SET007 Axioms: SET007+334.ax
%------------------------------------------------------------------------------
% File : SET007+334 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Transpose Matrices and Groups of Permutations
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : matrix_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 77 ( 8 unt; 0 def)
% Number of atoms : 578 ( 79 equ)
% Maximal formula atoms : 28 ( 7 avg)
% Number of connectives : 578 ( 77 ~; 7 |; 317 &)
% ( 15 <=>; 162 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 41 ( 39 usr; 1 prp; 0-4 aty)
% Number of functors : 57 ( 57 usr; 6 con; 0-5 aty)
% Number of variables : 211 ( 200 !; 11 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_matrix_2,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_matrix_2(A) ) ).
fof(fc1_matrix_2,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k11_matrix_2(A))
& v1_matrix_2(k11_matrix_2(A)) ) ) ).
fof(fc2_matrix_2,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k13_matrix_2(A))
& v1_group_1(k13_matrix_2(A)) ) ) ).
fof(fc3_matrix_2,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(k13_matrix_2(A))
& v1_group_1(k13_matrix_2(A))
& v2_group_1(k13_matrix_2(A))
& v3_group_1(k13_matrix_2(A))
& v4_group_1(k13_matrix_2(A)) ) ) ).
fof(d1_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] : k1_matrix_2(A,B,C) = k2_finseq_2(A,k2_finseq_2(B,C)) ) ) ).
fof(t1_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ~ v1_xboole_0(E)
=> ! [F] :
( m1_subset_1(F,E)
=> ( r2_hidden(k4_tarski(A,B),k2_matrix_1(k2_matrix_2(E,C,D,F)))
=> k3_matrix_1(E,k2_matrix_2(E,C,D,F),A,B) = F ) ) ) ) ) ) ) ).
fof(t2_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k13_matrix_1(B,A,k2_matrix_2(u1_struct_0(B),A,A,C),k2_matrix_2(u1_struct_0(B),A,A,D)) = k2_matrix_2(u1_struct_0(B),A,A,k4_rlvect_1(B,C,D)) ) ) ) ) ).
fof(d2_matrix_2,axiom,
! [A,B,C,D] : k3_matrix_2(A,B,C,D) = k10_finseq_1(k10_finseq_1(A,B),k10_finseq_1(C,D)) ).
fof(t3_matrix_2,axiom,
! [A,B,C,D] :
( k3_finseq_1(k3_matrix_2(A,B,C,D)) = np__2
& k1_matrix_1(k3_matrix_2(A,B,C,D)) = np__2
& k2_matrix_1(k3_matrix_2(A,B,C,D)) = k2_zfmisc_1(k2_finseq_1(np__2),k2_finseq_1(np__2)) ) ).
fof(t4_matrix_2,axiom,
! [A,B,C,D] :
( r2_hidden(k4_tarski(np__1,np__1),k2_matrix_1(k3_matrix_2(A,B,C,D)))
& r2_hidden(k4_tarski(np__1,np__2),k2_matrix_1(k3_matrix_2(A,B,C,D)))
& r2_hidden(k4_tarski(np__2,np__1),k2_matrix_1(k3_matrix_2(A,B,C,D)))
& r2_hidden(k4_tarski(np__2,np__2),k2_matrix_1(k3_matrix_2(A,B,C,D))) ) ).
fof(t5_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( r2_hidden(k4_tarski(np__1,np__1),k2_matrix_1(k5_matrix_2(A,np__1,k4_matrix_2(A,B))))
& k3_matrix_1(A,k5_matrix_2(A,np__1,k4_matrix_2(A,B)),np__1,np__1) = B ) ) ) ).
fof(t6_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),np__1,np__1) = B
& k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),np__1,np__2) = C
& k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),np__2,np__1) = D
& k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),np__2,np__2) = E ) ) ) ) ) ) ).
