src/ZF/OrdQuant.thy
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(*  Title:      ZF/AC/OrdQuant.thy
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    ID:         $Id$
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    Authors:    Krzysztof Grabczewski and L C Paulson
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*)
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header {*Special quantifiers*}
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theory OrdQuant = Ordinal:
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subsection {*Quantifiers and union operator for ordinals*}
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constdefs
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  (* Ordinal Quantifiers *)
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  oall :: "[i, i => o] => o"
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    "oall(A, P) == ALL x. x<A --> P(x)"
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  oex :: "[i, i => o] => o"
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    "oex(A, P)  == EX x. x<A & P(x)"
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  (* Ordinal Union *)
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  OUnion :: "[i, i => i] => i"
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    "OUnion(i,B) == {z: \<Union>x\<in>i. B(x). Ord(i)}"
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syntax
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  "@oall"     :: "[idt, i, o] => o"        ("(3ALL _<_./ _)" 10)
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  "@oex"      :: "[idt, i, o] => o"        ("(3EX _<_./ _)" 10)
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  "@OUNION"   :: "[idt, i, i] => i"        ("(3UN _<_./ _)" 10)
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translations
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  "ALL x<a. P"  == "oall(a, %x. P)"
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  "EX x<a. P"   == "oex(a, %x. P)"
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  "UN x<a. B"   == "OUnion(a, %x. B)"
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syntax (xsymbols)
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  "@oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
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  "@oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
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  "@OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
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syntax (HTML output)
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  "@oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
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  "@oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
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  "@OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
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subsubsection {*simplification of the new quantifiers*}
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(*MOST IMPORTANT that this is added to the simpset BEFORE Ord_atomize
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  is proved.  Ord_atomize would convert this rule to
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    x < 0 ==> P(x) == True, which causes dire effects!*)
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lemma [simp]: "(ALL x<0. P(x))"
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by (simp add: oall_def)
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lemma [simp]: "~(EX x<0. P(x))"
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by (simp add: oex_def)
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lemma [simp]: "(ALL x<succ(i). P(x)) <-> (Ord(i) --> P(i) & (ALL x<i. P(x)))"
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apply (simp add: oall_def le_iff)
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apply (blast intro: lt_Ord2)
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done
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lemma [simp]: "(EX x<succ(i). P(x)) <-> (Ord(i) & (P(i) | (EX x<i. P(x))))"
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apply (simp add: oex_def le_iff)
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apply (blast intro: lt_Ord2)
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done
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subsubsection {*Union over ordinals*}
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lemma Ord_OUN [intro,simp]:
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     "[| !!x. x<A ==> Ord(B(x)) |] ==> Ord(\<Union>x<A. B(x))"
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by (simp add: OUnion_def ltI Ord_UN)
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lemma OUN_upper_lt:
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     "[| a<A;  i < b(a);  Ord(\<Union>x<A. b(x)) |] ==> i < (\<Union>x<A. b(x))"
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by (unfold OUnion_def lt_def, blast )
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lemma OUN_upper_le:
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     "[| a<A;  i\<le>b(a);  Ord(\<Union>x<A. b(x)) |] ==> i \<le> (\<Union>x<A. b(x))"
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apply (unfold OUnion_def, auto)
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apply (rule UN_upper_le )
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apply (auto simp add: lt_def)
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done
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lemma Limit_OUN_eq: "Limit(i) ==> (\<Union>x<i. x) = i"
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by (simp add: OUnion_def Limit_Union_eq Limit_is_Ord)
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(* No < version; consider (\<Union>i\<in>nat.i)=nat *)
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lemma OUN_least:
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     "(!!x. x<A ==> B(x) \<subseteq> C) ==> (\<Union>x<A. B(x)) \<subseteq> C"
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by (simp add: OUnion_def UN_least ltI)
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(* No < version; consider (\<Union>i\<in>nat.i)=nat *)
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lemma OUN_least_le:
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     "[| Ord(i);  !!x. x<A ==> b(x) \<le> i |] ==> (\<Union>x<A. b(x)) \<le> i"
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by (simp add: OUnion_def UN_least_le ltI Ord_0_le)
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lemma le_implies_OUN_le_OUN:
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     "[| !!x. x<A ==> c(x) \<le> d(x) |] ==> (\<Union>x<A. c(x)) \<le> (\<Union>x<A. d(x))"
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by (blast intro: OUN_least_le OUN_upper_le le_Ord2 Ord_OUN)
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lemma OUN_UN_eq:
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     "(!!x. x:A ==> Ord(B(x)))
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      ==> (\<Union>z < (\<Union>x\<in>A. B(x)). C(z)) = (\<Union>x\<in>A. \<Union>z < B(x). C(z))"
