SET007 Axioms: SET007+300.ax
%------------------------------------------------------------------------------
% File : SET007+300 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Matrices. Abelian Group of Matrices
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 77 ( 6 unt; 0 def)
% Number of atoms : 506 ( 71 equ)
% Maximal formula atoms : 20 ( 6 avg)
% Number of connectives : 500 ( 71 ~; 2 |; 180 &)
% ( 17 <=>; 230 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 27 ( 25 usr; 1 prp; 0-4 aty)
% Number of functors : 49 ( 49 usr; 12 con; 0-4 aty)
% Number of variables : 251 ( 240 !; 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_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_matrix_1(A) ) ).
fof(rc2_matrix_1,axiom,
! [A] :
? [B] :
( m1_finseq_1(B,k3_finseq_2(A))
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v1_matrix_1(B) ) ).
fof(rc3_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_finseq_1(B,k3_finseq_2(A))
& v1_relat_1(B)
& ~ v3_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v1_matrix_1(B) ) ) ).
fof(fc1_matrix_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ~ v1_xboole_0(k9_matrix_1(A,B)) ) ).
fof(d1_matrix_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_matrix_1(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& ! [C] :
~ ( r2_hidden(C,k2_relat_1(A))
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ~ ( D = C
& k3_finseq_1(D) = B ) ) ) ) ) ) ).
fof(t1_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> v1_matrix_1(k9_finseq_1(k12_finseq_1(A,B))) ) ) ).
fof(t2_matrix_1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> v1_matrix_1(k2_finseq_2(B,k2_finseq_2(C,A))) ) ) ).
fof(t3_matrix_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> v1_matrix_1(k9_finseq_1(A)) ) ).
fof(t4_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( k3_finseq_1(B) = A
& k3_finseq_1(C) = A )
=> v1_matrix_1(k10_finseq_1(B,C)) ) ) ) ) ).
fof(t5_matrix_1,axiom,
v1_matrix_1(k1_xboole_0) ).
fof(t6_matrix_1,axiom,
v1_matrix_1(k10_finseq_1(k1_xboole_0,k1_xboole_0)) ).
fof(t7_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> v1_matrix_1(k10_finseq_1(k12_finseq_1(A,B),k12_finseq_1(A,C))) ) ) ) ).
fof(t8_matrix_1,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)
=> v1_matrix_1(k10_finseq_1(k10_finseq_1(B,C),k10_finseq_1(D,E))) ) ) ) ) ) ).
fof(t9_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
~ ( r2_hidden(D,k2_relat_1(B))
& ! [E] :
( m2_finseq_1(E,A)
=> ~ ( D = E
& k3_finseq_1(E) = C ) ) ) ) ) ) ) ).
fof(d2_matrix_1,axiom,
$true ).
fof(d3_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(A)) )
=> ( m1_matrix_1(D,A,B,C)
<=> ( k3_finseq_1(D) = B
& ! [E] :
( m2_finseq_1(E,A)
=> ( r2_hidden(E,k2_relat_1(D))
=> k3_finseq_1(E) = C ) ) ) ) ) ) ) ) ).
fof(t10_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,C)
=> m1_matrix_1(k4_finseqop(k4_finseq_2(B,C),A,k4_finseqop(C,B,D)),C,A,B) ) ) ) ) ).
fof(t11_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> m1_matrix_1(k9_finseq_1(B),A,np__1,k3_finseq_1(B)) ) ) ).
fof(t12_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( ( k3_finseq_1(C) = A
& k3_finseq_1(D) = A )
=> m1_matrix_1(k10_finseq_1(C,D),B,np__2,A) ) ) ) ) ) ).
fof(t13_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> m1_matrix_1(k1_xboole_0,B,np__0,A) ) ) ).
fof(t14_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m1_matrix_1(k9_finseq_1(k1_xboole_0),A,np__1,np__0) ) ).
fof(t15_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> m1_matrix_1(k9_finseq_1(k12_finseq_1(A,B)),A,np__1,np__1) ) ) ).
fof(t16_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m1_matrix_1(k10_finseq_1(k1_xboole_0,k1_xboole_0),A,np__2,np__0) ) ).
fof(t17_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> m1_matrix_1(k9_finseq_1(k10_finseq_1(B,C)),A,np__1,np__2) ) ) ) ).
