(* Title: ZF/quniv
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
For quniv.thy. A small universe for lazy recursive types
*)
open QUniv;
(** Introduction and elimination rules avoid tiresome folding/unfolding **)
goalw QUniv.thy [quniv_def]
"!!X A. X <= univ(eclose(A)) ==> X : quniv(A)";
by (etac PowI 1);
val qunivI = result();
goalw QUniv.thy [quniv_def]
"!!X A. X : quniv(A) ==> X <= univ(eclose(A))";
by (etac PowD 1);
val qunivD = result();
goalw QUniv.thy [quniv_def] "!!A B. A<=B ==> quniv(A) <= quniv(B)";
by (etac (eclose_mono RS univ_mono RS Pow_mono) 1);
val quniv_mono = result();
(*** Closure properties ***)
goalw QUniv.thy [quniv_def] "univ(eclose(A)) <= quniv(A)";
by (rtac (Transset_iff_Pow RS iffD1) 1);
by (rtac (Transset_eclose RS Transset_univ) 1);
val univ_eclose_subset_quniv = result();
goal QUniv.thy "univ(A) <= quniv(A)";
by (rtac (arg_subset_eclose RS univ_mono RS subset_trans) 1);
by (rtac univ_eclose_subset_quniv 1);
val univ_subset_quniv = result();
val univ_into_quniv = standard (univ_subset_quniv RS subsetD);
goalw QUniv.thy [quniv_def] "Pow(univ(A)) <= quniv(A)";
by (rtac (arg_subset_eclose RS univ_mono RS Pow_mono) 1);
val Pow_univ_subset_quniv = result();
val univ_subset_into_quniv = standard
(PowI RS (Pow_univ_subset_quniv RS subsetD));
val zero_in_quniv = standard (zero_in_univ RS univ_into_quniv);
val one_in_quniv = standard (one_in_univ RS univ_into_quniv);
val two_in_quniv = standard (two_in_univ RS univ_into_quniv);
val A_subset_quniv = standard
([A_subset_univ, univ_subset_quniv] MRS subset_trans);
val A_into_quniv = A_subset_quniv RS subsetD;
(*** univ(A) closure for Quine-inspired pairs and injections ***)
(*Quine ordered pairs*)
goalw QUniv.thy [QPair_def]
"!!A a. [| a <= univ(A); b <= univ(A) |] ==> <a;b> <= univ(A)";
by (REPEAT (ares_tac [sum_subset_univ] 1));
val QPair_subset_univ = result();
(** Quine disjoint sum **)
goalw QUniv.thy [QInl_def] "!!A a. a <= univ(A) ==> QInl(a) <= univ(A)";
by (etac (empty_subsetI RS QPair_subset_univ) 1);
val QInl_subset_univ = result();
val naturals_subset_nat =
rewrite_rule [Transset_def] (Ord_nat RS Ord_is_Transset)
RS bspec;
val naturals_subset_univ =
[naturals_subset_nat, nat_subset_univ] MRS subset_trans;
goalw QUniv.thy [QInr_def] "!!A a. a <= univ(A) ==> QInr(a) <= univ(A)";
by (etac (nat_1I RS naturals_subset_univ RS QPair_subset_univ) 1);
val QInr_subset_univ = result();
(*** Closure for Quine-inspired products and sums ***)
(*Quine ordered pairs*)
goalw QUniv.thy [quniv_def,QPair_def]
"!!A a. [| a: quniv(A); b: quniv(A) |] ==> <a;b> : quniv(A)";
by (REPEAT (dtac PowD 1));
by (REPEAT (ares_tac [PowI, sum_subset_univ] 1));
val QPair_in_quniv = result();
