TPTP Problem File: SET675+3.p
View Solutions
- Solve Problem
%--------------------------------------------------------------------------
% File : SET675+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : R o R^-1(Y) is the range of R & R^-1(R o X) is the domain of R
% Version : [Wor90] axioms : Reduced > Incomplete.
% English : A relation R from X to Y composed with the inverse of R applied
% to Y is the range of R; and the inverse of R applied to R
% composed with X is the domain of 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 (39) [Wor90]
% Status : Theorem
% Rating : 0.39 v8.1.0, 0.33 v7.5.0, 0.38 v7.4.0, 0.27 v7.3.0, 0.31 v7.1.0, 0.30 v6.4.0, 0.35 v6.3.0, 0.38 v6.2.0, 0.44 v6.1.0, 0.47 v6.0.0, 0.39 v5.5.0, 0.41 v5.4.0, 0.43 v5.3.0, 0.52 v5.2.0, 0.30 v5.1.0, 0.29 v5.0.0, 0.42 v4.1.0, 0.43 v4.0.0, 0.46 v3.7.0, 0.40 v3.5.0, 0.37 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 : 36 ( 2 unt; 0 def)
% Number of atoms : 135 ( 12 equ)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 103 ( 4 ~; 0 |; 11 &)
% ( 8 <=>; 80 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-2 aty)
% Number of functors : 16 ( 16 usr; 2 con; 0-4 aty)
% Number of variables : 84 ( 78 !; 6 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%---- line(relat_1 - th(146),1920038)
fof(p1,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> image(B,domain_of(B)) = range_of(B) ) ).
%---- line(relat_1 - th(169),1920434)
fof(p2,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> inverse2(B,range_of(B)) = domain_of(B) ) ).
%---- line(relset_1 - th(38),1916809)
fof(p3,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(B,C))
=> ( image4(B,C,D,B) = range(B,C,D)
& inverse4(B,C,D,C) = domain(B,C,D) ) ) ) ) ).
%---- line(relset_1 - df(1),1916080)
fof(p4,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(p5,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(p6,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( B = C
<=> ( subset(B,C)
& subset(C,B) ) ) ) ) ).
%---- declaration(op(inverse2,2,function))
fof(p7,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(inverse2(B,C),set_type) ) ) ).
%---- declaration(op(domain_of,1,function))
fof(p8,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(domain_of(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(range_of,1,function))
fof(p10,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(range_of(B),set_type) ) ).
%---- declaration(op(image,2,function))
fof(p11,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(image(B,C),set_type) ) ) ).
%---- line(relat_1 - axiom475,1917641)
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 - axiom475,1917641))
fof(p13,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(hidden - axiom476,1832648)
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 - axiom476,1832648))
fof(p15,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- line(tarski - df(3),1832749)
fof(p16,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(p17,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- line(hidden - axiom477,1832644)
fof(p18,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 - axiom477,1832644))
fof(p19,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom478,1832640)
fof(p20,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 - axiom478,1832640))
fof(p21,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- 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(axiom481,relation_like)
fof(p23,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) ) ) ) ).
%---- declaration(op(ordered_pair,2,function))
fof(p24,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(ordered_pair(B,C),set_type) ) ) ).
%---- line(hidden - axiom482,1832628)
fof(p25,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- conditional_cluster(axiom483,empty)
fof(p26,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- line(relset_1 - axiom487,1916330)
fof(p27,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 - axiom487,1916330))
fof(p28,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 - axiom488,1916334)
fof(p29,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 - axiom488,1916334))
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(range(B,C,D),subset_type(C)) ) ) ) ).
%---- line(relset_1 - axiom493,1916764)
fof(p31,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 - axiom493,1916764))
fof(p32,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)) ) ) ) ) ).
%---- line(relset_1 - axiom494,1916768)
fof(p33,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)
=> inverse4(B,C,D,E) = inverse2(D,E) ) ) ) ) ).
%---- declaration(line(relset_1 - axiom494,1916768))
fof(p34,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(inverse4(B,C,D,E),subset_type(B)) ) ) ) ) ).
%---- declaration(set)
fof(p35,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(39),1916819)
fof(prove_relset_1_39,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,relation_type(C,B))
=> ( image4(C,B,D,inverse4(C,B,D,B)) = range(C,B,D)
& inverse4(C,B,D,image4(C,B,D,C)) = domain(C,B,D) ) ) ) ) ).
%--------------------------------------------------------------------------