header {*Absoluteness Properties for Recursive Datatypes*}
theory Datatype_absolute = Formula + WF_absolute:
subsection{*The lfp of a continuous function can be expressed as a union*}
constdefs
directed :: "i=>o"
"directed(A) == A\<noteq>0 & (\<forall>x\<in>A. \<forall>y\<in>A. x \<union> y \<in> A)"
contin :: "(i=>i) => o"
"contin(h) == (\<forall>A. directed(A) --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))"
lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D"
apply (induct_tac n)
apply (simp_all add: bnd_mono_def, blast)
done
lemma bnd_mono_increasing [rule_format]:
"[|i \<in> nat; j \<in> nat; bnd_mono(D,h)|] ==> i \<le> j --> h^i(0) \<subseteq> h^j(0)"
apply (rule_tac m=i and n=j in diff_induct, simp_all)
apply (blast del: subsetI
intro: bnd_mono_iterates_subset bnd_monoD2 [of concl: h] )
done
lemma directed_iterates: "bnd_mono(D,h) ==> directed({h^n (0). n\<in>nat})"
apply (simp add: directed_def, clarify)
apply (rename_tac i j)
apply (rule_tac x="i \<union> j" in bexI)
apply (rule_tac i = i and j = j in Ord_linear_le)
apply (simp_all add: subset_Un_iff [THEN iffD1] le_imp_subset
subset_Un_iff2 [THEN iffD1])
apply (simp_all add: subset_Un_iff [THEN iff_sym] bnd_mono_increasing
subset_Un_iff2 [THEN iff_sym])
done
lemma contin_iterates_eq:
"[|bnd_mono(D, h); contin(h)|]
==> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))"
apply (simp add: contin_def directed_iterates)
apply (rule trans)
apply (rule equalityI)
apply (simp_all add: UN_subset_iff)
apply safe
apply (erule_tac [2] natE)
apply (rule_tac a="succ(x)" in UN_I)
apply simp_all
apply blast
done
lemma lfp_subset_Union:
"[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) <= (\<Union>n\<in>nat. h^n(0))"
apply (rule lfp_lowerbound)
apply (simp add: contin_iterates_eq)
apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff)
done
lemma Union_subset_lfp:
"bnd_mono(D,h) ==> (\<Union>n\<in>nat. h^n(0)) <= lfp(D,h)"
apply (simp add: UN_subset_iff)
apply (rule ballI)
apply (induct_tac n, simp_all)
apply (rule subset_trans [of _ "h(lfp(D,h))"])
apply (blast dest: bnd_monoD2 [OF _ _ lfp_subset] )
apply (erule lfp_lemma2)
done
lemma lfp_eq_Union:
"[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) = (\<Union>n\<in>nat. h^n(0))"
by (blast del: subsetI
intro: lfp_subset_Union Union_subset_lfp)
subsubsection{*Some Standard Datatype Constructions Preserve Continuity*}
lemma contin_imp_mono: "[|X\<subseteq>Y; contin(F)|] ==> F(X) \<subseteq> F(Y)"
apply (simp add: contin_def)
apply (drule_tac x="{X,Y}" in spec)
apply (simp add: directed_def subset_Un_iff2 Un_commute)
done
lemma sum_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) + G(X))"
by (simp add: contin_def, blast)
lemma prod_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) * G(X))"
apply (subgoal_tac "\<forall>B C. F(B) \<subseteq> F(B \<union> C)")
prefer 2 apply (simp add: Un_upper1 contin_imp_mono)
apply (subgoal_tac "\<forall>B C. G(C) \<subseteq> G(B \<union> C)")
prefer 2 apply (simp add: Un_upper2 contin_imp_mono)
apply (simp add: contin_def, clarify)
apply (rule equalityI)
prefer 2 apply blast
apply clarify
apply (rename_tac B C)
apply (rule_tac a="B \<union> C" in UN_I)
apply (simp add: directed_def, blast)
done
lemma const_contin: "contin(\<lambda>X. A)"
by (simp add: contin_def directed_def)
lemma id_contin: "contin(\<lambda>X. X)"
by (simp add: contin_def)
subsection {*Absoluteness for "Iterates"*}
constdefs
iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
"iterates_MH(M,isF,v,n,g,z) ==
is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
n, z)"
iterates_replacement :: "[i=>o, [i,i]=>o, i] => o"
"iterates_replacement(M,isF,v) ==
\<forall>n[M]. n\<in>nat -->
wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))"
lemma (in M_axioms) iterates_MH_abs:
"[| relativize1(M,isF,F); M(n); M(g); M(z) |]
==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, \<lambda>m. F(g`m), n)"
by (simp add: nat_case_abs [of _ "\<lambda>m. F(g ` m)"]
relativize1_def iterates_MH_def)
lemma (in M_axioms) iterates_imp_wfrec_replacement:
"[|relativize1(M,isF,F); n \<in> nat; iterates_replacement(M,isF,v)|]
==> wfrec_replacement(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n),
Memrel(succ(n)))"
by (simp add: iterates_replacement_def iterates_MH_abs)
theorem (in M_trancl) iterates_abs:
"[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
n \<in> nat; M(v); M(z); \<forall>x[M]. M(F(x)) |]
==> is_wfrec(M, iterates_MH(M,isF,v), Memrel(succ(n)), n, z) <->
z = iterates(F,n,v)"
apply (frule iterates_imp_wfrec_replacement, assumption+)
apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
relativize2_def iterates_MH_abs
iterates_nat_def recursor_def transrec_def
eclose_sing_Ord_eq nat_into_M
trans_wfrec_abs [of _ _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
done
lemma (in M_wfrank) iterates_closed [intro,simp]:
"[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
n \<in> nat; M(v); \<forall>x[M]. M(F(x)) |]
==> M(iterates(F,n,v))"
apply (frule iterates_imp_wfrec_replacement, assumption+)
apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
relativize2_def iterates_MH_abs
iterates_nat_def recursor_def transrec_def
eclose_sing_Ord_eq nat_into_M
trans_wfrec_closed [of _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
done
subsection {*lists without univ*}
lemmas datatype_univs = Inl_in_univ Inr_in_univ
Pair_in_univ nat_into_univ A_into_univ
lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"
apply (rule bnd_monoI)
apply (intro subset_refl zero_subset_univ A_subset_univ
sum_subset_univ Sigma_subset_univ)
apply (rule subset_refl sum_mono Sigma_mono | assumption)+
done
lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"
by (intro sum_contin prod_contin id_contin const_contin)
text{*Re-expresses lists using sum and product*}
lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"
apply (simp add: list_def)
apply (rule equalityI)
apply (rule lfp_lowerbound)
prefer 2 apply (rule lfp_subset)
apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
apply (simp add: Nil_def Cons_def)
apply blast
txt{*Opposite inclusion*}
apply (rule lfp_lowerbound)
prefer 2 apply (rule lfp_subset)
apply (clarify, subst lfp_unfold [OF list.bnd_mono])
apply (simp add: Nil_def Cons_def)
apply (blast intro: datatype_univs
dest: lfp_subset [THEN subsetD])
done
text{*Re-expresses lists using "iterates", no univ.*}
lemma list_eq_Union:
"list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"
by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)
constdefs
is_list_functor :: "[i=>o,i,i,i] => o"
"is_list_functor(M,A,X,Z) ==
\<exists>n1[M]. \<exists>AX[M].
number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)"
lemma (in M_axioms) list_functor_abs [simp]:
"[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) <-> (Z = {0} + A*X)"
by (simp add: is_list_functor_def singleton_0 nat_into_M)
subsection {*formulas without univ*}
lemma formula_fun_bnd_mono:
"bnd_mono(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
apply (rule bnd_monoI)
apply (intro subset_refl zero_subset_univ A_subset_univ
sum_subset_univ Sigma_subset_univ nat_subset_univ)
apply (rule subset_refl sum_mono Sigma_mono | assumption)+
done
lemma formula_fun_contin:
"contin(\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
by (intro sum_contin prod_contin id_contin const_contin)
text{*Re-expresses formulas using sum and product*}
lemma formula_eq_lfp2:
"formula = lfp(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))"
apply (simp add: formula_def)
apply (rule equalityI)
apply (rule lfp_lowerbound)
prefer 2 apply (rule lfp_subset)
apply (clarify, subst lfp_unfold [OF formula_fun_bnd_mono])
apply (simp add: Member_def Equal_def Neg_def And_def Forall_def)
apply blast
txt{*Opposite inclusion*}
apply (rule lfp_lowerbound)
prefer 2 apply (rule lfp_subset, clarify)
apply (subst lfp_unfold [OF formula.bnd_mono, simplified])
apply (simp add: Member_def Equal_def Neg_def And_def Forall_def)
apply (elim sumE SigmaE, simp_all)
apply (blast intro: datatype_univs dest: lfp_subset [THEN subsetD])+
done
text{*Re-expresses formulas using "iterates", no univ.*}
lemma formula_eq_Union:
"formula =
(\<Union>n\<in>nat. (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))) ^ n (0))"
by (simp add: formula_eq_lfp2 lfp_eq_Union formula_fun_bnd_mono
formula_fun_contin)
constdefs
is_formula_functor :: "[i=>o,i,i] => o"
"is_formula_functor(M,X,Z) ==
\<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M]. \<exists>X4[M].
