--- a/src/HOL/Library/Permutation.thy Tue Aug 27 22:23:40 2013 +0200
+++ b/src/HOL/Library/Permutation.thy Tue Aug 27 22:40:39 2013 +0200
@@ -8,13 +8,12 @@
imports Multiset
begin
-inductive
- perm :: "'a list => 'a list => bool" ("_ <~~> _" [50, 50] 50)
- where
- Nil [intro!]: "[] <~~> []"
- | swap [intro!]: "y # x # l <~~> x # y # l"
- | Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
- | trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
+inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ <~~> _" [50, 50] 50) (* FIXME proper infix, without ambiguity!? *)
+where
+ Nil [intro!]: "[] <~~> []"
+| swap [intro!]: "y # x # l <~~> x # y # l"
+| Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
+| trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
lemma perm_refl [iff]: "l <~~> l"
by (induct l) auto
@@ -22,7 +21,7 @@
subsection {* Some examples of rule induction on permutations *}
-lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
+lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
by (induct xs == "[]::'a list" ys pred: perm) simp_all
@@ -30,13 +29,13 @@
\medskip This more general theorem is easier to understand!
*}
-lemma perm_length: "xs <~~> ys ==> length xs = length ys"
+lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
by (induct pred: perm) simp_all
-lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
+lemma perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
by (drule perm_length) auto
-lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
+lemma perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
by (induct pred: perm) auto
@@ -64,10 +63,10 @@
apply (blast intro!: perm_append_single intro: perm_sym)
done
-lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
+lemma perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
by (induct l) auto
-lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
+lemma perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
by (blast intro!: perm_append_swap perm_append1)
@@ -81,7 +80,7 @@
apply (erule perm_sym [THEN perm_empty_imp])
done
-lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]"
+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])"
@@ -93,29 +92,26 @@
subsection {* Removing elements *}
-lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove1 x ys"
+lemma perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
by (induct ys) auto
text {* \medskip Congruence rule *}
-lemma perm_remove_perm: "xs <~~> ys ==> remove1 z xs <~~> remove1 z ys"
+lemma perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
by (induct pred: perm) auto
lemma remove_hd [simp]: "remove1 z (z # xs) = xs"
by auto
-lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
+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)"
by (blast intro: cons_perm_imp_perm)
-lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys"
- apply (induct zs arbitrary: xs ys rule: rev_induct)
- apply (simp_all (no_asm_use))
- apply blast
- done
+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)"
by (blast intro: append_perm_imp_perm perm_append1)
@@ -135,38 +131,38 @@
apply (induct_tac xs, auto)
apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)
apply (subgoal_tac "a \<in> set x")
- apply (drule_tac z=a in perm.Cons)
+ 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 \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
+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 (blast intro: sym)+
done
-lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys"
+lemma perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
by (metis multiset_of_eq_perm multiset_of_eq_setD)
-lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys"
+lemma perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
apply (induct pred: perm)
apply simp_all
apply fastforce
apply (metis perm_set_eq)
done
-lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys"
+lemma eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
apply (induct xs arbitrary: ys rule: length_induct)
- apply (case_tac "remdups xs", simp, simp)
- apply (subgoal_tac "a : set (remdups ys)")
+ apply (case_tac "remdups xs")
+ 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 simp
- apply (subgoal_tac "a#list <~~> a#ysa@zs")
+ 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)
@@ -180,21 +176,23 @@
apply (rule length_remdups_leq)
done
-lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (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
- case Nil then show ?case unfolding bij_betw_def by simp
+ case Nil
+ then show ?case unfolding bij_betw_def by simp
next
case (swap y x l)
show ?case
proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
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)"
+ fix i
+ 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
@@ -202,19 +200,21 @@
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
- let "?f i" = "case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
+ 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)
have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
"{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
by (simp_all add: lessThan_Suc_eq_insert_0)
- show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}" unfolding *
+ show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
+ unfolding *
proof (rule bij_betw_combine)
show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
using bij unfolding bij_betw_def
by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)
qed (auto simp: bij_betw_def)
- fix i assume "i < length (z#xs)"
+ fix i
+ assume "i < length (z#xs)"
then show "(z # xs) ! i = (z # ys) ! (?f i)"
using perm by (cases i) auto
qed
@@ -224,13 +224,13 @@
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)
+ 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
with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"
- using trans(1,3)[THEN perm_length] perm by force
+ using trans(1,3)[THEN perm_length] perm by auto
qed
qed