src/HOL/Library/Permutation.thy
author paulson
Fri Feb 16 13:37:21 2001 +0100 (2001-02-16)
changeset 11153 950ede59c05a
parent 11054 a5404c70982f
child 14706 71590b7733b7
permissions -rw-r--r--
Blast bug fix made old proof too slow
     1 (*  Title:      HOL/Library/Permutation.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1995  University of Cambridge
     5 
     6 TODO: it would be nice to prove (for "multiset", defined on
     7 HOL/ex/Sorting.thy) xs <~~> ys = (\<forall>x. multiset xs x = multiset ys x)
     8 *)
     9 
    10 header {*
    11  \title{Permutations}
    12  \author{Lawrence C Paulson and Thomas M Rasmussen}
    13 *}
    14 
    15 theory Permutation = Main:
    16 
    17 consts
    18   perm :: "('a list * 'a list) set"
    19 
    20 syntax
    21   "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
    22 translations
    23   "x <~~> y" == "(x, y) \<in> perm"
    24 
    25 inductive perm
    26   intros
    27     Nil  [intro!]: "[] <~~> []"
    28     swap [intro!]: "y # x # l <~~> x # y # l"
    29     Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
    30     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
    31 
    32 lemma perm_refl [iff]: "l <~~> l"
    33   apply (induct l)
    34    apply auto
    35   done
    36 
    37 
    38 subsection {* Some examples of rule induction on permutations *}
    39 
    40 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
    41     -- {* the form of the premise lets the induction bind @{term xs} and @{term ys} *}
    42   apply (erule perm.induct)
    43      apply (simp_all (no_asm_simp))
    44   done
    45 
    46 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
    47   apply (insert xperm_empty_imp_aux)
    48   apply blast
    49   done
    50 
    51 
    52 text {*
    53   \medskip This more general theorem is easier to understand!
    54   *}
    55 
    56 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
    57   apply (erule perm.induct)
    58      apply simp_all
    59   done
    60 
    61 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
    62   apply (drule perm_length)
    63   apply auto
    64   done
    65 
    66 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
    67   apply (erule perm.induct)
    68      apply auto
    69   done
    70 
    71 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
    72   apply (erule perm.induct)
    73      apply auto
    74   done
    75 
    76 
    77 subsection {* Ways of making new permutations *}
    78 
    79 text {*
    80   We can insert the head anywhere in the list.
    81 *}
    82 
    83 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
    84   apply (induct xs)
    85    apply auto
    86   done
    87 
    88 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
    89   apply (induct xs)
    90     apply simp_all
    91   apply (blast intro: perm_append_Cons)
    92   done
    93 
    94 lemma perm_append_single: "a # xs <~~> xs @ [a]"
    95   apply (rule perm.trans)
    96    prefer 2
    97    apply (rule perm_append_swap)
    98   apply simp
    99   done
   100 
   101 lemma perm_rev: "rev xs <~~> xs"
   102   apply (induct xs)
   103    apply simp_all
   104   apply (blast intro!: perm_append_single intro: perm_sym)
   105   done
   106 
   107 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
   108   apply (induct l)
   109    apply auto
   110   done
   111 
   112 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
   113   apply (blast intro!: perm_append_swap perm_append1)
   114   done
   115 
   116 
   117 subsection {* Further results *}
   118 
   119 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
   120   apply (blast intro: perm_empty_imp)
   121   done
   122 
   123 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
   124   apply auto
   125   apply (erule perm_sym [THEN perm_empty_imp])
   126   done
   127 
   128 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
   129   apply (erule perm.induct)
   130      apply auto
   131   done
   132 
   133 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
   134   apply (blast intro: perm_sing_imp)
   135   done
   136 
   137 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
   138   apply (blast dest: perm_sym)
   139   done
   140 
   141 
   142 subsection {* Removing elements *}
   143 
   144 consts
   145   remove :: "'a => 'a list => 'a list"
   146 primrec
   147   "remove x [] = []"
   148   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
   149 
   150 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
   151   apply (induct ys)
   152    apply auto
   153   done
   154 
   155 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
   156   apply (induct l)
   157    apply auto
   158   done
   159 
   160 
   161 text {* \medskip Congruence rule *}
   162 
   163 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
   164   apply (erule perm.induct)
   165      apply auto
   166   done
   167 
   168 lemma remove_hd [simp]: "remove z (z # xs) = xs"
   169   apply auto
   170   done
   171 
   172 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
   173   apply (drule_tac z = z in perm_remove_perm)
   174   apply auto
   175   done
   176 
   177 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
   178   apply (blast intro: cons_perm_imp_perm)
   179   done
   180 
   181 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
   182   apply (induct zs rule: rev_induct)
   183    apply (simp_all (no_asm_use))
   184   apply blast
   185   done
   186 
   187 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   188   apply (blast intro: append_perm_imp_perm perm_append1)
   189   done
   190 
   191 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
   192   apply (safe intro!: perm_append2)
   193   apply (rule append_perm_imp_perm)
   194   apply (rule perm_append_swap [THEN perm.trans])
   195     -- {* the previous step helps this @{text blast} call succeed quickly *}
   196   apply (blast intro: perm_append_swap)
   197   done
   198 
   199 end