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