# Theory Finite

```(*  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)
*)

section‹Finite Powerset Operator and Finite Function Space›

theory Finite imports Inductive Epsilon Nat 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 [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.intros

subsection ‹Finite Powerset Operator›

lemma Fin_mono: "A<=B ⟹ Fin(A) ⊆ Fin(B)"
unfolding Fin.defs
apply (rule lfp_mono)
apply (rule Fin.bnd_mono)+
apply blast
done

(* @{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 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 ∪ 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)) ⟹ ⋃(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 ∩ 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])
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"
unfolding 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]

(*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)
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) ⟹ ∀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) ⟹ (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 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
```