fof(t18_matrix_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> m1_matrix_1(k10_finseq_1(k12_finseq_1(A,B),k12_finseq_1(A,C)),A,np__2,np__1) ) ) ) ).
fof(t19_matrix_1,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)
=> m1_matrix_1(k10_finseq_1(k10_finseq_1(B,C),k10_finseq_1(D,E)),A,np__2,np__2) ) ) ) ) ) ).
fof(d4_matrix_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_matrix_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> ( B = k1_matrix_1(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C)
& r2_hidden(C,k2_relat_1(A))
& k3_finseq_1(C) = B ) ) )
& ( r1_xreal_0(k3_finseq_1(A),np__0)
=> ( B = k1_matrix_1(A)
<=> B = np__0 ) ) ) ) ) ).
fof(t20_matrix_1,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)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( m1_matrix_1(B,A,k3_finseq_1(B),C)
<=> C = k1_matrix_1(B) ) ) ) ) ) ).
fof(d5_matrix_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_matrix_1(A) )
=> k2_matrix_1(A) = k2_zfmisc_1(k4_finseq_1(A),k2_finseq_1(k1_matrix_1(A))) ) ).
fof(d6_matrix_1,axiom,
! [A,B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(C,D),k2_matrix_1(B))
=> ! [E] :
( m1_subset_1(E,A)
=> ( E = k3_matrix_1(A,B,C,D)
<=> ? [F] :
( m2_finseq_1(F,A)
& F = k1_funct_1(B,C)
& E = k1_funct_1(F,D) ) ) ) ) ) ) ) ).
fof(t21_matrix_1,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)) )
=> ( ( k3_finseq_1(B) = k3_finseq_1(C)
& k1_matrix_1(B) = k1_matrix_1(C)
& ! [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(B))
=> k3_matrix_1(A,B,D,E) = k3_matrix_1(A,C,D,E) ) ) ) )
=> B = C ) ) ) ) ).
fof(t22_matrix_1,axiom,
$true ).
fof(t23_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_matrix_1(C,B,np__0,A)
=> ( k3_finseq_1(C) = np__0
& k1_matrix_1(C) = np__0
& k2_matrix_1(C) = k1_xboole_0 ) ) ) ) ).
fof(t24_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [D] :
( m1_matrix_1(D,C,A,B)
=> ( k3_finseq_1(D) = A
& k1_matrix_1(D) = B
& k2_matrix_1(D) = k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)) ) ) ) ) ) ) ).
fof(t25_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_matrix_1(C,B,A,A)
=> ( k3_finseq_1(C) = A
& k1_matrix_1(C) = A
& k2_matrix_1(C) = k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(A)) ) ) ) ) ).
fof(t26_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_matrix_1(D,C,A,B)
=> ( k3_finseq_1(D) = A
& k2_matrix_1(D) = k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(k1_matrix_1(D))) ) ) ) ) ) ).
fof(t27_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_matrix_1(D,C,A,B)
=> ! [E] :
( m1_matrix_1(E,C,A,B)
=> k2_matrix_1(D) = k2_matrix_1(E) ) ) ) ) ) ).
fof(t28_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_matrix_1(D,C,A,B)
=> ! [E] :
( m1_matrix_1(E,C,A,B)
=> ( ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(F,G),k2_matrix_1(D))
=> k3_matrix_1(C,D,F,G) = k3_matrix_1(C,E,F,G) ) ) )
=> D = E ) ) ) ) ) ) ).
fof(t29_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_matrix_1(C,B,A,A)
=> ! [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))
=> r2_hidden(k4_tarski(E,D),k2_matrix_1(C)) ) ) ) ) ) ) ).
fof(d7_matrix_1,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)) )
=> ( C = k4_matrix_1(A,B)
<=> ( k3_finseq_1(C) = k1_matrix_1(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))
<=> r2_hidden(k4_tarski(E,D),k2_matrix_1(B)) ) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(E,D),k2_matrix_1(B))
=> k3_matrix_1(A,C,D,E) = k3_matrix_1(A,B,E,D) ) ) ) ) ) ) ) ) ).
fof(d8_matrix_1,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)
=> ! [D] :
( m2_finseq_1(D,A)
=> ( D = k5_matrix_1(A,B,C)
<=> ( k3_finseq_1(D) = k1_matrix_1(B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(k1_matrix_1(B)))
=> k1_funct_1(D,E) = k3_matrix_1(A,B,C,E) ) ) ) ) ) ) ) ) ).
