# HG changeset patch
# User wenzelm
# Date 1318704013 -7200
# Node ID 2dd44cd8f9633342e4097ffff9a7d048133c1ca2
# Parent  cf6a5de94bfcf5879cfb4d61e9058014e0f81068
misc tuning and modernization;

diff -r cf6a5de94bfc -r 2dd44cd8f963 src/HOL/Bali/Basis.thy
--- a/src/HOL/Bali/Basis.thy	Sat Oct 15 18:14:36 2011 +0200
+++ b/src/HOL/Bali/Basis.thy	Sat Oct 15 20:40:13 2011 +0200
@@ -3,8 +3,9 @@
 *)
 header {* Definitions extending HOL as logical basis of Bali *}
 
-theory Basis imports Main "~~/src/HOL/Library/Old_Recdef" begin
-
+theory Basis
+imports Main "~~/src/HOL/Library/Old_Recdef"
+begin
 
 section "misc"
 
@@ -14,162 +15,130 @@
 declare length_Suc_conv [iff]
 
 lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
-apply auto
-done
+  by auto
 
-lemma subset_insertD: 
-  "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
-apply (case_tac "x:A")
-apply (rule disjI2)
-apply (rule_tac x = "A-{x}" in exI)
-apply fast+
-done
+lemma subset_insertD: "A \<subseteq> insert x B \<Longrightarrow> A \<subseteq> B \<and> x \<notin> A \<or> (\<exists>B'. A = insert x B' \<and> B' \<subseteq> B)"
+  apply (case_tac "x \<in> A")
+   apply (rule disjI2)
+   apply (rule_tac x = "A - {x}" in exI)
+   apply fast+
+  done
 
-abbreviation nat3 :: nat  ("3") where "3 == Suc 2"
-abbreviation nat4 :: nat  ("4") where "4 == Suc 3"
-
-(*unused*)
-lemma range_bool_domain: "range f = {f True, f False}"
-apply auto
-apply (case_tac "xa")
-apply auto
-done
+abbreviation nat3 :: nat  ("3") where "3 \<equiv> Suc 2"
+abbreviation nat4 :: nat  ("4") where "4 \<equiv> Suc 3"
 
 (* irrefl_tranclI in Transitive_Closure.thy is more general *)
-lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
-by(blast elim: tranclE dest: trancl_into_rtrancl)
+lemma irrefl_tranclI': "r\<inverse> \<inter> r\<^sup>+ = {} \<Longrightarrow> \<forall>x. (x, x) \<notin> r\<^sup>+"
+  by (blast elim: tranclE dest: trancl_into_rtrancl)
 
 
-lemma trancl_rtrancl_trancl:
-"\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
-by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
+lemma trancl_rtrancl_trancl: "\<lbrakk>(x, y) \<in> r\<^sup>+; (y, z) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (x, z) \<in> r\<^sup>+"
+  by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
 
-lemma rtrancl_into_trancl3:
-"\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+" 
-apply (drule rtranclD)
-apply auto
-done
+lemma rtrancl_into_trancl3: "\<lbrakk>(a, b) \<in> r\<^sup>*; a \<noteq> b\<rbrakk> \<Longrightarrow> (a, b) \<in> r\<^sup>+"
+  apply (drule rtranclD)
+  apply auto
+  done
 
-lemma rtrancl_into_rtrancl2: 
-  "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
-by (auto intro: r_into_rtrancl rtrancl_trans)
+lemma rtrancl_into_rtrancl2: "\<lbrakk>(a, b) \<in>  r; (b, c) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (a, c) \<in> r\<^sup>*"
+  by (auto intro: rtrancl_trans)
 
