# HG changeset patch
# User paulson
# Date 1129106947 -7200
# Node ID b1f10b98430d1cd0545096fdb4be332449742888
# Parent  11bcd77cfa222643d2b6b8d3085adb91134be961
tidying

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
diff -r 11bcd77cfa22 -r b1f10b98430d src/HOL/ex/Tarski.thy
--- a/src/HOL/ex/Tarski.thy	Wed Oct 12 03:02:18 2005 +0200
+++ b/src/HOL/ex/Tarski.thy	Wed Oct 12 10:49:07 2005 +0200
@@ -66,8 +66,8 @@
 
   CompleteLattice :: "('a potype) set"
   "CompleteLattice == {cl. cl: PartialOrder &
-                        (\<forall>S. S <= pset cl --> (\<exists>L. isLub S cl L)) &
-                        (\<forall>S. S <= pset cl --> (\<exists>G. isGlb S cl G))}"
+                        (\<forall>S. S \<subseteq> pset cl --> (\<exists>L. isLub S cl L)) &
+                        (\<forall>S. S \<subseteq> pset cl --> (\<exists>G. isGlb S cl G))}"
 
   CLF :: "('a potype * ('a => 'a)) set"
   "CLF == SIGMA cl: CompleteLattice.
@@ -81,14 +81,14 @@
   sublattice :: "('a potype * 'a set)set"
   "sublattice ==
       SIGMA cl: CompleteLattice.
-          {S. S <= pset cl &
+          {S. S \<subseteq> pset cl &
            (| pset = S, order = induced S (order cl) |): CompleteLattice }"
 
 syntax
-  "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
+  "@SL"  :: "['a set, 'a potype] => bool" ("_ <\<subseteq> _" [51,50]50)
 
 translations
-  "S <<= cl" == "S : sublattice `` {cl}"
+  "S <\<subseteq> cl" == "S : sublattice `` {cl}"
 
 constdefs
   dual :: "'a potype => 'a potype"
@@ -117,7 +117,7 @@
     and intY1 :: "'a set"
     and v     :: "'a"
   assumes
-    Y_ss: "Y <= P"
+    Y_ss: "Y \<subseteq> P"
   defines
     intY1_def: "intY1 == interval r (lub Y cl) (Top cl)"
     and v_def: "v == glb {x. ((%x: intY1. f x) x, x): induced intY1 r &
@@ -163,7 +163,7 @@
 by (simp add: monotone_def)
 
 lemma (in PO) po_subset_po:
-     "S <= A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
+     "S \<subseteq> A ==> (| pset = S, order = induced S r |) \<in> PartialOrder"
 apply (simp (no_asm) add: PartialOrder_def)
 apply auto
 -- {* refl *}
@@ -177,13 +177,13 @@
 apply (blast intro: PO_imp_trans [THEN transE])
 done
 
-lemma (in PO) indE: "[| (x, y) \<in> induced S r; S <= A |] ==> (x, y) \<in> r"
+lemma (in PO) indE: "[| (x, y) \<in> induced S r; S \<subseteq> A |] ==> (x, y) \<in> r"
 by (simp add: add: induced_def)
 
 lemma (in PO) indI: "[| (x, y) \<in> r; x \<in> S; y \<in> S |] ==> (x, y) \<in> induced S r"
 by (simp add: add: induced_def)
 
-lemma (in CL) CL_imp_ex_isLub: "S <= A ==> \<exists>L. isLub S cl L"
+lemma (in CL) CL_imp_ex_isLub: "S \<subseteq> A ==> \<exists>L. isLub S cl L"
 apply (insert cl_co)
 apply (simp add: CompleteLattice_def A_def)
 done
@@ -209,8 +209,8 @@
 done
 
 lemma Rdual:
-     "\<forall>S. (S <= A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
-      ==> \<forall>S. (S <= A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
+     "\<forall>S. (S \<subseteq> A -->( \<exists>L. isLub S (| pset = A, order = r|) L))
+      ==> \<forall>S. (S \<subseteq> A --> (\<exists>G. isGlb S (| pset = A, order = r|) G))"
 apply safe
 apply (rule_tac x = "lub {y. y \<in> A & (\<forall>k \<in> S. (y, k) \<in> r)}
                       (|pset = A, order = r|) " in exI)
@@ -226,7 +226,7 @@
 lemma glb_dual_lub: "glb S cl = lub S (dual cl)"
 by (simp add: lub_def glb_def least_def greatest_def dual_def converse_def)
 