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by (simp add: OUnion_def)
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lemma OUN_Union_eq:
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     "(!!x. x:X ==> Ord(x))
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      ==> (\<Union>z < Union(X). C(z)) = (\<Union>x\<in>X. \<Union>z < x. C(z))"
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by (simp add: OUnion_def)
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(*So that rule_format will get rid of ALL x<A...*)
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lemma atomize_oall [symmetric, rulify]:
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     "(!!x. x<A ==> P(x)) == Trueprop (ALL x<A. P(x))"
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by (simp add: oall_def atomize_all atomize_imp)
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subsubsection {*universal quantifier for ordinals*}
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lemma oallI [intro!]:
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    "[| !!x. x<A ==> P(x) |] ==> ALL x<A. P(x)"
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by (simp add: oall_def)
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lemma ospec: "[| ALL x<A. P(x);  x<A |] ==> P(x)"
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by (simp add: oall_def)
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lemma oallE:
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    "[| ALL x<A. P(x);  P(x) ==> Q;  ~x<A ==> Q |] ==> Q"
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by (simp add: oall_def, blast)
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lemma rev_oallE [elim]:
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    "[| ALL x<A. P(x);  ~x<A ==> Q;  P(x) ==> Q |] ==> Q"
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by (simp add: oall_def, blast)
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(*Trival rewrite rule;   (ALL x<a.P)<->P holds only if a is not 0!*)
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lemma oall_simp [simp]: "(ALL x<a. True) <-> True"
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by blast
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(*Congruence rule for rewriting*)
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lemma oall_cong [cong]:
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    "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
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     ==> oall(a, %x. P(x)) <-> oall(a', %x. P'(x))"
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by (simp add: oall_def)
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subsubsection {*existential quantifier for ordinals*}
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lemma oexI [intro]:
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    "[| P(x);  x<A |] ==> EX x<A. P(x)"
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apply (simp add: oex_def, blast)
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done
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(*Not of the general form for such rules; ~EX has become ALL~ *)
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lemma oexCI:
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   "[| ALL x<A. ~P(x) ==> P(a);  a<A |] ==> EX x<A. P(x)"
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apply (simp add: oex_def, blast)
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done
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lemma oexE [elim!]:
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    "[| EX x<A. P(x);  !!x. [| x<A; P(x) |] ==> Q |] ==> Q"
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apply (simp add: oex_def, blast)
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done
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lemma oex_cong [cong]:
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    "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
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     ==> oex(a, %x. P(x)) <-> oex(a', %x. P'(x))"
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apply (simp add: oex_def cong add: conj_cong)
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done
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subsubsection {*Rules for Ordinal-Indexed Unions*}
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lemma OUN_I [intro]: "[| a<i;  b: B(a) |] ==> b: (\<Union>z<i. B(z))"
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by (unfold OUnion_def lt_def, blast)
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lemma OUN_E [elim!]:
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    "[| b : (\<Union>z<i. B(z));  !!a.[| b: B(a);  a<i |] ==> R |] ==> R"
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apply (unfold OUnion_def lt_def, blast)
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done
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lemma OUN_iff: "b : (\<Union>x<i. B(x)) <-> (EX x<i. b : B(x))"
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by (unfold OUnion_def oex_def lt_def, blast)
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lemma OUN_cong [cong]:
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    "[| i=j;  !!x. x<j ==> C(x)=D(x) |] ==> (\<Union>x<i. C(x)) = (\<Union>x<j. D(x))"
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by (simp add: OUnion_def lt_def OUN_iff)
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lemma lt_induct:
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    "[| i<k;  !!x.[| x<k;  ALL y<x. P(y) |] ==> P(x) |]  ==>  P(i)"
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apply (simp add: lt_def oall_def)
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apply (erule conjE)
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apply (erule Ord_induct, assumption, blast)
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done
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subsection {*Quantification over a class*}
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constdefs
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  "rall"     :: "[i=>o, i=>o] => o"
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    "rall(M, P) == ALL x. M(x) --> P(x)"
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  "rex"      :: "[i=>o, i=>o] => o"
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    "rex(M, P) == EX x. M(x) & P(x)"
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syntax
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  "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3ALL _[_]./ _)" 10)
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  "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3EX _[_]./ _)" 10)
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syntax (xsymbols)
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  "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
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  "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
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syntax (HTML output)
