TPTP Problem File: SET659+3.p
View Solutions
- Solve Problem
%--------------------------------------------------------------------------
% File : SET659+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : For every x in X ? y : <x,y> in R (X to Y) iff domain of R is X
% Version : [Wor90] axioms : Reduced > Incomplete.
% English : For every x in X there exists y such that <x,y> is in a
% relation R from X to Y iff the domain of R is X.
% 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 (22) [Wor90]
% Status : Theorem
% Rating : 0.56 v8.2.0, 0.58 v8.1.0, 0.56 v7.5.0, 0.53 v7.4.0, 0.50 v7.3.0, 0.55 v7.2.0, 0.52 v7.1.0, 0.48 v7.0.0, 0.47 v6.4.0, 0.50 v6.3.0, 0.54 v6.2.0, 0.56 v6.1.0, 0.70 v6.0.0, 0.65 v5.5.0, 0.70 v5.4.0, 0.71 v5.3.0, 0.74 v5.2.0, 0.60 v5.1.0, 0.62 v4.1.0, 0.61 v4.0.0, 0.58 v3.7.0, 0.60 v3.5.0, 0.58 v3.4.0, 0.74 v3.3.0, 0.71 v3.2.0, 0.64 v3.1.0, 0.78 v2.7.0, 0.67 v2.6.0, 0.71 v2.5.0, 0.75 v2.4.0, 0.75 v2.3.0, 0.67 v2.2.1
% Syntax : Number of formulae : 35 ( 2 unt; 0 def)
% Number of atoms : 141 ( 10 equ)
% Maximal formula atoms : 8 ( 4 avg)
% Number of connectives : 110 ( 4 ~; 0 |; 12 &)
% ( 14 <=>; 80 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-2 aty)
% Number of functors : 14 ( 14 usr; 2 con; 0-3 aty)
% Number of variables : 83 ( 75 !; 8 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(relat_1 - df(4),1917872)
fof(p1,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( member(C,domain_of(B))
<=> ? [D] :
( ilf_type(D,set_type)
& member(ordered_pair(C,D),B) ) ) ) ) ).
%---- declaration(line(relat_1 - df(4),1917872))
fof(p2,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(domain_of(B),set_type) ) ).
%---- line(relat_1 - th(20),1917986)
fof(p3,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,binary_relation_type)
=> ( member(ordered_pair(B,C),D)
=> ( member(B,domain_of(D))
& member(C,range_of(D)) ) ) ) ) ) ).
%---- line(tarski - th(2),1832736)
fof(p4,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
<=> member(D,C) ) )
=> B = C ) ) ) ).
%---- line(tarski - df(5),1832760)
fof(p5,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(p6,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(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(boole - df(8),1909359)
fof(p9,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(p10,axiom,
! [B] :
( ilf_type(B,set_type)
=> ilf_type(singleton(B),set_type) ) ).
%---- declaration(op(cross_product,2,function))
fof(p11,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(p12,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(range_of(B),set_type) ) ).
%---- declaration(op(unordered_pair,2,function))
fof(p13,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(p14,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> unordered_pair(B,C) = unordered_pair(C,B) ) ) ).
%---- line(relat_1 - axiom183,1917641)
fof(p15,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 - axiom183,1917641))
fof(p16,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(hidden - axiom184,1832648)
fof(p17,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 - axiom184,1832648))
fof(p18,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- line(hidden - axiom185,1832615)
fof(p19,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(p20,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(p21,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- line(hidden - axiom186,1832644)
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 - axiom186,1832644))
fof(p23,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom187,1832640)
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 - axiom187,1832640))
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(axiom189,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) ) ) ) ).
%---- line(hidden - axiom190,1832628)
fof(p28,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- conditional_cluster(axiom191,empty)
fof(p29,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- line(relset_1 - axiom195,1916330)
fof(p30,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 - axiom195,1916330))
fof(p31,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 - axiom196,1916334)
fof(p32,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 - axiom196,1916334))
fof(p33,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(p34,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(22),1916371)
fof(prove_relset_1_22,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(C,B))
=> ( ! [E] :
( ilf_type(E,set_type)
=> ( member(E,C)
=> ? [F] :
( ilf_type(F,set_type)
& member(ordered_pair(E,F),D) ) ) )
<=> domain(C,B,D) = C ) ) ) ) ).
%--------------------------------------------------------------------------