--- a/src/ZF/Finite.thy Fri May 31 12:27:24 2002 +0200
+++ b/src/ZF/Finite.thy Fri May 31 15:06:06 2002 +0200
@@ -3,36 +3,227 @@
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
-Finite powerset operator
+Finite powerset operator; finite function space
+
+prove X:Fin(A) ==> |X| < nat
+
+prove: b: Fin(A) ==> inj(b,b) <= surj(b,b)
*)
-Finite = Inductive + Epsilon + Nat +
+theory Finite = Inductive + Epsilon + Nat:
(*The natural numbers as a datatype*)
-rep_datatype
- elim natE
- induct nat_induct
- case_eqns nat_case_0, nat_case_succ
- recursor_eqns recursor_0, recursor_succ
+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)
+ Fin :: "i=>i"
+ FiniteFun :: "[i,i]=>i" ("(_ -||>/ _)" [61, 60] 60)
inductive
domains "Fin(A)" <= "Pow(A)"
- intrs
- emptyI "0 : Fin(A)"
- consI "[| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A)"
- type_intrs empty_subsetI, cons_subsetI, PowI
- type_elims "[make_elim PowD]"
+ 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)"
- intrs
- emptyI "0 : A -||> B"
- consI "[| a: A; b: B; h: A -||> B; a ~: domain(h)
- |] ==> cons(<a,b>,h) : A -||> B"
- type_intrs "Fin.intrs"
+ 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:
+ "[| 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]
+
+(*The union of two finite sets is finite.*)
+lemma Fin_UnI: "[| b: Fin(A); c: Fin(A) |] ==> b Un c : Fin(A)"
+apply (erule Fin_induct)
+apply (simp_all add: Un_cons)
+done
+
+declare Fin_UnI [simp]
+
+
+(*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
+
+
+(*** 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)"
+apply (erule FiniteFun.induct, simp)
+apply simp
+done
+
+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)
+apply clarify
+apply (case_tac "a:b")
+ apply (rotate_tac -1)
+ 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
+
+ML
+{*
+val Fin_intros = thms "Fin.intros";
+
+val Fin_mono = thm "Fin_mono";
+val FinD = thm "FinD";
+val Fin_induct = thm "Fin_induct";
+val Fin_UnI = thm "Fin_UnI";
+val Fin_UnionI = thm "Fin_UnionI";
+val Fin_subset = thm "Fin_subset";
+val Fin_IntI1 = thm "Fin_IntI1";
+val Fin_IntI2 = thm "Fin_IntI2";
+val Fin_0_induct = thm "Fin_0_induct";
+val nat_fun_subset_Fin = thm "nat_fun_subset_Fin";
+val FiniteFun_mono = thm "FiniteFun_mono";
+val FiniteFun_mono1 = thm "FiniteFun_mono1";
+val FiniteFun_is_fun = thm "FiniteFun_is_fun";
+val FiniteFun_domain_Fin = thm "FiniteFun_domain_Fin";
+val FiniteFun_apply_type = thm "FiniteFun_apply_type";
+val FiniteFun_subset = thm "FiniteFun_subset";
+val fun_FiniteFunI = thm "fun_FiniteFunI";
+val lam_FiniteFun = thm "lam_FiniteFun";
+val FiniteFun_Collect_iff = thm "FiniteFun_Collect_iff";
+*}
+
end