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  "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
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  "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
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translations
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  "ALL x[M]. P"  == "rall(M, %x. P)"
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  "EX x[M]. P"   == "rex(M, %x. P)"
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subsubsection{*Relativized universal quantifier*}
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lemma rallI [intro!]: "[| !!x. M(x) ==> P(x) |] ==> ALL x[M]. P(x)"
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by (simp add: rall_def)
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lemma rspec: "[| ALL x[M]. P(x); M(x) |] ==> P(x)"
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by (simp add: rall_def)
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(*Instantiates x first: better for automatic theorem proving?*)
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lemma rev_rallE [elim]:
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    "[| ALL x[M]. P(x);  ~ M(x) ==> Q;  P(x) ==> Q |] ==> Q"
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by (simp add: rall_def, blast)
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lemma rallE: "[| ALL x[M]. P(x);  P(x) ==> Q;  ~ M(x) ==> Q |] ==> Q"
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by blast
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(*Trival rewrite rule;   (ALL x[M].P)<->P holds only if A is nonempty!*)
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lemma rall_triv [simp]: "(ALL x[M]. P) <-> ((EX x. M(x)) --> P)"
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by (simp add: rall_def)
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(*Congruence rule for rewriting*)
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lemma rall_cong [cong]:
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    "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (ALL x[M]. P(x)) <-> (ALL x[M]. P'(x))"
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by (simp add: rall_def)
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subsubsection{*Relativized existential quantifier*}
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lemma rexI [intro]: "[| P(x); M(x) |] ==> EX x[M]. P(x)"
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by (simp add: rex_def, blast)
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(*The best argument order when there is only one M(x)*)
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lemma rev_rexI: "[| M(x);  P(x) |] ==> EX x[M]. P(x)"
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by blast
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(*Not of the general form for such rules; ~EX has become ALL~ *)
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lemma rexCI: "[| ALL x[M]. ~P(x) ==> P(a); M(a) |] ==> EX x[M]. P(x)"
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by blast
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lemma rexE [elim!]: "[| EX x[M]. P(x);  !!x. [| M(x); P(x) |] ==> Q |] ==> Q"
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by (simp add: rex_def, blast)
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(*We do not even have (EX x[M]. True) <-> True unless A is nonempty!!*)
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lemma rex_triv [simp]: "(EX x[M]. P) <-> ((EX x. M(x)) & P)"
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by (simp add: rex_def)
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lemma rex_cong [cong]:
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    "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (EX x[M]. P(x)) <-> (EX x[M]. P'(x))"
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by (simp add: rex_def cong: conj_cong)
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lemma rall_is_ball [simp]: "(\<forall>x[%z. z\<in>A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
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by blast
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lemma rex_is_bex [simp]: "(\<exists>x[%z. z\<in>A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
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by blast
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lemma atomize_rall: "(!!x. M(x) ==> P(x)) == Trueprop (ALL x[M]. P(x))";
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by (simp add: rall_def atomize_all atomize_imp)
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declare atomize_rall [symmetric, rulify]
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lemma rall_simps1:
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     "(ALL x[M]. P(x) & Q)   <-> (ALL x[M]. P(x)) & ((ALL x[M]. False) | Q)"
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     "(ALL x[M]. P(x) | Q)   <-> ((ALL x[M]. P(x)) | Q)"
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     "(ALL x[M]. P(x) --> Q) <-> ((EX x[M]. P(x)) --> Q)"
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     "(~(ALL x[M]. P(x))) <-> (EX x[M]. ~P(x))"
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by blast+
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lemma rall_simps2:
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     "(ALL x[M]. P & Q(x))   <-> ((ALL x[M]. False) | P) & (ALL x[M]. Q(x))"
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     "(ALL x[M]. P | Q(x))   <-> (P | (ALL x[M]. Q(x)))"
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     "(ALL x[M]. P --> Q(x)) <-> (P --> (ALL x[M]. Q(x)))"
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by blast+
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lemmas rall_simps [simp] = rall_simps1 rall_simps2
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lemma rall_conj_distrib:
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    "(ALL x[M]. P(x) & Q(x)) <-> ((ALL x[M]. P(x)) & (ALL x[M]. Q(x)))"
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by blast
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lemma rex_simps1:
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     "(EX x[M]. P(x) & Q) <-> ((EX x[M]. P(x)) & Q)"
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     "(EX x[M]. P(x) | Q) <-> (EX x[M]. P(x)) | ((EX x[M]. True) & Q)"
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     "(EX x[M]. P(x) --> Q) <-> ((ALL x[M]. P(x)) --> ((EX x[M]. True) & Q))"
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     "(~(EX x[M]. P(x))) <-> (ALL x[M]. ~P(x))"
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by blast+
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lemma rex_simps2:
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     "(EX x[M]. P & Q(x)) <-> (P & (EX x[M]. Q(x)))"
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     "(EX x[M]. P | Q(x)) <-> ((EX x[M]. True) & P) | (EX x[M]. Q(x))"
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     "(EX x[M]. P --> Q(x)) <-> (((ALL x[M]. False) | P) --> (EX x[M]. Q(x)))"
