src/ZF/Finite.thy
author wenzelm
Tue Jul 31 19:40:22 2007 +0200 (2007-07-31)
changeset 24091 109f19a13872
parent 16417 9bc16273c2d4
child 24893 b8ef7afe3a6b
permissions -rw-r--r--
added Tools/lin_arith.ML;
     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:  b: Fin(A) ==> inj(b,b) <= surj(b,b)
     7 *)
     8 
     9 header{*Finite Powerset Operator and Finite Function Space*}
    10 
    11 theory Finite imports Inductive Epsilon Nat begin
    12 
    13 (*The natural numbers as a datatype*)
    14 rep_datatype
    15   elimination    natE
    16   induction	 nat_induct
    17   case_eqns	 nat_case_0 nat_case_succ
    18   recursor_eqns  recursor_0 recursor_succ
    19 
    20 
    21 consts
    22   Fin       :: "i=>i"
    23   FiniteFun :: "[i,i]=>i"         ("(_ -||>/ _)" [61, 60] 60)
    24 
    25 inductive
    26   domains   "Fin(A)" <= "Pow(A)"
    27   intros
    28     emptyI:  "0 : Fin(A)"
    29     consI:   "[| a: A;  b: Fin(A) |] ==> cons(a,b) : Fin(A)"
    30   type_intros  empty_subsetI cons_subsetI PowI
    31   type_elims   PowD [THEN revcut_rl]
    32 
    33 inductive
    34   domains   "FiniteFun(A,B)" <= "Fin(A*B)"
    35   intros
    36     emptyI:  "0 : A -||> B"
    37     consI:   "[| a: A;  b: B;  h: A -||> B;  a ~: domain(h) |]
    38               ==> cons(<a,b>,h) : A -||> B"
    39   type_intros Fin.intros
    40 
    41 
    42 subsection {* Finite Powerset Operator *}
    43 
    44 lemma Fin_mono: "A<=B ==> Fin(A) <= Fin(B)"
    45 apply (unfold Fin.defs)
    46 apply (rule lfp_mono)
    47 apply (rule Fin.bnd_mono)+
    48 apply blast
    49 done
    50 
    51 (* A : Fin(B) ==> A <= B *)
    52 lemmas FinD = Fin.dom_subset [THEN subsetD, THEN PowD, standard]
    53 
    54 (** Induction on finite sets **)
    55 
    56 (*Discharging x~:y entails extra work*)
    57 lemma Fin_induct [case_names 0 cons, induct set: Fin]:
    58     "[| b: Fin(A);
    59         P(0);
    60         !!x y. [| x: A;  y: Fin(A);  x~:y;  P(y) |] ==> P(cons(x,y))
    61      |] ==> P(b)"
    62 apply (erule Fin.induct, simp)
    63 apply (case_tac "a:b")
    64  apply (erule cons_absorb [THEN ssubst], assumption) (*backtracking!*)
    65 apply simp
    66 done
    67 
    68 
    69 (** Simplification for Fin **)
    70 declare Fin.intros [simp]
    71 
    72 lemma Fin_0: "Fin(0) = {0}"
    73 by (blast intro: Fin.emptyI dest: FinD)
    74 
    75 (*The union of two finite sets is finite.*)
    76 lemma Fin_UnI [simp]: "[| b: Fin(A);  c: Fin(A) |] ==> b Un c : Fin(A)"
    77 apply (erule Fin_induct)
    78 apply (simp_all add: Un_cons)
    79 done
    80 
    81 
    82 (*The union of a set of finite sets is finite.*)
    83 lemma Fin_UnionI: "C : Fin(Fin(A)) ==> Union(C) : Fin(A)"
    84 by (erule Fin_induct, simp_all)
    85 
    86 (*Every subset of a finite set is finite.*)