fof(d9_matrix_1,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)
=> ! [D] :
( m2_finseq_1(D,A)
=> ( D = k6_matrix_1(A,B,C)
<=> ( k3_finseq_1(D) = k3_finseq_1(B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(B))
=> k1_funct_1(D,E) = k3_matrix_1(A,B,E,C) ) ) ) ) ) ) ) ) ).
fof(d10_matrix_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k9_matrix_1(A,B) = k4_finseq_2(B,k4_finseq_2(B,u1_struct_0(A))) ) ) ).
fof(d11_matrix_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k10_matrix_1(A,B) = k4_finseqop(k4_finseq_2(B,u1_struct_0(A)),B,k4_finseqop(u1_struct_0(A),B,k1_rlvect_1(A))) ) ) ).
fof(d12_matrix_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(A),B,B)
=> ( C = k11_matrix_1(A,B)
<=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(D,D),k2_matrix_1(C))
=> k3_matrix_1(u1_struct_0(A),C,D,D) = k2_group_1(A) ) )
& ! [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))
=> ( D = E
| k3_matrix_1(u1_struct_0(A),C,D,E) = k1_rlvect_1(A) ) ) ) ) ) ) ) ) ) ).
fof(d13_matrix_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(A),B,B)
=> ! [D] :
( m1_matrix_1(D,u1_struct_0(A),B,B)
=> ( D = k12_matrix_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(E,F),k2_matrix_1(C))
=> k3_matrix_1(u1_struct_0(A),D,E,F) = k5_rlvect_1(A,k3_matrix_1(u1_struct_0(A),C,E,F)) ) ) ) ) ) ) ) ) ).
fof(d14_matrix_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(A),B,B)
=> ! [D] :
( m1_matrix_1(D,u1_struct_0(A),B,B)
=> ! [E] :
( m1_matrix_1(E,u1_struct_0(A),B,B)
=> ( E = k13_matrix_1(A,B,C,D)
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(F,G),k2_matrix_1(C))
=> k3_matrix_1(u1_struct_0(A),E,F,G) = k2_rlvect_1(A,k3_matrix_1(u1_struct_0(A),C,F,G),k3_matrix_1(u1_struct_0(A),D,F,G)) ) ) ) ) ) ) ) ) ) ).
fof(t30_matrix_1,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)
& l3_vectsp_1(D) )
=> ( r2_hidden(k4_tarski(A,B),k2_matrix_1(k10_matrix_1(D,C)))
=> k3_matrix_1(u1_struct_0(D),k10_matrix_1(D,C),A,B) = k1_rlvect_1(D) ) ) ) ) ) ).
fof(t31_matrix_1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& l3_vectsp_1(C) )
=> ( m1_subset_1(A,k9_matrix_1(C,B))
<=> m1_matrix_1(A,u1_struct_0(C),B,B) ) ) ) ).
fof(d15_matrix_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(A),B,B)
=> ( m2_matrix_1(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))
=> ( k3_matrix_1(u1_struct_0(A),C,D,E) = k1_rlvect_1(A)
| D = E ) ) ) ) ) ) ) ) ).
fof(t32_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(B),A,A)
=> ! [D] :
( m1_matrix_1(D,u1_struct_0(B),A,A)
=> k13_matrix_1(B,A,C,D) = k13_matrix_1(B,A,D,C) ) ) ) ) ).
fof(t33_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(B),A,A)
=> ! [D] :
( m1_matrix_1(D,u1_struct_0(B),A,A)
=> ! [E] :
( m1_matrix_1(E,u1_struct_0(B),A,A)
=> k13_matrix_1(B,A,k13_matrix_1(B,A,C,D),E) = k13_matrix_1(B,A,C,k13_matrix_1(B,A,D,E)) ) ) ) ) ) ).
fof(t34_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(B),A,A)
=> k13_matrix_1(B,A,C,k10_matrix_1(B,A)) = C ) ) ) ).
fof(t35_matrix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_matrix_1(C,u1_struct_0(B),A,A)
=> k13_matrix_1(B,A,C,k12_matrix_1(B,A,C)) = k10_matrix_1(B,A) ) ) ) ).
