TPTP Problem File: SET685+3.p
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- Solve Problem
%--------------------------------------------------------------------------
% File : SET685+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : y in R (X to Y) o D1 iff ?x in D : <x,y> in R & x in D1
% Version : [Wor90] axioms : Reduced > Incomplete.
% English : y is in a relation R from X to Y composed with D1 iff there
% exists an element x in D such that <x,y> is in R and x is in D1.
% 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 (52) [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, 0.43 v7.0.0, 0.40 v6.4.0, 0.46 v6.3.0, 0.50 v6.2.0, 0.60 v6.0.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.68 v5.3.0, 0.70 v5.2.0, 0.55 v5.1.0, 0.57 v5.0.0, 0.62 v4.1.0, 0.61 v4.0.1, 0.57 v4.0.0, 0.58 v3.7.0, 0.55 v3.5.0, 0.53 v3.4.0, 0.58 v3.3.0, 0.50 v3.2.0, 0.45 v3.1.0, 0.56 v2.7.0, 0.50 v2.6.0, 0.43 v2.5.0, 0.50 v2.3.0, 0.33 v2.2.1
% Syntax : Number of formulae : 28 ( 2 unt; 0 def)
% Number of atoms : 123 ( 6 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 102 ( 7 ~; 0 |; 16 &)
% ( 11 <=>; 68 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 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; 0-4 aty)
% Number of variables : 74 ( 66 !; 8 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(relat_1 - th(142),1919963)
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,binary_relation_type)
=> ( member(C,image(D,B))
<=> ? [E] :
( ilf_type(E,set_type)
& member(ordered_pair(E,C),D)
& member(E,B) ) ) ) ) ) ).
%---- 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 - axiom693,1832640)
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 - axiom693,1832640))
fof(p8,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- line(hidden - axiom695,1832628)
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(image,2,function))
fof(p14,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(image(B,C),set_type) ) ) ).
%---- line(relat_1 - axiom696,1917641)
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 - axiom696,1917641))
fof(p16,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(hidden - axiom697,1832648)
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 - axiom697,1832648))
fof(p18,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- line(hidden - axiom698,1832615)
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(hidden - axiom700,1832644)
fof(p20,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 - axiom700,1832644))
fof(p21,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(p22,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(axiom701,relation_like)
fof(p23,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- conditional_cluster(axiom702,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(relset_1 - axiom712,1916764)
fof(p25,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,set_type)
=> image4(B,C,D,E) = image(D,E) ) ) ) ) ).
%---- declaration(line(relset_1 - axiom712,1916764))
fof(p26,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,set_type)
=> ilf_type(image4(B,C,D,E),subset_type(C)) ) ) ) ) ).
%---- declaration(set)
fof(p27,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(52),1916993)
fof(prove_relset_1_52,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(D,B))
=> ! [F] :
( ilf_type(F,member_type(B))
=> ( member(F,image4(D,B,E,C))
<=> ? [G] :
( ilf_type(G,member_type(D))
& member(ordered_pair(G,F),E)
& member(G,C) ) ) ) ) ) ) ) ).
%--------------------------------------------------------------------------