall> y. l \<^loc>< y \<and> y \<^loc>< u \<longrightarrow> P y )"
+  and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<^loc>\<le> x)"
+  and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<^loc>\<le> u)"
+  and mi: "\<exists>z. \<forall>x. x \<^loc>< z \<longrightarrow> (P x = MP)"  and pi: "\<exists>z. \<forall>x. z \<^loc>< x \<longrightarrow> (P x = PP)"
   shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
   (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
 proof-
  {
    assume px: "\<exists> x. P x"
    have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
    moreover {assume "MP \<or> PP" hence "?D" by blast}
    moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
      from npmibnd[OF nmibnd npibnd]
-     have nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')" .
+     have nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. \<exists> u' \<in> U. u \<^loc>\<le> x \<and> x \<^loc>\<le> u')" .
      from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast}
    ultimately have "?D" by blast}
  moreover
  { assume "?D"
    moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}

 lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
 lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
 lemmas lin_dense_thms = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P
 
 lemma ferrack_axiom: "constr_dense_linear_order less_eq less between" by fact
-lemma atoms: includes meta_term_syntax
-  shows "TERM (op \<sqsubset> :: 'a \<Rightarrow> _)" and "TERM (op \<sqsubseteq>)" and "TERM (op = :: 'a \<Rightarrow> _)" .
+lemma atoms:
+  includes meta_term_syntax
+  shows "TERM (less :: 'a \<Rightarrow> _)"
+    and "TERM (less_eq :: 'a \<Rightarrow> _)"
+    and "TERM (op = :: 'a \<Rightarrow> _)" .
 
 declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
     nmi: nmi_thms npi: npi_thms lindense:
     lin_dense_thms qe: fr_eq atoms: atoms]
 
 declaration {*
 let
 fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
 fun generic_whatis phi =
  let
-  val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
+  val [lt, le] = map (Morphism.term phi) [@{term "op \<^loc><"}, @{term "op \<^loc>\<le>"}]
   fun h x t =
    case term_of t of
      Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
                             else Ferrante_Rackoff_Data.Nox
    | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq