(* Title: ZF/AC/OrdQuant.thy ID: $Id: OrdQuant.thy,v 1.36 2007/10/07 19:19:32 wenzelm Exp $ Authors: Krzysztof Grabczewski and L C Paulson *) header {*Special quantifiers*} theory OrdQuant imports Ordinal begin subsection {*Quantifiers and union operator for ordinals*} definition (* Ordinal Quantifiers *) oall :: "[i, i => o] => o" where "oall(A, P) == ALL x. x<A --> P(x)" definition oex :: "[i, i => o] => o" where "oex(A, P) == EX x. x<A & P(x)" definition (* Ordinal Union *) OUnion :: "[i, i => i] => i" where "OUnion(i,B) == {z: \<Union>x∈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∀_<_./ _)" 10) "@oex" :: "[idt, i, o] => o" ("(3∃_<_./ _)" 10) "@OUNION" :: "[idt, i, i] => i" ("(3\<Union>_<_./ _)" 10) syntax (HTML output) "@oall" :: "[idt, i, o] => o" ("(3∀_<_./ _)" 10) "@oex" :: "[idt, i, o] => o" ("(3∃_<_./ _)" 10) "@OUNION" :: "[idt, i, i] => i" ("(3\<Union>_<_./ _)" 10) subsubsection {*simplification of the new quantifiers*} (*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]: "(ALL x<0. P(x))" by (simp add: oall_def) lemma [simp]: "~(EX x<0. P(x))" by (simp add: oex_def) lemma [simp]: "(ALL x<succ(i). P(x)) <-> (Ord(i) --> P(i) & (ALL x<i. P(x)))" apply (simp add: oall_def le_iff) apply (blast intro: lt_Ord2) done lemma [simp]: "(EX x<succ(i). P(x)) <-> (Ord(i) & (P(i) | (EX x<i. P(x))))" apply (simp add: oex_def le_iff) apply (blast intro: lt_Ord2) done subsubsection {*Union over ordinals*} 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≤b(a); Ord(\<Union>x<A. b(x)) |] ==> i ≤ (\<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; consider (\<Union>i∈nat.i)=nat *) lemma OUN_least: "(!!x. x<A ==> B(x) ⊆ C) ==> (\<Union>x<A. B(x)) ⊆ C" by (simp add: OUnion_def UN_least ltI) (* No < version; consider (\<Union>i∈nat.i)=nat *) lemma OUN_least_le: "[| Ord(i); !!x. x<A ==> b(x) ≤ i |] ==> (\<Union>x<A. b(x)) ≤ i" by (simp add: OUnion_def UN_least_le ltI Ord_0_le) lemma le_implies_OUN_le_OUN: "[| !!x. x<A ==> c(x) ≤ d(x) |] ==> (\<Union>x<A. c(x)) ≤ (\<Union>x<A. d(x))" by (blast intro: OUN_least_le OUN_upper_le le_Ord2 Ord_OUN) lemma OUN_UN_eq: "(!!x. x:A ==> Ord(B(x))) ==> (\<Union>z < (\<Union>x∈A. B(x)). C(z)) = (\<Union>x∈A. \<Union>z < B(x). C(z))" by (simp add: OUnion_def) lemma OUN_Union_eq: "(!!x. x:X ==> Ord(x)) ==> (\<Union>z < Union(X). C(z)) = (\<Union>x∈X. \<Union>z < x. C(z))" by (simp add: OUnion_def) (*So that rule_format will get rid of ALL x<A...*) lemma atomize_oall [symmetric, rulify]: "(!!x. x<A ==> P(x)) == Trueprop (ALL x<A. P(x))" by (simp add: oall_def atomize_all atomize_imp) subsubsection {*universal quantifier for ordinals*} lemma oallI [intro!]: "[| !!x. x<A ==> P(x) |] ==> ALL x<A. P(x)" by (simp add: oall_def) lemma ospec: "[| ALL x<A. P(x); x<A |] ==> P(x)" by (simp add: oall_def) lemma oallE: "[| ALL x<A. P(x); P(x) ==> Q; ~x<A ==> Q |] ==> Q" by (simp add: oall_def, blast) lemma rev_oallE [elim]: "[| ALL x<A. P(x); ~x<A ==> Q; P(x) ==> Q |] ==> Q" by (simp add: oall_def, blast) (*Trival rewrite rule; (ALL x<a.P)<->P holds only if a is not 0!*) lemma oall_simp [simp]: "(ALL 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 {*existential quantifier for ordinals*} lemma oexI [intro]: "[| P(x); x<A |] ==> EX x<A. P(x)" apply (simp add: oex_def, blast) done (*Not of the general form for such rules; ~EX has become ALL~ *) lemma oexCI: "[| ALL x<A. ~P(x) ==> P(a); a<A |] ==> EX x<A. P(x)" apply (simp add: oex_def, blast) done lemma oexE [elim!]: "[| EX 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 {*Rules for Ordinal-Indexed Unions*} lemma OUN_I [intro]: "[| a<i; b: B(a) |] ==> b: (\<Union>z<i. B(z))" by (unfold OUnion_def lt_def, blast) lemma OUN_E [elim!]: "[| b : (\<Union>z<i. B(z)); !!a.[| b: B(a); a<i |] ==> R |] ==> R" apply (unfold OUnion_def lt_def, blast) done lemma OUN_iff: "b : (\<Union>x<i. B(x)) <-> (EX x<i. b : 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; ALL 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 {*Quantification over a class*} definition "rall" :: "[i=>o, i=>o] => o" where "rall(M, P) == ALL x. M(x) --> P(x)" definition "rex" :: "[i=>o, i=>o] => o" where "rex(M, P) == EX 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∀_[_]./ _)" 10) "@rex" :: "[pttrn, i=>o, o] => o" ("(3∃_[_]./ _)" 10) syntax (HTML output) "@rall" :: "[pttrn, i=>o, o] => o" ("(3∀_[_]./ _)" 10) "@rex" :: "[pttrn, i=>o, o] => o" ("(3∃_[_]./ _)" 10) translations "ALL x[M]. P" == "CONST rall(M, %x. P)" "EX x[M]. P" == "CONST rex(M, %x. P)" subsubsection{*Relativized universal quantifier*} lemma rallI [intro!]: "[| !!x. M(x) ==> P(x) |] ==> ALL x[M]. P(x)" by (simp add: rall_def) lemma rspec: "[| ALL 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]: "[| ALL x[M]. P(x); ~ M(x) ==> Q; P(x) ==> Q |] ==> Q" by (simp add: rall_def, blast) lemma rallE: "[| ALL 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)) ==> (ALL x[M]. P(x)) <-> (ALL x[M]. P'(x))" by (simp add: rall_def) subsubsection{*Relativized existential quantifier*} lemma rexI [intro]: "[| P(x); M(x) |] ==> EX 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) |] ==> EX x[M]. P(x)" by blast (*Not of the general form for such rules; ~EX has become ALL~ *) lemma rexCI: "[| ALL x[M]. ~P(x) ==> P(a); M(a) |] ==> EX x[M]. P(x)" by blast lemma rexE [elim!]: "[| EX 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)) ==> (EX x[M]. P(x)) <-> (EX x[M]. P'(x))" by (simp add: rex_def cong: conj_cong) lemma rall_is_ball [simp]: "(∀x[%z. z∈A]. P(x)) <-> (∀x∈A. P(x))" by blast lemma rex_is_bex [simp]: "(∃x[%z. z∈A]. P(x)) <-> (∃x∈A. P(x))" by blast lemma atomize_rall: "(!!x. M(x) ==> P(x)) == Trueprop (ALL x[M]. P(x))"; by (simp add: rall_def atomize_all atomize_imp) declare atomize_rall [symmetric, rulify] lemma rall_simps1: "(ALL x[M]. P(x) & Q) <-> (ALL x[M]. P(x)) & ((ALL x[M]. False) | Q)" "(ALL x[M]. P(x) | Q) <-> ((ALL x[M]. P(x)) | Q)" "(ALL x[M]. P(x) --> Q) <-> ((EX x[M]. P(x)) --> Q)" "(~(ALL x[M]. P(x))) <-> (EX x[M]. ~P(x))" by blast+ lemma rall_simps2: "(ALL x[M]. P & Q(x)) <-> ((ALL x[M]. False) | P) & (ALL x[M]. Q(x))" "(ALL x[M]. P | Q(x)) <-> (P | (ALL x[M]. Q(x)))" "(ALL x[M]. P --> Q(x)) <-> (P --> (ALL x[M]. Q(x)))" by blast+ lemmas rall_simps [simp] = rall_simps1 rall_simps2 lemma rall_conj_distrib: "(ALL x[M]. P(x) & Q(x)) <-> ((ALL x[M]. P(x)) & (ALL x[M]. Q(x)))" by blast lemma rex_simps1: "(EX x[M]. P(x) & Q) <-> ((EX x[M]. P(x)) & Q)" "(EX x[M]. P(x) | Q) <-> (EX x[M]. P(x)) | ((EX x[M]. True) & Q)" "(EX x[M]. P(x) --> Q) <-> ((ALL x[M]. P(x)) --> ((EX x[M]. True) & Q))" "(~(EX x[M]. P(x))) <-> (ALL x[M]. ~P(x))" by blast+ lemma rex_simps2: "(EX x[M]. P & Q(x)) <-> (P & (EX x[M]. Q(x)))" "(EX x[M]. P | Q(x)) <-> ((EX x[M]. True) & P) | (EX x[M]. Q(x))" "(EX x[M]. P --> Q(x)) <-> (((ALL x[M]. False) | P) --> (EX x[M]. Q(x)))" by blast+ lemmas rex_simps [simp] = rex_simps1 rex_simps2 lemma rex_disj_distrib: "(EX x[M]. P(x) | Q(x)) <-> ((EX x[M]. P(x)) | (EX x[M]. Q(x)))" by blast subsubsection{*One-point rule for bounded quantifiers*} lemma rex_triv_one_point1 [simp]: "(EX x[M]. x=a) <-> ( M(a))" by blast lemma rex_triv_one_point2 [simp]: "(EX x[M]. a=x) <-> ( M(a))" by blast lemma rex_one_point1 [simp]: "(EX x[M]. x=a & P(x)) <-> ( M(a) & P(a))" by blast lemma rex_one_point2 [simp]: "(EX x[M]. a=x & P(x)) <-> ( M(a) & P(a))" by blast lemma rall_one_point1 [simp]: "(ALL x[M]. x=a --> P(x)) <-> ( M(a) --> P(a))" by blast lemma rall_one_point2 [simp]: "(ALL x[M]. a=x --> P(x)) <-> ( M(a) --> P(a))" by blast subsubsection{*Sets as Classes*} definition setclass :: "[i,i] => o" ("##_" [40] 40) where "setclass(A) == %x. x : A" lemma setclass_iff [simp]: "setclass(A,x) <-> x : A" by (simp add: setclass_def) lemma rall_setclass_is_ball [simp]: "(∀x[##A]. P(x)) <-> (∀x∈A. P(x))" by auto lemma rex_setclass_is_bex [simp]: "(∃x[##A]. P(x)) <-> (∃x∈A. P(x))" by auto ML {* val Ord_atomize = atomize ([("OrdQuant.oall", [@{thm ospec}]),("OrdQuant.rall", [@{thm rspec}])]@ ZF_conn_pairs, ZF_mem_pairs); change_simpset (fn ss => ss setmksimps (map mk_eq o Ord_atomize o gen_all)); *} text {* Setting up the one-point-rule simproc *} ML_setup {* local val unfold_rex_tac = unfold_tac [@{thm rex_def}]; fun prove_rex_tac ss = unfold_rex_tac ss THEN Quantifier1.prove_one_point_ex_tac; val rearrange_bex = Quantifier1.rearrange_bex