
diff -r 11bcd77cfa22 -r b1f10b98430d src/HOL/Induct/LList.thy
--- a/src/HOL/Induct/LList.thy	Wed Oct 12 03:02:18 2005 +0200
+++ b/src/HOL/Induct/LList.thy	Wed Oct 12 10:49:07 2005 +0200
@@ -143,8 +143,8 @@
 declare option.split [split]
 
 text{*This justifies using llist in other recursive type definitions*}
-lemma llist_mono: "A<=B ==> llist(A) <= llist(B)"
-apply (unfold llist.defs )
+lemma llist_mono: "A\<subseteq>B ==> llist(A) \<subseteq> llist(B)"
+apply (simp add: llist.defs)
 apply (rule gfp_mono)
 apply (assumption | rule basic_monos)+
 done
@@ -163,40 +163,36 @@
 *}
 
 lemma llist_coinduct: 
-    "[| M \<in> X;  X <= list_Fun A (X Un llist(A)) |] ==>  M \<in> llist(A)"
-apply (unfold list_Fun_def)
+    "[| M \<in> X;  X \<subseteq> list_Fun A (X Un llist(A)) |] ==>  M \<in> llist(A)"
+apply (simp add: list_Fun_def)
 apply (erule llist.coinduct)
-apply (erule subsetD [THEN CollectD], assumption)
+apply (blast intro: elim:); 
 done
 
 lemma list_Fun_NIL_I [iff]: "NIL \<in> list_Fun A X"
-by (unfold list_Fun_def NIL_def, fast)
+by (simp add: list_Fun_def NIL_def)
 
 lemma list_Fun_CONS_I [intro!,simp]: 
     "[| M \<in> A;  N \<in> X |] ==> CONS M N \<in> list_Fun A X"
-apply (unfold list_Fun_def CONS_def, fast)
-done
+by (simp add: list_Fun_def CONS_def)
 
 
 text{*Utilise the ``strong'' part, i.e. @{text "gfp(f)"}*}
 lemma list_Fun_llist_I: "M \<in> llist(A) ==> M \<in> list_Fun A (X Un llist(A))"
 apply (unfold llist.defs list_Fun_def)
 apply (rule gfp_fun_UnI2) 
-apply (rule monoI, blast) 
-apply assumption
+apply (rule monoI, auto)
 done
 
 subsection{* @{text LList_corec} satisfies the desired recurion equation *}
 
 text{*A continuity result?*}
 lemma CONS_UN1: "CONS M (\<Union>x. f(x)) = (\<Union>x. CONS M (f x))"
-apply (unfold CONS_def)
-apply (simp add: In1_UN1 Scons_UN1_y)
+apply (simp add: CONS_def In1_UN1 Scons_UN1_y)
 done
 
-lemma CONS_mono: "[| M<=M';  N<=N' |] ==> CONS M N <= CONS M' N'"
-apply (unfold CONS_def)
-apply (assumption | rule In1_mono Scons_mono)+
+lemma CONS_mono: "[| M\<subseteq>M';  N\<subseteq>N' |] ==> CONS M N \<subseteq> CONS M' N'"
+apply (simp add: CONS_def In1_mono Scons_mono)
 done
 
 declare LList_corec_fun_def [THEN def_nat_rec_0, simp]
@@ -205,19 +201,19 @@
 subsubsection{* The directions of the equality are proved separately *}
 
 lemma LList_corec_subset1: 
-    "LList_corec a f <=  
+    "LList_corec a f \<subseteq>  
      (case f a of None => NIL | Some(z,w) => CONS z (LList_corec w f))"
 apply (unfold LList_corec_def)
 apply (rule UN_least)
-apply (case_tac "k")
+apply (case_tac k) 
 apply (simp_all (no_asm_simp))
 apply (rule allI impI subset_refl [THEN CONS_mono] UNIV_I [THEN UN_upper])+
 done
 
