--- a/src/HOL/Predicate.thy Wed Dec 08 14:52:23 2010 +0100
+++ b/src/HOL/Predicate.thy Wed Dec 08 14:52:23 2010 +0100
@@ -16,6 +16,12 @@
top ("\<top>") and
bot ("\<bottom>")
+syntax (xsymbols)
+ "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
+ "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
+ "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
+ "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
+
subsection {* Predicates as (complete) lattices *}
@@ -179,61 +185,61 @@
subsubsection {* Unions of families *}
lemma SUP1_iff: "(SUP x:A. B x) b = (EX x:A. B x b)"
- by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
+ by (simp add: SUPR_apply)
lemma SUP2_iff: "(SUP x:A. B x) b c = (EX x:A. B x b c)"
- by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
+ by (simp add: SUPR_apply)
lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b"
- by (auto simp add: SUP1_iff)
+ by (auto simp add: SUPR_apply)
lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c"
- by (auto simp add: SUP2_iff)
+ by (auto simp add: SUPR_apply)
lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R"
- by (auto simp add: SUP1_iff)
+ by (auto simp add: SUPR_apply)
lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"
- by (auto simp add: SUP2_iff)
+ by (auto simp add: SUPR_apply)
lemma SUP_UN_eq: "(SUP i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (UN i. r i))"
- by (simp add: SUP1_iff fun_eq_iff)
+ by (simp add: SUPR_apply fun_eq_iff)
lemma SUP_UN_eq2: "(SUP i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (UN i. r i))"
- by (simp add: SUP2_iff fun_eq_iff)
+ by (simp add: SUPR_apply fun_eq_iff)
subsubsection {* Intersections of families *}
lemma INF1_iff: "(INF x:A. B x) b = (ALL x:A. B x b)"
- by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
+ by (simp add: INFI_apply)
lemma INF2_iff: "(INF x:A. B x) b c = (ALL x:A. B x b c)"
- by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
+ by (simp add: INFI_apply)
lemma INF1_I [intro!]: "(!!x. x : A ==> B x b) ==> (INF x:A. B x) b"
- by (auto simp add: INF1_iff)
+ by (auto simp add: INFI_apply)
lemma INF2_I [intro!]: "(!!x. x : A ==> B x b c) ==> (INF x:A. B x) b c"
- by (auto simp add: INF2_iff)
+ by (auto simp add: INFI_apply)
lemma INF1_D [elim]: "(INF x:A. B x) b ==> a : A ==> B a b"
- by (auto simp add: INF1_iff)
+ by (auto simp add: INFI_apply)
lemma INF2_D [elim]: "(INF x:A. B x) b c ==> a : A ==> B a b c"
- by (auto simp add: INF2_iff)
+ by (auto simp add: INFI_apply)
lemma INF1_E [elim]: "(INF x:A. B x) b ==> (B a b ==> R) ==> (a ~: A ==> R) ==> R"
- by (auto simp add: INF1_iff)
+ by (auto simp add: INFI_apply)
lemma INF2_E [elim]: "(INF x:A. B x) b c ==> (B a b c ==> R) ==> (a ~: A ==> R) ==> R"
- by (auto simp add: INF2_iff)
+ by (auto simp add: INFI_apply)
lemma INF_INT_eq: "(INF i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (INT i. r i))"
- by (simp add: INF1_iff fun_eq_iff)
+ by (simp add: INFI_apply fun_eq_iff)
lemma INF_INT_eq2: "(INF i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (INT i. r i))"
- by (simp add: INF2_iff fun_eq_iff)
+ by (simp add: INFI_apply fun_eq_iff)
subsection {* Predicates as relations *}
@@ -493,8 +499,7 @@
by (simp add: minus_pred_def)
instance proof
-qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def
- fun_Compl_def fun_diff_def bool_Compl_def bool_diff_def)
+qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def uminus_apply minus_apply)
end
@@ -514,10 +519,10 @@
by (simp add: single_def)
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
- "P \<guillemotright>= f = (SUPR {x. Predicate.eval P x} f)"
+ "P \<guillemotright>= f = (SUPR {x. eval P x} f)"
lemma eval_bind [simp]:
- "eval (P \<guillemotright>= f) = Predicate.eval (SUPR {x. Predicate.eval P x} f)"
+ "eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)"
by (simp add: bind_def)
lemma bind_bind:
@@ -822,7 +827,7 @@
"single x = Seq (\<lambda>u. Insert x \<bottom>)"
unfolding Seq_def by simp
-primrec "apply" :: "('a \<Rightarrow> 'b Predicate.pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
+primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
"apply f Empty = Empty"
| "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
| "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
@@ -972,7 +977,7 @@
"the A = (THE x. eval A x)"
lemma the_eqI:
- "(THE x. Predicate.eval P x) = x \<Longrightarrow> Predicate.the P = x"
+ "(THE x. eval P x) = x \<Longrightarrow> the P = x"
by (simp add: the_def)
lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A"
@@ -1030,6 +1035,12 @@
bot ("\<bottom>") and
bind (infixl "\<guillemotright>=" 70)
+no_syntax (xsymbols)
+ "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
+ "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
+ "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
+ "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
+
hide_type (open) pred seq
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the