 lemma triangle_lemma:
- "\<lbrakk> \<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c; (a,x)\<in>r\<^sup>*; (a,y)\<in>r\<^sup>*\<rbrakk> 
- \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
-proof -
-  assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
-  assume "(a,x)\<in>r\<^sup>*" 
-  then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
-  proof (induct rule: converse_rtrancl_induct)
-    assume "(x,y)\<in>r\<^sup>*"
-    then show ?thesis 
-      by blast
+  assumes unique: "\<And>a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b = c"
+    and ax: "(a,x)\<in>r\<^sup>*" and ay: "(a,y)\<in>r\<^sup>*"
+  shows "(x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
+  using ax ay
+proof (induct rule: converse_rtrancl_induct)
+  assume "(x,y)\<in>r\<^sup>*"
+  then show ?thesis by blast
+next
+  fix a v
+  assume a_v_r: "(a, v) \<in> r"
+    and v_x_rt: "(v, x) \<in> r\<^sup>*"
+    and a_y_rt: "(a, y) \<in> r\<^sup>*"
+    and hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
+  from a_y_rt show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
+  proof (cases rule: converse_rtranclE)
+    assume "a = y"
+    with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
+      by (auto intro: rtrancl_trans)
+    then show ?thesis by blast
   next
-    fix a v
-    assume a_v_r: "(a, v) \<in> r" and
-          v_x_rt: "(v, x) \<in> r\<^sup>*" and
-          a_y_rt: "(a, y) \<in> r\<^sup>*"  and
-             hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
-    from a_y_rt 
-    show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
-    proof (cases rule: converse_rtranclE)
-      assume "a=y"
-      with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
-        by (auto intro: r_into_rtrancl rtrancl_trans)
-      then show ?thesis 
-        by blast
-    next
-      fix w 
-      assume a_w_r: "(a, w) \<in> r" and
-            w_y_rt: "(w, y) \<in> r\<^sup>*"
-      from a_v_r a_w_r unique 
-      have "v=w" 
-        by auto
-      with w_y_rt hyp 
-      show ?thesis
-        by blast
-    qed
+    fix w
+    assume a_w_r: "(a, w) \<in> r"
+      and w_y_rt: "(w, y) \<in> r\<^sup>*"
+    from a_v_r a_w_r unique have "v=w" by auto
+    with w_y_rt hyp show ?thesis by blast
   qed
 qed
 
 
-lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
- "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
-apply (erule rtranclE)
-apply (auto dest: rtrancl_into_trancl1)
-done
+lemma rtrancl_cases:
+  assumes "(a,b)\<in>r\<^sup>*"
+  obtains (Refl) "a = b"
+    | (Trancl) "(a,b)\<in>r\<^sup>+"
+  apply (rule rtranclE [OF assms])
+   apply (auto dest: rtrancl_into_trancl1)
+  done
 
-(* context (theory "Set") *)
-lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
-by auto
+lemma Ball_weaken: "\<lbrakk>Ball s P; \<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
+  by auto
 
-(* context (theory "Finite") *)
-lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>  
-  finite {f y x |x y. P y}"
-apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
-prefer 2 apply  fast
-apply (erule ssubst)
-apply (erule finite_UN_I)
-apply fast
-done
-
+lemma finite_SetCompr2:
+  "finite (Collect P) \<Longrightarrow> \<forall>y. P y \<longrightarrow> finite (range (f y)) \<Longrightarrow>
+    finite {f y x |x y. P y}"
+  apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (\<lambda>y. range (f y))")
+   prefer 2 apply fast
+  apply (erule ssubst)
+  apply (erule finite_UN_I)
+  apply fast
+  done
 
-(* ### TO theory "List" *)
-lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
- \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
-apply (induct_tac "xs1")
-apply simp
-apply (rule allI)
-apply (induct_tac "xs2")
-apply simp
-apply (rule allI)
-apply (induct_tac "xs3")
-apply auto
-done
+lemma list_all2_trans: "\<forall>a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
+    \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
+  apply (induct_tac xs1)
+   apply simp
+  apply (rule allI)
+  apply (induct_tac xs2)
+   apply simp
+  apply (rule allI)
+  apply (induct_tac xs3)
+   apply auto
+  done
 
 
 section "pairs"
 
-lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))), 
-  snd (snd (snd (snd p))))"
-apply auto
-done
+lemma surjective_pairing5:
+  "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
+    snd (snd (snd (snd p))))"
+  by auto
 
-lemma fst_splitE [elim!]: 
-"[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
-by (cases s') auto
+lemma fst_splitE [elim!]:
+  assumes "fst s' = x'"
+  obtains x s where "s' = (x,s)" and "x = x'"
+  using assms by (cases s') auto
 
-lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
-apply (induct_tac "l")
-apply  auto
-done
+lemma fst_in_set_lemma: "(x, y) : set l \<Longrightarrow> x : fst ` set l"
+  by (induct l) auto
 
 
 section "quantifiers"
 
-lemma All_Ex_refl_eq2 [simp]: 
- "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
-apply auto
-done
+lemma All_Ex_refl_eq2 [simp]: "(\<forall>x. (\<exists>b. x = f b \<and> Q b) \<longrightarrow> P x) = (\<forall>b. Q b \<longrightarrow> P (f b))"
+  by auto
 
