TPTP Problem File: ALG216+1.p
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%------------------------------------------------------------------------------
% File : ALG216+1 : TPTP v9.0.0. Released v3.4.0.
% Domain : General Algebra
% Problem : Linear Independence in Right Module over Domain T06
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t6_rmod_5 [Urb08]
% Status : Theorem
% Rating : 0.27 v9.0.0, 0.31 v8.2.0, 0.25 v8.1.0, 0.28 v7.4.0, 0.13 v7.3.0, 0.24 v7.2.0, 0.21 v7.1.0, 0.22 v7.0.0, 0.17 v6.4.0, 0.23 v6.3.0, 0.25 v6.2.0, 0.32 v6.1.0, 0.40 v6.0.0, 0.35 v5.5.0, 0.56 v5.4.0, 0.57 v5.3.0, 0.59 v5.2.0, 0.50 v5.1.0, 0.48 v5.0.0, 0.50 v4.1.0, 0.57 v4.0.0, 0.58 v3.7.0, 0.55 v3.5.0, 0.58 v3.4.0
% Syntax : Number of formulae : 53 ( 21 unt; 0 def)
% Number of atoms : 151 ( 9 equ)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 130 ( 32 ~; 2 |; 62 &)
% ( 1 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 7 ( 7 usr; 1 con; 0-3 aty)
% Number of variables : 78 ( 60 !; 18 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t6_rmod_5,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( k1_rlvect_1(A) != k2_group_1(A)
=> ( ~ v1_rmod_5(k8_rlvect_2(B,C,k1_rlvect_1(B)),A,B)
& ~ v1_rmod_5(k8_rlvect_2(B,k1_rlvect_1(B),C),A,B) ) ) ) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(existence_l1_vectsp_1,axiom,
? [A] : l1_vectsp_1(A) ).
fof(dt_l1_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> l1_group_1(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_rlvect_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
fof(commutativity_k2_tarski,axiom,
! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ).
fof(existence_l1_group_1,axiom,
? [A] : l1_group_1(A) ).
fof(existence_l1_rlvect_1,axiom,
? [A] : l1_rlvect_1(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_l2_struct_0,axiom,
? [A] : l2_struct_0(A) ).
fof(existence_l2_vectsp_1,axiom,
? [A] : l2_vectsp_1(A) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_l1_group_1,axiom,
! [A] :
( l1_group_1(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_rlvect_1,axiom,
! [A] :
( l1_rlvect_1(A)
=> l2_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l2_struct_0,axiom,
! [A] :
( l2_struct_0(A)
=> l1_struct_0(A) ) ).
fof(dt_l2_vectsp_1,axiom,
! [A] :
( l2_vectsp_1(A)
=> ( l1_vectsp_1(A)
& l2_struct_0(A) ) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc3_subset_1,axiom,
! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ).
fof(rc3_struct_0,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc4_struct_0,axiom,
? [A] :
( l2_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc9_vectsp_2,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l1_vectsp_2(B,A)
& ~ v3_struct_0(B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(commutativity_k8_rlvect_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k8_rlvect_2(A,B,C) = k8_rlvect_2(A,C,B) ) ).
fof(existence_l1_vectsp_2,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] : l1_vectsp_2(B,A) ) ).
fof(existence_l3_vectsp_1,axiom,
? [A] : l3_vectsp_1(A) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(redefinition_k8_rlvect_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k8_rlvect_2(A,B,C) = k2_tarski(B,C) ) ).
fof(dt_k1_rlvect_1,axiom,
! [A] :
( l2_struct_0(A)
=> m1_subset_1(k1_rlvect_1(A),u1_struct_0(A)) ) ).
fof(dt_k2_group_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> m1_subset_1(k2_group_1(A),u1_struct_0(A)) ) ).
fof(dt_k8_rlvect_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( v1_finset_1(k8_rlvect_2(A,B,C))
& m1_subset_1(k8_rlvect_2(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_l1_vectsp_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_vectsp_2(B,A)
=> l1_rlvect_1(B) ) ) ).
fof(dt_l3_vectsp_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( l1_rlvect_1(A)
& l2_vectsp_1(A) ) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(t5_rmod_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( v1_rmod_5(k8_rlvect_2(B,C,D),A,B)
=> ( k1_rlvect_1(A) = k2_group_1(A)
| ( C != k1_rlvect_1(B)
& D != k1_rlvect_1(B) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------