Theory QUniv
section‹A Small Universe for Lazy Recursive Types›
theory QUniv imports Univ QPair begin
rep_datatype
elimination sumE
induction TrueI
case_eqns case_Inl case_Inr
rep_datatype
elimination qsumE
induction TrueI
case_eqns qcase_QInl qcase_QInr
definition
quniv :: "i ⇒ i" where
"quniv(A) ≡ Pow(univ(eclose(A)))"
subsection‹Properties involving Transset and Sum›
lemma Transset_includes_summands:
"⟦Transset(C); A+B ⊆ C⟧ ⟹ A ⊆ C ∧ B ⊆ C"
apply (simp add: sum_def Un_subset_iff)
apply (blast dest: Transset_includes_range)
done
lemma Transset_sum_Int_subset:
"Transset(C) ⟹ (A+B) ∩ C ⊆ (A ∩ C) + (B ∩ C)"
apply (simp add: sum_def Int_Un_distrib2)
apply (blast dest: Transset_Pair_D)
done
subsection‹Introduction and Elimination Rules›
lemma qunivI: "X ⊆ univ(eclose(A)) ⟹ X ∈ quniv(A)"
by (simp add: quniv_def)
lemma qunivD: "X ∈ quniv(A) ⟹ X ⊆ univ(eclose(A))"
by (simp add: quniv_def)
lemma quniv_mono: "A<=B ⟹ quniv(A) ⊆ quniv(B)"
unfolding quniv_def
apply (erule eclose_mono [THEN univ_mono, THEN Pow_mono])
done
subsection‹Closure Properties›
lemma univ_eclose_subset_quniv: "univ(eclose(A)) ⊆ quniv(A)"
apply (simp add: quniv_def Transset_iff_Pow [symmetric])
apply (rule Transset_eclose [THEN Transset_univ])
done
lemma univ_subset_quniv: "univ(A) ⊆ quniv(A)"
apply (rule arg_subset_eclose [THEN univ_mono, THEN subset_trans])
apply (rule univ_eclose_subset_quniv)
done
lemmas univ_into_quniv = univ_subset_quniv [THEN subsetD]
lemma Pow_univ_subset_quniv: "Pow(univ(A)) ⊆ quniv(A)"
unfolding quniv_def
apply (rule arg_subset_eclose [THEN univ_mono, THEN Pow_mono])
done
lemmas univ_subset_into_quniv =
PowI [THEN Pow_univ_subset_quniv [THEN subsetD]]
lemmas zero_in_quniv = zero_in_univ [THEN univ_into_quniv]
lemmas one_in_quniv = one_in_univ [THEN univ_into_quniv]
lemmas two_in_quniv = two_in_univ [THEN univ_into_quniv]
lemmas A_subset_quniv = subset_trans [OF A_subset_univ univ_subset_quniv]
lemmas A_into_quniv = A_subset_quniv [THEN subsetD]
lemma QPair_subset_univ:
"⟦a ⊆ univ(A); b ⊆ univ(A)⟧ ⟹ <a;b> ⊆ univ(A)"
by (simp add: QPair_def sum_subset_univ)
subsection‹Quine Disjoint Sum›
lemma QInl_subset_univ: "a ⊆ univ(A) ⟹ QInl(a) ⊆ univ(A)"
unfolding QInl_def
apply (erule empty_subsetI [THEN QPair_subset_univ])
done
lemmas naturals_subset_nat =
Ord_nat [THEN Ord_is_Transset, unfolded Transset_def, THEN bspec]
lemmas naturals_subset_univ =
subset_trans [OF naturals_subset_nat nat_subset_univ]
lemma QInr_subset_univ: "a ⊆ univ(A) ⟹ QInr(a) ⊆ univ(A)"
unfolding QInr_def
