clarified font_domain: strict excludes e.g. space character;
(* Title: ZF/Epsilon.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section\<open>Epsilon Induction and Recursion\<close>
theory Epsilon imports Nat begin
definition
eclose :: "i=>i" where
"eclose(A) == \<Union>n\<in>nat. nat_rec(n, A, %m r. \<Union>(r))"
definition
transrec :: "[i, [i,i]=>i] =>i" where
"transrec(a,H) == wfrec(Memrel(eclose({a})), a, H)"
definition
rank :: "i=>i" where
"rank(a) == transrec(a, %x f. \<Union>y\<in>x. succ(f`y))"
definition
transrec2 :: "[i, i, [i,i]=>i] =>i" where
"transrec2(k, a, b) ==
transrec(k,
%i r. if(i=0, a,
if(\<exists>j. i=succ(j),
b(THE j. i=succ(j), r`(THE j. i=succ(j))),
\<Union>j<i. r`j)))"
definition
recursor :: "[i, [i,i]=>i, i]=>i" where
"recursor(a,b,k) == transrec(k, %n f. nat_case(a, %m. b(m, f`m), n))"
definition
rec :: "[i, i, [i,i]=>i]=>i" where
"rec(k,a,b) == recursor(a,b,k)"
subsection\<open>Basic Closure Properties\<close>
lemma arg_subset_eclose: "A \<subseteq> eclose(A)"
apply (unfold eclose_def)
apply (rule nat_rec_0 [THEN equalityD2, THEN subset_trans])
apply (rule nat_0I [THEN UN_upper])
done
lemmas arg_into_eclose = arg_subset_eclose [THEN subsetD]
lemma Transset_eclose: "Transset(eclose(A))"
apply (unfold eclose_def Transset_def)
apply (rule subsetI [THEN ballI])
apply (erule UN_E)
apply (rule nat_succI [THEN UN_I], assumption)
apply (erule nat_rec_succ [THEN ssubst])
apply (erule UnionI, assumption)
done
(* @{term"x \<in> eclose(A) ==> x \<subseteq> eclose(A)"} *)
lemmas eclose_subset =
Transset_eclose [unfolded Transset_def, THEN bspec]
(* @{term"[| A \<in> eclose(B); c \<in> A |] ==> c \<in> eclose(B)"} *)
lemmas ecloseD = eclose_subset [THEN subsetD]
lemmas arg_in_eclose_sing = arg_subset_eclose [THEN singleton_subsetD]
lemmas arg_into_eclose_sing = arg_in_eclose_sing [THEN ecloseD]
(* This is epsilon-induction for eclose(A); see also eclose_induct_down...
[| a \<in> eclose(A); !!x. [| x \<in> eclose(A); \<forall>y\<in>x. P(y) |] ==> P(x)
|] ==> P(a)
*)
lemmas eclose_induct =
Transset_induct [OF _ Transset_eclose, induct set: eclose]
(*Epsilon induction*)
lemma eps_induct:
"[| !!x. \<forall>y\<in>x. P(y) ==> P(x) |] ==> P(a)"
by (rule arg_in_eclose_sing [THEN eclose_induct], blast)
subsection\<open>Leastness of @{term eclose}\<close>
(** eclose(A) is the least transitive set including A as a subset. **)
lemma eclose_least_lemma:
"[| Transset(X); A<=X; n \<in> nat |] ==> nat_rec(n, A, %m r. \<Union>(r)) \<subseteq> X"
apply (unfold Transset_def)
apply (erule nat_induct)
apply (simp add: nat_rec_0)
apply (simp add: nat_rec_succ, blast)
done
lemma eclose_least:
"[| Transset(X); A<=X |] ==> eclose(A) \<subseteq> X"
apply (unfold eclose_def)
apply (rule eclose_least_lemma [THEN UN_least], assumption+)
done
(*COMPLETELY DIFFERENT induction principle from eclose_induct!!*)
lemma eclose_induct_down [consumes 1]:
"[| a \<in> eclose(b);
!!y. [| y \<in> b |] ==> P(y);
!!y z. [| y \<in> eclose(b); P(y); z \<in> y |] ==> P(z)
|] ==> P(a)"
apply (rule eclose_least [THEN subsetD, THEN CollectD2, of "eclose(b)"])
prefer 3 apply assumption
apply (unfold Transset_def)
apply (blast intro: ecloseD)
apply (blast intro: arg_subset_eclose [THEN subsetD])
done
lemma Transset_eclose_eq_arg: "Transset(X) ==> eclose(X) = X"
apply (erule equalityI [OF eclose_least arg_subset_eclose])
apply (rule subset_refl)
done
text\<open>A transitive set either is empty or contains the empty set.\<close>
lemma Transset_0_lemma [rule_format]: "Transset(A) ==> x\<in>A \<longrightarrow> 0\<in>A"
apply (simp add: Transset_def)
apply (rule_tac a=x in eps_induct, clarify)
apply (drule bspec, assumption)
apply (case_tac "x=0", auto)
done
lemma Transset_0_disj: "Transset(A) ==> A=0 | 0\<in>A"
by (blast dest: Transset_0_lemma)
subsection\<open>Epsilon Recursion\<close>
(*Unused...*)
lemma mem_eclose_trans: "[| A \<in> eclose(B); B \<in> eclose(C) |] ==> A \<in> eclose(C)"
by (rule eclose_least [OF Transset_eclose eclose_subset, THEN subsetD],
assumption+)
(*Variant of the previous lemma in a useable form for the sequel*)
lemma mem_eclose_sing_trans:
"[| A \<in> eclose({B}); B \<in> eclose({C}) |] ==> A \<in> eclose({C})"
by (rule eclose_least [OF Transset_eclose singleton_subsetI, THEN subsetD],
assumption+)
lemma under_Memrel: "[| Transset(i); j \<in> i |] ==> Memrel(i)-``{j} = j"
by (unfold Transset_def, blast)
lemma lt_Memrel: "j < i ==> Memrel(i) -`` {j} = j"
by (simp add: lt_def Ord_def under_Memrel)
(* @{term"j \<in> eclose(A) ==> Memrel(eclose(A)) -`` j = j"} *)
lemmas under_Memrel_eclose = Transset_eclose [THEN under_Memrel]
lemmas wfrec_ssubst = wf_Memrel [THEN wfrec, THEN ssubst]
lemma wfrec_eclose_eq:
"[| k \<in> eclose({j}); j \<in> eclose({i}) |] ==>
wfrec(Memrel(eclose({i})), k, H) = wfrec(Memrel(eclose({j})), k, H)"
apply (erule eclose_induct)
apply (rule wfrec_ssubst)
apply (rule wfrec_ssubst)
apply (simp add: under_Memrel_eclose mem_eclose_sing_trans [of _ j i])
done
lemma wfrec_eclose_eq2:
"k \<in> i ==> wfrec(Memrel(eclose({i})),k,H) = wfrec(Memrel(eclose({k})),k,H)"
apply (rule arg_in_eclose_sing [THEN wfrec_eclose_eq])
apply (erule arg_into_eclose_sing)
done
lemma transrec: "transrec(a,H) = H(a, \<lambda>x\<in>a. transrec(x,H))"
apply (unfold transrec_def)
apply (rule wfrec_ssubst)
apply (simp add: wfrec_eclose_eq2 arg_in_eclose_sing under_Memrel_eclose)
done
(*Avoids explosions in proofs; resolve it with a meta-level definition.*)
lemma def_transrec:
"[| !!x. f(x)==transrec(x,H) |] ==> f(a) = H(a, \<lambda>x\<in>a. f(x))"
apply simp
apply (rule transrec)
done
lemma transrec_type:
"[| !!x u. [| x \<in> eclose({a}); u \<in> Pi(x,B) |] ==> H(x,u) \<in> B(x) |]
==> transrec(a,H) \<in> B(a)"
apply (rule_tac i = a in arg_in_eclose_sing [THEN eclose_induct])
apply (subst transrec)
apply (simp add: lam_type)
done
lemma eclose_sing_Ord: "Ord(i) ==> eclose({i}) \<subseteq> succ(i)"
apply (erule Ord_is_Transset [THEN Transset_succ, THEN eclose_least])
apply (rule succI1 [THEN singleton_subsetI])
done
lemma succ_subset_eclose_sing: "succ(i) \<subseteq> eclose({i})"
apply (insert arg_subset_eclose [of "{i}"], simp)
apply (frule eclose_subset, blast)
done
lemma eclose_sing_Ord_eq: "Ord(i) ==> eclose({i}) = succ(i)"
apply (rule equalityI)
apply (erule eclose_sing_Ord)
