--- a/src/HOL/Library/Permutation.thy Sun Sep 08 20:04:32 2019 +0200
+++ b/src/HOL/Library/Permutation.thy Tue Sep 10 14:40:00 2019 +0100
@@ -8,7 +8,7 @@
imports Multiset
begin
-inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (\<open>_ <~~> _\<close> [50, 50] 50) (* FIXME proper infix, without ambiguity!? *)
+inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr \<open><~~>\<close> 50)
where
Nil [intro!]: "[] <~~> []"
| swap [intro!]: "y # x # l <~~> x # y # l"
@@ -21,8 +21,8 @@
subsection \<open>Some examples of rule induction on permutations\<close>
-proposition xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
- by (induct xs == "[] :: 'a list" ys pred: perm) simp_all
+proposition perm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
+ by (induction "[] :: 'a list" ys pred: perm) simp_all
text \<open>\medskip This more general theorem is easier to understand!\<close>
@@ -30,9 +30,6 @@
proposition perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
by (induct pred: perm) simp_all
-proposition perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
- by (drule perm_length) auto
-
proposition perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
by (induct pred: perm) auto
@@ -66,9 +63,7 @@
by (blast intro: perm_empty_imp)
proposition perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
- apply auto
- apply (erule perm_sym [THEN perm_empty_imp])
- done
+ using perm_sym by auto
proposition perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
by (induct pred: perm) auto
@@ -98,7 +93,7 @@
by (drule perm_remove_perm [where z = z]) auto
proposition cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
- by (blast intro: cons_perm_imp_perm)
+ by (meson cons_perm_imp_perm perm.Cons)
proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
by (induct zs arbitrary: xs ys rule: rev_induct) auto
@@ -107,74 +102,52 @@
by (blast intro: append_perm_imp_perm perm_append1)
proposition 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])
- \<comment> \<open>the previous step helps this \<open>blast\<close> call succeed quickly\<close>
- apply (blast intro: perm_append_swap)
- done
+ by (meson perm.trans perm_append1_eq perm_append_swap)
theorem mset_eq_perm: "mset xs = mset ys \<longleftrightarrow> xs <~~> ys"
- apply (rule iffI)
- 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)
- apply (rule perm_sym)
- apply (erule perm_remove)
- apply (drule_tac f=set_mset in arg_cong)
- apply simp
- done
+proof
+ assume "mset xs = mset ys"
+ then show "xs <~~> ys"
+ proof (induction xs arbitrary: ys)
+ case (Cons x xs)
+ then have "x \<in> set ys"
+ using mset_eq_setD by fastforce
+ then show ?case
+ by (metis Cons.IH Cons.prems mset_remove1 perm.Cons perm.trans perm_remove perm_sym remove_hd)
+ qed auto
+next
+ assume "xs <~~> ys"
+ then show "mset xs = mset ys"
+ by induction (simp_all add: union_ac)
+qed
proposition mset_le_perm_append: "mset xs \<subseteq># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
- apply (auto simp: mset_eq_perm[THEN sym] mset_subset_eq_exists_conv)
- apply (insert surj_mset)
- apply (drule surjD)
- apply (blast intro: sym)+
- done
+ apply (rule iffI)
+ apply (metis mset_append mset_eq_perm mset_subset_eq_exists_conv surjD surj_mset)
+ by (metis mset_append mset_eq_perm mset_subset_eq_exists_conv)
proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
by (metis mset_eq_perm mset_eq_setD)
proposition 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
+ by (metis card_distinct distinct_card perm_length perm_set_eq)
theorem 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")
- apply simp_all
- apply (subgoal_tac "a \<in> set (remdups ys)")
- prefer 2 apply (metis list.set(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 (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) \<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)")
- apply simp
- apply (subgoal_tac "length (remdups xs) \<le> length xs")
- apply simp
- apply (rule length_remdups_leq)
- done
+proof (induction xs arbitrary: ys rule: length_induct)
+ case (1 xs)
+ show ?case
+ proof (cases "remdups xs")
+ case Nil
+ with "1.prems" show ?thesis
+ using "1.prems" by auto
+ next
+ case (Cons x us)
+ then have "x \<in> set (remdups ys)"
+ using "1.prems" set_remdups by fastforce
+ then show ?thesis
+ using "1.prems" mset_eq_perm set_eq_iff_mset_remdups_eq by blast
+ qed
+qed
proposition 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)
@@ -243,10 +216,7 @@
proposition perm_finite: "finite {B. B <~~> A}"
proof (rule finite_subset[where B="{xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"])
show "finite {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
- apply (cases A, simp)
- apply (rule card_ge_0_finite)
- apply (auto simp: card_lists_length_le)
- done
+ using finite_lists_length_le by blast
next
show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
by (clarsimp simp add: perm_length perm_set_eq)