prove_rex_tac; val unfold_rall_tac = unfold_tac [@{thm rall_def}]; fun prove_rall_tac ss = unfold_rall_tac ss THEN Quantifier1.prove_one_point_all_tac; val rearrange_ball = Quantifier1.rearrange_ball prove_rall_tac; in val defREX_regroup = Simplifier.simproc @{theory} "defined REX" ["EX x[M]. P(x) & Q(x)"] rearrange_bex; val defRALL_regroup = Simplifier.simproc @{theory} "defined RALL" ["ALL x[M]. P(x) --> Q(x)"] rearrange_ball; end; Addsimprocs [defRALL_regroup,defREX_regroup]; *} end
lemma
∀x<0. P(x)
lemma
¬ (∃x<0. P(x))
lemma
(∀x<succ(i). P(x)) <-> Ord(i) --> P(i) ∧ (∀x<i. P(x))
lemma
(∃x<succ(i). P(x)) <-> Ord(i) ∧ (P(i) ∨ (∃x<i. P(x)))
lemma Ord_OUN:
(!!x. x < A ==> Ord(B(x))) ==> Ord(\<Union>x<A. B(x))
lemma OUN_upper_lt:
[| a < A; i < b(a); Ord(\<Union>x<A. b(x)) |] ==> i < (\<Union>x<A. b(x))
lemma OUN_upper_le:
[| a < A; i ≤ b(a); Ord(\<Union>x<A. b(x)) |] ==> i ≤ (\<Union>x<A. b(x))
lemma Limit_OUN_eq:
Limit(i) ==> (\<Union>x<i. x) = i
lemma OUN_least:
(!!x. x < A ==> B(x) ⊆ C) ==> (\<Union>x<A. B(x)) ⊆ C
lemma OUN_least_le:
[| Ord(i); !!x. x < A ==> b(x) ≤ i |] ==> (\<Union>x<A. b(x)) ≤ i
lemma le_implies_OUN_le_OUN:
(!!x. x < A ==> c(x) ≤ d(x)) ==> (\<Union>x<A. c(x)) ≤ (\<Union>x<A. d(x))
lemma OUN_UN_eq:
(!!x. x ∈ A ==> Ord(B(x)))
==> (\<Union>z<\<Union>x∈A. B(x). C(z)) = (\<Union>x∈A. \<Union>z<B(x). C(z))
lemma OUN_Union_eq:
(!!x. x ∈ X ==> Ord(x))
==> (\<Union>z<\<Union>X. C(z)) = (\<Union>x∈X. \<Union>z<x. C(z))
lemma atomize_oall:
∀x<A. P(x) == (!!x. x < A ==> P(x))
lemma oallI:
(!!x. x < A ==> P(x)) ==> ∀x<A. P(x)
lemma ospec:
[| ∀x<A. P(x); x < A |] ==> P(x)
lemma oallE:
[| ∀x<A. P(x); P(x) ==> Q; ¬ x < A ==> Q |] ==> Q
lemma rev_oallE:
[| ∀x<A. P(x); ¬ x < A ==> Q; P(x) ==> Q |] ==> Q
lemma oall_simp:
(∀x<a. True) <-> True
lemma oall_cong:
[| a = a'; !!x. x < a' ==> P(x) <-> P'(x) |] ==> (∀x<a. P(x)) <-> (∀x<a'. P'(x))
lemma oexI:
[| P(x); x < A |] ==> ∃x<A. P(x)
lemma oexCI:
[| ∀x<A. ¬ P(x) ==> P(a); a < A |] ==> ∃x<A. P(x)
lemma oexE:
[| ∃x<A. P(x); !!x. [| x < A; P(x) |] ==> Q |] ==> Q
lemma oex_cong:
[| a = a'; !!x. x < a' ==> P(x) <-> P'(x) |] ==> (∃x<a. P(x)) <-> (∃x<a'. P'(x))
lemma OUN_I:
[| a < i; b ∈ B(a) |] ==> b ∈ (\<Union>z<i. B(z))
lemma OUN_E:
[| b ∈ (\<Union>z<i. B(z)); !!a. [| b ∈ B(a); a < i |] ==> R |] ==> R
lemma OUN_iff:
b ∈ (\<Union>x<i. B(x)) <-> (∃x<i. b ∈ B(x))
lemma OUN_cong:
[| i = j; !!x. x < j ==> C(x) = D(x) |]
==> (\<Union>x<i. C(x)) = (\<Union>x<j. D(x))
lemma lt_induct:
[| i < k; !!x. [| x < k; ∀y<x. P(y) |] ==> P(x) |] ==> P(i)
lemma rallI:
(!!x. M(x) ==> P(x)) ==> ∀x[M]. P(x)
lemma rspec:
[| ∀x[M]. P(x); M(x) |] ==> P(x)
lemma rev_rallE:
[| ∀x[M]. P(x); ¬ M(x) ==> Q; P(x) ==> Q |] ==> Q
lemma rallE:
[| ∀x[M]. P(x); P(x) ==> Q; ¬ M(x) ==> Q |] ==> Q
lemma rall_triv:
(∀x[M]. P) <-> (∃x. M(x)) --> P
lemma rall_cong:
(!!x. M(x) ==> P(x) <-> P'(x)) ==> (∀x[M]. P(x)) <-> (∀x[M]. P'(x))
lemma rexI:
[| P(x); M(x) |] ==> ∃x[M]. P(x)
lemma rev_rexI:
[| M(x); P(x) |] ==> ∃x[M]. P(x)
lemma rexCI:
[| ∀x[M]. ¬ P(x) ==> P(a); M(a) |] ==> ∃x[M]. P(x)
lemma rexE:
[| ∃x[M]. P(x); !!x. [| M(x); P(x) |] ==> Q |] ==> Q
lemma rex_triv:
(∃x[M]. P) <-> (∃x. M(x)) ∧ P
lemma rex_cong:
(!!x. M(x) ==> P(x) <-> P'(x)) ==> (∃x[M]. P(x)) <-> (∃x[M]. P'(x))
lemma rall_is_ball:
(∀x[λz. z ∈ A]. P(x)) <-> (∀x∈A. P(x))
lemma rex_is_bex:
(∃x[λz. z ∈ A]. P(x)) <-> (∃x∈A. P(x))
lemma atomize_rall:
(!!x. M(x) ==> P(x)) == ∀x[M]. P(x)
lemma rall_simps1:
(∀x[M]. P(x) ∧ Q) <-> (∀x[M]. P(x)) ∧ ((∀x[M]. False) ∨ Q)
(∀x[M]. P(x) ∨ Q) <-> (∀x[M]. P(x)) ∨ Q
(∀x[M]. P(x) --> Q) <-> (∃x[M]. P(x)) --> Q
¬ (∀x[M]. P(x)) <-> (∃x[M]. ¬ P(x))
lemma rall_simps2:
(∀x[M]. P ∧ Q(x)) <-> ((∀x[M]. False) ∨ P) ∧ (∀x[M]. Q(x))
(∀x[M]. P ∨ Q(x)) <-> P ∨ (∀x[M]. Q(x))
(∀x[M]. P --> Q(x)) <-> P --> (∀x[M]. Q(x))
lemma rall_simps:
(∀x[M]. P(x) ∧ Q) <-> (∀x[M]. P(x)) ∧ ((∀x[M]. False) ∨ Q)
(∀x[M]. P(x) ∨ Q) <-> (∀x[M]. P(x)) ∨ Q
(∀x[M]. P(x) --> Q) <-> (∃x[M]. P(x)) --> Q
¬ (∀x[M]. P(x)) <-> (∃x[M]. ¬ P(x))
(∀x[M]. P ∧ Q(x)) <-> ((∀x[M]. False) ∨ P) ∧ (∀x[M]. Q(x))
(∀x[M]. P ∨ Q(x)) <-> P ∨ (∀x[M]. Q(x))
(∀x[M]. P --> Q(x)) <-> P --> (∀x[M]. Q(x))
lemma rall_conj_distrib:
(∀x[M]. P(x) ∧ Q(x)) <-> (∀x[M]. P(x)) ∧ (∀x[M]. Q(x))
lemma rex_simps1:
(∃x[M]. P(x) ∧ Q) <-> (∃x[M]. P(x)) ∧ Q
(∃x[M]. P(x) ∨ Q) <-> (∃x[M]. P(x)) ∨ (∃x[M]. True) ∧ Q
(∃x[M]. P(x) --> Q) <-> (∀x[M]. P(x)) --> (∃x[M]. True) ∧ Q
¬ (∃x[M]. P(x)) <-> (∀x[M]. ¬ P(x))
lemma rex_simps2:
(∃x[M]. P ∧ Q(x)) <-> P ∧ (∃x[M]. Q(x))
(∃x[M]. P ∨ Q(x)) <-> (∃x[M]. True) ∧ P ∨ (∃x[M]. Q(x))
(∃x[M]. P --> Q(x)) <-> (∀x[M]. False) ∨ P --> (∃x[M]. Q(x))
lemma rex_simps:
(∃x[M]. P(x) ∧ Q) <-> (∃x[M]. P(x)) ∧ Q
(∃x[M]. P(x) ∨ Q) <-> (∃x[M]. P(x)) ∨ (∃x[M]. True) ∧ Q
(∃x[M]. P(x) --> Q) <-> (∀x[M]. P(x)) --> (∃x[M]. True) ∧ Q
¬ (∃x[M]. P(x)) <-> (∀x[M]. ¬ P(x))
(∃x[M]. P ∧ Q(x)) <-> P ∧ (∃x[M]. Q(x))
(∃x[M]. P ∨ Q(x)) <-> (∃x[M]. True) ∧ P ∨ (∃x[M]. Q(x))
(∃x[M]. P --> Q(x)) <-> (∀x[M]. False) ∨ P --> (∃x[M]. Q(x))
lemma rex_disj_distrib:
(∃x[M]. P(x) ∨ Q(x)) <-> (∃x[M]. P(x)) ∨ (∃x[M]. Q(x))
lemma rex_triv_one_point1:
(∃x[M]. x = a) <-> M(a)
lemma rex_triv_one_point2:
(∃x[M]. a = x) <-> M(a)
lemma rex_one_point1:
(∃x[M]. x = a ∧ P(x)) <-> M(a) ∧ P(a)
lemma rex_one_point2:
(∃x[M]. a = x ∧ P(x)) <-> M(a) ∧ P(a)
lemma rall_one_point1:
(∀x[M]. x = a --> P(x)) <-> M(a) --> P(a)
lemma rall_one_point2:
(∀x[M]. a = x --> P(x)) <-> M(a) --> P(a)
lemma setclass_iff:
(##A)(x) <-> x ∈ A
lemma rall_setclass_is_ball:
(∀x[##A]. P(x)) <-> (∀x∈A. P(x))
lemma rex_setclass_is_bex:
(∃x[##A]. P(x)) <-> (∃x∈A. P(x))