apply (rule succ_subset_eclose_sing)
done
lemma Ord_transrec_type:
assumes jini: "j \<in> i"
and ordi: "Ord(i)"
and minor: " !!x u. [| x \<in> i; u \<in> Pi(x,B) |] ==> H(x,u) \<in> B(x)"
shows "transrec(j,H) \<in> B(j)"
apply (rule transrec_type)
apply (insert jini ordi)
apply (blast intro!: minor
intro: Ord_trans
dest: Ord_in_Ord [THEN eclose_sing_Ord, THEN subsetD])
done
subsection\<open>Rank\<close>
(*NOT SUITABLE FOR REWRITING -- RECURSIVE!*)
lemma rank: "rank(a) = (\<Union>y\<in>a. succ(rank(y)))"
by (subst rank_def [THEN def_transrec], simp)
lemma Ord_rank [simp]: "Ord(rank(a))"
apply (rule_tac a=a in eps_induct)
apply (subst rank)
apply (rule Ord_succ [THEN Ord_UN])
apply (erule bspec, assumption)
done
lemma rank_of_Ord: "Ord(i) ==> rank(i) = i"
apply (erule trans_induct)
apply (subst rank)
apply (simp add: Ord_equality)
done
lemma rank_lt: "a \<in> b ==> rank(a) < rank(b)"
apply (rule_tac a1 = b in rank [THEN ssubst])
apply (erule UN_I [THEN ltI])
apply (rule_tac [2] Ord_UN, auto)
done
lemma eclose_rank_lt: "a \<in> eclose(b) ==> rank(a) < rank(b)"
apply (erule eclose_induct_down)
apply (erule rank_lt)
apply (erule rank_lt [THEN lt_trans], assumption)
done
lemma rank_mono: "a<=b ==> rank(a) \<le> rank(b)"
apply (rule subset_imp_le)
apply (auto simp add: rank [of a] rank [of b])
done
lemma rank_Pow: "rank(Pow(a)) = succ(rank(a))"
apply (rule rank [THEN trans])
apply (rule le_anti_sym)
apply (rule_tac [2] UN_upper_le)
apply (rule UN_least_le)
apply (auto intro: rank_mono simp add: Ord_UN)
done
lemma rank_0 [simp]: "rank(0) = 0"
by (rule rank [THEN trans], blast)
lemma rank_succ [simp]: "rank(succ(x)) = succ(rank(x))"
apply (rule rank [THEN trans])
apply (rule equalityI [OF UN_least succI1 [THEN UN_upper]])
apply (erule succE, blast)
apply (erule rank_lt [THEN leI, THEN succ_leI, THEN le_imp_subset])
done
lemma rank_Union: "rank(\<Union>(A)) = (\<Union>x\<in>A. rank(x))"
apply (rule equalityI)
apply (rule_tac [2] rank_mono [THEN le_imp_subset, THEN UN_least])
apply (erule_tac [2] Union_upper)
apply (subst rank)
apply (rule UN_least)
apply (erule UnionE)
apply (rule subset_trans)
apply (erule_tac [2] RepFunI [THEN Union_upper])
apply (erule rank_lt [THEN succ_leI, THEN le_imp_subset])
done
lemma rank_eclose: "rank(eclose(a)) = rank(a)"
apply (rule le_anti_sym)
apply (rule_tac [2] arg_subset_eclose [THEN rank_mono])
apply (rule_tac a1 = "eclose (a) " in rank [THEN ssubst])
apply (rule Ord_rank [THEN UN_least_le])
apply (erule eclose_rank_lt [THEN succ_leI])
done
lemma rank_pair1: "rank(a) < rank(<a,b>)"
apply (unfold Pair_def)
apply (rule consI1 [THEN rank_lt, THEN lt_trans])
apply (rule consI1 [THEN consI2, THEN rank_lt])
done
lemma rank_pair2: "rank(b) < rank(<a,b>)"
apply (unfold Pair_def)
apply (rule consI1 [THEN consI2, THEN rank_lt, THEN lt_trans])
apply (rule consI1 [THEN consI2, THEN rank_lt])
done
(*Not clear how to remove the P(a) condition, since the "then" part
must refer to "a"*)
lemma the_equality_if:
"P(a) ==> (THE x. P(x)) = (if (\<exists>!x. P(x)) then a else 0)"
by (simp add: the_0 the_equality2)
(*The first premise not only fixs i but ensures @{term"f\<noteq>0"}.
