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*)
  elimination    natE
  induction      nat_induct
  case_eqns      nat_case_0 nat_case_succ
  recursor_eqns  recursor_0 recursor_succ

  Fin       :: "ii"
  FiniteFun :: "[i,i]i"         ((_ -||>/ _) [61, 60] 60)

  domains   "Fin(A)"  "Pow(A)"
    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]

  domains   "FiniteFun(A,B)"  "Fin(A*B)"
    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

(* @{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);
        x y. x  A;  y  Fin(A);  xy;  P(y)  P(cons(x,y))
apply (erule Fin.induct, simp)
apply (case_tac "a ∈ b")
 apply (erule cons_absorb [THEN ssubst], assumption) (*backtracking!*)
apply simp

(** 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)

(*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)

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])

lemma Fin_0_induct:
    "b  Fin(A);
        x y. x  A;  y  Fin(A);  x  y;  P(y)  P(y-{x})
apply (rule Diff_cancel [THEN subst])
apply (blast intro: Fin_0_induct_lemma)

(*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)

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+)

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)

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)

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])

lemma lam_FiniteFun: "A  Fin(X)  (λxA. 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)  (xdomain(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)+

subsection‹The Contents of a Singleton Set›

  contents :: "ii"  where
   "contents(X)  THE x. X = {x}"

lemma contents_eq [simp]: "contents ({x}) = x"
by (simp add: contents_def)