src/HOL/Library/Permutation.thy
 author wenzelm Tue Aug 27 22:40:39 2013 +0200 (2013-08-27) changeset 53238 01ef0a103fc9 parent 51542 738598beeb26 child 55584 a879f14b6f95 permissions -rw-r--r--
tuned proofs;
1 (*  Title:      HOL/Library/Permutation.thy
2     Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
3 *)
5 header {* Permutations *}
7 theory Permutation
8 imports Multiset
9 begin
11 inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  ("_ <~~> _"  [50, 50] 50)  (* FIXME proper infix, without ambiguity!? *)
12 where
13   Nil [intro!]: "[] <~~> []"
14 | swap [intro!]: "y # x # l <~~> x # y # l"
15 | Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
16 | trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
18 lemma perm_refl [iff]: "l <~~> l"
19   by (induct l) auto
22 subsection {* Some examples of rule induction on permutations *}
24 lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
25   by (induct xs == "[]::'a list" ys pred: perm) simp_all
28 text {*
29   \medskip This more general theorem is easier to understand!
30   *}
32 lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
33   by (induct pred: perm) simp_all
35 lemma perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
36   by (drule perm_length) auto
38 lemma perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
39   by (induct pred: perm) auto
42 subsection {* Ways of making new permutations *}
44 text {*
45   We can insert the head anywhere in the list.
46 *}
48 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
49   by (induct xs) auto
51 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
52   apply (induct xs)
53     apply simp_all
54   apply (blast intro: perm_append_Cons)
55   done
57 lemma perm_append_single: "a # xs <~~> xs @ [a]"
58   by (rule perm.trans [OF _ perm_append_swap]) simp
60 lemma perm_rev: "rev xs <~~> xs"
61   apply (induct xs)
62    apply simp_all
63   apply (blast intro!: perm_append_single intro: perm_sym)
64   done
66 lemma perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
67   by (induct l) auto
69 lemma perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
70   by (blast intro!: perm_append_swap perm_append1)
73 subsection {* Further results *}
75 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
76   by (blast intro: perm_empty_imp)
78 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
79   apply auto
80   apply (erule perm_sym [THEN perm_empty_imp])
81   done
83 lemma perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
84   by (induct pred: perm) auto
86 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
87   by (blast intro: perm_sing_imp)
89 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
90   by (blast dest: perm_sym)
93 subsection {* Removing elements *}
95 lemma perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
96   by (induct ys) auto
99 text {* \medskip Congruence rule *}
101 lemma perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
102   by (induct pred: perm) auto
104 lemma remove_hd [simp]: "remove1 z (z # xs) = xs"
105   by auto
107 lemma cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
108   by (drule_tac z = z in perm_remove_perm) auto
110 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
111   by (blast intro: cons_perm_imp_perm)
113 lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
114   by (induct zs arbitrary: xs ys rule: rev_induct) auto
116 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
117   by (blast intro: append_perm_imp_perm perm_append1)
119 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
120   apply (safe intro!: perm_append2)
121   apply (rule append_perm_imp_perm)
122   apply (rule perm_append_swap [THEN perm.trans])
123     -- {* the previous step helps this @{text blast} call succeed quickly *}
124   apply (blast intro: perm_append_swap)
125   done
127 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
128   apply (rule iffI)
129   apply (erule_tac  perm.induct, simp_all add: union_ac)
130   apply (erule rev_mp, rule_tac x=ys in spec)
131   apply (induct_tac xs, auto)
132   apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)
133   apply (subgoal_tac "a \<in> set x")
134   apply (drule_tac z = a in perm.Cons)
135   apply (erule perm.trans, rule perm_sym, erule perm_remove)
136   apply (drule_tac f=set_of in arg_cong, simp)
137   done
139 lemma multiset_of_le_perm_append: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
140   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
141   apply (insert surj_multiset_of, drule surjD)
142   apply (blast intro: sym)+
143   done
145 lemma perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
146   by (metis multiset_of_eq_perm multiset_of_eq_setD)
148 lemma perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
149   apply (induct pred: perm)
150      apply simp_all
151    apply fastforce
152   apply (metis perm_set_eq)
153   done
155 lemma eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
156   apply (induct xs arbitrary: ys rule: length_induct)
157   apply (case_tac "remdups xs")
158    apply simp_all
159   apply (subgoal_tac "a \<in> set (remdups ys)")
160    prefer 2 apply (metis set.simps(2) insert_iff set_remdups)
161   apply (drule split_list) apply(elim exE conjE)
162   apply (drule_tac x=list in spec) apply(erule impE) prefer 2
163    apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
164     apply simp
165     apply (subgoal_tac "a # list <~~> a # ysa @ zs")
166      apply (metis Cons_eq_appendI perm_append_Cons trans)
167     apply (metis Cons Cons_eq_appendI distinct.simps(2)
168       distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
169    apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")
170     apply (fastforce simp add: insert_ident)
171    apply (metis distinct_remdups set_remdups)
172    apply (subgoal_tac "length (remdups xs) < Suc (length xs)")
173    apply simp
174    apply (subgoal_tac "length (remdups xs) \<le> length xs")
175    apply simp
176    apply (rule length_remdups_leq)
177   done
179 lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> (set x = set y)"
180   by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
182 lemma permutation_Ex_bij:
183   assumes "xs <~~> ys"
184   shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
185 using assms proof induct
186   case Nil
187   then show ?case unfolding bij_betw_def by simp
188 next
189   case (swap y x l)
190   show ?case
191   proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
192     show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
193       by (auto simp: bij_betw_def)
194     fix i
195     assume "i < length(y#x#l)"
196     show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
197       by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
198   qed
199 next
200   case (Cons xs ys z)
201   then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and
202     perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast
203   let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
204   show ?case
205   proof (intro exI[of _ ?f] allI conjI impI)
206     have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
207             "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
208       by (simp_all add: lessThan_Suc_eq_insert_0)
209     show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
210       unfolding *
211     proof (rule bij_betw_combine)
212       show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
213         using bij unfolding bij_betw_def
214         by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)
215     qed (auto simp: bij_betw_def)
216     fix i
217     assume "i < length (z#xs)"
218     then show "(z # xs) ! i = (z # ys) ! (?f i)"
219       using perm by (cases i) auto
220   qed
221 next
222   case (trans xs ys zs)
223   then obtain f g where
224     bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and
225     perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast
226   show ?case
227   proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
228     show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
229       using bij by (rule bij_betw_trans)
230     fix i assume "i < length xs"
231     with bij have "f i < length ys" unfolding bij_betw_def by force
232     with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"
233       using trans(1,3)[THEN perm_length] perm by auto
234   qed
235 qed
237 end