src/CCL/Gfp.thy

--- a/src/CCL/Gfp.thy	Tue Nov 11 13:50:56 2014 +0100
+++ b/src/CCL/Gfp.thy	Tue Nov 11 15:55:31 2014 +0100
@@ -10,37 +10,35 @@
 begin
 
 definition
-  gfp :: "['a set=>'a set] => 'a set" where -- "greatest fixed point"
+  gfp :: "['a set\<Rightarrow>'a set] \<Rightarrow> 'a set" where -- "greatest fixed point"
   "gfp(f) == Union({u. u <= f(u)})"
 
 (* gfp(f) is the least upper bound of {u. u <= f(u)} *)
 
-lemma gfp_upperbound: "[| A <= f(A) |] ==> A <= gfp(f)"
+lemma gfp_upperbound: "A <= f(A) \<Longrightarrow> A <= gfp(f)"
   unfolding gfp_def by blast
 
-lemma gfp_least: "[| !!u. u <= f(u) ==> u<=A |] ==> gfp(f) <= A"
+lemma gfp_least: "(\<And>u. u <= f(u) \<Longrightarrow> u <= A) \<Longrightarrow> gfp(f) <= A"
   unfolding gfp_def by blast
 
-lemma gfp_lemma2: "mono(f) ==> gfp(f) <= f(gfp(f))"
+lemma gfp_lemma2: "mono(f) \<Longrightarrow> gfp(f) <= f(gfp(f))"
   by (rule gfp_least, rule subset_trans, assumption, erule monoD,
     rule gfp_upperbound, assumption)
 
-lemma gfp_lemma3: "mono(f) ==> f(gfp(f)) <= gfp(f)"
+lemma gfp_lemma3: "mono(f) \<Longrightarrow> f(gfp(f)) <= gfp(f)"
   by (rule gfp_upperbound, frule monoD, rule gfp_lemma2, assumption+)
 
-lemma gfp_Tarski: "mono(f) ==> gfp(f) = f(gfp(f))"
+lemma gfp_Tarski: "mono(f) \<Longrightarrow> gfp(f) = f(gfp(f))"
   by (rule equalityI gfp_lemma2 gfp_lemma3 | assumption)+
 
 
 (*** Coinduction rules for greatest fixed points ***)
 
 (*weak version*)
-lemma coinduct: "[| a: A;  A <= f(A) |] ==> a : gfp(f)"
+lemma coinduct: "\<lbrakk>a: A;  A <= f(A)\<rbrakk> \<Longrightarrow> a : gfp(f)"
   by (blast dest: gfp_upperbound)
 
-lemma coinduct2_lemma:
-  "[| A <= f(A) Un gfp(f);  mono(f) |] ==>   
-    A Un gfp(f) <= f(A Un gfp(f))"
+lemma coinduct2_lemma: "\<lbrakk>A <= f(A) Un gfp(f); mono(f)\<rbrakk> \<Longrightarrow> A Un gfp(f) <= f(A Un gfp(f))"
   apply (rule subset_trans)
    prefer 2
    apply (erule mono_Un)
@@ -50,8 +48,7 @@
   done
 
 (*strong version, thanks to Martin Coen*)
-lemma coinduct2:
-  "[| a: A;  A <= f(A) Un gfp(f);  mono(f) |] ==> a : gfp(f)"
+lemma coinduct2: "\<lbrakk>a: A; A <= f(A) Un gfp(f); mono(f)\<rbrakk> \<Longrightarrow> a : gfp(f)"
   apply (rule coinduct)
    prefer 2
    apply (erule coinduct2_lemma, assumption)
@@ -62,13 +59,13 @@
          - instead of the condition  A <= f(A)
                            consider  A <= (f(A) Un f(f(A)) ...) Un gfp(A) ***)
 
-lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un A Un B)"
+lemma coinduct3_mono_lemma: "mono(f) \<Longrightarrow> mono(\<lambda>x. f(x) Un A Un B)"
   by (rule monoI) (blast dest: monoD)
 
 lemma coinduct3_lemma:
-  assumes prem: "A <= f(lfp(%x. f(x) Un A Un gfp(f)))"
+  assumes prem: "A <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))"
     and mono: "mono(f)"
-  shows "lfp(%x. f(x) Un A Un gfp(f)) <= f(lfp(%x. f(x) Un A Un gfp(f)))"
+  shows "lfp(\<lambda>x. f(x) Un A Un gfp(f)) <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))"
   apply (rule subset_trans)
    apply (rule mono [THEN coinduct3_mono_lemma, THEN lfp_lemma3])
   apply (rule Un_least [THEN Un_least])
@@ -82,7 +79,7 @@
 
 lemma coinduct3:
   assumes 1: "a:A"
-    and 2: "A <= f(lfp(%x. f(x) Un A Un gfp(f)))"
+    and 2: "A <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))"
     and 3: "mono(f)"
   shows "a : gfp(f)"
   apply (rule coinduct)
@@ -95,25 +92,25 @@
 
 subsection {* Definition forms of @{text "gfp_Tarski"}, to control unfolding *}
 
-lemma def_gfp_Tarski: "[| h==gfp(f);  mono(f) |] ==> h = f(h)"
+lemma def_gfp_Tarski: "\<lbrakk>h == gfp(f); mono(f)\<rbrakk> \<Longrightarrow> h = f(h)"
   apply unfold
   apply (erule gfp_Tarski)
   done
 
-lemma def_coinduct: "[| h==gfp(f);  a:A;  A <= f(A) |] ==> a: h"
+lemma def_coinduct: "\<lbrakk>h == gfp(f); a:A; A <= f(A)\<rbrakk> \<Longrightarrow> a: h"
   apply unfold
   apply (erule coinduct)
   apply assumption
   done
 
-lemma def_coinduct2: "[| h==gfp(f);  a:A;  A <= f(A) Un h; mono(f) |] ==> a: h"
+lemma def_coinduct2: "\<lbrakk>h == gfp(f); a:A; A <= f(A) Un h; mono(f)\<rbrakk> \<Longrightarrow> a: h"
   apply unfold
   apply (erule coinduct2)
    apply assumption
   apply assumption
   done
 
-lemma def_coinduct3: "[| h==gfp(f);  a:A;  A <= f(lfp(%x. f(x) Un A Un h)); mono(f) |] ==> a: h"
+lemma def_coinduct3: "\<lbrakk>h == gfp(f); a:A; A <= f(lfp(\<lambda>x. f(x) Un A Un h)); mono(f)\<rbrakk> \<Longrightarrow> a: h"
   apply unfold
   apply (erule coinduct3)
    apply assumption
@@ -121,7 +118,7 @@
   done
 
 (*Monotonicity of gfp!*)
-lemma gfp_mono: "[| mono(f);  !!Z. f(Z)<=g(Z) |] ==> gfp(f) <= gfp(g)"
+lemma gfp_mono: "\<lbrakk>mono(f); \<And>Z. f(Z) <= g(Z)\<rbrakk> \<Longrightarrow> gfp(f) <= gfp(g)"
   apply (rule gfp_upperbound)
   apply (rule subset_trans)
    apply (rule gfp_lemma2)