src/CCL/Lfp.thy
changeset 58977 9576b510f6a2
parent 58889 5b7a9633cfa8
child 60770 240563fbf41d
--- 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)