    87 lemma Fin_subset_lemma [rule_format]: "b: Fin(A) ==> \<forall>z. z<=b --> z: Fin(A)"
    88 apply (erule Fin_induct)
    89 apply (simp add: subset_empty_iff)
    90 apply (simp add: subset_cons_iff distrib_simps, safe)
    91 apply (erule_tac b = z in cons_Diff [THEN subst], simp)
    92 done
    93 
    94 lemma Fin_subset: "[| c<=b;  b: Fin(A) |] ==> c: Fin(A)"
    95 by (blast intro: Fin_subset_lemma)
    96 
    97 lemma Fin_IntI1 [intro,simp]: "b: Fin(A) ==> b Int c : Fin(A)"
    98 by (blast intro: Fin_subset)
    99 
   100 lemma Fin_IntI2 [intro,simp]: "c: Fin(A) ==> b Int c : Fin(A)"
   101 by (blast intro: Fin_subset)
   102 
   103 lemma Fin_0_induct_lemma [rule_format]:
   104     "[| c: Fin(A);  b: Fin(A); P(b);
   105         !!x y. [| x: A;  y: Fin(A);  x:y;  P(y) |] ==> P(y-{x})
   106      |] ==> c<=b --> P(b-c)"
   107 apply (erule Fin_induct, simp)
   108 apply (subst Diff_cons)
   109 apply (simp add: cons_subset_iff Diff_subset [THEN Fin_subset])
   110 done
   111 
   112 lemma Fin_0_induct:
   113     "[| b: Fin(A);
   114         P(b);
   115         !!x y. [| x: A;  y: Fin(A);  x:y;  P(y) |] ==> P(y-{x})
   116      |] ==> P(0)"
   117 apply (rule Diff_cancel [THEN subst])
   118 apply (blast intro: Fin_0_induct_lemma) 
   119 done
   120 
   121 (*Functions from a finite ordinal*)
   122 lemma nat_fun_subset_Fin: "n: nat ==> n->A <= Fin(nat*A)"
   123 apply (induct_tac "n")
   124 apply (simp add: subset_iff)
   125 apply (simp add: succ_def mem_not_refl [THEN cons_fun_eq])
   126 apply (fast intro!: Fin.consI)
   127 done
   128 
   129 
   130 subsection{*Finite Function Space*}
   131 
   132 lemma FiniteFun_mono:
   133     "[| A<=C;  B<=D |] ==> A -||> B  <=  C -||> D"
   134 apply (unfold FiniteFun.defs)
   135 apply (rule lfp_mono)
   136 apply (rule FiniteFun.bnd_mono)+
   137 apply (intro Fin_mono Sigma_mono basic_monos, assumption+)
   138 done
   139 
   140 lemma FiniteFun_mono1: "A<=B ==> A -||> A  <=  B -||> B"
   141 by (blast dest: FiniteFun_mono)
   142 
   143 lemma FiniteFun_is_fun: "h: A -||>B ==> h: domain(h) -> B"
   144 apply (erule FiniteFun.induct, simp)
   145 apply (simp add: fun_extend3)
   146 done
   147 
   148 lemma FiniteFun_domain_Fin: "h: A -||>B ==> domain(h) : Fin(A)"
   149 by (erule FiniteFun.induct, simp, simp)
   150 
   151 lemmas FiniteFun_apply_type = FiniteFun_is_fun [THEN apply_type, standard]
   152 
   153 (*Every subset of a finite function is a finite function.*)
   154 lemma FiniteFun_subset_lemma [rule_format]:
   155      "b: A-||>B ==> ALL z. z<=b --> z: A-||>B"
   156 apply (erule FiniteFun.induct)
   157 apply (simp add: subset_empty_iff FiniteFun.intros)
   158 apply (simp add: subset_cons_iff distrib_simps, safe)
   159 apply (erule_tac b = z in cons_Diff [THEN subst])
   160 apply (drule spec [THEN mp], assumption)
   161 apply (fast intro!: FiniteFun.intros)
   162 done
   163 
