src/ZF/Constructible/Reflection.thy
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(*  Title:      ZF/Constructible/Reflection.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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*)
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header {* The Reflection Theorem*}
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theory Reflection imports Normal begin
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lemma all_iff_not_ex_not: "(\<forall>x. P(x)) \<longleftrightarrow> (~ (\<exists>x. ~ P(x)))";
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by blast
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lemma ball_iff_not_bex_not: "(\<forall>x\<in>A. P(x)) \<longleftrightarrow> (~ (\<exists>x\<in>A. ~ P(x)))";
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by blast
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text{*From the notes of A. S. Kechris, page 6, and from
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      Andrzej Mostowski, \emph{Constructible Sets with Applications},
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      North-Holland, 1969, page 23.*}
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subsection{*Basic Definitions*}
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text{*First part: the cumulative hierarchy defining the class @{text M}.
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To avoid handling multiple arguments, we assume that @{text "Mset(l)"} is
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closed under ordered pairing provided @{text l} is limit.  Possibly this
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could be avoided: the induction hypothesis @{term Cl_reflects}
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(in locale @{text ex_reflection}) could be weakened to
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@{term "\<forall>y\<in>Mset(a). \<forall>z\<in>Mset(a). P(<y,z>) \<longleftrightarrow> Q(a,<y,z>)"}, removing most
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uses of @{term Pair_in_Mset}.  But there isn't much point in doing so, since
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ultimately the @{text ex_reflection} proof is packaged up using the
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predicate @{text Reflects}.
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*}
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locale reflection =
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  fixes Mset and M and Reflects
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  assumes Mset_mono_le : "mono_le_subset(Mset)"
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      and Mset_cont    : "cont_Ord(Mset)"
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      and Pair_in_Mset : "[| x \<in> Mset(a); y \<in> Mset(a); Limit(a) |]
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                          ==> <x,y> \<in> Mset(a)"
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  defines "M(x) == \<exists>a. Ord(a) & x \<in> Mset(a)"
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      and "Reflects(Cl,P,Q) == Closed_Unbounded(Cl) &
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                              (\<forall>a. Cl(a) \<longrightarrow> (\<forall>x\<in>Mset(a). P(x) \<longleftrightarrow> Q(a,x)))"
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  fixes F0 --{*ordinal for a specific value @{term y}*}
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  fixes FF --{*sup over the whole level, @{term "y\<in>Mset(a)"}*}
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  fixes ClEx --{*Reflecting ordinals for the formula @{term "\<exists>z. P"}*}
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  defines "F0(P,y) == \<mu> b. (\<exists>z. M(z) & P(<y,z>)) \<longrightarrow>
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                               (\<exists>z\<in>Mset(b). P(<y,z>))"
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      and "FF(P)   == \<lambda>a. \<Union>y\<in>Mset(a). F0(P,y)"
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      and "ClEx(P,a) == Limit(a) & normalize(FF(P),a) = a"
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lemma (in reflection) Mset_mono: "i\<le>j ==> Mset(i) \<subseteq> Mset(j)"
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apply (insert Mset_mono_le)
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apply (simp add: mono_le_subset_def leI)
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done
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text{*Awkward: we need a version of @{text ClEx_def} as an equality
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      at the level of classes, which do not really exist*}
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lemma (in reflection) ClEx_eq:
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     "ClEx(P) == \<lambda>a. Limit(a) & normalize(FF(P),a) = a"
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by (simp add: ClEx_def [symmetric])
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subsection{*Easy Cases of the Reflection Theorem*}
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theorem (in reflection) Triv_reflection [intro]:
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     "Reflects(Ord, P, \<lambda>a x. P(x))"
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by (simp add: Reflects_def)
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theorem (in reflection) Not_reflection [intro]:
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     "Reflects(Cl,P,Q) ==> Reflects(Cl, \<lambda>x. ~P(x), \<lambda>a x. ~Q(a,x))"
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by (simp add: Reflects_def)
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theorem (in reflection) And_reflection [intro]:
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     "[| Reflects(Cl,P,Q); Reflects(C',P',Q') |]
