# Theory Finite

Up to index of Isabelle/ZF

theory Finite
imports Inductive_ZF Epsilon
`(*  Title:      ZF/Finite.thy    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory    Copyright   1994  University of Cambridgeprove:  b ∈ Fin(A) ==> inj(b,b) ⊆ surj(b,b)*)header{*Finite Powerset Operator and Finite Function Space*}theory Finite imports Inductive_ZF Epsilon Nat_ZF begin(*The natural numbers as a datatype*)rep_datatype  elimination    natE  induction      nat_induct  case_eqns      nat_case_0 nat_case_succ  recursor_eqns  recursor_0 recursor_succconsts  Fin       :: "i=>i"  FiniteFun :: "[i,i]=>i"         ("(_ -||>/ _)" [61, 60] 60)inductive  domains   "Fin(A)" ⊆ "Pow(A)"  intros    emptyI:  "0 ∈ Fin(A)"    consI:   "[| a ∈ A;  b ∈ Fin(A) |] ==> cons(a,b) ∈ Fin(A)"  type_intros  empty_subsetI cons_subsetI PowI  type_elims   PowD [elim_format]inductive  domains   "FiniteFun(A,B)" ⊆ "Fin(A*B)"  intros    emptyI:  "0 ∈ A -||> B"    consI:   "[| a ∈ A;  b ∈ B;  h ∈ A -||> B;  a ∉ domain(h) |]              ==> cons(<a,b>,h) ∈ A -||> B"  type_intros Fin.introssubsection {* Finite Powerset Operator *}lemma Fin_mono: "A<=B ==> Fin(A) ⊆ Fin(B)"apply (unfold Fin.defs)apply (rule lfp_mono)apply (rule Fin.bnd_mono)+apply blastdone(* @{term"A ∈ Fin(B) ==> A ⊆ B"} *)lemmas FinD = Fin.dom_subset [THEN subsetD, THEN PowD](** Induction on finite sets **)(*Discharging @{term"x∉y"} entails extra work*)lemma Fin_induct [case_names 0 cons, induct set: Fin]:    "[| b ∈ Fin(A);        P(0);        !!x y. [| x ∈ A;  y ∈ Fin(A);  x∉y;  P(y) |] ==> P(cons(x,y))     |] ==> P(b)"apply (erule Fin.induct, simp)apply (case_tac "a ∈ b") apply (erule cons_absorb [THEN ssubst], assumption) (*backtracking!*)apply simpdone(** Simplification for Fin **)declare Fin.intros [simp]lemma Fin_0: "Fin(0) = {0}"by (blast intro: Fin.emptyI dest: FinD)(*The union of two finite sets is finite.*)lemma Fin_UnI [simp]: "[| b ∈ Fin(A);  c ∈ Fin(A) |] ==> b ∪ c ∈ Fin(A)"apply (erule Fin_induct)apply (simp_all add: Un_cons)done(*The union of a set of finite sets is finite.*)lemma Fin_UnionI: "C ∈ Fin(Fin(A)) ==> \<Union>(C) ∈ Fin(A)"by (erule Fin_induct, simp_all)(*Every subset of a finite set is finite.*)lemma Fin_subset_lemma [rule_format]: "b ∈ Fin(A) ==> ∀z. z<=b --> z ∈ Fin(A)"apply (erule Fin_induct)apply (simp add: subset_empty_iff)apply (simp add: subset_cons_iff distrib_simps, safe)apply (erule_tac b = z in cons_Diff [THEN subst], simp)donelemma Fin_subset: "[| c<=b;  b ∈ Fin(A) |] ==> c ∈ Fin(A)"by (blast intro: Fin_subset_lemma)lemma Fin_IntI1 [intro,simp]: "b ∈ Fin(A) ==> b ∩ c ∈ Fin(A)"by (blast intro: Fin_subset)lemma Fin_IntI2 [intro,simp]: "c ∈ Fin(A) ==> b ∩ c ∈ Fin(A)"by (blast intro: Fin_subset)lemma Fin_0_induct_lemma [rule_format]:    "[| c ∈ Fin(A);  b ∈ Fin(A); P(b);        !!x y. [| x ∈ A;  y ∈ Fin(A);  x ∈ y;  P(y) |] ==> P(y-{x})     |] ==> c<=b --> P(b-c)"apply (erule Fin_induct, simp)apply (subst Diff_cons)apply (simp add: cons_subset_iff Diff_subset [THEN Fin_subset])donelemma