--- a/src/CCL/Lfp.thy Tue Nov 11 13:50:56 2014 +0100
+++ b/src/CCL/Lfp.thy Tue Nov 11 15:55:31 2014 +0100
@@ -10,24 +10,24 @@
begin
definition
- lfp :: "['a set=>'a set] => 'a set" where -- "least fixed point"
+ lfp :: "['a set\<Rightarrow>'a set] \<Rightarrow> 'a set" where -- "least fixed point"
"lfp(f) == Inter({u. f(u) <= u})"
(* lfp(f) is the greatest lower bound of {u. f(u) <= u} *)
-lemma lfp_lowerbound: "[| f(A) <= A |] ==> lfp(f) <= A"
+lemma lfp_lowerbound: "f(A) <= A \<Longrightarrow> lfp(f) <= A"
unfolding lfp_def by blast
-lemma lfp_greatest: "[| !!u. f(u) <= u ==> A<=u |] ==> A <= lfp(f)"
+lemma lfp_greatest: "(\<And>u. f(u) <= u \<Longrightarrow> A<=u) \<Longrightarrow> A <= lfp(f)"
unfolding lfp_def by blast
-lemma lfp_lemma2: "mono(f) ==> f(lfp(f)) <= lfp(f)"
+lemma lfp_lemma2: "mono(f) \<Longrightarrow> f(lfp(f)) <= lfp(f)"
by (rule lfp_greatest, rule subset_trans, drule monoD, rule lfp_lowerbound, assumption+)
-lemma lfp_lemma3: "mono(f) ==> lfp(f) <= f(lfp(f))"
+lemma lfp_lemma3: "mono(f) \<Longrightarrow> lfp(f) <= f(lfp(f))"
by (rule lfp_lowerbound, frule monoD, drule lfp_lemma2, assumption+)
-lemma lfp_Tarski: "mono(f) ==> lfp(f) = f(lfp(f))"
+lemma lfp_Tarski: "mono(f) \<Longrightarrow> lfp(f) = f(lfp(f))"
by (rule equalityI lfp_lemma2 lfp_lemma3 | assumption)+
@@ -36,7 +36,7 @@
lemma induct:
assumes lfp: "a: lfp(f)"
and mono: "mono(f)"
- and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)"
+ and indhyp: "\<And>x. \<lbrakk>x: f(lfp(f) Int {x. P(x)})\<rbrakk> \<Longrightarrow> P(x)"
shows "P(a)"
apply (rule_tac a = a in Int_lower2 [THEN subsetD, THEN CollectD])
apply (rule lfp [THEN [2] lfp_lowerbound [THEN subsetD]])
@@ -46,16 +46,13 @@
(** Definition forms of lfp_Tarski and induct, to control unfolding **)
-lemma def_lfp_Tarski: "[| h==lfp(f); mono(f) |] ==> h = f(h)"
+lemma def_lfp_Tarski: "\<lbrakk>h == lfp(f); mono(f)\<rbrakk> \<Longrightarrow> h = f(h)"
apply unfold
apply (drule lfp_Tarski)
apply assumption
done
-lemma def_induct:
- "[| A == lfp(f); a:A; mono(f);
- !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)
- |] ==> P(a)"
+lemma def_induct: "\<lbrakk>A == lfp(f); a:A; mono(f); \<And>x. x: f(A Int {x. P(x)}) \<Longrightarrow> P(x)\<rbrakk> \<Longrightarrow> P(a)"
apply (rule induct [of concl: P a])
apply simp
apply assumption
@@ -63,7 +60,7 @@
done
(*Monotonicity of lfp!*)
-lemma lfp_mono: "[| mono(g); !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)"
+lemma lfp_mono: "\<lbrakk>mono(g); \<And>Z. f(Z) <= g(Z)\<rbrakk> \<Longrightarrow> lfp(f) <= lfp(g)"
apply (rule lfp_lowerbound)
apply (rule subset_trans)
apply (erule meta_spec)