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      ==> Reflects(\<lambda>a. Cl(a) & C'(a), \<lambda>x. P(x) & P'(x),
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                                      \<lambda>a x. Q(a,x) & Q'(a,x))"
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by (simp add: Reflects_def Closed_Unbounded_Int, blast)
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theorem (in reflection) Or_reflection [intro]:
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     "[| Reflects(Cl,P,Q); Reflects(C',P',Q') |]
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      ==> Reflects(\<lambda>a. Cl(a) & C'(a), \<lambda>x. P(x) | P'(x),
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                                      \<lambda>a x. Q(a,x) | Q'(a,x))"
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by (simp add: Reflects_def Closed_Unbounded_Int, blast)
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theorem (in reflection) Imp_reflection [intro]:
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     "[| Reflects(Cl,P,Q); Reflects(C',P',Q') |]
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      ==> Reflects(\<lambda>a. Cl(a) & C'(a),
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                   \<lambda>x. P(x) \<longrightarrow> P'(x),
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                   \<lambda>a x. Q(a,x) \<longrightarrow> Q'(a,x))"
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by (simp add: Reflects_def Closed_Unbounded_Int, blast)
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theorem (in reflection) Iff_reflection [intro]:
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     "[| Reflects(Cl,P,Q); Reflects(C',P',Q') |]
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      ==> Reflects(\<lambda>a. Cl(a) & C'(a),
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                   \<lambda>x. P(x) \<longleftrightarrow> P'(x),
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                   \<lambda>a x. Q(a,x) \<longleftrightarrow> Q'(a,x))"
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by (simp add: Reflects_def Closed_Unbounded_Int, blast)
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subsection{*Reflection for Existential Quantifiers*}
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lemma (in reflection) F0_works:
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     "[| y\<in>Mset(a); Ord(a); M(z); P(<y,z>) |] ==> \<exists>z\<in>Mset(F0(P,y)). P(<y,z>)"
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apply (unfold F0_def M_def, clarify)
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apply (rule LeastI2)
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  apply (blast intro: Mset_mono [THEN subsetD])
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 apply (blast intro: lt_Ord2, blast)
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done
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lemma (in reflection) Ord_F0 [intro,simp]: "Ord(F0(P,y))"
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by (simp add: F0_def)
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lemma (in reflection) Ord_FF [intro,simp]: "Ord(FF(P,y))"
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by (simp add: FF_def)
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lemma (in reflection) cont_Ord_FF: "cont_Ord(FF(P))"
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apply (insert Mset_cont)
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apply (simp add: cont_Ord_def FF_def, blast)
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done
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text{*Recall that @{term F0} depends upon @{term "y\<in>Mset(a)"},
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while @{term FF} depends only upon @{term a}. *}
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lemma (in reflection) FF_works:
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     "[| M(z); y\<in>Mset(a); P(<y,z>); Ord(a) |] ==> \<exists>z\<in>Mset(FF(P,a)). P(<y,z>)"
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apply (simp add: FF_def)
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apply (simp_all add: cont_Ord_Union [of concl: Mset]
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                     Mset_cont Mset_mono_le not_emptyI Ord_F0)
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apply (blast intro: F0_works)
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done
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lemma (in reflection) FFN_works:
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     "[| M(z); y\<in>Mset(a); P(<y,z>); Ord(a) |]
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      ==> \<exists>z\<in>Mset(normalize(FF(P),a)). P(<y,z>)"
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apply (drule FF_works [of concl: P], assumption+)
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apply (blast intro: cont_Ord_FF le_normalize [THEN Mset_mono, THEN subsetD])
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done
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text{*Locale for the induction hypothesis*}
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locale ex_reflection = reflection +
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  fixes P  --"the original formula"
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  fixes Q  --"the reflected formula"
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  fixes Cl --"the class of reflecting ordinals"
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  assumes Cl_reflects: "[| Cl(a); Ord(a) |] ==> \<forall>x\<in>Mset(a). P(x) \<longleftrightarrow> Q(a,x)"
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lemma (in ex_reflection) ClEx_downward:
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     "[| M(z); y\<in>Mset(a); P(<y,z>); Cl(a); ClEx(P,a) |]
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      ==> \<exists>z\<in>Mset(a). Q(a,<y,z>)"
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apply (simp add: ClEx_def, clarify)
