--- a/src/HOL/Predicate.thy Wed Dec 08 13:34:50 2010 +0100
+++ b/src/HOL/Predicate.thy Wed Dec 08 13:34:50 2010 +0100
@@ -87,16 +87,16 @@
subsubsection {* Top and bottom elements *}
lemma top1I [intro!]: "top x"
- by (simp add: top_fun_eq top_bool_eq)
+ by (simp add: top_fun_def top_bool_def)
lemma top2I [intro!]: "top x y"
- by (simp add: top_fun_eq top_bool_eq)
+ by (simp add: top_fun_def top_bool_def)
lemma bot1E [no_atp, elim!]: "bot x \<Longrightarrow> P"
- by (simp add: bot_fun_eq bot_bool_eq)
+ by (simp add: bot_fun_def bot_bool_def)
lemma bot2E [elim!]: "bot x y \<Longrightarrow> P"
- by (simp add: bot_fun_eq bot_bool_eq)
+ by (simp add: bot_fun_def bot_bool_def)
lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
by (auto simp add: fun_eq_iff)
@@ -108,22 +108,22 @@
subsubsection {* Binary union *}
lemma sup1I1 [elim?]: "A x \<Longrightarrow> sup A B x"
- by (simp add: sup_fun_eq sup_bool_eq)
+ by (simp add: sup_fun_def sup_bool_def)
lemma sup2I1 [elim?]: "A x y \<Longrightarrow> sup A B x y"
- by (simp add: sup_fun_eq sup_bool_eq)
+ by (simp add: sup_fun_def sup_bool_def)
lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x"
- by (simp add: sup_fun_eq sup_bool_eq)
+ by (simp add: sup_fun_def sup_bool_def)
lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"
- by (simp add: sup_fun_eq sup_bool_eq)
+ by (simp add: sup_fun_def sup_bool_def)
lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P"
- by (simp add: sup_fun_eq sup_bool_eq) iprover
+ by (simp add: sup_fun_def sup_bool_def) iprover
lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P"
- by (simp add: sup_fun_eq sup_bool_eq) iprover
+ by (simp add: sup_fun_def sup_bool_def) iprover
text {*
\medskip Classical introduction rule: no commitment to @{text A} vs
@@ -131,49 +131,49 @@
*}
lemma sup1CI [intro!]: "(~ B x ==> A x) ==> sup A B x"
- by (auto simp add: sup_fun_eq sup_bool_eq)
+ by (auto simp add: sup_fun_def sup_bool_def)
lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y"
- by (auto simp add: sup_fun_eq sup_bool_eq)
+ by (auto simp add: sup_fun_def sup_bool_def)
lemma sup_Un_eq: "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
- by (simp add: sup_fun_eq sup_bool_eq mem_def)
+ by (simp add: sup_fun_def sup_bool_def mem_def)
lemma sup_Un_eq2 [pred_set_conv]: "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
- by (simp add: sup_fun_eq sup_bool_eq mem_def)
+ by (simp add: sup_fun_def sup_bool_def mem_def)
subsubsection {* Binary intersection *}
lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
lemma inf1D1: "inf A B x ==> A x"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
lemma inf2D1: "inf A B x y ==> A x y"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
lemma inf1D2: "inf A B x ==> B x"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
lemma inf2D2: "inf A B x y ==> B x y"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
lemma inf_Int_eq: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
- by (simp add: inf_fun_eq inf_bool_eq mem_def)
+ by (simp add: inf_fun_def inf_bool_def mem_def)
lemma inf_Int_eq2 [pred_set_conv]: "inf (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
- by (simp add: inf_fun_eq inf_bool_eq mem_def)
+ by (simp add: inf_fun_def inf_bool_def mem_def)
subsubsection {* Unions of families *}
@@ -286,11 +286,11 @@
elim: pred_compE dest: conversepD)
lemma converse_meet: "(inf r s)^--1 = inf r^--1 s^--1"
- by (simp add: inf_fun_eq inf_bool_eq)
+ by (simp add: inf_fun_def inf_bool_def)
(iprover intro: conversepI ext dest: conversepD)
lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"
- by (simp add: sup_fun_eq sup_bool_eq)
+ by (simp add: sup_fun_def sup_bool_def)
(iprover intro: conversepI ext dest: conversepD)
lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="