src/HOL/Library/Permutation.thy
changeset 15005 546c8e7e28d4
parent 14706 71590b7733b7
child 15072 4861bf6af0b4
     1.1 --- a/src/HOL/Library/Permutation.thy	Thu Jun 24 17:52:55 2004 +0200
     1.2 +++ b/src/HOL/Library/Permutation.thy	Thu Jun 24 17:54:53 2004 +0200
     1.3 @@ -1,15 +1,10 @@
     1.4  (*  Title:      HOL/Library/Permutation.thy
     1.5 -    ID:         $Id$
     1.6 -    Author:     Lawrence C Paulson and Thomas M Rasmussen
     1.7 -    Copyright   1995  University of Cambridge
     1.8 -
     1.9 -TODO: it would be nice to prove (for "multiset", defined on
    1.10 -HOL/ex/Sorting.thy) xs <~~> ys = (\<forall>x. multiset xs x = multiset ys x)
    1.11 +    Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
    1.12  *)
    1.13  
    1.14  header {* Permutations *}
    1.15  
    1.16 -theory Permutation = Main:
    1.17 +theory Permutation = Multiset:
    1.18  
    1.19  consts
    1.20    perm :: "('a list * 'a list) set"
    1.21 @@ -27,9 +22,7 @@
    1.22      trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
    1.23  
    1.24  lemma perm_refl [iff]: "l <~~> l"
    1.25 -  apply (induct l)
    1.26 -   apply auto
    1.27 -  done
    1.28 +by (induct l, auto)
    1.29  
    1.30  
    1.31  subsection {* Some examples of rule induction on permutations *}
    1.32 @@ -41,9 +34,7 @@
    1.33    done
    1.34  
    1.35  lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
    1.36 -  apply (insert xperm_empty_imp_aux)
    1.37 -  apply blast
    1.38 -  done
    1.39 +by (insert xperm_empty_imp_aux, blast)
    1.40  
    1.41  
    1.42  text {*
    1.43 @@ -51,24 +42,16 @@
    1.44    *}
    1.45  
    1.46  lemma perm_length: "xs <~~> ys ==> length xs = length ys"
    1.47 -  apply (erule perm.induct)
    1.48 -     apply simp_all
    1.49 -  done
    1.50 +by (erule perm.induct, simp_all)
    1.51  
    1.52  lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
    1.53 -  apply (drule perm_length)
    1.54 -  apply auto
    1.55 -  done
    1.56 +by (drule perm_length, auto)
    1.57  
    1.58  lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
    1.59 -  apply (erule perm.induct)
    1.60 -     apply auto
    1.61 -  done
    1.62 +by (erule perm.induct, auto)
    1.63  
    1.64  lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
    1.65 -  apply (erule perm.induct)
    1.66 -     apply auto
    1.67 -  done
    1.68 +by (erule perm.induct, auto)
    1.69  
    1.70  
    1.71  subsection {* Ways of making new permutations *}
    1.72 @@ -78,44 +61,35 @@
    1.73  *}
    1.74  
    1.75  lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
    1.76 -  apply (induct xs)
    1.77 -   apply auto
    1.78 -  done
    1.79 +by (induct xs, auto)
    1.80  
    1.81  lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
    1.82 -  apply (induct xs)
    1.83 -    apply simp_all
    1.84 +  apply (induct xs, simp_all)
    1.85    apply (blast intro: perm_append_Cons)
    1.86    done
    1.87  
    1.88  lemma perm_append_single: "a # xs <~~> xs @ [a]"
    1.89    apply (rule perm.trans)
    1.90     prefer 2
    1.91 -   apply (rule perm_append_swap)
    1.92 -  apply simp
    1.93 +   apply (rule perm_append_swap, simp)
    1.94    done
    1.95  
    1.96  lemma perm_rev: "rev xs <~~> xs"
    1.97 -  apply (induct xs)
    1.98 -   apply simp_all
    1.99 +  apply (induct xs, simp_all)
   1.100    apply (blast intro!: perm_append_single intro: perm_sym)
   1.101    done
   1.102  
   1.103  lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
   1.104 -  apply (induct l)
   1.105 -   apply auto
   1.106 -  done
   1.107 +by (induct l, auto)
   1.108  
   1.109  lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
   1.110 -  apply (blast intro!: perm_append_swap perm_append1)
   1.111 -  done
   1.112 +by (blast intro!: perm_append_swap perm_append1)
   1.113  
   1.114  
   1.115  subsection {* Further results *}
   1.116  
   1.117  lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
   1.118 -  apply (blast intro: perm_empty_imp)
   1.119 -  done
   1.120 +by (blast intro: perm_empty_imp)
   1.121  
   1.122  lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
   1.123    apply auto
   1.124 @@ -123,17 +97,13 @@
   1.125    done
   1.126  
   1.127  lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
   1.128 -  apply (erule perm.induct)