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apply (frule Limit_is_Ord)
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apply (frule FFN_works [of concl: P], assumption+)
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apply (drule Cl_reflects, assumption+)
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apply (auto simp add: Limit_is_Ord Pair_in_Mset)
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done
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lemma (in ex_reflection) ClEx_upward:
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     "[| z\<in>Mset(a); y\<in>Mset(a); Q(a,<y,z>); Cl(a); ClEx(P,a) |]
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      ==> \<exists>z. M(z) & P(<y,z>)"
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apply (simp add: ClEx_def M_def)
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apply (blast dest: Cl_reflects
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             intro: Limit_is_Ord Pair_in_Mset)
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done
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text{*Class @{text ClEx} indeed consists of reflecting ordinals...*}
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lemma (in ex_reflection) ZF_ClEx_iff:
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     "[| y\<in>Mset(a); Cl(a); ClEx(P,a) |]
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      ==> (\<exists>z. M(z) & P(<y,z>)) \<longleftrightarrow> (\<exists>z\<in>Mset(a). Q(a,<y,z>))"
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by (blast intro: dest: ClEx_downward ClEx_upward)
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text{*...and it is closed and unbounded*}
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lemma (in ex_reflection) ZF_Closed_Unbounded_ClEx:
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     "Closed_Unbounded(ClEx(P))"
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apply (simp add: ClEx_eq)
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apply (fast intro: Closed_Unbounded_Int Normal_imp_fp_Closed_Unbounded
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                   Closed_Unbounded_Limit Normal_normalize)
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done
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text{*The same two theorems, exported to locale @{text reflection}.*}
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text{*Class @{text ClEx} indeed consists of reflecting ordinals...*}
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lemma (in reflection) ClEx_iff:
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     "[| y\<in>Mset(a); Cl(a); ClEx(P,a);
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        !!a. [| Cl(a); Ord(a) |] ==> \<forall>x\<in>Mset(a). P(x) \<longleftrightarrow> Q(a,x) |]
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      ==> (\<exists>z. M(z) & P(<y,z>)) \<longleftrightarrow> (\<exists>z\<in>Mset(a). Q(a,<y,z>))"
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apply (unfold ClEx_def FF_def F0_def M_def)
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apply (rule ex_reflection.ZF_ClEx_iff
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  [OF ex_reflection.intro, OF reflection.intro ex_reflection_axioms.intro,
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    of Mset Cl])
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apply (simp_all add: Mset_mono_le Mset_cont Pair_in_Mset)
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done
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(*Alternative proof, less unfolding:
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apply (rule Reflection.ZF_ClEx_iff [of Mset _ _ Cl, folded M_def])
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apply (fold ClEx_def FF_def F0_def)
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apply (rule ex_reflection.intro, assumption)
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apply (simp add: ex_reflection_axioms.intro, assumption+)
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*)
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lemma (in reflection) Closed_Unbounded_ClEx:
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     "(!!a. [| Cl(a); Ord(a) |] ==> \<forall>x\<in>Mset(a). P(x) \<longleftrightarrow> Q(a,x))
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      ==> Closed_Unbounded(ClEx(P))"
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apply (unfold ClEx_eq FF_def F0_def M_def)
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apply (rule ex_reflection.ZF_Closed_Unbounded_ClEx [of Mset _ _ Cl])
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apply (rule ex_reflection.intro, rule reflection_axioms)
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apply (blast intro: ex_reflection_axioms.intro)
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done
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subsection{*Packaging the Quantifier Reflection Rules*}
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lemma (in reflection) Ex_reflection_0:
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     "Reflects(Cl,P0,Q0)
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      ==> Reflects(\<lambda>a. Cl(a) & ClEx(P0,a),
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                   \<lambda>x. \<exists>z. M(z) & P0(<x,z>),
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                   \<lambda>a x. \<exists>z\<in>Mset(a). Q0(a,<x,z>))"
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apply (simp add: Reflects_def)
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apply (intro conjI Closed_Unbounded_Int)
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  apply blast
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 apply (rule Closed_Unbounded_ClEx [of Cl P0 Q0], blast, clarify)
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apply (rule_tac Cl=Cl in  ClEx_iff, assumption+, blast)
