src/ZF/Finite.thy
author wenzelm
Tue Aug 27 11:03:05 2002 +0200 (2002-08-27)
changeset 13524 604d0f3622d6
parent 13356 c9cfe1638bf2
child 13615 449a70d88b38
permissions -rw-r--r--
*** empty log message ***
     1 (*  Title:      ZF/Finite.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 prove X:Fin(A) ==> |X| < nat
     7 
     8 prove:  b: Fin(A) ==> inj(b,b) <= surj(b,b)
     9 *)
    10 
    11 header{*Finite Powerset Operator and Finite Function Space*}
    12 
    13 theory Finite = Inductive + Epsilon + Nat:
    14 
    15 (*The natural numbers as a datatype*)
    16 rep_datatype
    17   elimination    natE
    18   induction	 nat_induct
    19   case_eqns	 nat_case_0 nat_case_succ
    20   recursor_eqns  recursor_0 recursor_succ
    21 
    22 
    23 consts
    24   Fin       :: "i=>i"
    25   FiniteFun :: "[i,i]=>i"         ("(_ -||>/ _)" [61, 60] 60)
    26 
    27 inductive
    28   domains   "Fin(A)" <= "Pow(A)"
    29   intros
    30     emptyI:  "0 : Fin(A)"
    31     consI:   "[| a: A;  b: Fin(A) |] ==> cons(a,b) : Fin(A)"
    32   type_intros  empty_subsetI cons_subsetI PowI
    33   type_elims   PowD [THEN revcut_rl]
    34 
    35 inductive
    36   domains   "FiniteFun(A,B)" <= "Fin(A*B)"
    37   intros
    38     emptyI:  "0 : A -||> B"
    39     consI:   "[| a: A;  b: B;  h: A -||> B;  a ~: domain(h) |]
    40               ==> cons(<a,b>,h) : A -||> B"
    41   type_intros Fin.intros
    42 
    43 
    44 subsection {* Finite Powerset Operator *}
    45 
    46 lemma Fin_mono: "A<=B ==> Fin(A) <= Fin(B)"
    47 apply (unfold Fin.defs)
    48 apply (rule lfp_mono)
    49 apply (rule Fin.bnd_mono)+
    50 apply blast
    51 done
    52 
    53 (* A : Fin(B) ==> A <= B *)
    54 lemmas FinD = Fin.dom_subset [THEN subsetD, THEN PowD, standard]
    55 
    56 (** Induction on finite sets **)
    57 
    58 (*Discharging x~:y entails extra work*)
    59 lemma Fin_induct [case_names 0 cons, induct set: Fin]:
    60     "[| b: Fin(A);
    61         P(0);
    62         !!x y. [| x: A;  y: Fin(A);  x~:y;  P(y) |] ==> P(cons(x,y))
    63      |] ==> P(b)"
    64 apply (erule Fin.induct, simp)
    65 apply (case_tac "a:b")
    66  apply (erule cons_absorb [THEN ssubst], assumption) (*backtracking!*)
    67 apply simp
    68 done
    69 
    70 
    71 (** Simplification for Fin **)
    72 declare Fin.intros [simp]
    73 
    74 lemma Fin_0: "Fin(0) = {0}"
    75 by (blast intro: Fin.emptyI dest: FinD)
    76 
    77 (*The union of two finite sets is finite.*)
    78 lemma Fin_UnI [simp]: "[| b: Fin(A);  c: Fin(A) |] ==> b Un c : Fin(A)"
    79 apply (erule Fin_induct)
    80 apply (simp_all add: Un_cons)
    81 done
    82 
    83 
    84 (*The union of a set of finite sets is finite.*)
    85 lemma Fin_UnionI: "C : Fin(Fin(A)) ==> Union(C) : Fin(A)"
    86 by (erule Fin_induct, simp_all)
    87 
    88 (*Every subset of a finite set is finite.*)
    89 lemma Fin_subset_lemma [rule_format]: "b: Fin(A) ==> \<forall>z. z<=b --> z: Fin(A)"
    90 apply (erule Fin_induct)
    91 apply (simp add: subset_empty_iff)
    92 apply (simp add: subset_cons_iff distrib_simps, safe)
    93 apply (erule_tac b = "z" in cons_Diff [THEN subst], simp)
    94 done
    95 
    96 lemma Fin_subset: "[| c<=b;  b: Fin(A) |] ==> c: Fin(A)"
    97 by (blast intro: Fin_subset_lemma)
    98 
    99 lemma Fin_IntI1 [intro,simp]: "b: Fin(A) ==> b Int c : Fin(A)"
   100 by (blast intro: Fin_subset)
   101 
   102 lemma Fin_IntI2 [intro,simp]: "c: Fin(A) ==> b Int c : Fin(A)"
   103 by (blast intro: Fin_subset)
   104 
   105 lemma Fin_0_induct_lemma [rule_format]:
   106     "[| c: Fin(A);  b: Fin(A); P(b);
   107         !!x y. [| x: A;  y: Fin(A);  x:y;  P(y) |] ==> P(y-{x})
   108      |] ==> c<=b --> P(b-c)"
   109 apply (erule Fin_induct, simp)
   110 apply (subst Diff_cons)
   111 apply (simp add: cons_subset_iff Diff_subset [THEN Fin_subset])
   112 done
   113 
   114 lemma Fin_0_induct:
   115     "[| b: Fin(A);
   116         P(b);
   117         !!x y. [| x: A;  y: Fin(A);  x:y;  P(y) |] ==> P(y-{x})
   118      |] ==> P(0)"
   119 apply (rule Diff_cancel [THEN subst])
   120 apply (blast intro: Fin_0_induct_lemma) 
   121 done
   122 
   123 (*Functions from a finite ordinal*)