-lemma CL_subset_PO: "CompleteLattice <= PartialOrder"
+lemma CL_subset_PO: "CompleteLattice \<subseteq> PartialOrder"
 by (simp add: PartialOrder_def CompleteLattice_def, fast)
 
 lemmas CL_imp_PO = CL_subset_PO [THEN subsetD]
@@ -245,8 +245,8 @@
 by (rule PO_imp_trans)
 
 lemma CompleteLatticeI:
-     "[| po \<in> PartialOrder; (\<forall>S. S <= pset po --> (\<exists>L. isLub S po L));
-         (\<forall>S. S <= pset po --> (\<exists>G. isGlb S po G))|]
+     "[| po \<in> PartialOrder; (\<forall>S. S \<subseteq> pset po --> (\<exists>L. isLub S po L));
+         (\<forall>S. S \<subseteq> pset po --> (\<exists>G. isGlb S po G))|]
       ==> po \<in> CompleteLattice"
 apply (unfold CompleteLattice_def, blast)
 done
@@ -303,24 +303,24 @@
 subsection {* sublattice *}
 
 lemma (in PO) sublattice_imp_CL:
-     "S <<= cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
+     "S <\<subseteq> cl  ==> (| pset = S, order = induced S r |) \<in> CompleteLattice"
 by (simp add: sublattice_def CompleteLattice_def A_def r_def)
 
 lemma (in CL) sublatticeI:
-     "[| S <= A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
-      ==> S <<= cl"
+     "[| S \<subseteq> A; (| pset = S, order = induced S r |) \<in> CompleteLattice |]
+      ==> S <\<subseteq> cl"
 by (simp add: sublattice_def A_def r_def)
 
 
 subsection {* lub *}
 
-lemma (in CL) lub_unique: "[| S <= A; isLub S cl x; isLub S cl L|] ==> x = L"
+lemma (in CL) lub_unique: "[| S \<subseteq> A; isLub S cl x; isLub S cl L|] ==> x = L"
 apply (rule antisymE)
 apply (rule CO_antisym)
 apply (auto simp add: isLub_def r_def)
 done
 
-lemma (in CL) lub_upper: "[|S <= A; x \<in> S|] ==> (x, lub S cl) \<in> r"
+lemma (in CL) lub_upper: "[|S \<subseteq> A; x \<in> S|] ==> (x, lub S cl) \<in> r"
 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
 apply (unfold lub_def least_def)
 apply (rule some_equality [THEN ssubst])
@@ -330,7 +330,7 @@
 done
 
 lemma (in CL) lub_least:
-     "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
+     "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r |] ==> (lub S cl, L) \<in> r"
 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
 apply (unfold lub_def least_def)
 apply (rule_tac s=x in some_equality [THEN ssubst])
@@ -339,7 +339,7 @@
 apply (simp add: isLub_def r_def A_def)
 done
 
-lemma (in CL) lub_in_lattice: "S <= A ==> lub S cl \<in> A"
+lemma (in CL) lub_in_lattice: "S \<subseteq> A ==> lub S cl \<in> A"
 apply (rule CL_imp_ex_isLub [THEN exE], assumption)
 apply (unfold lub_def least_def)
 apply (subst some_equality)
@@ -349,7 +349,7 @@
 done
 
 lemma (in CL) lubI:
-     "[| S <= A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
+     "[| S \<subseteq> A; L \<in> A; \<forall>x \<in> S. (x,L) \<in> r;
          \<forall>z \<in> A. (\<forall>y \<in> S. (y,z) \<in> r) --> (L,z) \<in> r |] ==> L = lub S cl"
 apply (rule lub_unique, assumption)
 apply (simp add: isLub_def A_def r_def)
@@ -360,7 +360,7 @@
 apply (simp add: lub_upper lub_least)
 done
 
-lemma (in CL) lubIa: "[| S <= A; isLub S cl L |] ==> L = lub S cl"
+lemma (in CL) lubIa: "[| S \<subseteq> A; isLub S cl L |] ==> L = lub S cl"
 by (simp add: lubI isLub_def A_def r_def)
 
 lemma (in CL) isLub_in_lattice: "isLub S cl L ==> L \<in> A"
@@ -381,7 +381,7 @@
 
 subsection {* glb *}
 
-lemma (in CL) glb_in_lattice: "S <= A ==> glb S cl \<in> A"
+lemma (in CL) glb_in_lattice: "S \<subseteq> A ==> glb S cl \<in> A"
 apply (subst glb_dual_lub)
 apply (simp add: A_def)
 apply (rule dualA_iff [THEN subst])
@@ -391,11 +391,11 @@
 apply (simp add: dualA_iff)
 done
 