fof(d3_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(B),A,A)
=> ( m1_matrix_2(C,A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(D,E),k2_matrix_1(C))
=> ( r1_xreal_0(D,E)
| k3_matrix_1(u1_struct_0(B),C,D,E) = k1_rlvect_1(B) ) ) ) ) ) ) ) ) ).
fof(d4_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(B),A,A)
=> ( m2_matrix_2(C,A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(D,E),k2_matrix_1(C))
=> ( r1_xreal_0(E,D)
| k3_matrix_1(u1_struct_0(B),C,D,E) = k1_rlvect_1(B) ) ) ) ) ) ) ) ) ).
fof(t7_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(B)) )
=> ( k3_finseq_1(C) = A
=> m1_matrix_1(C,B,A,k1_matrix_1(C)) ) ) ) ) ).
fof(t8_matrix_2,axiom,
$true ).
fof(t9_matrix_2,axiom,
$true ).
fof(t10_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_matrix_1(D,u1_struct_0(C),A,B)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k1_funct_1(D,E) = k7_matrix_1(u1_struct_0(C),D,E) ) ) ) ) ) ) ).
fof(d5_matrix_2,axiom,
$true ).
fof(d6_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(B))) )
=> ( r2_hidden(A,k2_finseq_1(k1_matrix_1(C)))
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(B))) )
=> ( D = k7_matrix_2(A,B,C)
<=> ( k3_finseq_1(D) = k3_finseq_1(C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(C))
=> k1_funct_1(D,E) = k2_finseq_3(A,k7_matrix_1(u1_struct_0(B),C,E)) ) ) ) ) ) ) ) ) ) ).
fof(t11_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(A)) )
=> ( ( k4_matrix_1(A,B) = k4_matrix_1(A,C)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> B = C ) ) ) ) ).
fof(t12_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ( ~ r1_xreal_0(k1_matrix_1(B),np__0)
=> ( k3_finseq_1(k4_matrix_1(A,B)) = k1_matrix_1(B)
& k1_matrix_1(k4_matrix_1(A,B)) = k3_finseq_1(B) ) ) ) ) ).
fof(t13_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(A)) )
=> ( ( k4_matrix_1(A,B) = k4_matrix_1(A,C)
& k1_matrix_1(k4_matrix_1(A,B)) = k1_matrix_1(k4_matrix_1(A,C)) )
=> ( r1_xreal_0(k1_matrix_1(B),np__0)
| r1_xreal_0(k1_matrix_1(C),np__0)
| B = C ) ) ) ) ) ).
fof(t14_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(A)) )
=> ~ ( ~ r1_xreal_0(k1_matrix_1(B),np__0)
& ~ r1_xreal_0(k1_matrix_1(C),np__0)
& ~ ( B = C
<=> ( k4_matrix_1(A,B) = k4_matrix_1(A,C)
& k1_matrix_1(B) = k1_matrix_1(C) ) ) ) ) ) ) ).
fof(t15_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ~ ( ~ r1_xreal_0(k3_finseq_1(B),np__0)
& ~ r1_xreal_0(k1_matrix_1(B),np__0)
& k4_matrix_1(A,k4_matrix_1(A,B)) != B ) ) ) ).
fof(t16_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> k7_matrix_1(A,B,C) = k8_matrix_1(A,k4_matrix_1(A,B),C) ) ) ) ) ).
fof(t17_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(k1_matrix_1(B)))
=> k7_matrix_1(A,k4_matrix_1(A,B),C) = k8_matrix_1(A,B,C) ) ) ) ) ).
fof(t18_matrix_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> k1_funct_1(B,C) = k7_matrix_1(A,B,C) ) ) ) ) ).
fof(d7_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(u1_struct_0(B))) )
=> ( r2_hidden(A,k4_finseq_1(C))
=> ( r1_xreal_0(k1_matrix_1(C),np__0)
| ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(u1_struct_0(B))) )
=> ( ( k3_finseq_1(C) = np__1
=> ( D = k8_matrix_2(A,B,C)
<=> D = k1_xboole_0 ) )
& ( k3_finseq_1(C) != np__1
=> ( D = k8_matrix_2(A,B,C)
<=> ( k1_matrix_1(D) = k1_matrix_1(C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(k1_matrix_1(C)))
=> k8_matrix_1(u1_struct_0(B),D,E) = k2_finseq_3(A,k8_matrix_1(u1_struct_0(B),C,E)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d8_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( ~ v3_struct_0(D)
& v4_group_1(D)
& v7_group_1(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_vectsp_1(D)
& v8_vectsp_1(D)
& v9_vectsp_1(D)
& ~ v10_vectsp_1(D)
& l3_vectsp_1(D) )
=> ! [E] :
( m1_matrix_1(E,u1_struct_0(D),C,C)
=> ( ( C = np__1
=> k9_matrix_2(A,B,C,D,E) = k1_xboole_0 )
& ( C != np__1
=> k9_matrix_2(A,B,C,D,E) = k7_matrix_2(B,D,k8_matrix_2(A,D,E)) ) ) ) ) ) ) ) ).
fof(d9_matrix_2,axiom,
! [A] :
( v1_matrix_2(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
( r2_hidden(C,A)
=> ( v1_funct_1(C)
& v1_funct_2(C,k2_finseq_1(B),k2_finseq_1(B))
& v3_funct_2(C,k2_finseq_1(B),k2_finseq_1(B))
& m2_relset_1(C,k2_finseq_1(B),k2_finseq_1(B)) ) ) ) ) ).
fof(d10_matrix_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_matrix_2(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k10_matrix_2(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& r2_hidden(C,A)
& B = k3_finseq_1(C) ) ) ) ) ).
fof(t19_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( ~ v1_xboole_0(B)
& v1_matrix_2(B)
& k10_matrix_2(B) = A ) ) ).
fof(d11_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( B = k11_matrix_2(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v1_funct_1(C)
& v1_funct_2(C,k2_finseq_1(A),k2_finseq_1(A))
& v3_funct_2(C,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(C,k2_finseq_1(A),k2_finseq_1(A)) ) ) ) ) ).
fof(t20_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k10_matrix_2(k11_matrix_2(A)) = A ) ).
fof(t21_matrix_2,axiom,
k11_matrix_2(np__1) = k1_tarski(k1_finseq_2(np__1)) ).
fof(d12_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m3_matrix_2(B,k11_matrix_2(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k12_matrix_2(A,B)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D)
& D = B
& C = k3_finseq_1(D) ) ) ) ) ) ).
fof(t22_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m3_matrix_2(B,k11_matrix_2(A))
=> k12_matrix_2(A,B) = A ) ) ).
fof(d13_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_group_1(B)
& l1_group_1(B) )
=> ( B = k13_matrix_2(A)
<=> ( u1_struct_0(B) = k11_matrix_2(A)
& ! [C] :
( m3_matrix_2(C,k11_matrix_2(A))
=> ! [D] :
( m3_matrix_2(D,k11_matrix_2(A))
=> k1_binop_1(u1_group_1(B),C,D) = k5_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),C,D) ) ) ) ) ) ) ).
fof(t23_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> m1_subset_1(k1_finseq_2(A),u1_struct_0(k13_matrix_2(A))) ) ).
fof(t24_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m3_matrix_2(B,k11_matrix_2(A))
=> ( k5_relat_1(k1_finseq_2(A),B) = B
& k5_relat_1(B,k1_finseq_2(A)) = B ) ) ) ).
fof(t25_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m3_matrix_2(B,k11_matrix_2(A))
=> ( k5_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),k6_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),B),B) = k1_finseq_2(A)
& k5_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),B,k6_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),B)) = k1_finseq_2(A) ) ) ) ).
fof(t26_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m3_matrix_2(B,k11_matrix_2(A))
=> m1_subset_1(k6_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),B),u1_struct_0(k13_matrix_2(A))) ) ) ).