   124 lemma nat_fun_subset_Fin: "n: nat ==> n->A <= Fin(nat*A)"
   125 apply (induct_tac "n")
   126 apply (simp add: subset_iff)
   127 apply (simp add: succ_def mem_not_refl [THEN cons_fun_eq])
   128 apply (fast intro!: Fin.consI)
   129 done
   130 
   131 
   132 subsection{*Finite Function Space*}
   133 
   134 lemma FiniteFun_mono:
   135     "[| A<=C;  B<=D |] ==> A -||> B  <=  C -||> D"
   136 apply (unfold FiniteFun.defs)
   137 apply (rule lfp_mono)
   138 apply (rule FiniteFun.bnd_mono)+
   139 apply (intro Fin_mono Sigma_mono basic_monos, assumption+)
   140 done
   141 
   142 lemma FiniteFun_mono1: "A<=B ==> A -||> A  <=  B -||> B"
   143 by (blast dest: FiniteFun_mono)
   144 
   145 lemma FiniteFun_is_fun: "h: A -||>B ==> h: domain(h) -> B"
   146 apply (erule FiniteFun.induct, simp)
   147 apply (simp add: fun_extend3)
   148 done
   149 
   150 lemma FiniteFun_domain_Fin: "h: A -||>B ==> domain(h) : Fin(A)"
   151 by (erule FiniteFun.induct, simp, simp)
   152 
   153 lemmas FiniteFun_apply_type = FiniteFun_is_fun [THEN apply_type, standard]
   154 
   155 (*Every subset of a finite function is a finite function.*)
   156 lemma FiniteFun_subset_lemma [rule_format]:
   157      "b: A-||>B ==> ALL z. z<=b --> z: A-||>B"
   158 apply (erule FiniteFun.induct)
   159 apply (simp add: subset_empty_iff FiniteFun.intros)
   160 apply (simp add: subset_cons_iff distrib_simps, safe)
   161 apply (erule_tac b = "z" in cons_Diff [THEN subst])
   162 apply (drule spec [THEN mp], assumption)
   163 apply (fast intro!: FiniteFun.intros)
   164 done
   165 
   166 lemma FiniteFun_subset: "[| c<=b;  b: A-||>B |] ==> c: A-||>B"
   167 by (blast intro: FiniteFun_subset_lemma)
   168 
   169 (** Some further results by Sidi O. Ehmety **)
   170 
   171 lemma fun_FiniteFunI [rule_format]: "A:Fin(X) ==> ALL f. f:A->B --> f:A-||>B"
   172 apply (erule Fin.induct)
   173  apply (simp add: FiniteFun.intros, clarify)
   174 apply (case_tac "a:b")
   175  apply (rotate_tac -1)
   176  apply (simp add: cons_absorb)
   177 apply (subgoal_tac "restrict (f,b) : b -||> B")
   178  prefer 2 apply (blast intro: restrict_type2)
   179 apply (subst fun_cons_restrict_eq, assumption)
   180 apply (simp add: restrict_def lam_def)
   181 apply (blast intro: apply_funtype FiniteFun.intros 
   182                     FiniteFun_mono [THEN [2] rev_subsetD])
   183 done
   184 
   185 lemma lam_FiniteFun: "A: Fin(X) ==> (lam x:A. b(x)) : A -||> {b(x). x:A}"
   186 by (blast intro: fun_FiniteFunI lam_funtype)
   187 
   188 lemma FiniteFun_Collect_iff:
   189      "f : FiniteFun(A, {y:B. P(y)})
   190       <-> f : FiniteFun(A,B) & (ALL x:domain(f). P(f`x))"
   191 apply auto
   192 apply (blast intro: FiniteFun_mono [THEN [2] rev_subsetD])
   193 apply (blast dest: Pair_mem_PiD FiniteFun_is_fun)
   194 apply (rule_tac A1="domain(f)" in 
   195        subset_refl [THEN [2] FiniteFun_mono, THEN subsetD])
   196  apply (fast dest: FiniteFun_domain_Fin Fin.dom_subset [THEN subsetD])
   197 apply (rule fun_FiniteFunI)
   198 apply (erule FiniteFun_domain_Fin)
   199 apply (rule_tac B = "range (f) " in fun_weaken_type)
   200  apply (blast dest: FiniteFun_is_fun range_of_fun range_type apply_equality)+
   201 done
   202 
   203 ML
   204 {*
   205 val Fin_intros = thms "Fin.intros";
   206 
   207 val Fin_mono = thm "Fin_mono";
   208 val FinD = thm "FinD";
   209 val Fin_induct = thm "Fin_induct";
   210 val Fin_UnI = thm "Fin_UnI";
   211 val Fin_UnionI = thm "Fin_UnionI";
   212 val Fin_subset = thm "Fin_subset";
   213 val Fin_IntI1 = thm "Fin_IntI1";
   214 val Fin_IntI2 = thm "Fin_IntI2";
   215 val Fin_0_induct = thm "Fin_0_induct";
   216 val nat_fun_subset_Fin = thm "nat_fun_subset_Fin";
   217 val FiniteFun_mono = thm "FiniteFun_mono";
   218 val FiniteFun_mono1 = thm "FiniteFun_mono1";
   219 val FiniteFun_is_fun = thm "FiniteFun_is_fun";
   220 val FiniteFun_domain_Fin = thm "FiniteFun_domain_Fin";
   221 val FiniteFun_apply_type = thm "FiniteFun_apply_type";
   222 val FiniteFun_subset = thm "FiniteFun_subset";
   223 val fun_FiniteFunI = thm "fun_FiniteFunI";
   224 val lam_FiniteFun = thm "lam_FiniteFun";
   225 val FiniteFun_Collect_iff = thm "FiniteFun_Collect_iff";
   226 *}
   227 
   228 end