TPTP Problem File: SET673+3.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SET673+3 : TPTP v9.0.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : Y a subset of Y1 => Y1 restricted to R (X to Y) is R
% Version : [Wor90] axioms : Reduced > Incomplete.
% English : If Y is a subset of Y1 then Y1 restricted to a relation R from
% X to Y is 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 (36) [Wor90]
% Status : Theorem
% Rating : 0.64 v9.0.0, 0.67 v8.1.0, 0.72 v7.4.0, 0.63 v7.3.0, 0.69 v7.2.0, 0.66 v7.1.0, 0.57 v7.0.0, 0.63 v6.4.0, 0.62 v6.3.0, 0.71 v6.2.0, 0.80 v6.1.0, 0.90 v6.0.0, 0.83 v5.5.0, 0.89 v5.3.0, 0.81 v5.2.0, 0.75 v5.1.0, 0.76 v5.0.0, 0.79 v4.1.0, 0.83 v4.0.0, 0.79 v3.7.0, 0.80 v3.5.0, 0.84 v3.3.0, 0.71 v3.2.0, 0.82 v3.1.0, 0.78 v2.7.0, 0.83 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 : 28 ( 2 unt; 0 def)
% Number of atoms : 117 ( 7 equ)
% Maximal formula atoms : 8 ( 4 avg)
% Number of connectives : 93 ( 4 ~; 0 |; 10 &)
% ( 10 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 7 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-2 aty)
% Number of functors : 10 ( 10 usr; 2 con; 0-4 aty)
% Number of variables : 70 ( 64 !; 6 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
%---- line(relat_1 - df(2),1917780)
fof(p1,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) ) ) ) ) ) ) ).
%---- line(relat_1 - th(114),1919569)
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,binary_relation_type)
=> ( member(ordered_pair(C,D),restrict(B,E))
<=> ( member(D,B)
& member(ordered_pair(C,D),E) ) ) ) ) ) ) ).
%---- line(relset_1 - th(7),1916125)
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,relation_type(B,C))
=> ( member(ordered_pair(D,E),F)
=> ( member(D,B)
& member(E,C) ) ) ) ) ) ) ) ).
%---- 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(tarski - df(3),1832749)
fof(p6,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) ) ) ) ) ) ).
%---- declaration(op(cross_product,2,function))
fof(p7,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(cross_product(B,C),set_type) ) ) ).
%---- declaration(op(ordered_pair,2,function))
fof(p8,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(ordered_pair(B,C),set_type) ) ) ).
%---- declaration(op(restrict,2,function))
fof(p9,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,binary_relation_type)
=> ilf_type(restrict(B,C),binary_relation_type) ) ) ).
%---- line(relat_1 - axiom428,1917641)
fof(p10,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 - axiom428,1917641))
fof(p11,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(hidden - axiom429,1832648)
fof(p12,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 - axiom429,1832648))
fof(p13,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- property(symmetry,op(=,2,predicate))
fof(p14,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(p15,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> B = B ) ).
%---- property(reflexivity,op(subset,2,predicate))
fof(p16,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- line(hidden - axiom431,1832644)
fof(p17,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 - axiom431,1832644))
fof(p18,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom432,1832640)
fof(p19,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 - axiom432,1832640))
fof(p20,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- line(relat_1 - df(1),1917627)
fof(p21,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(axiom434,relation_like)
fof(p22,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 - axiom435,1832628)
fof(p23,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- conditional_cluster(axiom436,empty)
fof(p24,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- line(relset_1 - axiom445,1916652)
fof(p25,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))
=> restrict4(B,C,D,E) = restrict(D,E) ) ) ) ) ).
%---- declaration(line(relset_1 - axiom445,1916652))
fof(p26,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))
=> ilf_type(restrict4(B,C,D,E),relation_type(B,C)) ) ) ) ) ).
%---- declaration(set)
fof(p27,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(36),1916736)
fof(prove_relset_1_36,conjecture,
! [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))
=> ( subset(C,D)
=> restrict4(B,C,D,E) = E ) ) ) ) ) ).
%------------------------------------------------------------------------------