 lemma LList_corec_subset2: 
-    "(case f a of None => NIL | Some(z,w) => CONS z (LList_corec w f)) <=  
+    "(case f a of None => NIL | Some(z,w) => CONS z (LList_corec w f)) \<subseteq>  
      LList_corec a f"
-apply (unfold LList_corec_def)
+apply (simp add: LList_corec_def)
 apply (simp add: CONS_UN1, safe) 
 apply (rule_tac a="Suc(?k)" in UN_I, simp, simp)+ 
 done
@@ -237,7 +233,7 @@
 text{*A typical use of co-induction to show membership in the @{text gfp}. 
   Bisimulation is @{text "range(%x. LList_corec x f)"} *}
 lemma LList_corec_type: "LList_corec a f \<in> llist UNIV"
-apply (rule_tac X = "range (%x. LList_corec x ?g) " in llist_coinduct)
+apply (rule_tac X = "range (%x. LList_corec x ?g)" in llist_coinduct)
 apply (rule rangeI, safe)
 apply (subst LList_corec, simp)
 done
@@ -252,7 +248,7 @@
 
 lemma LListD_implies_ntrunc_equality [rule_format]:
      "\<forall>M N. (M,N) \<in> LListD(diag A) --> ntrunc k M = ntrunc k N"
-apply (induct_tac "k" rule: nat_less_induct)
+apply (induct_tac "k" rule: nat_less_induct) 
 apply (safe del: equalityI)
 apply (erule LListD.cases)
 apply (safe del: equalityI)
@@ -264,20 +260,19 @@
 
 text{*The domain of the @{text LListD} relation*}
 lemma Domain_LListD: 
-    "Domain (LListD(diag A)) <= llist(A)"
-apply (unfold llist.defs NIL_def CONS_def)
+    "Domain (LListD(diag A)) \<subseteq> llist(A)"
+apply (simp add: llist.defs NIL_def CONS_def)
 apply (rule gfp_upperbound)
 txt{*avoids unfolding @{text LListD} on the rhs*}
-apply (rule_tac P = "%x. Domain x <= ?B" in LListD_unfold [THEN ssubst], simp)
-apply blast 
+apply (rule_tac P = "%x. Domain x \<subseteq> ?B" in LListD_unfold [THEN ssubst], auto) 
 done
 
 text{*This inclusion justifies the use of coinduction to show @{text "M = N"}*}
-lemma LListD_subset_diag: "LListD(diag A) <= diag(llist(A))"
+lemma LListD_subset_diag: "LListD(diag A) \<subseteq> diag(llist(A))"
 apply (rule subsetI)
-apply (rule_tac p = "x" in PairE, safe)
+apply (rule_tac p = x in PairE, safe)
 apply (rule diag_eqI)
-apply (rule LListD_implies_ntrunc_equality [THEN ntrunc_equality], assumption)
+apply (rule LListD_implies_ntrunc_equality [THEN ntrunc_equality], assumption) 
 apply (erule DomainI [THEN Domain_LListD [THEN subsetD]])
 done
 
@@ -286,38 +281,36 @@
 
 text {* THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS! *}
 
-lemma LListD_Fun_mono: "A<=B ==> LListD_Fun r A <= LListD_Fun r B"
-apply (unfold LListD_Fun_def)
+lemma LListD_Fun_mono: "A\<subseteq>B ==> LListD_Fun r A \<subseteq> LListD_Fun r B"
+apply (simp add: LListD_Fun_def)
 apply (assumption | rule basic_monos)+
 done
 
 lemma LListD_coinduct: 
-    "[| M \<in> X;  X <= LListD_Fun r (X Un LListD(r)) |] ==>  M \<in> LListD(r)"
-apply (unfold LListD_Fun_def)
+    "[| M \<in> X;  X \<subseteq> LListD_Fun r (X Un LListD(r)) |] ==>  M \<in> LListD(r)"
+apply (simp add: LListD_Fun_def)
 apply (erule LListD.coinduct)
-apply (erule subsetD [THEN CollectD], assumption)
+apply (auto ); 
 done
 