fof(t27_matrix_2,axiom,
$true ).
fof(t28_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_finseq_2(A) = k2_group_1(k13_matrix_2(A)) ) ).
fof(d14_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_finseq_1(A),k2_finseq_1(A))
& v3_funct_2(B,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(B,k2_finseq_1(A),k2_finseq_1(A)) )
=> ( v2_matrix_2(B,A)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r2_hidden(C,k1_relat_1(B))
& r2_hidden(D,k1_relat_1(B))
& C != D
& k1_funct_1(B,C) = D
& k1_funct_1(B,D) = C
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k1_relat_1(B))
=> ( E = C
| E = D
| k1_funct_1(B,E) = E ) ) ) ) ) ) ) ) ).
fof(d15_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_finseq_1(A),k2_finseq_1(A))
& v3_funct_2(B,k2_finseq_1(A),k2_finseq_1(A))
& m2_relset_1(B,k2_finseq_1(A),k2_finseq_1(A)) )
=> ( v3_matrix_2(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_struct_0(k13_matrix_2(A)))
& k4_nat_1(k3_finseq_1(C),np__2) = np__0
& B = k3_group_4(k13_matrix_2(A),C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(C))
& ! [E] :
( m3_matrix_2(E,k11_matrix_2(A))
=> ~ ( k1_funct_1(C,D) = E
& v2_matrix_2(E,k10_matrix_2(k11_matrix_2(A))) ) ) ) ) ) ) ) ) ).
fof(t29_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v3_matrix_2(k6_partfun1(k2_finseq_1(A)),A) ) ).
fof(d16_matrix_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m3_matrix_2(D,k11_matrix_2(B))
=> ( ( v3_matrix_2(D,k10_matrix_2(k11_matrix_2(B)))
=> k14_matrix_2(A,B,C,D) = C )
& ( ~ v3_matrix_2(D,k10_matrix_2(k11_matrix_2(B)))
=> k14_matrix_2(A,B,C,D) = k5_rlvect_1(A,C) ) ) ) ) ) ) ).
fof(d17_matrix_2,axiom,
! [A] :
( v1_finset_1(A)
=> k15_matrix_2(A) = A ) ).
fof(t30_matrix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v1_finset_1(k11_matrix_2(A)) ) ).
fof(dt_m1_matrix_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_matrix_2(C,A,B)
=> m1_matrix_1(C,u1_struct_0(B),A,A) ) ) ).
fof(existence_m1_matrix_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ? [C] : m1_matrix_2(C,A,B) ) ).
fof(dt_m2_matrix_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m2_matrix_2(C,A,B)
=> m1_matrix_1(C,u1_struct_0(B),A,A) ) ) ).
fof(existence_m2_matrix_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ? [C] : m2_matrix_2(C,A,B) ) ).
fof(dt_m3_matrix_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_matrix_2(A) )
=> ! [B] :
( m3_matrix_2(B,A)
=> ( v1_funct_1(B)
& v1_funct_2(B,k2_finseq_1(k10_matrix_2(A)),k2_finseq_1(k10_matrix_2(A)))
& v3_funct_2(B,k2_finseq_1(k10_matrix_2(A)),k2_finseq_1(k10_matrix_2(A)))
& m2_relset_1(B,k2_finseq_1(k10_matrix_2(A)),k2_finseq_1(k10_matrix_2(A))) ) ) ) ).
fof(existence_m3_matrix_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_matrix_2(A) )
=> ? [B] : m3_matrix_2(B,A) ) ).
fof(redefinition_m3_matrix_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_matrix_2(A) )
=> ! [B] :
( m3_matrix_2(B,A)
<=> m1_subset_1(B,A) ) ) ).
fof(dt_k1_matrix_2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k1_matrix_2(A,B,C))
& v1_funct_1(k1_matrix_2(A,B,C))
& v1_finseq_1(k1_matrix_2(A,B,C))
& v1_matrix_1(k1_matrix_2(A,B,C)) ) ) ).
