(* Title: ZF/Finite.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
prove: 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_succ
consts
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 [THEN revcut_rl]
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.intros
subsection {* 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 blast
done
(* A : Fin(B) ==> A <= B *)
lemmas FinD = Fin.dom_subset [THEN subsetD, THEN PowD, standard]
(** Induction on finite sets **)
(*Discharging 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 simp
done
(** 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 Un 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) ==> \<forall>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)
done
lemma 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 Int c : Fin(A)"
by (blast intro: Fin_subset)
lemma Fin_IntI2 [intro,simp]: "c: Fin(A) ==> b Int 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])
done
lemma 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)
done
subsection{*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+)
done
lemma 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)
done
lemma 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, standard]
(*Every subset of a finite function is a finite function.*)
lemma FiniteFun_subset_lemma [rule_format]:
"b: A-||>B ==> ALL 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)
done
lemma 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) ==> ALL 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])
done
lemma lam_FiniteFun: "A: Fin(X) ==> (lam x: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) & (ALL x:domain(f). P(f`x))"
apply auto
apply (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)+
done
subsection{*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