 lemma LListD_Fun_NIL_I: "(NIL,NIL) \<in> LListD_Fun r s"
-by (unfold LListD_Fun_def NIL_def, fast)
+by (simp add: LListD_Fun_def NIL_def)
 
 lemma LListD_Fun_CONS_I: 
      "[| x\<in>A;  (M,N):s |] ==> (CONS x M, CONS x N) \<in> LListD_Fun (diag A) s"
-apply (unfold LListD_Fun_def CONS_def, blast)
-done
+by (simp add: LListD_Fun_def CONS_def, blast)
 
 text{*Utilise the "strong" part, i.e. @{text "gfp(f)"}*}
 lemma LListD_Fun_LListD_I:
      "M \<in> LListD(r) ==> M \<in> LListD_Fun r (X Un LListD(r))"
 apply (unfold LListD.defs LListD_Fun_def)
 apply (rule gfp_fun_UnI2) 
-apply (rule monoI, blast) 
-apply assumption
+apply (rule monoI, auto)
 done
 
 
 text{*This converse inclusion helps to strengthen @{text LList_equalityI}*}
-lemma diag_subset_LListD: "diag(llist(A)) <= LListD(diag A)"
+lemma diag_subset_LListD: "diag(llist(A)) \<subseteq> LListD(diag A)"
 apply (rule subsetI)
 apply (erule LListD_coinduct)
 apply (rule subsetI)
@@ -342,7 +335,7 @@
       [also admits true equality]
    Replace @{text A} by some particular set, like @{text "{x. True}"}??? *}
 lemma LList_equalityI:
-     "[| (M,N) \<in> r;  r <= LListD_Fun (diag A) (r Un diag(llist(A))) |] 
+     "[| (M,N) \<in> r;  r \<subseteq> LListD_Fun (diag A) (r Un diag(llist(A))) |] 
       ==>  M=N"
 apply (rule LListD_subset_diag [THEN subsetD, THEN diagE])
 apply (erule LListD_coinduct)
@@ -365,7 +358,7 @@
   ==> h1=h2"
 apply (rule ext)
 txt{*next step avoids an unknown (and flexflex pair) in simplification*}
-apply (rule_tac A = "UNIV" and r = "range(%u. (h1 u, h2 u))" 
+apply (rule_tac A = UNIV and r = "range(%u. (h1 u, h2 u))" 
        in LList_equalityI)
 apply (rule rangeI, safe)
 apply (simp add: LListD_Fun_NIL_I UNIV_I [THEN LListD_Fun_CONS_I])
@@ -395,7 +388,7 @@
           "!!x. h2 x = (case f x of None => NIL | Some(z,w) => CONS z (h2 w))"
  shows "h1=h2"
 apply (rule ntrunc_equality [THEN ext])
-apply (rule_tac x = "x" in spec)
+apply (rule_tac x = x in spec)
 apply (induct_tac "k" rule: nat_less_induct)
 apply (rename_tac "n")
 apply (rule allI)
@@ -404,8 +397,7 @@
 apply (intro strip) 
 apply (case_tac "n") 
 apply (rename_tac [2] "m")
-apply (case_tac [2] "m")
-apply simp_all
+apply (case_tac [2] "m", simp_all)
 done
 
 
@@ -444,16 +436,16 @@
 subsection{* Isomorphisms *}
 
 lemma LListI: "x \<in> llist (range Leaf) ==> x \<in> LList"
-by (unfold LList_def, simp)
+by (simp add: LList_def)
 
 lemma LListD: "x \<in> LList ==> x \<in> llist (range Leaf)"
-by (unfold LList_def, simp)
+by (simp add: LList_def)
 
 
 subsubsection{* Distinctness of constructors *}
 
 lemma LCons_not_LNil [iff]: "~ LCons x xs = LNil"
-apply (unfold LNil_def LCons_def)
+apply (simp add: LNil_def LCons_def)
 apply (subst Abs_LList_inject)
 apply (rule llist.intros CONS_not_NIL rangeI LListI Rep_LList [THEN LListD])+
 done
@@ -464,12 +456,12 @@
 subsubsection{* llist constructors *}
 
