library theories for debugging and parallel computing using code generation towards Isabelle/ML
1 (* Title: HOL/Library/Permutation.thy
2 Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
5 header {* Permutations *}
12 perm :: "'a list => 'a list => bool" ("_ <~~> _" [50, 50] 50)
14 Nil [intro!]: "[] <~~> []"
15 | swap [intro!]: "y # x # l <~~> x # y # l"
16 | Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
17 | trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
19 lemma perm_refl [iff]: "l <~~> l"
23 subsection {* Some examples of rule induction on permutations *}
25 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
26 by (induct xs == "[]::'a list" ys pred: perm) simp_all
30 \medskip This more general theorem is easier to understand!
33 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
34 by (induct pred: perm) simp_all
36 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
37 by (drule perm_length) auto
39 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
40 by (induct pred: perm) auto
43 subsection {* Ways of making new permutations *}
46 We can insert the head anywhere in the list.
49 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
52 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
55 apply (blast intro: perm_append_Cons)
58 lemma perm_append_single: "a # xs <~~> xs @ [a]"
59 by (rule perm.trans [OF _ perm_append_swap]) simp
61 lemma perm_rev: "rev xs <~~> xs"
64 apply (blast intro!: perm_append_single intro: perm_sym)
67 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
70 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
71 by (blast intro!: perm_append_swap perm_append1)
74 subsection {* Further results *}
76 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
77 by (blast intro: perm_empty_imp)
79 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
81 apply (erule perm_sym [THEN perm_empty_imp])
84 lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]"
85 by (induct pred: perm) auto
87 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
88 by (blast intro: perm_sing_imp)
90 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
91 by (blast dest: perm_sym)
94 subsection {* Removing elements *}
96 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove1 x ys"
100 text {* \medskip Congruence rule *}
102 lemma perm_remove_perm: "xs <~~> ys ==> remove1 z xs <~~> remove1 z ys"
103 by (induct pred: perm) auto
105 lemma remove_hd [simp]: "remove1 z (z # xs) = xs"
108 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
109 by (drule_tac z = z in perm_remove_perm) auto
111 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
112 by (blast intro: cons_perm_imp_perm)
114 lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys"
115 apply (induct zs arbitrary: xs ys rule: rev_induct)
116 apply (simp_all (no_asm_use))
120 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
121 by (blast intro: append_perm_imp_perm perm_append1)
123 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
124 apply (safe intro!: perm_append2)
125 apply (rule append_perm_imp_perm)
126 apply (rule perm_append_swap [THEN perm.trans])
127 -- {* the previous step helps this @{text blast} call succeed quickly *}
128 apply (blast intro: perm_append_swap)
131 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
133 apply (erule_tac [2] perm.induct, simp_all add: union_ac)
134 apply (erule rev_mp, rule_tac x=ys in spec)
135 apply (induct_tac xs, auto)
136 apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)
137 apply (subgoal_tac "a \<in> set x")
138 apply (drule_tac z=a in perm.Cons)
139 apply (erule perm.trans, rule perm_sym, erule perm_remove)
140 apply (drule_tac f=set_of in arg_cong, simp)
143 lemma multiset_of_le_perm_append:
144 "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
145 apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
146 apply (insert surj_multiset_of, drule surjD)
147 apply (blast intro: sym)+
150 lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys"
151 by (metis multiset_of_eq_perm multiset_of_eq_setD)
153 lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys"
154 apply (induct pred: perm)
157 apply (metis perm_set_eq)
160 lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys"
161 apply (induct xs arbitrary: ys rule: length_induct)
162 apply (case_tac "remdups xs", simp, simp)
163 apply (subgoal_tac "a : set (remdups ys)")
164 prefer 2 apply (metis set.simps(2) insert_iff set_remdups)
165 apply (drule split_list) apply(elim exE conjE)
166 apply (drule_tac x=list in spec) apply(erule impE) prefer 2
167 apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
169 apply (subgoal_tac "a#list <~~> a#ysa@zs")
170 apply (metis Cons_eq_appendI perm_append_Cons trans)
171 apply (metis Cons Cons_eq_appendI distinct.simps(2)
172 distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
173 apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")
174 apply (fastforce simp add: insert_ident)
175 apply (metis distinct_remdups set_remdups)
176 apply (subgoal_tac "length (remdups xs) < Suc (length xs)")
178 apply (subgoal_tac "length (remdups xs) \<le> length xs")
180 apply (rule length_remdups_leq)
183 lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (set x = set y)"
184 by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
186 lemma permutation_Ex_bij:
188 shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
189 using assms proof induct
190 case Nil then show ?case unfolding bij_betw_def by simp
194 proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
195 show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
196 by (auto simp: bij_betw_def bij_betw_swap_iff)
197 fix i assume "i < length(y#x#l)"
198 show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
199 by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
203 then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and
204 perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast
205 let "?f i" = "case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
207 proof (intro exI[of _ ?f] allI conjI impI)
208 have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
209 "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
210 by (simp_all add: lessThan_Suc_eq_insert_0)
211 show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}" unfolding *
212 proof (rule bij_betw_combine)
213 show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
214 using bij unfolding bij_betw_def
215 by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)
216 qed (auto simp: bij_betw_def)
217 fix i assume "i < length (z#xs)"
218 then show "(z # xs) ! i = (z # ys) ! (?f i)"
219 using perm by (cases i) auto
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
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 force