goal QUniv.thy "quniv(A) <*> quniv(A) <= quniv(A)";
by (REPEAT (ares_tac [subsetI, QPair_in_quniv] 1
ORELSE eresolve_tac [QSigmaE, ssubst] 1));
val QSigma_quniv = result();
val QSigma_subset_quniv = standard
(QSigma_mono RS (QSigma_quniv RSN (2,subset_trans)));
(*The opposite inclusion*)
goalw QUniv.thy [quniv_def,QPair_def]
"!!A a b. <a;b> : quniv(A) ==> a: quniv(A) & b: quniv(A)";
by (etac ([Transset_eclose RS Transset_univ, PowD] MRS
Transset_includes_summands RS conjE) 1);
by (REPEAT (ares_tac [conjI,PowI] 1));
val quniv_QPair_D = result();
val quniv_QPair_E = standard (quniv_QPair_D RS conjE);
goal QUniv.thy "<a;b> : quniv(A) <-> a: quniv(A) & b: quniv(A)";
by (REPEAT (ares_tac [iffI, QPair_in_quniv, quniv_QPair_D] 1
ORELSE etac conjE 1));
val quniv_QPair_iff = result();
(** Quine disjoint sum **)
goalw QUniv.thy [QInl_def] "!!A a. a: quniv(A) ==> QInl(a) : quniv(A)";
by (REPEAT (ares_tac [zero_in_quniv,QPair_in_quniv] 1));
val QInl_in_quniv = result();
goalw QUniv.thy [QInr_def] "!!A b. b: quniv(A) ==> QInr(b) : quniv(A)";
by (REPEAT (ares_tac [one_in_quniv, QPair_in_quniv] 1));
val QInr_in_quniv = result();
goal QUniv.thy "quniv(C) <+> quniv(C) <= quniv(C)";
by (REPEAT (ares_tac [subsetI, QInl_in_quniv, QInr_in_quniv] 1
ORELSE eresolve_tac [qsumE, ssubst] 1));
val qsum_quniv = result();
val qsum_subset_quniv = standard
(qsum_mono RS (qsum_quniv RSN (2,subset_trans)));
(*** The natural numbers ***)
val nat_subset_quniv = standard
([nat_subset_univ, univ_subset_quniv] MRS subset_trans);
(* n:nat ==> n:quniv(A) *)
val nat_into_quniv = standard (nat_subset_quniv RS subsetD);
val bool_subset_quniv = standard
([bool_subset_univ, univ_subset_quniv] MRS subset_trans);
val bool_into_quniv = standard (bool_subset_quniv RS subsetD);
(**** Properties of Vfrom analogous to the "take-lemma" ****)
(*** Intersecting a*b with Vfrom... ***)
(*This version says a, b exist one level down, in the smaller set Vfrom(X,i)*)
goal Univ.thy
"!!X. [| {a,b} : Vfrom(X,succ(i)); Transset(X) |] ==> \
\ a: Vfrom(X,i) & b: Vfrom(X,i)";
by (dtac (Transset_Vfrom_succ RS equalityD1 RS subsetD RS PowD) 1);
by (assume_tac 1);
by (fast_tac ZF_cs 1);
val doubleton_in_Vfrom_D = result();
(*This weaker version says a, b exist at the same level*)
val Vfrom_doubleton_D = standard (Transset_Vfrom RS Transset_doubleton_D);
(** Using only the weaker theorem would prove <a,b> : Vfrom(X,i)
implies a, b : Vfrom(X,i), which is useless for induction.
Using only the stronger theorem would prove <a,b> : Vfrom(X,succ(succ(i)))
implies a, b : Vfrom(X,i), leaving the succ(i) case untreated.
The combination gives a reduction by precisely one level, which is
most convenient for proofs.