-lemma ex_ex_miniscope1 [simp]:
-  "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
-apply auto
-done
+lemma ex_ex_miniscope1 [simp]: "(\<exists>w v. P w v \<and> Q v) = (\<exists>v. (\<exists>w. P w v) \<and> Q v)"
+  by auto
 
-lemma ex_miniscope2 [simp]:
-  "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))" 
-apply auto
-done
+lemma ex_miniscope2 [simp]: "(\<exists>v. P v \<and> Q \<and> R v) = (Q \<and> (\<exists>v. P v \<and> R v))"
+  by auto
 
 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
-apply auto
-done
+  by auto
 
-lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
-apply auto
-done
+lemma All_Ex_refl_eq1 [simp]: "(\<forall>x. (\<exists>b. x = f b) \<longrightarrow> P x) = (\<forall>b. P (f b))"
+  by auto
 
 
 section "sums"
@@ -178,38 +147,38 @@
 
 notation sum_case  (infixr "'(+')"80)
 
-primrec the_Inl  :: "'a + 'b \<Rightarrow> 'a"
+primrec the_Inl :: "'a + 'b \<Rightarrow> 'a"
   where "the_Inl (Inl a) = a"
 
-primrec the_Inr  :: "'a + 'b \<Rightarrow> 'b"
+primrec the_Inr :: "'a + 'b \<Rightarrow> 'b"
   where "the_Inr (Inr b) = b"
 
 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
 
-primrec the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
+primrec the_In1 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
   where "the_In1 (In1 a) = a"
 
-primrec the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
+primrec the_In2 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
   where "the_In2 (In2 b) = b"
 
-primrec the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
+primrec the_In3 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
   where "the_In3 (In3 c) = c"
 
-abbreviation In1l   :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
-  where "In1l e == In1 (Inl e)"
+abbreviation In1l :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
+  where "In1l e \<equiv> In1 (Inl e)"
 
-abbreviation In1r   :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
-  where "In1r c == In1 (Inr c)"
+abbreviation In1r :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
+  where "In1r c \<equiv> In1 (Inr c)"
 
 abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"
-  where "the_In1l == the_Inl \<circ> the_In1"
+  where "the_In1l \<equiv> the_Inl \<circ> the_In1"
 
 abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"
-  where "the_In1r == the_Inr \<circ> the_In1"
+  where "the_In1r \<equiv> the_Inr \<circ> the_In1"
 
 ML {*
 fun sum3_instantiate ctxt thm = map (fn s =>
-  simplify (simpset_of ctxt delsimps[@{thm not_None_eq}])
+  simplify (simpset_of ctxt delsimps [@{thm not_None_eq}])
     (read_instantiate ctxt [(("t", 0), "In" ^ s ^ " ?x")] thm)) ["1l","2","3","1r"]
 *}
 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
@@ -218,107 +187,83 @@
 section "quantifiers for option type"
 
 syntax
-  "_Oall" :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
-  "_Oex"  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
+  "_Oall" :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3! _:_:/ _)" [0,0,10] 10)
+  "_Oex"  :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3? _:_:/ _)" [0,0,10] 10)
 
 syntax (symbols)
-  "_Oall" :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
-  "_Oex"  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
+  "_Oall" :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
+  "_Oex"  :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
 
 translations
-  "! x:A: P"    == "! x:CONST Option.set A. P"
-  "? x:A: P"    == "? x:CONST Option.set A. P"
+  "\<forall>x\<in>A: P" \<rightleftharpoons> "\<forall>x\<in>CONST Option.set A. P"
+  "\<exists>x\<in>A: P" \<rightleftharpoons> "\<exists>x\<in>CONST Option.set A. P"
+
 
 section "Special map update"
 
 text{* Deemed too special for theory Map. *}
 
-definition
-  chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
-  where "chg_map f a m = (case m a of None => m | Some b => m(a|->f b))"
+definition chg_map :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)"
+  where "chg_map f a m = (case m a of None \<Rightarrow> m | Some b \<Rightarrow> m(a\<mapsto>f b))"
 
-lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
-by (unfold chg_map_def, auto)
+lemma chg_map_new[simp]: "m a = None \<Longrightarrow> chg_map f a m = m"
+  unfolding chg_map_def by auto
 
-lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
-by (unfold chg_map_def, auto)
+lemma chg_map_upd[simp]: "m a = Some b \<Longrightarrow> chg_map f a m = m(a\<mapsto>f b)"
+  unfolding chg_map_def by auto
 
 lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
-by (auto simp: chg_map_def split add: option.split)
+  by (auto simp: chg_map_def)
 
 
 section "unique association lists"
 
-definition
-  unique :: "('a \<times> 'b) list \<Rightarrow> bool"
+definition unique :: "('a \<times> 'b) list \<Rightarrow> bool"
   where "unique = distinct \<circ> map fst"
 
-lemma uniqueD [rule_format (no_asm)]: 
-"unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
-apply (unfold unique_def o_def)
-apply (induct_tac "l")
-apply  (auto dest: fst_in_set_lemma)
-done
+lemma uniqueD: "unique l \<Longrightarrow> (x, y) \<in> set l \<Longrightarrow> (x', y') \<in> set l \<Longrightarrow> x = x' \<Longrightarrow> y = y'"
+  unfolding unique_def o_def
+  by (induct l) (auto dest: fst_in_set_lemma)
 
 lemma unique_Nil [simp]: "unique []"
-apply (unfold unique_def)
-apply (simp (no_asm))
-done
+  by (simp add: unique_def)
 
-lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
-apply (unfold unique_def)
-apply  (auto dest: fst_in_set_lemma)
-done
+lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l \<and> (\<forall>y. (x,y) \<notin> set l))"
+  by (auto simp: unique_def dest: fst_in_set_lemma)
 
-lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
-
-lemma unique_single [simp]: "!!p. unique [p]"
-apply auto
-done
+lemma unique_ConsD: "unique (x#xs) \<Longrightarrow> unique xs"
+  by (simp add: unique_def)
 
-lemma unique_ConsD: "unique (x#xs) ==> unique xs"
-apply (simp add: unique_def)
-done
-
-lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->  
-  (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
-apply (induct_tac "l")
-apply  (auto dest: fst_in_set_lemma)
-done
+lemma unique_single [simp]: "\<And>p. unique [p]"
+  by simp
 
-lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
-apply (induct_tac "l")
-apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
-done
+lemma unique_append [rule_format (no_asm)]: "unique l' \<Longrightarrow> unique l \<Longrightarrow>
+    (\<forall>(x,y)\<in>set l. \<forall>(x',y')\<in>set l'. x' \<noteq> x) \<longrightarrow> unique (l @ l')"
+  by (induct l) (auto dest: fst_in_set_lemma)
 
-lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
-apply (induct_tac "l")
-apply auto
-done
+lemma unique_map_inj: "unique l \<Longrightarrow> inj f \<Longrightarrow> unique (map (\<lambda>(k,x). (f k, g k x)) l)"
+  by (induct l) (auto dest: fst_in_set_lemma simp add: inj_eq)
+
+lemma map_of_SomeI: "unique l \<Longrightarrow> (k, x) : set l \<Longrightarrow> map_of l k = Some x"
+  by (induct l) auto
 
 
 section "list patterns"
 
-definition
-  lsplit :: "[['a, 'a list] => 'b, 'a list] => 'b" where
-  "lsplit = (\<lambda>f l. f (hd l) (tl l))"
+definition lsplit :: "[['a, 'a list] \<Rightarrow> 'b, 'a list] \<Rightarrow> 'b"
+  where "lsplit = (\<lambda>f l. f (hd l) (tl l))"
 
 text {* list patterns -- extends pre-defined type "pttrn" used in abstractions *}
 syntax
-  "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
+  "_lpttrn" :: "[pttrn, pttrn] \<Rightarrow> pttrn"    ("_#/_" [901,900] 900)
 translations
-  "%y#x#xs. b"  == "CONST lsplit (%y x#xs. b)"
-  "%x#xs  . b"  == "CONST lsplit (%x xs  . b)"
+  "\<lambda>y # x # xs. b" \<rightleftharpoons> "CONST lsplit (\<lambda>y x # xs. b)"
+  "\<lambda>x # xs. b" \<rightleftharpoons> "CONST lsplit (\<lambda>x xs. b)"
 
 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
-apply (unfold lsplit_def)
-apply (simp (no_asm))
-done
+  by (simp add: lsplit_def)
 
 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
-apply (unfold lsplit_def)
-apply simp
-done 
-
+  by (simp add: lsplit_def)
 
 end