# HG changeset patch
# User haftmann
# Date 1252923902 -7200
# Node ID 8df26038caa9d9a35ae5ab38d2d73fba7f3aff4a
# Parent  de411627a98511dcd07d7611ba4549f36529af85# Parent  c4c12ef9d4d1fbf4c1bc95203d50712fee82fec8
merged

diff -r de411627a985 -r 8df26038caa9 src/HOL/Lattices.thy
--- a/src/HOL/Lattices.thy	Sun Sep 13 02:10:41 2009 +0200
+++ b/src/HOL/Lattices.thy	Mon Sep 14 12:25:02 2009 +0200
@@ -12,7 +12,9 @@
 
 notation
   less_eq  (infix "\<sqsubseteq>" 50) and
-  less  (infix "\<sqsubset>" 50)
+  less  (infix "\<sqsubset>" 50) and
+  top ("\<top>") and
+  bot ("\<bottom>")
 
 class lower_semilattice = order +
   fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
@@ -220,6 +222,46 @@
 
 end
 
+subsubsection {* Strict order *}
+
+context lower_semilattice
+begin
+
+lemma less_infI1:
+  "a \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
+  by (auto simp add: less_le intro: le_infI1)
+
+lemma less_infI2:
+  "b \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
+  by (auto simp add: less_le intro: le_infI2)
+
+end
+
+context upper_semilattice
+begin
+
+lemma less_supI1:
+  "x < a \<Longrightarrow> x < a \<squnion> b"
+proof -
+  interpret dual: lower_semilattice "op \<ge>" "op >" sup
+    by (fact dual_semilattice)
+  assume "x < a"
+  then show "x < a \<squnion> b"
+    by (fact dual.less_infI1)
+qed
+
+lemma less_supI2:
+  "x < b \<Longrightarrow> x < a \<squnion> b"
+proof -
+  interpret dual: lower_semilattice "op \<ge>" "op >" sup
+    by (fact dual_semilattice)
+  assume "x < b"
+  then show "x < a \<squnion> b"
+    by (fact dual.less_infI2)
+qed
+
+end
+
 
 subsection {* Distributive lattices *}
 
@@ -306,6 +348,40 @@
   "x \<squnion> bot = x"
   by (rule sup_absorb1) simp
 
+lemma inf_eq_top_eq1:
+  assumes "A \<sqinter> B = \<top>"
+  shows "A = \<top>"
+proof (cases "B = \<top>")
+  case True with assms show ?thesis by simp
+next
+  case False with top_greatest have "B < \<top>" by (auto intro: neq_le_trans)
+  then have "A \<sqinter> B < \<top>" by (rule less_infI2)
+  with assms show ?thesis by simp
+qed
+
+lemma inf_eq_top_eq2:
+  assumes "A \<sqinter> B = \<top>"
+  shows "B = \<top>"
+  by (rule inf_eq_top_eq1, unfold inf_commute [of B]) (fact assms)
+
+lemma sup_eq_bot_eq1:
+  assumes "A \<squnion> B = \<bottom>"
+  shows "A = \<bottom>"
+proof -
+  interpret dual: boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" top bot
+    by (rule dual_boolean_algebra)
+  from dual.inf_eq_top_eq1 assms show ?thesis .
+qed
+
+lemma sup_eq_bot_eq2:
+  assumes "A \<squnion> B = \<bottom>"
+  shows "B = \<bottom>"
+proof -
+  interpret dual: boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" top bot
+    by (rule dual_boolean_algebra)
+  from dual.inf_eq_top_eq2 assms show ?thesis .
+qed
+
 lemma compl_unique:
   assumes "x \<sqinter> y = bot"
     and "x \<squnion> y = top"
@@ -517,6 +593,8 @@
   less_eq  (infix "\<sqsubseteq>" 50) and
   less (infix "\<sqsubset>" 50) and
   inf  (infixl "\<sqinter>" 70) and
-  sup  (infixl "\<squnion>" 65)
+  sup  (infixl "\<squnion>" 65) and
+  top ("\<top>") and
+  bot ("\<bottom>")
 
 end
diff -r de411627a985 -r 8df26038caa9 src/HOL/Tools/Function/fundef_core.ML
--- a/src/HOL/Tools/Function/fundef_core.ML	Sun Sep 13 02:10:41 2009 +0200
+++ b/src/HOL/Tools/Function/fundef_core.ML	Mon Sep 14 12:25:02 2009 +0200
@@ -478,7 +478,7 @@
 fun define_function fdefname (fname, mixfix) domT ranT G default lthy =
     let
       val f_def =
-          Abs ("x", domT, Const ("FunDef.THE_default", ranT --> (ranT --> boolT) --> ranT) $ (default $ Bound 0) $
+          Abs ("x", domT, Const (@{const_name FunDef.THE_default}, ranT --> (ranT --> boolT) --> ranT) $ (default $ Bound 0) $
                                 Abs ("y", ranT, G $ Bound 1 $ Bound 0))
               |> Syntax.check_term lthy
 
@@ -767,7 +767,7 @@
       val R' = Free ("R", fastype_of R)
 
       val Rrel = Free ("R", HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)))
-      val inrel_R = Const ("FunDef.in_rel", HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)) --> fastype_of R) $ Rrel
+      val inrel_R = Const (@{const_name FunDef.in_rel}, HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)) --> fastype_of R) $ Rrel
 
       val wfR' = cterm_of thy (HOLogic.mk_Trueprop (Const (@{const_name "Wellfounded.wfP"}, (domT --> domT --> boolT) --> boolT) $ R')) (* "wf R'" *)