src/ZF/QUniv.thy
author paulson
Tue Jul 02 22:46:23 2002 +0200 (2002-07-02)
changeset 13285 28d1823ce0f2
parent 13220 62c899c77151
child 13356 c9cfe1638bf2
permissions -rw-r--r--
conversion of QPair to Isar
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(*  Title:      ZF/QUniv.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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A small universe for lazy recursive types
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*)
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(** Properties involving Transset and Sum **)
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theory QUniv = Univ + QPair + mono + equalities:
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(*Disjoint sums as a datatype*)
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rep_datatype 
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  elimination	sumE
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  induction	TrueI
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  case_eqns	case_Inl case_Inr
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(*Variant disjoint sums as a datatype*)
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rep_datatype 
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  elimination	qsumE
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  induction	TrueI
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  case_eqns	qcase_QInl qcase_QInr
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constdefs
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  quniv :: "i => i"
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   "quniv(A) == Pow(univ(eclose(A)))"
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lemma Transset_includes_summands:
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     "[| Transset(C); A+B <= C |] ==> A <= C & B <= C"
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apply (simp add: sum_def Un_subset_iff) 
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apply (blast dest: Transset_includes_range)
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done
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lemma Transset_sum_Int_subset:
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     "Transset(C) ==> (A+B) Int C <= (A Int C) + (B Int C)"
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apply (simp add: sum_def Int_Un_distrib2) 
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apply (blast dest: Transset_Pair_D)
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done
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(** Introduction and elimination rules avoid tiresome folding/unfolding **)
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lemma qunivI: "X <= univ(eclose(A)) ==> X : quniv(A)"
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by (simp add: quniv_def)
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lemma qunivD: "X : quniv(A) ==> X <= univ(eclose(A))"
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by (simp add: quniv_def)
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lemma quniv_mono: "A<=B ==> quniv(A) <= quniv(B)"
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apply (unfold quniv_def)
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apply (erule eclose_mono [THEN univ_mono, THEN Pow_mono])
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done
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(*** Closure properties ***)
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lemma univ_eclose_subset_quniv: "univ(eclose(A)) <= quniv(A)"
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apply (simp add: quniv_def Transset_iff_Pow [symmetric]) 
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apply (rule Transset_eclose [THEN Transset_univ])
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done
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(*Key property for proving A_subset_quniv; requires eclose in def of quniv*)
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lemma univ_subset_quniv: "univ(A) <= quniv(A)"
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apply (rule arg_subset_eclose [THEN univ_mono, THEN subset_trans])
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apply (rule univ_eclose_subset_quniv)
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done
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lemmas univ_into_quniv = univ_subset_quniv [THEN subsetD, standard]
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lemma Pow_univ_subset_quniv: "Pow(univ(A)) <= quniv(A)"
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apply (unfold quniv_def)
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apply (rule arg_subset_eclose [THEN univ_mono, THEN Pow_mono])
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done
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lemmas univ_subset_into_quniv =
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    PowI [THEN Pow_univ_subset_quniv [THEN subsetD], standard]
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lemmas zero_in_quniv = zero_in_univ [THEN univ_into_quniv, standard]
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lemmas one_in_quniv = one_in_univ [THEN univ_into_quniv, standard]
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lemmas two_in_quniv = two_in_univ [THEN univ_into_quniv, standard]
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lemmas A_subset_quniv =  subset_trans [OF A_subset_univ univ_subset_quniv]
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lemmas A_into_quniv = A_subset_quniv [THEN subsetD, standard]
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(*** univ(A) closure for Quine-inspired pairs and injections ***)
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(*Quine ordered pairs*)
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lemma QPair_subset_univ: 
