TPTP Problem File: SET664+3.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : SET664+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : R (X to Y) is (X to empty set) => R is empty set
% Version : [Wor90] axioms : Reduced > Incomplete.
% English : If a relation R from X to Y is a relation from X to the empty
% set then R is the empty set.
% 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 (27) [Wor90]
% Status : Theorem
% Rating : 0.33 v8.2.0, 0.31 v7.5.0, 0.34 v7.4.0, 0.20 v7.3.0, 0.28 v7.1.0, 0.30 v7.0.0, 0.20 v6.4.0, 0.23 v6.3.0, 0.33 v6.2.0, 0.28 v6.1.0, 0.30 v5.4.0, 0.32 v5.3.0, 0.37 v5.2.0, 0.15 v5.1.0, 0.14 v5.0.0, 0.25 v4.1.0, 0.30 v4.0.1, 0.39 v4.0.0, 0.38 v3.7.0, 0.35 v3.5.0, 0.37 v3.4.0, 0.26 v3.3.0, 0.29 v3.2.0, 0.27 v3.1.0, 0.22 v2.7.0, 0.17 v2.6.0, 0.14 v2.5.0, 0.12 v2.4.0, 0.25 v2.3.0, 0.33 v2.2.1
% Syntax : Number of formulae : 35 ( 4 unt; 0 def)
% Number of atoms : 124 ( 12 equ)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 94 ( 5 ~; 1 |; 9 &)
% ( 9 <=>; 70 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-2 aty)
% Number of functors : 13 ( 13 usr; 3 con; 0-3 aty)
% Number of variables : 70 ( 64 !; 6 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(boole - th(30),1909435)
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( subset(B,empty_set)
=> B = empty_set ) ) ).
%---- line(relat_1 - th(64),1918818)
fof(p2,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ( ( domain_of(B) = empty_set
| range_of(B) = empty_set )
=> B = empty_set ) ) ).
%---- line(relset_1 - th(12),1916203)
fof(p3,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> ( subset(domain_of(D),B)
& subset(range_of(D),C) ) ) ) ) ).
%---- line(hidden - axiom266,1832636)
fof(p4,axiom,
! [B] :
( ilf_type(B,set_type)
=> ~ member(B,empty_set) ) ).
%---- declaration(line(hidden - axiom266,1832636)) Part 1
fof(p5a,axiom,
empty(empty_set) ).
%---- declaration(line(hidden - axiom266,1832636)) Part 2
fof(p5b,axiom,
type(empty_set,set_type) ).
%---- 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,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(p7,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ? [D] : ilf_type(D,relation_type(C,B)) ) ) ).
%---- line(relat_1 - df(2),1917780)
fof(p8,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,binary_relation_type)
=> ( B = C
<=> ! [D] :
( ilf_type(D,set_type)
=> ! [E] :
( ilf_type(E,set_type)
=> ( member(ordered_pair(D,E),B)
<=> member(ordered_pair(D,E),C) ) ) ) ) ) ) ).
%---- declaration(op(domain_of,1,function))
fof(p9,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(domain_of(B),set_type) ) ).
%---- declaration(op(cross_product,2,function))
fof(p10,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(p11,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(range_of(B),set_type) ) ).
%---- declaration(op(ordered_pair,2,function))
fof(p12,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(ordered_pair(B,C),set_type) ) ) ).
%---- line(relat_1 - axiom267,1917641)
fof(p13,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 - axiom267,1917641))
fof(p14,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(hidden - axiom268,1832648)
fof(p15,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 - axiom268,1832648))
fof(p16,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- property(symmetry,op(=,2,predicate))
fof(p17,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(p18,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> B = B ) ).
%---- line(tarski - df(3),1832749)
fof(p19,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(p20,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- line(hidden - axiom270,1832628)
fof(p21,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- line(hidden - axiom272,1832644)
fof(p22,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 - axiom272,1832644))
fof(p23,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom273,1832640)
fof(p24,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 - axiom273,1832640))
fof(p25,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- 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(axiom274,relation_like)
fof(p27,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- conditional_cluster(axiom275,relation_like)
fof(p28,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(relset_1 - axiom279,1916330)
fof(p29,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 - axiom279,1916330))
fof(p30,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 - axiom280,1916334)
fof(p31,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 - axiom280,1916334))
fof(p32,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(p33,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(27),1916521)
fof(prove_relset_1_27,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(C,B))
=> ( ilf_type(D,relation_type(C,empty_set))
=> D = empty_set ) ) ) ) ).
%--------------------------------------------------------------------------