Fin_0_induct:    "[| b ∈ Fin(A);        P(b);        !!x y. [| x ∈ A;  y ∈ Fin(A);  x ∈ y;  P(y) |] ==> P(y-{x})     |] ==> P(0)"apply (rule Diff_cancel [THEN subst])apply (blast intro: Fin_0_induct_lemma)done(*Functions from a finite ordinal*)lemma nat_fun_subset_Fin: "n ∈ nat ==> n->A ⊆ Fin(nat*A)"apply (induct_tac "n")apply (simp add: subset_iff)apply (simp add: succ_def mem_not_refl [THEN cons_fun_eq])apply (fast intro!: Fin.consI)donesubsection{*Finite Function Space*}lemma FiniteFun_mono:    "[| A<=C;  B<=D |] ==> A -||> B  ⊆  C -||> D"apply (unfold FiniteFun.defs)apply (rule lfp_mono)apply (rule FiniteFun.bnd_mono)+apply (intro Fin_mono Sigma_mono basic_monos, assumption+)donelemma FiniteFun_mono1: "A<=B ==> A -||> A  ⊆  B -||> B"by (blast dest: FiniteFun_mono)lemma FiniteFun_is_fun: "h ∈ A -||>B ==> h ∈ domain(h) -> B"apply (erule FiniteFun.induct, simp)apply (simp add: fun_extend3)donelemma FiniteFun_domain_Fin: "h ∈ A -||>B ==> domain(h) ∈ Fin(A)"by (erule FiniteFun.induct, simp, simp)lemmas FiniteFun_apply_type = FiniteFun_is_fun [THEN apply_type](*Every subset of a finite function is a finite function.*)lemma FiniteFun_subset_lemma [rule_format]:     "b ∈ A-||>B ==> ∀z. z<=b --> z ∈ A-||>B"apply (erule FiniteFun.induct)apply (simp add: subset_empty_iff FiniteFun.intros)apply (simp add: subset_cons_iff distrib_simps, safe)apply (erule_tac b = z in cons_Diff [THEN subst])apply (drule spec [THEN mp], assumption)apply (fast intro!: FiniteFun.intros)donelemma FiniteFun_subset: "[| c<=b;  b ∈ A-||>B |] ==> c ∈ A-||>B"by (blast intro: FiniteFun_subset_lemma)(** Some further results by Sidi O. Ehmety **)lemma fun_FiniteFunI [rule_format]: "A ∈ Fin(X) ==> ∀f. f ∈ A->B --> f ∈ A-||>B"apply (erule Fin.induct) apply (simp add: FiniteFun.intros, clarify)apply (case_tac "a ∈ b") apply (simp add: cons_absorb)apply (subgoal_tac "restrict (f,b) ∈ b -||> B") prefer 2 apply (blast intro: restrict_type2)apply (subst fun_cons_restrict_eq, assumption)apply (simp add: restrict_def lam_def)apply (blast intro: apply_funtype FiniteFun.intros                    FiniteFun_mono [THEN [2] rev_subsetD])donelemma lam_FiniteFun: "A ∈ Fin(X) ==> (λx∈A. b(x)) ∈ A -||> {b(x). x ∈ A}"by (blast intro: fun_FiniteFunI lam_funtype)lemma FiniteFun_Collect_iff:     "f ∈ FiniteFun(A, {y ∈ B. P(y)})      <-> f ∈ FiniteFun(A,B) & (∀x∈domain(f). P(f`x))"apply autoapply (blast intro: FiniteFun_mono [THEN [2] rev_subsetD])apply (blast dest: Pair_mem_PiD FiniteFun_is_fun)apply (rule_tac A1="domain(f)" in       subset_refl [THEN [2] FiniteFun_mono, THEN subsetD]) apply (fast dest: FiniteFun_domain_Fin Fin.dom_subset [THEN subsetD])apply (rule fun_FiniteFunI)apply (erule FiniteFun_domain_Fin)apply (rule_tac B = "range (f) " in fun_weaken_type) apply (blast dest: FiniteFun_is_fun range_of_fun range_type apply_equality)+donesubsection{*The Contents of a Singleton Set*}definition  contents :: "i=>i"  where   "contents(X) == THE x. X = {x}"lemma contents_eq [simp]: "contents ({x}) = x"by (simp add: contents_def)end`