src/HOL/Library/Permutation.thy

--- a/src/HOL/Library/Permutation.thy	Wed Aug 31 15:46:36 2005 +0200
+++ b/src/HOL/Library/Permutation.thy	Wed Aug 31 15:46:37 2005 +0200
@@ -24,20 +24,20 @@
     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
 
 lemma perm_refl [iff]: "l <~~> l"
-by (induct l, auto)
+  by (induct l) auto
 
 
 subsection {* Some examples of rule induction on permutations *}
 
 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
-    -- {*the form of the premise lets the induction bind @{term xs} 
+    -- {*the form of the premise lets the induction bind @{term xs}
          and @{term ys} *}
   apply (erule perm.induct)
      apply (simp_all (no_asm_simp))
   done
 
 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
-by (insert xperm_empty_imp_aux, blast)
+  using xperm_empty_imp_aux by blast
 
 
 text {*
@@ -45,16 +45,16 @@
   *}
 
 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
-by (erule perm.induct, simp_all)
+  by (erule perm.induct) simp_all
 
 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
-by (drule perm_length, auto)
+  by (drule perm_length) auto
 
 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
-by (erule perm.induct, auto)
+  by (erule perm.induct) auto
 
 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
-by (erule perm.induct, auto)
+  by (erule perm.induct) auto
 
 
 subsection {* Ways of making new permutations *}
@@ -64,32 +64,34 @@
 *}
 
 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
-by (induct xs, auto)
+  by (induct xs) auto
 
 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
-  apply (induct xs, simp_all)
+  apply (induct xs)
+    apply simp_all
   apply (blast intro: perm_append_Cons)
   done
 
 lemma perm_append_single: "a # xs <~~> xs @ [a]"
-  by (rule perm.trans [OF _ perm_append_swap], simp)
+  by (rule perm.trans [OF _ perm_append_swap]) simp
 
 lemma perm_rev: "rev xs <~~> xs"
-  apply (induct xs, simp_all)
+  apply (induct xs)
+   apply simp_all
   apply (blast intro!: perm_append_single intro: perm_sym)
   done
 
 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
-by (induct l, auto)
+  by (induct l) auto
 
 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
-by (blast intro!: perm_append_swap perm_append1)
+  by (blast intro!: perm_append_swap perm_append1)
 
 
 subsection {* Further results *}
 
 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
-by (blast intro: perm_empty_imp)
+  by (blast intro: perm_empty_imp)
 
 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
   apply auto
@@ -97,13 +99,13 @@
   done
 
 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
-by (erule perm.induct, auto)
+  by (erule perm.induct) auto
 
 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
-by (blast intro: perm_sing_imp)
+  by (blast intro: perm_sing_imp)
 
 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
-by (blast dest: perm_sym)
+  by (blast dest: perm_sym)
 
 
 subsection {* Removing elements *}
@@ -115,29 +117,31 @@
   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
 
 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
-by (induct ys, auto)
+  by (induct ys) auto
 
 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
-by (induct l, auto)
+  by (induct l) auto
 
-lemma multiset_of_remove[simp]: 
-  "multiset_of (remove a x) = multiset_of x - {#a#}"
-  by (induct_tac x, auto simp: multiset_eq_conv_count_eq) 
+lemma multiset_of_remove[simp]:
+    "multiset_of (remove a x) = multiset_of x - {#a#}"
+  apply (induct x)
+   apply (auto simp: multiset_eq_conv_count_eq)
+  done
 
 
 text {* \medskip Congruence rule *}
 
 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
-by (erule perm.induct, auto)
+  by (erule perm.induct) auto
 
 lemma remove_hd [simp]: "remove z (z # xs) = xs"
   by auto
 
 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
-by (drule_tac z = z in perm_remove_perm, auto)
+  by (drule_tac z = z in perm_remove_perm) auto
 
 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
-by (blast intro: cons_perm_imp_perm)
+  by (blast intro: cons_perm_imp_perm)
 
 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
   apply (induct zs rule: rev_induct)
@@ -146,7 +150,7 @@
   done
 
 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
-by (blast intro: append_perm_imp_perm perm_append1)
+  by (blast intro: append_perm_imp_perm perm_append1)
 
 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
   apply (safe intro!: perm_append2)
@@ -157,20 +161,20 @@
   done
 
 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
-  apply (rule iffI) 
-  apply (erule_tac [2] perm.induct, simp_all add: union_ac) 
-  apply (erule rev_mp, rule_tac x=ys in spec) 
-  apply (induct_tac xs, auto) 
-  apply (erule_tac x = "remove a x" in allE, drule sym, simp) 
-  apply (subgoal_tac "a \<in> set x") 
-  apply (drule_tac z=a in perm.Cons) 
-  apply (erule perm.trans, rule perm_sym, erule perm_remove) 
+  apply (rule iffI)
+  apply (erule_tac [2] perm.induct, simp_all add: union_ac)
+  apply (erule rev_mp, rule_tac x=ys in spec)
+  apply (induct_tac xs, auto)
+  apply (erule_tac x = "remove a x" in allE, drule sym, simp)
+  apply (subgoal_tac "a \<in> set x")
+  apply (drule_tac z=a in perm.Cons)
+  apply (erule perm.trans, rule perm_sym, erule perm_remove)
   apply (drule_tac f=set_of in arg_cong, simp)
   done
 
-lemma multiset_of_le_perm_append: 
-  "(multiset_of xs \<le># multiset_of ys) = (\<exists> zs. xs @ zs <~~> ys)"; 
-  apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) 
+lemma multiset_of_le_perm_append:
+    "(multiset_of xs \<le># multiset_of ys) = (\<exists>zs. xs @ zs <~~> ys)";
+  apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
   apply (insert surj_multiset_of, drule surjD)
   apply (blast intro: sym)+
   done