src/ZF/Finite.thy
 author paulson Tue Jul 02 13:28:08 2002 +0200 (2002-07-02) changeset 13269 3ba9be497c33 parent 13203 fac77a839aa2 child 13328 703de709a64b permissions -rw-r--r--
Tidying and introduction of various new theorems
```     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 Finite powerset operator; finite function space
```
```     7
```
```     8 prove X:Fin(A) ==> |X| < nat
```
```     9
```
```    10 prove:  b: Fin(A) ==> inj(b,b) <= surj(b,b)
```
```    11 *)
```
```    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:
```
```    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 (*fixed up for induct method*)
```
```    71 lemmas Fin_induct = Fin_induct [case_names 0 cons, induct set: Fin]
```
```    72
```
```    73
```
```    74 (** Simplification for Fin **)
```
```    75 declare Fin.intros [simp]
```
```    76
```
```    77 lemma Fin_0: "Fin(0) = {0}"
```
```    78 by (blast intro: Fin.emptyI dest: FinD)
```
```    79
```
```    80 (*The union of two finite sets is finite.*)
```
```    81 lemma Fin_UnI [simp]: "[| b: Fin(A);  c: Fin(A) |] ==> b Un c : Fin(A)"
```
```    82 apply (erule Fin_induct)
```
```    83 apply (simp_all add: Un_cons)
```
```    84 done
```
```    85
```
```    86
```
```    87 (*The union of a set of finite sets is finite.*)
```
```    88 lemma Fin_UnionI: "C : Fin(Fin(A)) ==> Union(C) : Fin(A)"
```
```    89 by (erule Fin_induct, simp_all)
```
```    90
```
```    91 (*Every subset of a finite set is finite.*)
```
```    92 lemma Fin_subset_lemma [rule_format]: "b: Fin(A) ==> \<forall>z. z<=b --> z: Fin(A)"
```
```    93 apply (erule Fin_induct)
```
```    94 apply (simp add: subset_empty_iff)
```
```    95 apply (simp add: subset_cons_iff distrib_simps, safe)
```
```    96 apply (erule_tac b = "z" in cons_Diff [THEN subst], simp)
```
```    97 done
```
```    98
```
```    99 lemma Fin_subset: "[| c<=b;  b: Fin(A) |] ==> c: Fin(A)"
```
```   100 by (blast intro: Fin_subset_lemma)
```
```   101
```
```   102 lemma Fin_IntI1 [intro,simp]: "b: Fin(A) ==> b Int c : Fin(A)"
```
```   103 by (blast intro: Fin_subset)
```
```   104
```
```   105 lemma Fin_IntI2 [intro,simp]: "c: Fin(A) ==> b Int c : Fin(A)"
```
```   106 by (blast intro: Fin_subset)
```
```   107
```
```   108 lemma Fin_0_induct_lemma [rule_format]:
```
```   109     "[| c: Fin(A);  b: Fin(A); P(b);
```
```   110         !!x y. [| x: A;  y: Fin(A);  x:y;  P(y) |] ==> P(y-{x})
```
```   111      |] ==> c<=b --> P(b-c)"
```
```   112 apply (erule Fin_induct, simp)
```
```   113 apply (subst Diff_cons)
```
```   114 apply (simp add: cons_subset_iff Diff_subset [THEN Fin_subset])
```
```   115 done
```
```   116
```
```   117 lemma Fin_0_induct:
```
```   118     "[| b: Fin(A);
```
```   119         P(b);
```
```   120         !!x y. [| x: A;  y: Fin(A);  x:y;  P(y) |] ==> P(y-{x})
```
```   121      |] ==> P(0)"
```
```   122 apply (rule Diff_cancel [THEN subst])
```
```   123 apply (blast intro: Fin_0_induct_lemma)
```
```   124 done
```
```   125
```
```   126 (*Functions from a finite ordinal*)
```
```   127 lemma nat_fun_subset_Fin: "n: nat ==> n->A <= Fin(nat*A)"
```
```   128 apply (induct_tac "n")
```
```   129 apply (simp add: subset_iff)
```
```   130 apply (simp add: succ_def mem_not_refl [THEN cons_fun_eq])
```
```   131 apply (fast intro!: Fin.consI)
```
```   132 done
```
```   133
```
```   134
```
```   135 (*** Finite function space ***)
```
```   136
```
```   137 lemma FiniteFun_mono:
```
```   138     "[| A<=C;  B<=D |] ==> A -||> B  <=  C -||> D"
```
```   139 apply (unfold FiniteFun.defs)
```
```   140 apply (rule lfp_mono)
```
```   141 apply (rule FiniteFun.bnd_mono)+
```
```   142 apply (intro Fin_mono Sigma_mono basic_monos, assumption+)
```
```   143 done
```
```   144
```
```   145 lemma FiniteFun_mono1: "A<=B ==> A -||> A  <=  B -||> B"
```
```   146 by (blast dest: FiniteFun_mono)
```
```   147
```
```   148 lemma FiniteFun_is_fun: "h: A -||>B ==> h: domain(h) -> B"
```
```   149 apply (erule FiniteFun.induct, simp)