omega(M,nat') & cartprod(M,nat',nat',natnat) &
is_sum(M,natnat,natnat,natnatsum) &
cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & is_sum(M,X,X3,X4) &
is_sum(M,natnatsum,X4,Z)"
lemma (in M_axioms) formula_functor_abs [simp]:
"[| M(X); M(Z) |]
==> is_formula_functor(M,X,Z) <->
Z = ((nat*nat) + (nat*nat)) + (X + (X*X + X))"
by (simp add: is_formula_functor_def)
subsection{*@{term M} Contains the List and Formula Datatypes*}
constdefs
is_list_n :: "[i=>o,i,i,i] => o"
"is_list_n(M,A,n,Z) ==
\<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M].
empty(M,zero) &
successor(M,n,sn) & membership(M,sn,msn) &
is_wfrec(M, iterates_MH(M, is_list_functor(M,A),zero), msn, n, Z)"
mem_list :: "[i=>o,i,i] => o"
"mem_list(M,A,l) ==
\<exists>n[M]. \<exists>listn[M].
finite_ordinal(M,n) & is_list_n(M,A,n,listn) & l \<in> listn"
is_list :: "[i=>o,i,i] => o"
"is_list(M,A,Z) == \<forall>l[M]. l \<in> Z <-> mem_list(M,A,l)"
constdefs
is_formula_n :: "[i=>o,i,i] => o"
"is_formula_n(M,n,Z) ==
\<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M].
empty(M,zero) &
successor(M,n,sn) & membership(M,sn,msn) &
is_wfrec(M, iterates_MH(M, is_formula_functor(M),zero), msn, n, Z)"
mem_formula :: "[i=>o,i] => o"
"mem_formula(M,p) ==
\<exists>n[M]. \<exists>formn[M].
finite_ordinal(M,n) & is_formula_n(M,n,formn) & p \<in> formn"
is_formula :: "[i=>o,i] => o"
"is_formula(M,Z) == \<forall>p[M]. p \<in> Z <-> mem_formula(M,p)"
locale (open) M_datatypes = M_wfrank +
assumes list_replacement1:
"M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)"
and list_replacement2:
"M(A) ==> strong_replacement(M,
\<lambda>n y. n\<in>nat &
(\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
is_wfrec(M, iterates_MH(M,is_list_functor(M,A), 0),
msn, n, y)))"
and formula_replacement1:
"iterates_replacement(M, is_formula_functor(M), 0)"
and formula_replacement2:
"strong_replacement(M,
\<lambda>n y. n\<in>nat &
(\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
is_wfrec(M, iterates_MH(M,is_formula_functor(M), 0),
msn, n, y)))"
subsubsection{*Absoluteness of the List Construction*}
lemma (in M_datatypes) list_replacement2':
"M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"
apply (insert list_replacement2 [of A])
apply (rule strong_replacement_cong [THEN iffD1])
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]])
apply (simp_all add: list_replacement1 relativize1_def)
done
lemma (in M_datatypes) list_closed [intro,simp]:
"M(A) ==> M(list(A))"
apply (insert list_replacement1)
by (simp add: RepFun_closed2 list_eq_Union
list_replacement2' relativize1_def
iterates_closed [of "is_list_functor(M,A)"])
lemma (in M_datatypes) is_list_n_abs [simp]:
"[|M(A); n\<in>nat; M(Z)|]
==> is_list_n(M,A,n,Z) <-> Z = (\<lambda>X. {0} + A * X)^n (0)"
apply (insert list_replacement1)
apply (simp add: is_list_n_def relativize1_def nat_into_M
iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"])
done
lemma (in M_datatypes) mem_list_abs [simp]:
"M(A) ==> mem_list(M,A,l) <-> l \<in> list(A)"
apply (insert list_replacement1)
apply (simp add: mem_list_def relativize1_def list_eq_Union
iterates_closed [of "is_list_functor(M,A)"])
done
lemma (in M_datatypes) list_abs [simp]:
"[|M(A); M(Z)|] ==> is_list(M,A,Z) <-> Z = list(A)"
apply (simp add: is_list_def, safe)
apply (rule M_equalityI, simp_all)
done
subsubsection{*Absoluteness of Formulas*}
lemma (in M_datatypes) formula_replacement2':
"strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0))"
apply (insert formula_replacement2)
apply (rule strong_replacement_cong [THEN iffD1])
apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]])
apply (simp_all add: formula_replacement1 relativize1_def)
done
lemma (in M_datatypes) formula_closed [intro,simp]:
"M(formula)"
apply (insert formula_replacement1)
apply (simp add: RepFun_closed2 formula_eq_Union
formula_replacement2' relativize1_def
iterates_closed [of "is_formula_functor(M)"])
done
lemma (in M_datatypes) is_formula_n_abs [simp]:
"[|n\<in>nat; M(Z)|]
==> is_formula_n(M,n,Z) <->
Z = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0)"
apply (insert formula_replacement1)
apply (simp add: is_formula_n_def relativize1_def nat_into_M
iterates_abs [of "is_formula_functor(M)" _
"\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))"])
done
lemma (in M_datatypes) mem_formula_abs [simp]:
"mem_formula(M,l) <-> l \<in> formula"
apply (insert formula_replacement1)
apply (simp add: mem_formula_def relativize1_def formula_eq_Union
iterates_closed [of "is_formula_functor(M)"])
done
lemma (in M_datatypes) formula_abs [simp]:
"[|M(Z)|] ==> is_formula(M,Z) <-> Z = formula"
apply (simp add: is_formula_def, safe)
apply (rule M_equalityI, simp_all)
done
subsection{*Absoluteness for @{text \<epsilon>}-Closure: the @{term eclose} Operator*}
text{*Re-expresses eclose using "iterates"*}
lemma eclose_eq_Union:
"eclose(A) = (\<Union>n\<in>nat. Union^n (A))"
apply (simp add: eclose_def)
apply (rule UN_cong)
apply (rule refl)
apply (induct_tac n)
apply (simp add: nat_rec_0)
apply (simp add: nat_rec_succ)
done
constdefs
is_eclose_n :: "[i=>o,i,i,i] => o"
"is_eclose_n(M,A,n,Z) ==
\<exists>sn[M]. \<exists>msn[M].