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done
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lemma (in reflection) All_reflection_0:
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     "Reflects(Cl,P0,Q0)
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      ==> Reflects(\<lambda>a. Cl(a) & ClEx(\<lambda>x.~P0(x), a),
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                   \<lambda>x. \<forall>z. M(z) \<longrightarrow> P0(<x,z>),
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                   \<lambda>a x. \<forall>z\<in>Mset(a). Q0(a,<x,z>))"
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apply (simp only: all_iff_not_ex_not ball_iff_not_bex_not)
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apply (rule Not_reflection, drule Not_reflection, simp)
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apply (erule Ex_reflection_0)
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done
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theorem (in reflection) Ex_reflection [intro]:
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     "Reflects(Cl, \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
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      ==> Reflects(\<lambda>a. Cl(a) & ClEx(\<lambda>x. P(fst(x),snd(x)), a),
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                   \<lambda>x. \<exists>z. M(z) & P(x,z),
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                   \<lambda>a x. \<exists>z\<in>Mset(a). Q(a,x,z))"
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by (rule Ex_reflection_0 [of _ " \<lambda>x. P(fst(x),snd(x))"
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                               "\<lambda>a x. Q(a,fst(x),snd(x))", simplified])
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theorem (in reflection) All_reflection [intro]:
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     "Reflects(Cl,  \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
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      ==> Reflects(\<lambda>a. Cl(a) & ClEx(\<lambda>x. ~P(fst(x),snd(x)), a),
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                   \<lambda>x. \<forall>z. M(z) \<longrightarrow> P(x,z),
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                   \<lambda>a x. \<forall>z\<in>Mset(a). Q(a,x,z))"
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by (rule All_reflection_0 [of _ "\<lambda>x. P(fst(x),snd(x))"
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                                "\<lambda>a x. Q(a,fst(x),snd(x))", simplified])
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text{*And again, this time using class-bounded quantifiers*}
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theorem (in reflection) Rex_reflection [intro]:
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     "Reflects(Cl, \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
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      ==> Reflects(\<lambda>a. Cl(a) & ClEx(\<lambda>x. P(fst(x),snd(x)), a),
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                   \<lambda>x. \<exists>z[M]. P(x,z),
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                   \<lambda>a x. \<exists>z\<in>Mset(a). Q(a,x,z))"
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by (unfold rex_def, blast)
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theorem (in reflection) Rall_reflection [intro]:
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     "Reflects(Cl,  \<lambda>x. P(fst(x),snd(x)), \<lambda>a x. Q(a,fst(x),snd(x)))
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      ==> Reflects(\<lambda>a. Cl(a) & ClEx(\<lambda>x. ~P(fst(x),snd(x)), a),
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                   \<lambda>x. \<forall>z[M]. P(x,z),
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                   \<lambda>a x. \<forall>z\<in>Mset(a). Q(a,x,z))"
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by (unfold rall_def, blast)
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text{*No point considering bounded quantifiers, where reflection is trivial.*}
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subsection{*Simple Examples of Reflection*}
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text{*Example 1: reflecting a simple formula.  The reflecting class is first
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given as the variable @{text ?Cl} and later retrieved from the final
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proof state.*}
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schematic_lemma (in reflection)
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     "Reflects(?Cl,
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               \<lambda>x. \<exists>y. M(y) & x \<in> y,
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               \<lambda>a x. \<exists>y\<in>Mset(a). x \<in> y)"
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by fast
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text{*Problem here: there needs to be a conjunction (class intersection)
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in the class of reflecting ordinals.  The @{term "Ord(a)"} is redundant,
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though harmless.*}
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lemma (in reflection)
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     "Reflects(\<lambda>a. Ord(a) & ClEx(\<lambda>x. fst(x) \<in> snd(x), a),
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               \<lambda>x. \<exists>y. M(y) & x \<in> y,
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               \<lambda>a x. \<exists>y\<in>Mset(a). x \<in> y)"
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text{*Example 2*}
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schematic_lemma (in reflection)
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     "Reflects(?Cl,