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by blast+
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lemmas rex_simps [simp] = rex_simps1 rex_simps2
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lemma rex_disj_distrib:
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    "(EX x[M]. P(x) | Q(x)) <-> ((EX x[M]. P(x)) | (EX x[M]. Q(x)))"
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by blast
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subsubsection{*One-point rule for bounded quantifiers*}
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lemma rex_triv_one_point1 [simp]: "(EX x[M]. x=a) <-> ( M(a))"
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by blast
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lemma rex_triv_one_point2 [simp]: "(EX x[M]. a=x) <-> ( M(a))"
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by blast
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lemma rex_one_point1 [simp]: "(EX x[M]. x=a & P(x)) <-> ( M(a) & P(a))"
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by blast
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lemma rex_one_point2 [simp]: "(EX x[M]. a=x & P(x)) <-> ( M(a) & P(a))"
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by blast
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lemma rall_one_point1 [simp]: "(ALL x[M]. x=a --> P(x)) <-> ( M(a) --> P(a))"
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by blast
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lemma rall_one_point2 [simp]: "(ALL x[M]. a=x --> P(x)) <-> ( M(a) --> P(a))"
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by blast
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subsubsection{*Sets as Classes*}
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constdefs setclass :: "[i,i] => o"       ("##_" [40] 40)
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   "setclass(A) == %x. x : A"
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lemma setclass_iff [simp]: "setclass(A,x) <-> x : A"
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by (simp add: setclass_def)
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lemma rall_setclass_is_ball [simp]: "(\<forall>x[##A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
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by auto
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lemma rex_setclass_is_bex [simp]: "(\<exists>x[##A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
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by auto
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ML
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{*
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val oall_def = thm "oall_def"
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val oex_def = thm "oex_def"
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val OUnion_def = thm "OUnion_def"
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val oallI = thm "oallI";
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val ospec = thm "ospec";
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val oallE = thm "oallE";
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val rev_oallE = thm "rev_oallE";
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val oall_simp = thm "oall_simp";
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val oall_cong = thm "oall_cong";
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val oexI = thm "oexI";
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val oexCI = thm "oexCI";
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val oexE = thm "oexE";
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val oex_cong = thm "oex_cong";
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val OUN_I = thm "OUN_I";
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val OUN_E = thm "OUN_E";
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val OUN_iff = thm "OUN_iff";
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val OUN_cong = thm "OUN_cong";
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val lt_induct = thm "lt_induct";
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val rall_def = thm "rall_def"
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val rex_def = thm "rex_def"
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val rallI = thm "rallI";
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val rspec = thm "rspec";
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val rallE = thm "rallE";
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val rev_oallE = thm "rev_oallE";
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val rall_cong = thm "rall_cong";
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val rexI = thm "rexI";
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val rexCI = thm "rexCI";
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val rexE = thm "rexE";
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val rex_cong = thm "rex_cong";
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val Ord_atomize =
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    atomize ([("OrdQuant.oall", [ospec]),("OrdQuant.rall", [rspec])]@
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                 ZF_conn_pairs,
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             ZF_mem_pairs);
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simpset_ref() := simpset() setmksimps (map mk_eq o Ord_atomize o gen_all);
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*}
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text {* Setting up the one-point-rule simproc *}
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ML_setup {*
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local
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   402
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val prove_rex_tac = rewtac rex_def THEN Quantifier1.prove_one_point_ex_tac;
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val rearrange_bex = Quantifier1.rearrange_bex prove_rex_tac;
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   405
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val prove_rall_tac = rewtac rall_def THEN Quantifier1.prove_one_point_all_tac;
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val rearrange_ball = Quantifier1.rearrange_ball prove_rall_tac;
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   408
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in
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val defREX_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
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  "defined REX" ["EX x[M]. P(x) & Q(x)"] rearrange_bex;
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val defRALL_regroup = Simplifier.simproc (Theory.sign_of (the_context ()))
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  "defined RALL" ["ALL x[M]. P(x) --> Q(x)"] rearrange_ball;
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end;
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Addsimprocs [defRALL_regroup,defREX_regroup];
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*}
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end