fof(dt_k2_matrix_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,A) )
=> m1_matrix_1(k2_matrix_2(A,B,C,D),A,B,C) ) ).
fof(redefinition_k2_matrix_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,A) )
=> k2_matrix_2(A,B,C,D) = k1_matrix_2(B,C,D) ) ).
fof(dt_k3_matrix_2,axiom,
! [A,B,C,D] :
( v1_relat_1(k3_matrix_2(A,B,C,D))
& v1_funct_1(k3_matrix_2(A,B,C,D))
& v1_finseq_1(k3_matrix_2(A,B,C,D))
& v1_matrix_1(k3_matrix_2(A,B,C,D)) ) ).
fof(dt_k4_matrix_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m2_finseq_2(k4_matrix_2(A,B),A,k4_finseq_2(np__1,A)) ) ).
fof(redefinition_k4_matrix_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k4_matrix_2(A,B) = k5_finseq_1(B) ) ).
fof(dt_k5_matrix_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k4_finseq_2(B,A)) )
=> m1_matrix_1(k5_matrix_2(A,B,C),A,np__1,B) ) ).
fof(redefinition_k5_matrix_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k4_finseq_2(B,A)) )
=> k5_matrix_2(A,B,C) = k5_finseq_1(C) ) ).
fof(dt_k6_matrix_2,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,A)
& m1_subset_1(E,A) )
=> m1_matrix_1(k6_matrix_2(A,B,C,D,E),A,np__2,np__2) ) ).
fof(redefinition_k6_matrix_2,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,A)
& m1_subset_1(E,A) )
=> k6_matrix_2(A,B,C,D,E) = k3_matrix_2(B,C,D,E) ) ).
fof(dt_k7_matrix_2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B)
& v1_matrix_1(C)
& m1_finseq_1(C,k3_finseq_2(u1_struct_0(B))) )
=> ( v1_matrix_1(k7_matrix_2(A,B,C))
& m2_finseq_1(k7_matrix_2(A,B,C),k3_finseq_2(u1_struct_0(B))) ) ) ).
fof(dt_k8_matrix_2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B)
& v1_matrix_1(C)
& m1_finseq_1(C,k3_finseq_2(u1_struct_0(B))) )
=> ( v1_matrix_1(k8_matrix_2(A,B,C))
& m2_finseq_1(k8_matrix_2(A,B,C),k3_finseq_2(u1_struct_0(B))) ) ) ).
fof(dt_k9_matrix_2,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& ~ v3_struct_0(D)
& v4_group_1(D)
& v7_group_1(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_vectsp_1(D)
& v8_vectsp_1(D)
& v9_vectsp_1(D)
& ~ v10_vectsp_1(D)
& l3_vectsp_1(D)
& m1_matrix_1(E,u1_struct_0(D),C,C) )
=> ( v1_matrix_1(k9_matrix_2(A,B,C,D,E))
& m2_finseq_1(k9_matrix_2(A,B,C,D,E),k3_finseq_2(u1_struct_0(D))) ) ) ).
fof(dt_k10_matrix_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_matrix_2(A) )
=> m2_subset_1(k10_matrix_2(A),k1_numbers,k5_numbers) ) ).
fof(dt_k11_matrix_2,axiom,
$true ).
fof(dt_k12_matrix_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k11_matrix_2(A)) )
=> m2_subset_1(k12_matrix_2(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k13_matrix_2,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_group_1(k13_matrix_2(A))
& l1_group_1(k13_matrix_2(A)) ) ) ).
fof(dt_k14_matrix_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k11_matrix_2(B)) )
=> m1_subset_1(k14_matrix_2(A,B,C,D),u1_struct_0(A)) ) ).
fof(dt_k15_matrix_2,axiom,
! [A] : m1_subset_1(k15_matrix_2(A),k5_finsub_1(A)) ).
%------------------------------------------------------------------------------