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    "[| a <= univ(A);  b <= univ(A) |] ==> <a;b> <= univ(A)"
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by (simp add: QPair_def sum_subset_univ)
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(** Quine disjoint sum **)
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lemma QInl_subset_univ: "a <= univ(A) ==> QInl(a) <= univ(A)"
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apply (unfold QInl_def)
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apply (erule empty_subsetI [THEN QPair_subset_univ])
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done
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lemmas naturals_subset_nat = 
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    Ord_nat [THEN Ord_is_Transset, unfolded Transset_def, THEN bspec, standard]
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lemmas naturals_subset_univ =
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    subset_trans [OF naturals_subset_nat nat_subset_univ]
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lemma QInr_subset_univ: "a <= univ(A) ==> QInr(a) <= univ(A)"
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apply (unfold QInr_def)
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apply (erule nat_1I [THEN naturals_subset_univ, THEN QPair_subset_univ])
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done
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(*** Closure for Quine-inspired products and sums ***)
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(*Quine ordered pairs*)
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lemma QPair_in_quniv: 
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    "[| a: quniv(A);  b: quniv(A) |] ==> <a;b> : quniv(A)"
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by (simp add: quniv_def QPair_def sum_subset_univ) 
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lemma QSigma_quniv: "quniv(A) <*> quniv(A) <= quniv(A)" 
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by (blast intro: QPair_in_quniv)
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lemmas QSigma_subset_quniv =  subset_trans [OF QSigma_mono QSigma_quniv]
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(*The opposite inclusion*)
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lemma quniv_QPair_D: 
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    "<a;b> : quniv(A) ==> a: quniv(A) & b: quniv(A)"
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apply (unfold quniv_def QPair_def)
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apply (rule Transset_includes_summands [THEN conjE]) 
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apply (rule Transset_eclose [THEN Transset_univ])
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apply (erule PowD, blast) 
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done
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lemmas quniv_QPair_E = quniv_QPair_D [THEN conjE, standard]
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lemma quniv_QPair_iff: "<a;b> : quniv(A) <-> a: quniv(A) & b: quniv(A)"
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by (blast intro: QPair_in_quniv dest: quniv_QPair_D)
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(** Quine disjoint sum **)
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lemma QInl_in_quniv: "a: quniv(A) ==> QInl(a) : quniv(A)"
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by (simp add: QInl_def zero_in_quniv QPair_in_quniv)
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lemma QInr_in_quniv: "b: quniv(A) ==> QInr(b) : quniv(A)"
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by (simp add: QInr_def one_in_quniv QPair_in_quniv)
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lemma qsum_quniv: "quniv(C) <+> quniv(C) <= quniv(C)"
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by (blast intro: QInl_in_quniv QInr_in_quniv)
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lemmas qsum_subset_quniv = subset_trans [OF qsum_mono qsum_quniv]
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(*** The natural numbers ***)
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lemmas nat_subset_quniv =  subset_trans [OF nat_subset_univ univ_subset_quniv]
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(* n:nat ==> n:quniv(A) *)
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lemmas nat_into_quniv = nat_subset_quniv [THEN subsetD, standard]
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lemmas bool_subset_quniv = subset_trans [OF bool_subset_univ univ_subset_quniv]
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lemmas bool_into_quniv = bool_subset_quniv [THEN subsetD, standard]
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(*** Intersecting <a;b> with Vfrom... ***)
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lemma QPair_Int_Vfrom_succ_subset: 
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 "Transset(X) ==>           
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       <a;b> Int Vfrom(X, succ(i))  <=  <a Int Vfrom(X,i);  b Int Vfrom(X,i)>"
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by (simp add: QPair_def sum_def Int_Un_distrib2 Un_mono
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              product_Int_Vfrom_subset [THEN subset_trans]
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              Sigma_mono [OF Int_lower1 subset_refl])
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(**** "Take-lemma" rules for proving a=b by coinduction and c: quniv(A) ****)
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(*Rule for level i -- preserving the level, not decreasing it*)
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lemma QPair_Int_Vfrom_subset: 
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 "Transset(X) ==>           