 lemma Rep_LList_LNil: "Rep_LList LNil = NIL"
-apply (unfold LNil_def)
+apply (simp add: LNil_def)
 apply (rule llist.NIL_I [THEN LListI, THEN Abs_LList_inverse])
 done
 
 lemma Rep_LList_LCons: "Rep_LList(LCons x l) = CONS (Leaf x) (Rep_LList l)"
-apply (unfold LCons_def)
+apply (simp add: LCons_def)
 apply (rule llist.CONS_I [THEN LListI, THEN Abs_LList_inverse] 
             rangeI Rep_LList [THEN LListD])+
 done
@@ -477,8 +469,7 @@
 subsubsection{* Injectiveness of @{text CONS} and @{text LCons} *}
 
 lemma CONS_CONS_eq2: "(CONS M N=CONS M' N') = (M=M' & N=N')"
-apply (unfold CONS_def)
-apply (fast elim!: Scons_inject)
+apply (simp add: CONS_def)
 done
 
 lemmas CONS_inject = CONS_CONS_eq [THEN iffD1, THEN conjE, standard]
@@ -490,7 +481,7 @@
 declare llist.intros [intro]
 
 lemma LCons_LCons_eq [iff]: "(LCons x xs=LCons y ys) = (x=y & xs=ys)"
-apply (unfold LCons_def)
+apply (simp add: LCons_def)
 apply (subst Abs_LList_inject)
 apply (auto simp add: Rep_LList_inject)
 done
@@ -546,7 +537,7 @@
 subsubsection{* Two easy results about @{text Lmap} *}
 
 lemma Lmap_compose: "M \<in> llist(A) ==> Lmap (f o g) M = Lmap f (Lmap g M)"
-apply (unfold o_def)
+apply (simp add: o_def)
 apply (drule imageI)
 apply (erule LList_equalityI, safe)
 apply (erule llist.cases, simp_all)
@@ -566,19 +557,19 @@
 subsection{* @{text Lappend} -- its two arguments cause some complications! *}
 
 lemma Lappend_NIL_NIL [simp]: "Lappend NIL NIL = NIL"
-apply (unfold Lappend_def)
+apply (simp add: Lappend_def)
 apply (rule LList_corec [THEN trans], simp)
 done
 
 lemma Lappend_NIL_CONS [simp]: 
     "Lappend NIL (CONS N N') = CONS N (Lappend NIL N')"
-apply (unfold Lappend_def)
+apply (simp add: Lappend_def)
 apply (rule LList_corec [THEN trans], simp)
 done
 
 lemma Lappend_CONS [simp]: 
     "Lappend (CONS M M') N = CONS M (Lappend M' N)"
-apply (unfold Lappend_def)
+apply (simp add: Lappend_def)
 apply (rule LList_corec [THEN trans], simp)
 done
 
@@ -600,17 +591,16 @@
 apply (rule_tac X = "\<Union>u\<in>llist (A) . \<Union>v \<in> llist (A) . {Lappend u v}" in llist_coinduct)
 apply fast
 apply safe
-apply (erule_tac aa = "u" in llist.cases)
-apply (erule_tac aa = "v" in llist.cases, simp_all)
-apply blast
+apply (erule_tac aa = u in llist.cases)
+apply (erule_tac aa = v in llist.cases, simp_all, blast)
 done
 
 text{*strong co-induction: bisimulation and case analysis on one variable*}
 lemma Lappend_type': "[| M \<in> llist(A); N \<in> llist(A) |] ==> Lappend M N \<in> llist(A)"
-apply (rule_tac X = " (%u. Lappend u N) `llist (A) " in llist_coinduct)
+apply (rule_tac X = "(%u. Lappend u N) `llist (A)" in llist_coinduct)
 apply (erule imageI)
 apply (rule image_subsetI)
-apply (erule_tac aa = "x" in llist.cases)
+apply (erule_tac aa = x in llist.cases)
 apply (simp add: list_Fun_llist_I, simp)
 done
 