```
```   150 apply (simp add: fun_extend3)
```
```   151 done
```
```   152
```
```   153 lemma FiniteFun_domain_Fin: "h: A -||>B ==> domain(h) : Fin(A)"
```
```   154 by (erule FiniteFun.induct, simp, simp)
```
```   155
```
```   156 lemmas FiniteFun_apply_type = FiniteFun_is_fun [THEN apply_type, standard]
```
```   157
```
```   158 (*Every subset of a finite function is a finite function.*)
```
```   159 lemma FiniteFun_subset_lemma [rule_format]:
```
```   160      "b: A-||>B ==> ALL z. z<=b --> z: A-||>B"
```
```   161 apply (erule FiniteFun.induct)
```
```   162 apply (simp add: subset_empty_iff FiniteFun.intros)
```
```   163 apply (simp add: subset_cons_iff distrib_simps, safe)
```
```   164 apply (erule_tac b = "z" in cons_Diff [THEN subst])
```
```   165 apply (drule spec [THEN mp], assumption)
```
```   166 apply (fast intro!: FiniteFun.intros)
```
```   167 done
```
```   168
```
```   169 lemma FiniteFun_subset: "[| c<=b;  b: A-||>B |] ==> c: A-||>B"
```
```   170 by (blast intro: FiniteFun_subset_lemma)
```
```   171
```
```   172 (** Some further results by Sidi O. Ehmety **)
```
```   173
```
```   174 lemma fun_FiniteFunI [rule_format]: "A:Fin(X) ==> ALL f. f:A->B --> f:A-||>B"
```
```   175 apply (erule Fin.induct)
```
```   176  apply (simp add: FiniteFun.intros, clarify)
```
```   177 apply (case_tac "a:b")
```
```   178  apply (rotate_tac -1)
```
```   179  apply (simp add: cons_absorb)
```
```   180 apply (subgoal_tac "restrict (f,b) : b -||> B")
```
```   181  prefer 2 apply (blast intro: restrict_type2)
```
```   182 apply (subst fun_cons_restrict_eq, assumption)
```
```   183 apply (simp add: restrict_def lam_def)
```
```   184 apply (blast intro: apply_funtype FiniteFun.intros
```
```   185                     FiniteFun_mono [THEN [2] rev_subsetD])
```
```   186 done
```
```   187
```
```   188 lemma lam_FiniteFun: "A: Fin(X) ==> (lam x:A. b(x)) : A -||> {b(x). x:A}"
```
```   189 by (blast intro: fun_FiniteFunI lam_funtype)
```
```   190
```
```   191 lemma FiniteFun_Collect_iff:
```
```   192      "f : FiniteFun(A, {y:B. P(y)})
```
```   193       <-> f : FiniteFun(A,B) & (ALL x:domain(f). P(f`x))"
```
```   194 apply auto
```
```   195 apply (blast intro: FiniteFun_mono [THEN [2] rev_subsetD])
```
```   196 apply (blast dest: Pair_mem_PiD FiniteFun_is_fun)
```
```   197 apply (rule_tac A1="domain(f)" in
```
```   198        subset_refl [THEN [2] FiniteFun_mono, THEN subsetD])
```
```   199  apply (fast dest: FiniteFun_domain_Fin Fin.dom_subset [THEN subsetD])
```
```   200 apply (rule fun_FiniteFunI)
```
```   201 apply (erule FiniteFun_domain_Fin)
```
```   202 apply (rule_tac B = "range (f) " in fun_weaken_type)
```
```   203  apply (blast dest: FiniteFun_is_fun range_of_fun range_type apply_equality)+
```
```   204 done
```
```   205
```
```   206 ML
```
```   207 {*
```
```   208 val Fin_intros = thms "Fin.intros";
```
```   209
```
```   210 val Fin_mono = thm "Fin_mono";
```
```   211 val FinD = thm "FinD";
```
```   212 val Fin_induct = thm "Fin_induct";
```
```   213 val Fin_UnI = thm "Fin_UnI";
```
```   214 val Fin_UnionI = thm "Fin_UnionI";
```
```   215 val Fin_subset = thm "Fin_subset";
```
```   216 val Fin_IntI1 = thm "Fin_IntI1";
```
```   217 val Fin_IntI2 = thm "Fin_IntI2";
```
```   218 val Fin_0_induct = thm "Fin_0_induct";
```
```   219 val nat_fun_subset_Fin = thm "nat_fun_subset_Fin";
```
```   220 val FiniteFun_mono = thm "FiniteFun_mono";
```
```   221 val FiniteFun_mono1 = thm "FiniteFun_mono1";
```
```   222 val FiniteFun_is_fun = thm "FiniteFun_is_fun";
```
```   223 val FiniteFun_domain_Fin = thm "FiniteFun_domain_Fin";
```
```   224 val FiniteFun_apply_type = thm "FiniteFun_apply_type";
```
```   225 val FiniteFun_subset = thm "FiniteFun_subset";
```
```   226 val fun_FiniteFunI = thm "fun_FiniteFunI";
```
```   227 val lam_FiniteFun = thm "lam_FiniteFun";
```
```   228 val FiniteFun_Collect_iff = thm "FiniteFun_Collect_iff";
```
```   229 *}
```
```   230
```
```   231 end
```