TPTP Problem File: SET644+3.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : SET644+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : a in R (X to Y) => ? x, y : a is <x,y> & x in X & y in Y
% Version : [Wor90] axioms : Reduced > Incomplete.
% English : If a is in a relation R from X to Y then there exist x, y such
% that a is <x,y> and x is in X and y is in Y.
% 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 (6) [Wor90]
% Status : Theorem
% Rating : 0.53 v8.1.0, 0.50 v7.4.0, 0.43 v7.3.0, 0.48 v7.2.0, 0.45 v7.1.0, 0.43 v6.4.0, 0.42 v6.2.0, 0.52 v6.1.0, 0.70 v6.0.0, 0.52 v5.5.0, 0.59 v5.4.0, 0.61 v5.3.0, 0.67 v5.2.0, 0.50 v5.1.0, 0.48 v5.0.0, 0.50 v4.1.0, 0.48 v4.0.0, 0.46 v3.7.0, 0.45 v3.5.0, 0.47 v3.4.0, 0.53 v3.3.0, 0.50 v3.2.0, 0.27 v3.1.0, 0.44 v2.7.0, 0.33 v2.6.0, 0.29 v2.5.0, 0.25 v2.4.0, 0.25 v2.3.0, 0.33 v2.2.1
% Syntax : Number of formulae : 28 ( 2 unt; 0 def)
% Number of atoms : 119 ( 8 equ)
% Maximal formula atoms : 10 ( 4 avg)
% Number of connectives : 95 ( 4 ~; 0 |; 16 &)
% ( 12 <=>; 63 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-2 aty)
% Number of functors : 10 ( 10 usr; 2 con; 0-2 aty)
% Number of variables : 69 ( 59 !; 10 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(relat_1 - th(2),1917645)
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,binary_relation_type)
=> ( member(B,C)
=> ? [D] :
( ilf_type(D,set_type)
& ? [E] :
( ilf_type(E,set_type)
& B = ordered_pair(D,E) ) ) ) ) ) ).
%---- line(zfmisc_1 - th(106),1905180)
fof(p2,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)
=> ( member(ordered_pair(B,C),cross_product(D,E))
<=> ( member(B,D)
& member(C,E) ) ) ) ) ) ) ).
%---- line(tarski - df(5),1832760)
fof(p3,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)
=> ! [F] :
( ilf_type(F,set_type)
=> ( F = ordered_pair(D,E)
<=> F = unordered_pair(unordered_pair(D,E),singleton(D)) ) ) ) ) ) ) ).
%---- declaration(line(tarski - df(5),1832760))
fof(p4,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(ordered_pair(B,C),set_type) ) ) ).
%---- line(relset_1 - df(1),1916080)
fof(p5,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(p6,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ? [D] : ilf_type(D,relation_type(C,B)) ) ) ).
%---- line(boole - df(8),1909359)
fof(p7,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( B = C
<=> ( subset(B,C)
& subset(C,B) ) ) ) ) ).
%---- declaration(op(singleton,1,function))
fof(p8,axiom,
! [B] :
( ilf_type(B,set_type)
=> ilf_type(singleton(B),set_type) ) ).
%---- declaration(op(cross_product,2,function))
fof(p9,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(cross_product(B,C),set_type) ) ) ).
%---- declaration(op(unordered_pair,2,function))
fof(p10,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(unordered_pair(B,C),set_type) ) ) ).
%---- property(commutativity,op(unordered_pair,2,function))
fof(p11,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> unordered_pair(B,C) = unordered_pair(C,B) ) ) ).
%---- line(relat_1 - axiom34,1917641)
fof(p12,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 - axiom34,1917641))
fof(p13,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(hidden - axiom35,1832648)
fof(p14,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 - axiom35,1832648))
fof(p15,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- line(hidden - axiom36,1832615)
fof(p16,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(tarski - df(3),1832749)
fof(p17,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(reflexivity,op(subset,2,predicate))
fof(p18,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- line(hidden - axiom37,1832644)
fof(p19,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 - axiom37,1832644))
fof(p20,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom38,1832640)
fof(p21,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 - axiom38,1832640))
fof(p22,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- line(relat_1 - df(1),1917627)
fof(p23,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(axiom40,relation_like)
fof(p24,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) ) ) ) ).
%---- line(hidden - axiom41,1832628)
fof(p25,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- conditional_cluster(axiom42,empty)
fof(p26,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- declaration(set)
fof(p27,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(6),1916124)
fof(prove_relset_1_6,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,relation_type(B,C))
=> ( member(D,E)
=> ? [F] :
( ilf_type(F,set_type)
& ? [G] :
( ilf_type(G,set_type)
& D = ordered_pair(F,G)
& member(F,B)
& member(G,C) ) ) ) ) ) ) ) ).
%--------------------------------------------------------------------------