**)
goalw Univ.thy [Pair_def]
"!!X. [| <a,b> : Vfrom(X,succ(i)); Transset(X) |] ==> \
\ a: Vfrom(X,i) & b: Vfrom(X,i)";
by (fast_tac (ZF_cs addSDs [doubleton_in_Vfrom_D, Vfrom_doubleton_D]) 1);
val Pair_in_Vfrom_D = result();
goal Univ.thy
"!!X. Transset(X) ==> \
\ (a*b) Int Vfrom(X, succ(i)) <= (a Int Vfrom(X,i)) * (b Int Vfrom(X,i))";
by (fast_tac (ZF_cs addSDs [Pair_in_Vfrom_D]) 1);
val product_Int_Vfrom_subset = result();
(*** Intersecting <a;b> with Vfrom... ***)
goalw QUniv.thy [QPair_def,sum_def]
"!!X. Transset(X) ==> \
\ <a;b> Int Vfrom(X, succ(i)) <= <a Int Vfrom(X,i); b Int Vfrom(X,i)>";
by (rtac (Int_Un_distrib RS ssubst) 1);
by (rtac Un_mono 1);
by (REPEAT (ares_tac [product_Int_Vfrom_subset RS subset_trans,
[Int_lower1, subset_refl] MRS Sigma_mono] 1));
val QPair_Int_Vfrom_succ_subset = result();
(** Pairs in quniv -- for handling the base case **)
goal QUniv.thy "!!X. <a,b> : quniv(X) ==> <a,b> : univ(eclose(X))";
by (etac ([qunivD, Transset_eclose] MRS Transset_Pair_subset_univ) 1);
val Pair_in_quniv_D = result();
goal QUniv.thy "a*b Int quniv(A) = a*b Int univ(eclose(A))";
by (rtac equalityI 1);
by (rtac ([subset_refl, univ_eclose_subset_quniv] MRS Int_mono) 2);
by (fast_tac (ZF_cs addSEs [Pair_in_quniv_D]) 1);
val product_Int_quniv_eq = result();
goalw QUniv.thy [QPair_def,sum_def]
"<a;b> Int quniv(A) = <a;b> Int univ(eclose(A))";
by (simp_tac (ZF_ss addsimps [Int_Un_distrib, product_Int_quniv_eq]) 1);
val QPair_Int_quniv_eq = result();
(**** "Take-lemma" rules for proving c: quniv(A) ****)
goalw QUniv.thy [quniv_def] "Transset(quniv(A))";
by (rtac (Transset_eclose RS Transset_univ RS Transset_Pow) 1);
val Transset_quniv = result();
val [aprem, iprem] = goal QUniv.thy
"[| a: quniv(quniv(X)); \
\ !!i. i:nat ==> a Int Vfrom(quniv(X),i) : quniv(A) \
\ |] ==> a : quniv(A)";
by (rtac (univ_Int_Vfrom_subset RS qunivI) 1);
by (rtac (aprem RS qunivD) 1);
by (rtac (Transset_quniv RS Transset_eclose_eq_arg RS ssubst) 1);
by (etac (iprem RS qunivD) 1);
val quniv_Int_Vfrom = result();
(** Rules for level 0 **)
goal QUniv.thy "<a;b> Int quniv(A) : quniv(A)";
by (rtac (QPair_Int_quniv_eq RS ssubst) 1);
by (rtac (Int_lower2 RS qunivI) 1);
val QPair_Int_quniv_in_quniv = result();
(*Unused; kept as an example. QInr rule is similar*)
goalw QUniv.thy [QInl_def] "QInl(a) Int quniv(A) : quniv(A)";
by (rtac QPair_Int_quniv_in_quniv 1);
val QInl_Int_quniv_in_quniv = result();
goal QUniv.thy "!!a A X. a : quniv(A) ==> a Int X : quniv(A)";
by (etac ([Int_lower1, qunivD] MRS subset_trans RS qunivI) 1);
val Int_quniv_in_quniv = result();
goal QUniv.thy
"!!X. a Int X : quniv(A) ==> a Int Vfrom(X, 0) : quniv(A)";
by (etac (Vfrom_0 RS ssubst) 1);
val Int_Vfrom_0_in_quniv = result();
(** Rules for level succ(i), decreasing it to i **)
goal QUniv.thy
