TPTP Problem File: SET684+3.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SET684+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : <x,z> in P(DtoE) o R(EtoF) iff ?y in E:<x,y> in P & <y,z> in R
% Version : [Wor90] axioms : Reduced > Incomplete.
% English : Let P be a relation from D to E, R a relation from E to F, x an
% element of D, and z in F. Then <x,z> is in P composed with R if
% and only if there exists an element y in E such that <x,y> is in
% P and <y,z> is in R.
% 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 (51) [Wor90]
% Status : Theorem
% Rating : 0.58 v8.2.0, 0.61 v7.5.0, 0.56 v7.4.0, 0.53 v7.3.0, 0.59 v7.2.0, 0.55 v7.1.0, 0.52 v7.0.0, 0.50 v6.4.0, 0.58 v6.3.0, 0.62 v6.2.0, 0.64 v6.1.0, 0.77 v6.0.0, 0.65 v5.5.0, 0.78 v5.4.0, 0.75 v5.3.0, 0.74 v5.2.0, 0.65 v5.1.0, 0.62 v5.0.0, 0.67 v4.1.0, 0.65 v4.0.1, 0.70 v4.0.0, 0.71 v3.7.0, 0.70 v3.5.0, 0.74 v3.3.0, 0.64 v3.1.0, 0.78 v2.7.0, 0.83 v2.6.0, 0.86 v2.5.0, 1.00 v2.4.0, 0.75 v2.3.0, 0.67 v2.2.1
% Syntax : Number of formulae : 30 ( 2 unt; 0 def)
% Number of atoms : 134 ( 9 equ)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 111 ( 7 ~; 0 |; 16 &)
% ( 11 <=>; 77 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 5 ( 4 usr; 0 prp; 1-2 aty)
% Number of functors : 12 ( 12 usr; 2 con; 0-5 aty)
% Number of variables : 82 ( 74 !; 8 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
%---- line(relat_1 - th(43),1918466)
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,binary_relation_type)
=> ! [E] :
( ilf_type(E,binary_relation_type)
=> ( member(ordered_pair(B,C),compose(D,E))
<=> ? [F] :
( ilf_type(F,set_type)
& member(ordered_pair(B,F),D)
& member(ordered_pair(F,C),E) ) ) ) ) ) ) ).
%---- line(relset_1 - th(7),1916125)
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)
=> ! [F] :
( ilf_type(F,relation_type(B,C))
=> ( member(ordered_pair(D,E),F)
=> ( member(D,B)
& member(E,C) ) ) ) ) ) ) ) ).
%---- 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(hidden - axiom671,1832640)
fof(p7,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 - axiom671,1832640))
fof(p8,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- line(hidden - axiom673,1832628)
fof(p9,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(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(unordered_pair,2,function))
fof(p12,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(p13,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> unordered_pair(B,C) = unordered_pair(C,B) ) ) ).
%---- declaration(op(compose,2,function))
fof(p14,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,binary_relation_type)
=> ilf_type(compose(B,C),binary_relation_type) ) ) ).
%---- line(relat_1 - axiom674,1917641)
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 - axiom674,1917641))
fof(p16,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(hidden - axiom675,1832648)
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 - axiom675,1832648))
fof(p18,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- line(hidden - axiom676,1832615)
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) ) ) ) ) ) ).
%---- property(symmetry,op(=,2,predicate))
fof(p20,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(p21,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> B = B ) ).
%---- line(hidden - axiom678,1832644)
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 - axiom678,1832644))
fof(p23,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(relat_1 - df(1),1917627)
fof(p24,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(axiom679,relation_like)
fof(p25,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- conditional_cluster(axiom680,relation_like)
fof(p26,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 - axiom687,1916444)
fof(p27,axiom,
! [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))
=> ! [F] :
( ilf_type(F,relation_type(C,D))
=> compose5(B,C,D,E,F) = compose(E,F) ) ) ) ) ) ).
%---- declaration(line(relset_1 - axiom687,1916444))
fof(p28,axiom,
! [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))
=> ! [F] :
( ilf_type(F,relation_type(C,D))
=> ilf_type(compose5(B,C,D,E,F),relation_type(B,D)) ) ) ) ) ) ).
%---- declaration(set)
fof(p29,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(51),1916968)
fof(prove_relset_1_51,conjecture,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ! [C] :
( ( ~ empty(C)
& ilf_type(C,set_type) )
=> ! [D] :
( ( ~ empty(D)
& ilf_type(D,set_type) )
=> ! [E] :
( ilf_type(E,relation_type(B,C))
=> ! [F] :
( ilf_type(F,relation_type(C,D))
=> ! [G] :
( ilf_type(G,member_type(B))
=> ! [H] :
( ilf_type(H,member_type(D))
=> ( member(ordered_pair(G,H),compose5(B,C,D,E,F))
<=> ? [I] :
( ilf_type(I,member_type(C))
& member(ordered_pair(G,I),E)
& member(ordered_pair(I,H),F) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------