@@ -624,18 +614,16 @@
 declare rangeI [simp] inj_Leaf [simp] 
 
 lemma llist_case_LNil [simp]: "llist_case c d LNil = c"
-by (unfold llist_case_def LNil_def, simp)
+by (simp add: llist_case_def LNil_def)
 
-lemma llist_case_LCons [simp]: 
-    "llist_case c d (LCons M N) = d M N"
-apply (unfold llist_case_def LCons_def, simp)
-done
+lemma llist_case_LCons [simp]: "llist_case c d (LCons M N) = d M N"
+by (simp add: llist_case_def LCons_def)
+
 
 text{*Elimination is case analysis, not induction.*}
 lemma llistE: "[| l=LNil ==> P;  !!x l'. l=LCons x l' ==> P |] ==> P"
 apply (rule Rep_LList [THEN LListD, THEN llist.cases])
- apply (simp add: Rep_LList_LNil [symmetric] Rep_LList_inject)
- apply blast 
+ apply (simp add: Rep_LList_LNil [symmetric] Rep_LList_inject, blast) 
 apply (erule LListI [THEN Rep_LList_cases], clarify)
 apply (simp add: Rep_LList_LCons [symmetric] Rep_LList_inject, blast) 
 done
@@ -647,7 +635,7 @@
      "LList_corec a  
            (%z. case f z of None => None | Some(v,w) => Some(Leaf(v),w)) 
        \<in> llist(range Leaf)"
-apply (rule_tac X = "range (%x. LList_corec x ?g) " in llist_coinduct)
+apply (rule_tac X = "range (%x. LList_corec x ?g)" in llist_coinduct)
 apply (rule rangeI, safe)
 apply (subst LList_corec, force)
 done
@@ -673,18 +661,18 @@
 subsection{* Deriving @{text llist_equalityI} -- @{text llist} equality is a bisimulation *}
 
 lemma LListD_Fun_subset_Times_llist: 
-    "r <= (llist A) <*> (llist A) 
-     ==> LListD_Fun (diag A) r <= (llist A) <*> (llist A)"
+    "r \<subseteq> (llist A) <*> (llist A) 
+     ==> LListD_Fun (diag A) r \<subseteq> (llist A) <*> (llist A)"
 by (auto simp add: LListD_Fun_def)
 
 lemma subset_Times_llist:
-     "prod_fun Rep_LList Rep_LList ` r <=  
+     "prod_fun Rep_LList Rep_LList ` r \<subseteq>  
      (llist(range Leaf)) <*> (llist(range Leaf))"
 by (blast intro: Rep_LList [THEN LListD])
 
 lemma prod_fun_lemma:
-     "r <= (llist(range Leaf)) <*> (llist(range Leaf)) 
-      ==> prod_fun (Rep_LList o Abs_LList) (Rep_LList o Abs_LList) ` r <= r"
+     "r \<subseteq> (llist(range Leaf)) <*> (llist(range Leaf)) 
+      ==> prod_fun (Rep_LList o Abs_LList) (Rep_LList o Abs_LList) ` r \<subseteq> r"
 apply safe
 apply (erule subsetD [THEN SigmaE2], assumption)
 apply (simp add: LListI [THEN Abs_LList_inverse])
@@ -699,8 +687,8 @@
 
