(* Title: ZF/Finite.thy ID: $Id: Finite.thy,v 1.23 2008/02/11 14:40:21 krauss Exp $ 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) ==> ∀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
lemma Fin_mono:
A ⊆ B ==> Fin(A) ⊆ Fin(B)
lemma FinD:
A ∈ Fin(B) ==> A ⊆ B
lemma Fin_induct:
[| b ∈ Fin(A); P(0);
!!x y. [| x ∈ A; y ∈ Fin(A); x ∉ y; P(y) |] ==> P(cons(x, y)) |]
==> P(b)
lemma Fin_0:
Fin(0) = {0}
lemma Fin_UnI:
[| b ∈ Fin(A); c ∈ Fin(A) |] ==> b ∪ c ∈ Fin(A)
lemma Fin_UnionI:
C ∈ Fin(Fin(A)) ==> \<Union>C ∈ Fin(A)
lemma Fin_subset_lemma:
[| b ∈ Fin(A); z ⊆ b |] ==> z ∈ Fin(A)
lemma Fin_subset:
[| c ⊆ b; b ∈ Fin(A) |] ==> c ∈ Fin(A)
lemma Fin_IntI1:
b ∈ Fin(A) ==> b ∩ c ∈ Fin(A)
lemma Fin_IntI2:
c ∈ Fin(A) ==> b ∩ c ∈ Fin(A)
lemma Fin_0_induct_lemma:
[| 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)
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)
lemma nat_fun_subset_Fin:
n ∈ nat ==> n -> A ⊆ Fin(nat × A)
lemma FiniteFun_mono:
[| A ⊆ C; B ⊆ D |] ==> A -||> B ⊆ C -||> D
lemma FiniteFun_mono1:
A ⊆ B ==> A -||> A ⊆ B -||> B
lemma FiniteFun_is_fun:
h ∈ A -||> B ==> h ∈ domain(h) -> B
lemma FiniteFun_domain_Fin:
h ∈ A -||> B ==> domain(h) ∈ Fin(A)
lemma FiniteFun_apply_type:
[| f ∈ A -||> B; a ∈ domain(f) |] ==> f ` a ∈ B
lemma FiniteFun_subset_lemma:
[| b ∈ A -||> B; z ⊆ b |] ==> z ∈ A -||> B
lemma FiniteFun_subset:
[| c ⊆ b; b ∈ A -||> B |] ==> c ∈ A -||> B
lemma fun_FiniteFunI:
[| A ∈ Fin(X); f ∈ A -> B |] ==> f ∈ A -||> B
lemma lam_FiniteFun:
A ∈ Fin(X) ==> (λx∈A. b(x)) ∈ A -||> {b(x) . x ∈ A}
lemma FiniteFun_Collect_iff:
f ∈ A -||> {y ∈ B . P(y)} <-> f ∈ A -||> B ∧ (∀x∈domain(f). P(f ` x))
lemma contents_eq:
contents({x}) = x