src/ZF/Finite.thy
 author krauss Mon Feb 11 15:40:21 2008 +0100 (2008-02-11) changeset 26056 6a0801279f4c parent 24893 b8ef7afe3a6b child 32960 69916a850301 permissions -rw-r--r--
in a single session (but not merge them).
```     1 (*  Title:      ZF/Finite.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     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*}
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```    10
```
```    11 theory Finite imports Inductive_ZF Epsilon Nat_ZF 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
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```    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 **)
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```    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
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```   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)
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```   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*}
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```   202
```
```   203 definition
```
```   204   contents :: "i=>i"  where
```
```   205    "contents(X) == THE x. X = {x}"
```
```   206
```
```   207 lemma contents_eq [simp]: "contents ({x}) = x"
```
```   208 by (simp add: contents_def)
```
```   209
```
```   210 end
```