TPTP Problem File: SET657+3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SET657+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : The field of a relation R from X to Y is a subset of X union Y
% Version : [Wor90] axioms : Reduced > Incomplete.
% English :
% Refs : [ILF] The ILF Group (1998), The ILF System: A Tool for the Int
% : [Wor90] Woronowicz (1990), Relations Defined on Sets
% Source : [ILF]
% Names : RELSET_1 (19) [Wor90]
% Status : Theorem
% Rating : 0.53 v8.2.0, 0.50 v8.1.0, 0.47 v7.4.0, 0.43 v7.3.0, 0.55 v7.2.0, 0.52 v7.1.0, 0.39 v7.0.0, 0.40 v6.4.0, 0.42 v6.3.0, 0.50 v6.2.0, 0.52 v6.1.0, 0.60 v6.0.0, 0.52 v5.5.0, 0.63 v5.4.0, 0.64 v5.3.0, 0.70 v5.2.0, 0.60 v5.1.0, 0.57 v5.0.0, 0.67 v4.1.0, 0.74 v4.0.1, 0.70 v4.0.0, 0.71 v3.7.0, 0.70 v3.5.0, 0.68 v3.4.0, 0.63 v3.3.0, 0.57 v3.2.0, 0.55 v3.1.0, 0.67 v2.7.0, 0.50 v2.6.0, 0.43 v2.5.0, 0.50 v2.4.0, 0.75 v2.3.0, 0.67 v2.2.1
% Syntax : Number of formulae : 38 ( 2 unt; 0 def)
% Number of atoms : 141 ( 11 equ)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 107 ( 4 ~; 1 |; 9 &)
% ( 10 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-2 aty)
% Number of functors : 15 ( 15 usr; 2 con; 0-4 aty)
% Number of variables : 85 ( 79 !; 6 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
%---- line(boole - th(34),1909498)
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,set_type)
=> ( ( subset(B,C)
& subset(D,E) )
=> subset(union(B,D),union(C,E)) ) ) ) ) ) ).
%---- line(relat_1 - th(29),1918207)
fof(p2,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> field_of(B) = union(domain_of(B),range_of(B)) ) ).
%---- line(boole - df(2),1909298)
fof(p3,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,union(B,C))
<=> ( member(D,B)
| member(D,C) ) ) ) ) ) ).
%---- declaration(line(boole - df(2),1909298))
fof(p4,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(union(B,C),set_type) ) ) ).
%---- line(relat_1 - df(6),1918204)
fof(p5,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> field_of(B) = union(domain_of(B),range_of(B)) ) ).
%---- declaration(line(relat_1 - df(6),1918204))
fof(p6,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(field_of(B),set_type) ) ).
%---- line(relset_1 - df(1),1916080)
fof(p7,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ! [D] :
( ilf_type(D,subset_type(cross_product(B,C)))
=> ilf_type(D,relation_type(B,C)) )
& ! [E] :
( ilf_type(E,relation_type(B,C))
=> ilf_type(E,subset_type(cross_product(B,C))) ) ) ) ) ).
%---- type_nonempty(line(relset_1 - df(1),1916080))
fof(p8,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ? [D] : ilf_type(D,relation_type(C,B)) ) ) ).
%---- line(tarski - df(3),1832749)
fof(p9,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( subset(B,C)
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
=> member(D,C) ) ) ) ) ) ).
%---- property(commutativity,op(union,2,function))
fof(p10,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> union(B,C) = union(C,B) ) ) ).
%---- declaration(op(domain_of,1,function))
fof(p11,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(domain_of(B),set_type) ) ).
%---- declaration(op(cross_product,2,function))
fof(p12,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(cross_product(B,C),set_type) ) ) ).
%---- declaration(op(range_of,1,function))
fof(p13,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(range_of(B),set_type) ) ).
%---- line(relat_1 - axiom152,1917641)
fof(p14,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ilf_type(B,binary_relation_type)
<=> ( relation_like(B)
& ilf_type(B,set_type) ) ) ) ).
%---- type_nonempty(line(relat_1 - axiom152,1917641))
fof(p15,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(hidden - axiom153,1832648)
fof(p16,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ilf_type(C,subset_type(B))
<=> ilf_type(C,member_type(power_set(B))) ) ) ) ).
%---- type_nonempty(line(hidden - axiom153,1832648))
fof(p17,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- property(symmetry,op(=,2,predicate))
fof(p18,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,binary_relation_type)
=> ( B = C
=> C = B ) ) ) ).
%---- property(reflexivity,op(=,2,predicate))
fof(p19,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> B = B ) ).
%---- property(reflexivity,op(subset,2,predicate))
fof(p20,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- line(hidden - axiom155,1832644)
fof(p21,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( member(B,power_set(C))
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
=> member(D,C) ) ) ) ) ) ).
%---- declaration(line(hidden - axiom155,1832644))
fof(p22,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom156,1832640)
fof(p23,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ( ~ empty(C)
& ilf_type(C,set_type) )
=> ( ilf_type(B,member_type(C))
<=> member(B,C) ) ) ) ).
%---- type_nonempty(line(hidden - axiom156,1832640))
fof(p24,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- line(hidden - axiom157,1832615)
fof(p25,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( B = C
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
<=> member(D,C) ) ) ) ) ) ).
%---- line(relat_1 - df(1),1917627)
fof(p26,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( relation_like(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ( member(C,B)
=> ? [D] :
( ilf_type(D,set_type)
& ? [E] :
( ilf_type(E,set_type)
& C = ordered_pair(D,E) ) ) ) ) ) ) ).
%---- conditional_cluster(axiom159,relation_like)
fof(p27,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,subset_type(cross_product(B,C)))
=> relation_like(D) ) ) ) ).
%---- declaration(op(ordered_pair,2,function))
fof(p28,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(ordered_pair(B,C),set_type) ) ) ).
%---- line(hidden - axiom160,1832628)
fof(p29,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- conditional_cluster(axiom161,empty)
fof(p30,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- line(relset_1 - axiom162,1916314)
fof(p31,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> ! [E] :
( ilf_type(E,relation_type(B,C))
=> union4(B,C,D,E) = union(D,E) ) ) ) ) ).
%---- declaration(line(relset_1 - axiom162,1916314))
fof(p32,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> ! [E] :
( ilf_type(E,relation_type(B,C))
=> ilf_type(union4(B,C,D,E),relation_type(B,C)) ) ) ) ) ).
%---- line(relset_1 - axiom165,1916330)
fof(p33,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> domain(B,C,D) = domain_of(D) ) ) ) ).
%---- declaration(line(relset_1 - axiom165,1916330))
fof(p34,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> ilf_type(domain(B,C,D),subset_type(B)) ) ) ) ).
%---- line(relset_1 - axiom166,1916334)
fof(p35,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> range(B,C,D) = range_of(D) ) ) ) ).
%---- declaration(line(relset_1 - axiom166,1916334))
fof(p36,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> ilf_type(range(B,C,D),subset_type(C)) ) ) ) ).
%---- declaration(set)
fof(p37,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(19),1916346)
fof(prove_relset_1_19,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> subset(field_of(D),union(B,C)) ) ) ) ).
%------------------------------------------------------------------------------