"!!X. [| a Int Vfrom(X,i) : quniv(A); \
\ b Int Vfrom(X,i) : quniv(A); \
\ Transset(X) \
\ |] ==> <a;b> Int Vfrom(X, succ(i)) : quniv(A)";
by (rtac (QPair_Int_Vfrom_succ_subset RS subset_trans RS qunivI) 1);
by (rtac (QPair_in_quniv RS qunivD) 2);
by (REPEAT (assume_tac 1));
val QPair_Int_Vfrom_succ_in_quniv = result();
val zero_Int_in_quniv = standard
([Int_lower1,empty_subsetI] MRS subset_trans RS qunivI);
val one_Int_in_quniv = standard
([Int_lower1, one_in_quniv RS qunivD] MRS subset_trans RS qunivI);
(*Unused; kept as an example. QInr rule is similar*)
goalw QUniv.thy [QInl_def]
"!!X. [| a Int Vfrom(X,i) : quniv(A); Transset(X) \
\ |] ==> QInl(a) Int Vfrom(X, succ(i)) : quniv(A)";
by (rtac QPair_Int_Vfrom_succ_in_quniv 1);
by (REPEAT (ares_tac [zero_Int_in_quniv] 1));
val QInl_Int_Vfrom_succ_in_quniv = result();
(** Rules for level i -- preserving the level, not decreasing it **)
goalw QUniv.thy [QPair_def]
"!!X. Transset(X) ==> \
\ <a;b> Int Vfrom(X,i) <= <a Int Vfrom(X,i); b Int Vfrom(X,i)>";
by (etac (Transset_Vfrom RS Transset_sum_Int_subset) 1);
val QPair_Int_Vfrom_subset = result();
goal QUniv.thy
"!!X. [| a Int Vfrom(X,i) : quniv(A); \
\ b Int Vfrom(X,i) : quniv(A); \
\ Transset(X) \
\ |] ==> <a;b> Int Vfrom(X,i) : quniv(A)";
by (rtac (QPair_Int_Vfrom_subset RS subset_trans RS qunivI) 1);
by (rtac (QPair_in_quniv RS qunivD) 2);
by (REPEAT (assume_tac 1));
val QPair_Int_Vfrom_in_quniv = result();
(**** "Take-lemma" rules for proving a=b by coinduction ****)
(** Unfortunately, the technique used above does not apply here, since
the base case appears impossible to prove: it involves an intersection
with eclose(X) for arbitrary X. So a=b is proved by transfinite induction
over ALL ordinals, using Vset(i) instead of Vfrom(X,i).
**)
(*Rule for level 0*)
goal QUniv.thy "a Int Vset(0) <= b";
by (rtac (Vfrom_0 RS ssubst) 1);
by (fast_tac ZF_cs 1);
val Int_Vset_0_subset = result();
(*Rule for level succ(i), decreasing it to i*)
goal QUniv.thy
"!!i. [| a Int Vset(i) <= c; \
\ b Int Vset(i) <= d \
\ |] ==> <a;b> Int Vset(succ(i)) <= <c;d>";
by (rtac ([Transset_0 RS QPair_Int_Vfrom_succ_subset, QPair_mono]
MRS subset_trans) 1);
by (REPEAT (assume_tac 1));
val QPair_Int_Vset_succ_subset_trans = result();
(*Unused; kept as an example. QInr rule is similar*)
goalw QUniv.thy [QInl_def]
"!!i. a Int Vset(i) <= b ==> QInl(a) Int Vset(succ(i)) <= QInl(b)";
by (etac (Int_lower1 RS QPair_Int_Vset_succ_subset_trans) 1);
val QInl_Int_Vset_succ_subset_trans = result();
(*Rule for level i -- preserving the level, not decreasing it*)
goal QUniv.thy
"!!i. [| a Int Vset(i) <= c; \
\ b Int Vset(i) <= d \
\ |] ==> <a;b> Int Vset(i) <= <c;d>";
by (rtac ([Transset_0 RS QPair_Int_Vfrom_subset, QPair_mono]
MRS subset_trans) 1);
by (REPEAT (assume_tac 1));
val QPair_Int_Vset_subset_trans = result();