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               \<lambda>x. \<exists>y. M(y) & (\<forall>z. M(z) \<longrightarrow> z \<subseteq> x \<longrightarrow> z \<in> y),
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               \<lambda>a x. \<exists>y\<in>Mset(a). \<forall>z\<in>Mset(a). z \<subseteq> x \<longrightarrow> z \<in> y)"
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by fast
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text{*Example 2'.  We give the reflecting class explicitly. *}
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lemma (in reflection)
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  "Reflects
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    (\<lambda>a. (Ord(a) &
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          ClEx(\<lambda>x. ~ (snd(x) \<subseteq> fst(fst(x)) \<longrightarrow> snd(x) \<in> snd(fst(x))), a)) &
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          ClEx(\<lambda>x. \<forall>z. M(z) \<longrightarrow> z \<subseteq> fst(x) \<longrightarrow> z \<in> snd(x), a),
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            \<lambda>x. \<exists>y. M(y) & (\<forall>z. M(z) \<longrightarrow> z \<subseteq> x \<longrightarrow> z \<in> y),
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            \<lambda>a x. \<exists>y\<in>Mset(a). \<forall>z\<in>Mset(a). z \<subseteq> x \<longrightarrow> z \<in> y)"
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by fast
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text{*Example 2''.  We expand the subset relation.*}
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schematic_lemma (in reflection)
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  "Reflects(?Cl,
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        \<lambda>x. \<exists>y. M(y) & (\<forall>z. M(z) \<longrightarrow> (\<forall>w. M(w) \<longrightarrow> w\<in>z \<longrightarrow> w\<in>x) \<longrightarrow> z\<in>y),
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        \<lambda>a x. \<exists>y\<in>Mset(a). \<forall>z\<in>Mset(a). (\<forall>w\<in>Mset(a). w\<in>z \<longrightarrow> w\<in>x) \<longrightarrow> z\<in>y)"
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text{*Example 2'''.  Single-step version, to reveal the reflecting class.*}
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schematic_lemma (in reflection)
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     "Reflects(?Cl,
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               \<lambda>x. \<exists>y. M(y) & (\<forall>z. M(z) \<longrightarrow> z \<subseteq> x \<longrightarrow> z \<in> y),
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               \<lambda>a x. \<exists>y\<in>Mset(a). \<forall>z\<in>Mset(a). z \<subseteq> x \<longrightarrow> z \<in> y)"
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apply (rule Ex_reflection)
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txt{*
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@{goals[display,indent=0,margin=60]}
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*}
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apply (rule All_reflection)
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txt{*
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@{goals[display,indent=0,margin=60]}
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*}
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apply (rule Triv_reflection)
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txt{*
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@{goals[display,indent=0,margin=60]}
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*}
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done
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text{*Example 3.  Warning: the following examples make sense only
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if @{term P} is quantifier-free, since it is not being relativized.*}
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     "Reflects(?Cl,
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               \<lambda>x. \<exists>y. M(y) & (\<forall>z. M(z) \<longrightarrow> z \<in> y \<longleftrightarrow> z \<in> x & P(z)),
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               \<lambda>a x. \<exists>y\<in>Mset(a). \<forall>z\<in>Mset(a). z \<in> y \<longleftrightarrow> z \<in> x & P(z))"
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by fast
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text{*Example 3'*}
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     "Reflects(?Cl,
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               \<lambda>x. \<exists>y. M(y) & y = Collect(x,P),
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               \<lambda>a x. \<exists>y\<in>Mset(a). y = Collect(x,P))";
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text{*Example 3''*}
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     "Reflects(?Cl,
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               \<lambda>x. \<exists>y. M(y) & y = Replace(x,P),
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               \<lambda>a x. \<exists>y\<in>Mset(a). y = Replace(x,P))";
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text{*Example 4: Axiom of Choice.  Possibly wrong, since @{text \<Pi>} needs
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to be relativized.*}
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schematic_lemma (in reflection)
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     "Reflects(?Cl,
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               \<lambda>A. 0\<notin>A \<longrightarrow> (\<exists>f. M(f) & f \<in> (\<Pi> X \<in> A. X)),
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               \<lambda>a A. 0\<notin>A \<longrightarrow> (\<exists>f\<in>Mset(a). f \<in> (\<Pi> X \<in> A. X)))"
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end
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