successor(M,n,sn) & membership(M,sn,msn) &
is_wfrec(M, iterates_MH(M, big_union(M), A), msn, n, Z)"
mem_eclose :: "[i=>o,i,i] => o"
"mem_eclose(M,A,l) ==
\<exists>n[M]. \<exists>eclosen[M].
finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen"
is_eclose :: "[i=>o,i,i] => o"
"is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z <-> mem_eclose(M,A,u)"
locale (open) M_eclose = M_wfrank +
assumes eclose_replacement1:
"M(A) ==> iterates_replacement(M, big_union(M), A)"
and eclose_replacement2:
"M(A) ==> strong_replacement(M,
\<lambda>n y. n\<in>nat &
(\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
is_wfrec(M, iterates_MH(M,big_union(M), A),
msn, n, y)))"
lemma (in M_eclose) eclose_replacement2':
"M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))"
apply (insert eclose_replacement2 [of A])
apply (rule strong_replacement_cong [THEN iffD1])
apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]])
apply (simp_all add: eclose_replacement1 relativize1_def)
done
lemma (in M_eclose) eclose_closed [intro,simp]:
"M(A) ==> M(eclose(A))"
apply (insert eclose_replacement1)
by (simp add: RepFun_closed2 eclose_eq_Union
eclose_replacement2' relativize1_def
iterates_closed [of "big_union(M)"])
lemma (in M_eclose) is_eclose_n_abs [simp]:
"[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) <-> Z = Union^n (A)"
apply (insert eclose_replacement1)
apply (simp add: is_eclose_n_def relativize1_def nat_into_M
iterates_abs [of "big_union(M)" _ "Union"])
done
lemma (in M_eclose) mem_eclose_abs [simp]:
"M(A) ==> mem_eclose(M,A,l) <-> l \<in> eclose(A)"
apply (insert eclose_replacement1)
apply (simp add: mem_eclose_def relativize1_def eclose_eq_Union
iterates_closed [of "big_union(M)"])
done
lemma (in M_eclose) eclose_abs [simp]:
"[|M(A); M(Z)|] ==> is_eclose(M,A,Z) <-> Z = eclose(A)"
apply (simp add: is_eclose_def, safe)
apply (rule M_equalityI, simp_all)
done
subsection {*Absoluteness for @{term transrec}*}
text{* @{term "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"} *}
constdefs
is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o"
"is_transrec(M,MH,a,z) ==
\<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
is_wfrec(M,MH,mesa,a,z)"
transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o"
"transrec_replacement(M,MH,a) ==
\<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
wfrec_replacement(M,MH,mesa)"
(*????????????????Ordinal.thy*)
lemma Transset_trans_Memrel:
"\<forall>j\<in>i. Transset(j) ==> trans(Memrel(i))"
by (unfold Transset_def trans_def, blast)
text{*The condition @{term "Ord(i)"} lets us use the simpler
@{text "trans_wfrec_abs"} rather than @{text "trans_wfrec_abs"},
which I haven't even proved yet. *}
theorem (in M_eclose) transrec_abs:
"[|Ord(i); M(i); M(z);
transrec_replacement(M,MH,i); relativize2(M,MH,H);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> is_transrec(M,MH,i,z) <-> z = transrec(i,H)"
by (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def
transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
theorem (in M_eclose) transrec_closed:
"[|Ord(i); M(i); M(z);
transrec_replacement(M,MH,i); relativize2(M,MH,H);
\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
==> M(transrec(i,H))"
by (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def
transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
end