 text{*Used with @{text lfilter}*}
 lemma llistD_Fun_mono: 
-    "A<=B ==> llistD_Fun A <= llistD_Fun B"
-apply (unfold llistD_Fun_def prod_fun_def, auto)
+    "A\<subseteq>B ==> llistD_Fun A \<subseteq> llistD_Fun B"
+apply (simp add: llistD_Fun_def prod_fun_def, auto)
 apply (rule image_eqI)
  prefer 2 apply (blast intro: rev_subsetD [OF _ LListD_Fun_mono], force)
 done
@@ -708,10 +696,10 @@
 subsubsection{* To show two llists are equal, exhibit a bisimulation! 
       [also admits true equality] *}
 lemma llist_equalityI: 
-    "[| (l1,l2) \<in> r;  r <= llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2"
-apply (unfold llistD_Fun_def)
+    "[| (l1,l2) \<in> r;  r \<subseteq> llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2"
+apply (simp add: llistD_Fun_def)
 apply (rule Rep_LList_inject [THEN iffD1])
-apply (rule_tac r = "prod_fun Rep_LList Rep_LList `r" and A = "range (Leaf) " in LList_equalityI)
+apply (rule_tac r = "prod_fun Rep_LList Rep_LList `r" and A = "range (Leaf)" in LList_equalityI)
 apply (erule prod_fun_imageI)
 apply (erule image_mono [THEN subset_trans])
 apply (rule image_compose [THEN subst])
@@ -725,20 +713,20 @@
 
 subsubsection{* Rules to prove the 2nd premise of @{text llist_equalityI} *}
 lemma llistD_Fun_LNil_I [simp]: "(LNil,LNil) \<in> llistD_Fun(r)"
-apply (unfold llistD_Fun_def LNil_def)
+apply (simp add: llistD_Fun_def LNil_def)
 apply (rule LListD_Fun_NIL_I [THEN prod_fun_imageI])
 done
 
 lemma llistD_Fun_LCons_I [simp]: 
     "(l1,l2):r ==> (LCons x l1, LCons x l2) \<in> llistD_Fun(r)"
-apply (unfold llistD_Fun_def LCons_def)
+apply (simp add: llistD_Fun_def LCons_def)
 apply (rule rangeI [THEN LListD_Fun_CONS_I, THEN prod_fun_imageI])
 apply (erule prod_fun_imageI)
 done
 
 text{*Utilise the "strong" part, i.e. @{text "gfp(f)"}*}
 lemma llistD_Fun_range_I: "(l,l) \<in> llistD_Fun(r Un range(%x.(x,x)))"
-apply (unfold llistD_Fun_def)
+apply (simp add: llistD_Fun_def)
 apply (rule Rep_LList_inverse [THEN subst])
 apply (rule prod_fun_imageI)
 apply (subst image_Un)
@@ -752,9 +740,9 @@
          !!x l. (f(LCons x l),g(LCons x l)) 
                 \<in> llistD_Fun(range(%u. (f(u),g(u))) Un range(%v. (v,v)))    
       |] ==> f(l) = (g(l :: 'a llist) :: 'b llist)"
-apply (rule_tac r = "range (%u. (f (u),g (u))) " in llist_equalityI)
+apply (rule_tac r = "range (%u. (f (u),g (u)))" in llist_equalityI)
 apply (rule rangeI, clarify) 
-apply (rule_tac l = "u" in llistE)
+apply (rule_tac l = u in llistE)
 apply (simp_all add: llistD_Fun_range_I) 
 done
 
@@ -771,10 +759,10 @@
 subsubsection{* Two easy results about @{text lmap} *}
 
 lemma lmap_compose [simp]: "lmap (f o g) l = lmap f (lmap g l)"
-by (rule_tac l = "l" in llist_fun_equalityI, simp_all)
+by (rule_tac l = l in llist_fun_equalityI, simp_all)
 
 lemma lmap_ident [simp]: "lmap (%x. x) l = l"
-by (rule_tac l = "l" in llist_fun_equalityI, simp_all)
+by (rule_tac l = l in llist_fun_equalityI, simp_all)
 
 
 subsection{* iterates -- @{text llist_fun_equalityI} cannot be used! *}
@@ -783,10 +771,10 @@
 by (rule iterates_def [THEN def_llist_corec, THEN trans], simp)
 