The second premise is now essential. Consider otherwise the relation
r = {<0,0>,<0,1>,<0,2>,...}. Then f`0 = \<Union>(f``{0}) = \<Union>(nat) = nat,
whose rank equals that of r.*)
lemma rank_apply: "[|i \<in> domain(f); function(f)|] ==> rank(f`i) < rank(f)"
apply clarify
apply (simp add: function_apply_equality)
apply (blast intro: lt_trans rank_lt rank_pair2)
done
subsection\<open>Corollaries of Leastness\<close>
lemma mem_eclose_subset: "A \<in> B ==> eclose(A)<=eclose(B)"
apply (rule Transset_eclose [THEN eclose_least])
apply (erule arg_into_eclose [THEN eclose_subset])
done
lemma eclose_mono: "A<=B ==> eclose(A) \<subseteq> eclose(B)"
apply (rule Transset_eclose [THEN eclose_least])
apply (erule subset_trans)
apply (rule arg_subset_eclose)
done
(** Idempotence of eclose **)
lemma eclose_idem: "eclose(eclose(A)) = eclose(A)"
apply (rule equalityI)
apply (rule eclose_least [OF Transset_eclose subset_refl])
apply (rule arg_subset_eclose)
done
(** Transfinite recursion for definitions based on the
three cases of ordinals **)
lemma transrec2_0 [simp]: "transrec2(0,a,b) = a"
by (rule transrec2_def [THEN def_transrec, THEN trans], simp)
lemma transrec2_succ [simp]: "transrec2(succ(i),a,b) = b(i, transrec2(i,a,b))"
apply (rule transrec2_def [THEN def_transrec, THEN trans])
apply (simp add: the_equality if_P)
done
lemma transrec2_Limit:
"Limit(i) ==> transrec2(i,a,b) = (\<Union>j<i. transrec2(j,a,b))"
apply (rule transrec2_def [THEN def_transrec, THEN trans])
apply (auto simp add: OUnion_def)
done
lemma def_transrec2:
"(!!x. f(x)==transrec2(x,a,b))
==> f(0) = a &
f(succ(i)) = b(i, f(i)) &
(Limit(K) \<longrightarrow> f(K) = (\<Union>j<K. f(j)))"
by (simp add: transrec2_Limit)
(** recursor -- better than nat_rec; the succ case has no type requirement! **)
(*NOT suitable for rewriting*)
lemmas recursor_lemma = recursor_def [THEN def_transrec, THEN trans]
lemma recursor_0: "recursor(a,b,0) = a"
by (rule nat_case_0 [THEN recursor_lemma])
lemma recursor_succ: "recursor(a,b,succ(m)) = b(m, recursor(a,b,m))"
by (rule recursor_lemma, simp)
(** rec: old version for compatibility **)
lemma rec_0 [simp]: "rec(0,a,b) = a"
apply (unfold rec_def)
apply (rule recursor_0)
done
lemma rec_succ [simp]: "rec(succ(m),a,b) = b(m, rec(m,a,b))"
apply (unfold rec_def)
apply (rule recursor_succ)
done
lemma rec_type:
"[| n \<in> nat;
a \<in> C(0);
!!m z. [| m \<in> nat; z \<in> C(m) |] ==> b(m,z): C(succ(m)) |]
==> rec(n,a,b) \<in> C(n)"
by (erule nat_induct, auto)
end