   164 lemma FiniteFun_subset: "[| c<=b;  b: A-||>B |] ==> c: A-||>B"
   165 by (blast intro: FiniteFun_subset_lemma)
   166 
   167 (** Some further results by Sidi O. Ehmety **)
   168 
   169 lemma fun_FiniteFunI [rule_format]: "A:Fin(X) ==> ALL f. f:A->B --> f:A-||>B"
   170 apply (erule Fin.induct)
   171  apply (simp add: FiniteFun.intros, clarify)
   172 apply (case_tac "a:b")
   173  apply (simp add: cons_absorb)
   174 apply (subgoal_tac "restrict (f,b) : b -||> B")
   175  prefer 2 apply (blast intro: restrict_type2)
   176 apply (subst fun_cons_restrict_eq, assumption)
   177 apply (simp add: restrict_def lam_def)
   178 apply (blast intro: apply_funtype FiniteFun.intros 
   179                     FiniteFun_mono [THEN [2] rev_subsetD])
   180 done
   181 
   182 lemma lam_FiniteFun: "A: Fin(X) ==> (lam x:A. b(x)) : A -||> {b(x). x:A}"
   183 by (blast intro: fun_FiniteFunI lam_funtype)
   184 
   185 lemma FiniteFun_Collect_iff:
   186      "f : FiniteFun(A, {y:B. P(y)})
   187       <-> f : FiniteFun(A,B) & (ALL x:domain(f). P(f`x))"
   188 apply auto
   189 apply (blast intro: FiniteFun_mono [THEN [2] rev_subsetD])
   190 apply (blast dest: Pair_mem_PiD FiniteFun_is_fun)
   191 apply (rule_tac A1="domain(f)" in 
   192        subset_refl [THEN [2] FiniteFun_mono, THEN subsetD])
   193  apply (fast dest: FiniteFun_domain_Fin Fin.dom_subset [THEN subsetD])
   194 apply (rule fun_FiniteFunI)
   195 apply (erule FiniteFun_domain_Fin)
   196 apply (rule_tac B = "range (f) " in fun_weaken_type)
   197  apply (blast dest: FiniteFun_is_fun range_of_fun range_type apply_equality)+
   198 done
   199 
   200 
   201 subsection{*The Contents of a Singleton Set*}
   202 
   203 constdefs
   204   contents :: "i=>i"
   205    "contents(X) == THE x. X = {x}"
   206 
   207 lemma contents_eq [simp]: "contents ({x}) = x"
   208 by (simp add: contents_def)
   209 
   210 
   211 ML
   212 {*
   213 val Fin_intros = thms "Fin.intros";
   214 
   215 val Fin_mono = thm "Fin_mono";
   216 val FinD = thm "FinD";
   217 val Fin_induct = thm "Fin_induct";
   218 val Fin_UnI = thm "Fin_UnI";
   219 val Fin_UnionI = thm "Fin_UnionI";
   220 val Fin_subset = thm "Fin_subset";
   221 val Fin_IntI1 = thm "Fin_IntI1";
   222 val Fin_IntI2 = thm "Fin_IntI2";
   223 val Fin_0_induct = thm "Fin_0_induct";
   224 val nat_fun_subset_Fin = thm "nat_fun_subset_Fin";
   225 val FiniteFun_mono = thm "FiniteFun_mono";
   226 val FiniteFun_mono1 = thm "FiniteFun_mono1";
   227 val FiniteFun_is_fun = thm "FiniteFun_is_fun";
   228 val FiniteFun_domain_Fin = thm "FiniteFun_domain_Fin";
   229 val FiniteFun_apply_type = thm "FiniteFun_apply_type";
   230 val FiniteFun_subset = thm "FiniteFun_subset";
   231 val fun_FiniteFunI = thm "fun_FiniteFunI";
   232 val lam_FiniteFun = thm "lam_FiniteFun";
   233 val FiniteFun_Collect_iff = thm "FiniteFun_Collect_iff";
   234 *}
   235 
   236 end