 lemma lmap_iterates [simp]: "lmap f (iterates f x) = iterates f (f x)"
-apply (rule_tac r = "range (%u. (lmap f (iterates f u),iterates f (f u))) " in llist_equalityI)
+apply (rule_tac r = "range (%u. (lmap f (iterates f u),iterates f (f u)))" in llist_equalityI)
 apply (rule rangeI, safe)
-apply (rule_tac x1 = "f (u) " in iterates [THEN ssubst])
-apply (rule_tac x1 = "u" in iterates [THEN ssubst], simp)
+apply (rule_tac x1 = "f (u)" in iterates [THEN ssubst])
+apply (rule_tac x1 = u in iterates [THEN ssubst], simp)
 done
 
 lemma iterates_lmap: "iterates f x = LCons x (lmap f (iterates f x))"
@@ -801,8 +789,7 @@
 
 lemma fun_power_lmap: "nat_rec (LCons b l) (%m. lmap(f)) n =       
      LCons (nat_rec b (%m. f) n) (nat_rec l (%m. lmap(f)) n)"
-apply (induct_tac "n", simp_all)
-done
+by (induct_tac "n", simp_all)
 
 lemma fun_power_Suc: "nat_rec (g x) (%m. g) n = nat_rec x (%m. g) (Suc n)"
 by (induct_tac "n", simp_all)
@@ -836,35 +823,34 @@
 subsection{* @{text lappend} -- its two arguments cause some complications! *}
 
 lemma lappend_LNil_LNil [simp]: "lappend LNil LNil = LNil"
-apply (unfold lappend_def)
+apply (simp add: lappend_def)
 apply (rule llist_corec [THEN trans], simp)
 done
 
 lemma lappend_LNil_LCons [simp]: 
     "lappend LNil (LCons l l') = LCons l (lappend LNil l')"
-apply (unfold lappend_def)
+apply (simp add: lappend_def)
 apply (rule llist_corec [THEN trans], simp)
 done
 
 lemma lappend_LCons [simp]: 
     "lappend (LCons l l') N = LCons l (lappend l' N)"
-apply (unfold lappend_def)
+apply (simp add: lappend_def)
 apply (rule llist_corec [THEN trans], simp)
 done
 
 lemma lappend_LNil [simp]: "lappend LNil l = l"
-by (rule_tac l = "l" in llist_fun_equalityI, simp_all)
+by (rule_tac l = l in llist_fun_equalityI, simp_all)
 
 lemma lappend_LNil2 [simp]: "lappend l LNil = l"
-by (rule_tac l = "l" in llist_fun_equalityI, simp_all)
+by (rule_tac l = l in llist_fun_equalityI, simp_all)
 
 
 text{*The infinite first argument blocks the second*}
 lemma lappend_iterates [simp]: "lappend (iterates f x) N = iterates f x"
 apply (rule_tac r = "range (%u. (lappend (iterates f u) N,iterates f u))" 
        in llist_equalityI)
- apply (rule rangeI)
-apply (safe)
+ apply (rule rangeI, safe)
 apply (subst (1 2) iterates)
 apply simp 
 done
@@ -879,22 +865,18 @@
        in llist_equalityI)
 apply (rule UN1_I)
 apply (rule rangeI, safe)
-apply (rule_tac l = "l" in llistE)
-apply (rule_tac l = "n" in llistE, simp_all)
+apply (rule_tac l = l in llistE)
+apply (rule_tac l = n in llistE, simp_all)
 apply (blast intro: llistD_Fun_LCons_I) 
 done
 
 text{*Shorter proof of theorem above using @{text llist_equalityI} as strong coinduction*}
 lemma lmap_lappend_distrib':
      "lmap f (lappend l n) = lappend (lmap f l) (lmap f n)"
-apply (rule_tac l = "l" in llist_fun_equalityI, simp)
-apply simp
-done
+by (rule_tac l = l in llist_fun_equalityI, auto)
 
 text{*Without strong coinduction, three case analyses might be needed*}
 lemma lappend_assoc': "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)"
-apply (rule_tac l = "l1" in llist_fun_equalityI, simp)
-apply simp
-done
+by (rule_tac l = l1 in llist_fun_equalityI, auto)
 
 end