   1.129 -     apply auto
   1.130 -  done
   1.131 +by (erule perm.induct, auto)
   1.132  
   1.133  lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
   1.134 -  apply (blast intro: perm_sing_imp)
   1.135 -  done
   1.136 +by (blast intro: perm_sing_imp)
   1.137  
   1.138  lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
   1.139 -  apply (blast dest: perm_sym)
   1.140 -  done
   1.141 +by (blast dest: perm_sym)
   1.142  
   1.143  
   1.144  subsection {* Removing elements *}
   1.145 @@ -145,35 +115,26 @@
   1.146    "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
   1.147  
   1.148  lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
   1.149 -  apply (induct ys)
   1.150 -   apply auto
   1.151 -  done
   1.152 +by (induct ys, auto)
   1.153  
   1.154  lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
   1.155 -  apply (induct l)
   1.156 -   apply auto
   1.157 -  done
   1.158 +by (induct l, auto)
   1.159  
   1.160  
   1.161  text {* \medskip Congruence rule *}
   1.162  
   1.163  lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
   1.164 -  apply (erule perm.induct)
   1.165 -     apply auto
   1.166 -  done
   1.167 +by (erule perm.induct, auto)
   1.168  
   1.169  lemma remove_hd [simp]: "remove z (z # xs) = xs"
   1.170    apply auto
   1.171    done
   1.172  
   1.173  lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
   1.174 -  apply (drule_tac z = z in perm_remove_perm)
   1.175 -  apply auto
   1.176 -  done
   1.177 +by (drule_tac z = z in perm_remove_perm, auto)
   1.178  
   1.179  lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
   1.180 -  apply (blast intro: cons_perm_imp_perm)
   1.181 -  done
   1.182 +by (blast intro: cons_perm_imp_perm)
   1.183  
   1.184  lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
   1.185    apply (induct zs rule: rev_induct)
   1.186 @@ -182,8 +143,7 @@
   1.187    done
   1.188  
   1.189  lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   1.190 -  apply (blast intro: append_perm_imp_perm perm_append1)
   1.191 -  done
   1.192 +by (blast intro: append_perm_imp_perm perm_append1)
   1.193  
   1.194  lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
   1.195    apply (safe intro!: perm_append2)
   1.196 @@ -193,4 +153,47 @@
   1.197    apply (blast intro: perm_append_swap)
   1.198    done
   1.199  
   1.200 +(****************** Norbert Voelker 17 June 2004 **************) 
   1.201 +
   1.202 +consts 
   1.203 +  multiset_of :: "'a list \<Rightarrow> 'a multiset"
   1.204 +primrec
   1.205 +  "multiset_of [] = {#}"
   1.206 +  "multiset_of (a # x) = multiset_of x + {# a #}"
   1.207 +
   1.208 +lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
   1.209 +  by (induct_tac x, auto) 
   1.210 +
   1.211 +lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
   1.212 +  by (induct_tac x, auto)
   1.213 +
   1.214 +lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
   1.215 + by (induct_tac x, auto) 
   1.216 +
   1.217 +lemma multiset_of_remove[simp]: 
   1.218 +  "multiset_of (remove a x) = multiset_of x - {#a#}"
   1.219 +  by (induct_tac x, auto simp: multiset_eq_conv_count_eq) 
   1.220 +
   1.221 +lemma multiset_of_eq_perm:  "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
   1.222 +  apply (rule iffI) 
   1.223 +  apply (erule_tac [2] perm.induct, simp_all add: union_ac) 
   1.224 +  apply (erule rev_mp, rule_tac x=ys in spec, induct_tac xs, auto) 
   1.225 +  apply (erule_tac x = "remove a x" in allE, drule sym, simp) 
   1.226 +  apply (subgoal_tac "a \<in> set x") 
   1.227 +  apply (drule_tac z=a in perm.Cons) 
   1.228 +  apply (erule perm.trans, rule perm_sym, erule perm_remove) 
   1.229 +  apply (drule_tac f=set_of in arg_cong, simp)
   1.230 +  done
   1.231 +
   1.232 +lemma set_count_multiset_of: "set x = {a. 0 < count (multiset_of x) a}"
   1.233 +  by (induct_tac x, auto)  
   1.234 +
   1.235 +lemma distinct_count_multiset_of: 
   1.236 +   "distinct x \<Longrightarrow> count (multiset_of x) a = (if a \<in> set x then 1 else 0)"
   1.237 +  by (erule rev_mp, induct_tac x, auto) 
   1.238 +
   1.239 +lemma distinct_set_eq_iff_multiset_of_eq: 
   1.240 +  "\<lbrakk>distinct x; distinct y\<rbrakk> \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
   1.241 +  by (auto simp: multiset_eq_conv_count_eq distinct_count_multiset_of) 
   1.242 +
   1.243  end