apply (erule nat_1I [THEN naturals_subset_univ, THEN QPair_subset_univ])
done
subsection‹Closure for Quine-Inspired Products and Sums›
lemma QPair_in_quniv:
"⟦a: quniv(A); b: quniv(A)⟧ ⟹ <a;b> ∈ quniv(A)"
by (simp add: quniv_def QPair_def sum_subset_univ)
lemma QSigma_quniv: "quniv(A) <*> quniv(A) ⊆ quniv(A)"
by (blast intro: QPair_in_quniv)
lemmas QSigma_subset_quniv = subset_trans [OF QSigma_mono QSigma_quniv]
lemma quniv_QPair_D:
"<a;b> ∈ quniv(A) ⟹ a: quniv(A) ∧ b: quniv(A)"
unfolding quniv_def QPair_def
apply (rule Transset_includes_summands [THEN conjE])
apply (rule Transset_eclose [THEN Transset_univ])
apply (erule PowD, blast)
done
lemmas quniv_QPair_E = quniv_QPair_D [THEN conjE]
lemma quniv_QPair_iff: "<a;b> ∈ quniv(A) ⟷ a: quniv(A) ∧ b: quniv(A)"
by (blast intro: QPair_in_quniv dest: quniv_QPair_D)
subsection‹Quine Disjoint Sum›
lemma QInl_in_quniv: "a: quniv(A) ⟹ QInl(a) ∈ quniv(A)"
by (simp add: QInl_def zero_in_quniv QPair_in_quniv)
lemma QInr_in_quniv: "b: quniv(A) ⟹ QInr(b) ∈ quniv(A)"
by (simp add: QInr_def one_in_quniv QPair_in_quniv)
lemma qsum_quniv: "quniv(C) <+> quniv(C) ⊆ quniv(C)"
by (blast intro: QInl_in_quniv QInr_in_quniv)
lemmas qsum_subset_quniv = subset_trans [OF qsum_mono qsum_quniv]
subsection‹The Natural Numbers›
lemmas nat_subset_quniv = subset_trans [OF nat_subset_univ univ_subset_quniv]
lemmas nat_into_quniv = nat_subset_quniv [THEN subsetD]
lemmas bool_subset_quniv = subset_trans [OF bool_subset_univ univ_subset_quniv]
lemmas bool_into_quniv = bool_subset_quniv [THEN subsetD]
lemma QPair_Int_Vfrom_succ_subset:
"Transset(X) ⟹
<a;b> ∩ Vfrom(X, succ(i)) ⊆ <a ∩ Vfrom(X,i); b ∩ Vfrom(X,i)>"
by (simp add: QPair_def sum_def Int_Un_distrib2 Un_mono
product_Int_Vfrom_subset [THEN subset_trans]
Sigma_mono [OF Int_lower1 subset_refl])
subsection‹"Take-Lemma" Rules›
lemma QPair_Int_Vfrom_subset:
"Transset(X) ⟹
<a;b> ∩ Vfrom(X,i) ⊆ <a ∩ Vfrom(X,i); b ∩ Vfrom(X,i)>"
unfolding QPair_def
apply (erule Transset_Vfrom [THEN Transset_sum_Int_subset])
done
lemmas QPair_Int_Vset_subset_trans =
subset_trans [OF Transset_0 [THEN QPair_Int_Vfrom_subset] QPair_mono]
lemma QPair_Int_Vset_subset_UN:
"Ord(i) ⟹ <a;b> ∩ Vset(i) ⊆ (⋃j∈i. <a ∩ Vset(j); b ∩ Vset(j)>)"
apply (erule Ord_cases)
apply (simp add: Vfrom_0)
apply (erule ssubst)
apply (rule Transset_0 [THEN QPair_Int_Vfrom_succ_subset, THEN subset_trans])
apply (rule succI1 [THEN UN_upper])
apply (simp del: UN_simps
add: Limit_Vfrom_eq Int_UN_distrib UN_mono QPair_Int_Vset_subset_trans)
done
end