-lemma (in CL) glb_lower: "[|S <= A; x \<in> S|] ==> (glb S cl, x) \<in> r"
+lemma (in CL) glb_lower: "[|S \<subseteq> A; x \<in> S|] ==> (glb S cl, x) \<in> r"
 apply (subst glb_dual_lub)
 apply (simp add: r_def)
 apply (rule dualr_iff [THEN subst])
-apply (rule Tarski.lub_upper [rule_format])
+apply (rule Tarski.lub_upper)
 apply (rule dualPO)
 apply (rule CL_dualCL)
 apply (simp add: dualA_iff A_def, assumption)
@@ -429,16 +429,15 @@
 
 subsection {* fixed points *}
 
-lemma fix_subset: "fix f A <= A"
+lemma fix_subset: "fix f A \<subseteq> A"
 by (simp add: fix_def, fast)
 
 lemma fix_imp_eq: "x \<in> fix f A ==> f x = x"
 by (simp add: fix_def)
 
 lemma fixf_subset:
-     "[| A <= B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
-apply (simp add: fix_def, auto)
-done
+     "[| A \<subseteq> B; x \<in> fix (%y: A. f y) A |] ==> x \<in> fix f B"
+by (simp add: fix_def, auto)
 
 
 subsection {* lemmas for Tarski, lub *}
@@ -547,42 +546,39 @@
 apply (simp add: refl_def, blast)
 done
 
-lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b <= A"
+lemma (in CLF) interval_subset: "[| a \<in> A; b \<in> A |] ==> interval r a b \<subseteq> A"
 apply (simp add: interval_def)
 apply (blast intro: rel_imp_elem)
 done
 
 lemma (in CLF) intervalI:
      "[| (a, x) \<in> r; (x, b) \<in> r |] ==> x \<in> interval r a b"
-apply (simp add: interval_def)
-done
+by (simp add: interval_def)
 
 lemma (in CLF) interval_lemma1:
-     "[| S <= interval r a b; x \<in> S |] ==> (a, x) \<in> r"
-apply (unfold interval_def, fast)
-done
+     "[| S \<subseteq> interval r a b; x \<in> S |] ==> (a, x) \<in> r"
+by (unfold interval_def, fast)
 
 lemma (in CLF) interval_lemma2:
-     "[| S <= interval r a b; x \<in> S |] ==> (x, b) \<in> r"
-apply (unfold interval_def, fast)
-done
+     "[| S \<subseteq> interval r a b; x \<in> S |] ==> (x, b) \<in> r"
+by (unfold interval_def, fast)
 
 lemma (in CLF) a_less_lub:
-     "[| S <= A; S \<noteq> {};
+     "[| S \<subseteq> A; S \<noteq> {};
          \<forall>x \<in> S. (a,x) \<in> r; \<forall>y \<in> S. (y, L) \<in> r |] ==> (a,L) \<in> r"
 by (blast intro: transE PO_imp_trans)
 
 lemma (in CLF) glb_less_b:
-     "[| S <= A; S \<noteq> {};
+     "[| S \<subseteq> A; S \<noteq> {};
          \<forall>x \<in> S. (x,b) \<in> r; \<forall>y \<in> S. (G, y) \<in> r |] ==> (G,b) \<in> r"
 by (blast intro: transE PO_imp_trans)
 
 lemma (in CLF) S_intv_cl:
-     "[| a \<in> A; b \<in> A; S <= interval r a b |]==> S <= A"
+     "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b |]==> S \<subseteq> A"
 by (simp add: subset_trans [OF _ interval_subset])
 
 lemma (in CLF) L_in_interval:
-     "[| a \<in> A; b \<in> A; S <= interval r a b;
+     "[| a \<in> A; b \<in> A; S \<subseteq> interval r a b;
          S \<noteq> {}; isLub S cl L; interval r a b \<noteq> {} |] ==> L \<in> interval r a b"
 apply (rule intervalI)
 apply (rule a_less_lub)
@@ -596,7 +592,7 @@
 done
 
