--- a/src/HOL/Library/Permutation.thy Tue Apr 29 21:54:26 2014 +0200
+++ b/src/HOL/Library/Permutation.thy Tue Apr 29 22:50:55 2014 +0200
@@ -22,12 +22,10 @@
subsection {* Some examples of rule induction on permutations *}
lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
- by (induct xs == "[]::'a list" ys pred: perm) simp_all
+ by (induct xs == "[] :: 'a list" ys pred: perm) simp_all
-text {*
- \medskip This more general theorem is easier to understand!
- *}
+text {* \medskip This more general theorem is easier to understand! *}
lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
by (induct pred: perm) simp_all
@@ -41,9 +39,7 @@
subsection {* Ways of making new permutations *}
-text {*
- We can insert the head anywhere in the list.
-*}
+text {* We can insert the head anywhere in the list. *}
lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
by (induct xs) auto
@@ -72,10 +68,10 @@
subsection {* Further results *}
-lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
+lemma perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []"
by (blast intro: perm_empty_imp)
-lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
+lemma perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
apply auto
apply (erule perm_sym [THEN perm_empty_imp])
done
@@ -83,10 +79,10 @@
lemma perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
by (induct pred: perm) auto
-lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
+lemma perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]"
by (blast intro: perm_sing_imp)
-lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
+lemma perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]"
by (blast dest: perm_sym)
@@ -107,16 +103,16 @@
lemma cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
by (drule_tac z = z in perm_remove_perm) auto
-lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
+lemma cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
by (blast intro: cons_perm_imp_perm)
lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
by (induct zs arbitrary: xs ys rule: rev_induct) auto
-lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
+lemma perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys"
by (blast intro: append_perm_imp_perm perm_append1)
-lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
+lemma perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys"
apply (safe intro!: perm_append2)
apply (rule append_perm_imp_perm)
apply (rule perm_append_swap [THEN perm.trans])
@@ -124,21 +120,30 @@
apply (blast intro: perm_append_swap)
done
-lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
+lemma multiset_of_eq_perm: "multiset_of xs = multiset_of ys \<longleftrightarrow> 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 = "remove1 a x" in allE, drule sym, simp)
+ apply (erule_tac [2] perm.induct)
+ apply (simp_all add: union_ac)
+ apply (erule rev_mp)
+ apply (rule_tac x=ys in spec)
+ apply (induct_tac xs)
+ apply auto
+ apply (erule_tac x = "remove1 a x" in allE)
+ apply (drule sym)
+ apply 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)
+ apply (erule perm.trans)
+ apply (rule perm_sym)
+ apply (erule perm_remove)
+ apply (drule_tac f=set_of in arg_cong)
+ apply simp
done
lemma multiset_of_le_perm_append: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<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 (insert surj_multiset_of)
+ apply (drule surjD)
apply (blast intro: sym)+
done
@@ -158,15 +163,16 @@
apply simp_all
apply (subgoal_tac "a \<in> set (remdups ys)")
prefer 2 apply (metis set_simps(2) insert_iff set_remdups)
- apply (drule split_list) apply(elim exE conjE)
- apply (drule_tac x=list in spec) apply(erule impE) prefer 2
- apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
+ apply (drule split_list) apply (elim exE conjE)
+ apply (drule_tac x = list in spec) apply (erule impE) prefer 2
+ apply (drule_tac x = "ysa @ zs" in spec) apply (erule impE) prefer 2
apply simp
apply (subgoal_tac "a # list <~~> a # ysa @ zs")
apply (metis Cons_eq_appendI perm_append_Cons trans)
apply (metis Cons Cons_eq_appendI distinct.simps(2)
distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
- apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")
+ apply (subgoal_tac "set (a # list) =
+ set (ysa @ a # zs) \<and> distinct (a # list) \<and> distinct (ysa @ a # zs)")
apply (fastforce simp add: insert_ident)
apply (metis distinct_remdups set_remdups)
apply (subgoal_tac "length (remdups xs) < Suc (length xs)")
@@ -176,15 +182,17 @@
apply (rule length_remdups_leq)
done
-lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> (set x = set y)"
+lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> set x = set y"
by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
lemma permutation_Ex_bij:
assumes "xs <~~> ys"
shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
-using assms proof induct
+ using assms
+proof induct
case Nil
- then show ?case unfolding bij_betw_def by simp
+ then show ?case
+ unfolding bij_betw_def by simp
next
case (swap y x l)
show ?case
@@ -192,14 +200,15 @@
show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
by (auto simp: bij_betw_def)
fix i
- assume "i < length(y#x#l)"
+ assume "i < length (y # x # l)"
show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
qed
next
case (Cons xs ys z)
- then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and
- perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast
+ then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}"
+ and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)"
+ by blast
let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
show ?case
proof (intro exI[of _ ?f] allI conjI impI)
@@ -214,21 +223,24 @@
by (auto intro!: inj_onI imageI dest: inj_onD simp: image_comp comp_def)
qed (auto simp: bij_betw_def)
fix i
- assume "i < length (z#xs)"
+ assume "i < length (z # xs)"
then show "(z # xs) ! i = (z # ys) ! (?f i)"
using perm by (cases i) auto
qed
next
case (trans xs ys zs)
- then obtain f g where
- bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and
- perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast
+ then obtain f g
+ where bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}"
+ and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)"
+ by blast
show ?case
proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
using bij by (rule bij_betw_trans)
- fix i assume "i < length xs"
- with bij have "f i < length ys" unfolding bij_betw_def by force
+ fix i
+ assume "i < length xs"
+ with bij have "f i < length ys"
+ unfolding bij_betw_def by force
with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"
using trans(1,3)[THEN perm_length] perm by auto
qed