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       <a;b> Int Vfrom(X,i)  <=  <a Int Vfrom(X,i);  b Int Vfrom(X,i)>"
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apply (unfold QPair_def)
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apply (erule Transset_Vfrom [THEN Transset_sum_Int_subset])
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done
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(*[| a Int Vset(i) <= c; b Int Vset(i) <= d |] ==> <a;b> Int Vset(i) <= <c;d>*)
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lemmas QPair_Int_Vset_subset_trans =
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     subset_trans [OF Transset_0 [THEN QPair_Int_Vfrom_subset] QPair_mono]
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lemma QPair_Int_Vset_subset_UN:
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     "Ord(i) ==> <a;b> Int Vset(i) <= (UN j:i. <a Int Vset(j); b Int Vset(j)>)"
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apply (erule Ord_cases)
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(*0 case*)
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apply (simp add: Vfrom_0)
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(*succ(j) case*)
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apply (erule ssubst) 
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apply (rule Transset_0 [THEN QPair_Int_Vfrom_succ_subset, THEN subset_trans])
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apply (rule succI1 [THEN UN_upper])
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(*Limit(i) case*)
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apply (simp del: UN_simps 
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        add: Limit_Vfrom_eq Int_UN_distrib UN_mono QPair_Int_Vset_subset_trans)
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done
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ML
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{*
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val Transset_includes_summands = thm "Transset_includes_summands";
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val Transset_sum_Int_subset = thm "Transset_sum_Int_subset";
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val qunivI = thm "qunivI";
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val qunivD = thm "qunivD";
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val quniv_mono = thm "quniv_mono";
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val univ_eclose_subset_quniv = thm "univ_eclose_subset_quniv";
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val univ_subset_quniv = thm "univ_subset_quniv";
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val univ_into_quniv = thm "univ_into_quniv";
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val Pow_univ_subset_quniv = thm "Pow_univ_subset_quniv";
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val univ_subset_into_quniv = thm "univ_subset_into_quniv";
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val zero_in_quniv = thm "zero_in_quniv";
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val one_in_quniv = thm "one_in_quniv";
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val two_in_quniv = thm "two_in_quniv";
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val A_subset_quniv = thm "A_subset_quniv";
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val A_into_quniv = thm "A_into_quniv";
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val QPair_subset_univ = thm "QPair_subset_univ";
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val QInl_subset_univ = thm "QInl_subset_univ";
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val naturals_subset_nat = thm "naturals_subset_nat";
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val naturals_subset_univ = thm "naturals_subset_univ";
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val QInr_subset_univ = thm "QInr_subset_univ";
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val QPair_in_quniv = thm "QPair_in_quniv";
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val QSigma_quniv = thm "QSigma_quniv";
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val QSigma_subset_quniv = thm "QSigma_subset_quniv";
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val quniv_QPair_D = thm "quniv_QPair_D";
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val quniv_QPair_E = thm "quniv_QPair_E";
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val quniv_QPair_iff = thm "quniv_QPair_iff";
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val QInl_in_quniv = thm "QInl_in_quniv";
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val QInr_in_quniv = thm "QInr_in_quniv";
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val qsum_quniv = thm "qsum_quniv";
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val qsum_subset_quniv = thm "qsum_subset_quniv";
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val nat_subset_quniv = thm "nat_subset_quniv";
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val nat_into_quniv = thm "nat_into_quniv";
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val bool_subset_quniv = thm "bool_subset_quniv";
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val bool_into_quniv = thm "bool_into_quniv";
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val QPair_Int_Vfrom_succ_subset = thm "QPair_Int_Vfrom_succ_subset";
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val QPair_Int_Vfrom_subset = thm "QPair_Int_Vfrom_subset";
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val QPair_Int_Vset_subset_trans = thm "QPair_Int_Vset_subset_trans";
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val QPair_Int_Vset_subset_UN = thm "QPair_Int_Vset_subset_UN";
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*}
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end