 lemma (in CLF) G_in_interval:
-     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S <= interval r a b; isGlb S cl G;
+     "[| a \<in> A; b \<in> A; interval r a b \<noteq> {}; S \<subseteq> interval r a b; isGlb S cl G;
          S \<noteq> {} |] ==> G \<in> interval r a b"
 apply (simp add: interval_dual)
 apply (simp add: Tarski.L_in_interval [of _ f]
@@ -613,7 +609,7 @@
 
 lemma (in CLF) intv_CL_lub:
  "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
-  ==> \<forall>S. S <= interval r a b -->
+  ==> \<forall>S. S \<subseteq> interval r a b -->
           (\<exists>L. isLub S (| pset = interval r a b,
                           order = induced (interval r a b) r |)  L)"
 apply (intro strip)
@@ -667,7 +663,7 @@
 
 lemma (in CLF) interval_is_sublattice:
      "[| a \<in> A; b \<in> A; interval r a b \<noteq> {} |]
-        ==> interval r a b <<= cl"
+        ==> interval r a b <\<subseteq> cl"
 apply (rule sublatticeI)
 apply (simp add: interval_subset)
 apply (rule CompleteLatticeI)
@@ -689,14 +685,9 @@
 
 lemma (in CLF) Bot_in_lattice: "Bot cl \<in> A"
 apply (simp add: Bot_def least_def)
-apply (rule someI2)
-apply (fold A_def)
-apply (erule_tac [2] conjunct1)
-apply (rule conjI)
-apply (rule glb_in_lattice)
-apply (rule subset_refl)
-apply (fold r_def)
-apply (simp add: glb_lower)
+apply (rule_tac a="glb A cl" in someI2)
+apply (simp_all add: glb_in_lattice glb_lower 
+                     r_def [symmetric] A_def [symmetric])
 done
 
 lemma (in CLF) Top_in_lattice: "Top cl \<in> A"
@@ -707,12 +698,10 @@
 
 lemma (in CLF) Top_prop: "x \<in> A ==> (x, Top cl) \<in> r"
 apply (simp add: Top_def greatest_def)
+apply (rule_tac a="lub A cl" in someI2)
 apply (rule someI2)
-apply (fold r_def  A_def)
-prefer 2 apply fast
-apply (intro conjI ballI)
-apply (rule_tac [2] lub_upper)
-apply (auto simp add: lub_in_lattice)
+apply (simp_all add: lub_in_lattice lub_upper 
+                     r_def [symmetric] A_def [symmetric])
 done
 
 lemma (in CLF) Bot_prop: "x \<in> A ==> (Bot cl, x) \<in> r"
@@ -746,7 +735,7 @@
 lemma (in CLF) fixf_po: "(| pset = P, order = induced P r|) \<in> PartialOrder"
 by (simp add: P_def fix_subset po_subset_po)
 
-lemma (in Tarski) Y_subset_A: "Y <= A"
+lemma (in Tarski) Y_subset_A: "Y \<subseteq> A"
 apply (rule subset_trans [OF _ fix_subset])
 apply (rule Y_ss [simplified P_def])
 done
@@ -759,7 +748,7 @@
 apply (rule Y_subset_A)
 apply (rule f_in_funcset [THEN funcset_mem])
 apply (rule lubY_in_A)
--- {* @{text "Y <= P ==> f x = x"} *}
+-- {* @{text "Y \<subseteq> P ==> f x = x"} *}
 apply (rule ballI)
 apply (rule_tac t = "x" in fix_imp_eq [THEN subst])
 apply (erule Y_ss [simplified P_def, THEN subsetD])
@@ -771,7 +760,7 @@
 apply (simp add: lub_upper Y_subset_A)
 done
 
-lemma (in Tarski) intY1_subset: "intY1 <= A"
+lemma (in Tarski) intY1_subset: "intY1 \<subseteq> A"
 apply (unfold intY1_def)
 apply (rule interval_subset)
 apply (rule lubY_in_A)
@@ -887,7 +876,7 @@
 
 lemma CompleteLatticeI_simp:
      "[| (| pset = A, order = r |) \<in> PartialOrder;
-         \<forall>S. S <= A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
+         \<forall>S. S \<subseteq> A --> (\<exists>L. isLub S (| pset = A, order = r |)  L) |]
     ==> (| pset = A, order = r |) \<in> CompleteLattice"
 by (simp add: CompleteLatticeI Rdual)