fof(d16_matrix_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_rlvect_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& l1_rlvect_1(C) )
=> ( C = k14_matrix_1(A,B)
<=> ( u1_struct_0(C) = k9_matrix_1(A,B)
& ! [D] :
( m1_matrix_1(D,u1_struct_0(A),B,B)
=> ! [E] :
( m1_matrix_1(E,u1_struct_0(A),B,B)
=> k1_binop_1(u1_rlvect_1(C),D,E) = k13_matrix_1(A,B,D,E) ) )
& u2_struct_0(C) = k10_matrix_1(A,B) ) ) ) ) ) ).
fof(s1_matrix_1,axiom,
? [A] :
( m1_matrix_1(A,f1_s1_matrix_1,f2_s1_matrix_1,f3_s1_matrix_1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(B,C),k2_matrix_1(A))
=> k3_matrix_1(f1_s1_matrix_1,A,B,C) = f4_s1_matrix_1(B,C) ) ) ) ) ).
fof(s2_matrix_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(k2_finseq_1(f2_s2_matrix_1),k2_finseq_1(f3_s2_matrix_1)))
=> ! [C] :
( m1_subset_1(C,f1_s2_matrix_1)
=> ! [D] :
( m1_subset_1(D,f1_s2_matrix_1)
=> ( ( p1_s2_matrix_1(A,B,C)
& p1_s2_matrix_1(A,B,D) )
=> C = D ) ) ) ) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(k2_finseq_1(f2_s2_matrix_1),k2_finseq_1(f3_s2_matrix_1)))
& ! [C] :
( m1_subset_1(C,f1_s2_matrix_1)
=> ~ p1_s2_matrix_1(A,B,C) ) ) ) ) )
=> ? [A] :
( m1_matrix_1(A,f1_s2_matrix_1,f2_s2_matrix_1,f3_s2_matrix_1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(B,C),k2_matrix_1(A))
=> p1_s2_matrix_1(B,C,k3_matrix_1(f1_s2_matrix_1,A,B,C)) ) ) ) ) ) ).
fof(dt_m1_matrix_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers) )
=> ! [D] :
( m1_matrix_1(D,A,B,C)
=> ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(A)) ) ) ) ).
fof(existence_m1_matrix_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers) )
=> ? [D] : m1_matrix_1(D,A,B,C) ) ).
fof(dt_m2_matrix_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ! [C] :
( m2_matrix_1(C,A,B)
=> m1_matrix_1(C,u1_struct_0(A),B,B) ) ) ).
fof(existence_m2_matrix_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ? [C] : m2_matrix_1(C,A,B) ) ).
fof(dt_k1_matrix_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_matrix_1(A) )
=> m2_subset_1(k1_matrix_1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k2_matrix_1,axiom,
$true ).
fof(dt_k3_matrix_1,axiom,
! [A,B,C,D] :
( ( v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(A))
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers) )
=> m1_subset_1(k3_matrix_1(A,B,C,D),A) ) ).
fof(dt_k4_matrix_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(A)) )
=> ( v1_matrix_1(k4_matrix_1(A,B))
& m2_finseq_1(k4_matrix_1(A,B),k3_finseq_2(A)) ) ) ).
fof(dt_k5_matrix_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(A))
& m1_subset_1(C,k5_numbers) )
=> m2_finseq_1(k5_matrix_1(A,B,C),A) ) ).
fof(dt_k6_matrix_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(A))
& m1_subset_1(C,k5_numbers) )
=> m2_finseq_1(k6_matrix_1(A,B,C),A) ) ).
fof(dt_k7_matrix_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(A))
& m1_subset_1(C,k5_numbers) )
=> m2_finseq_2(k7_matrix_1(A,B,C),A,k4_finseq_2(k1_matrix_1(B),A)) ) ).
fof(redefinition_k7_matrix_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(A))
& m1_subset_1(C,k5_numbers) )
=> k7_matrix_1(A,B,C) = k5_matrix_1(A,B,C) ) ).
fof(dt_k8_matrix_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(A))
& m1_subset_1(C,k5_numbers) )
=> m2_finseq_2(k8_matrix_1(A,B,C),A,k4_finseq_2(k3_finseq_1(B),A)) ) ).
fof(redefinition_k8_matrix_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_matrix_1(B)
& m1_finseq_1(B,k3_finseq_2(A))
& m1_subset_1(C,k5_numbers) )
=> k8_matrix_1(A,B,C) = k6_matrix_1(A,B,C) ) ).
fof(dt_k9_matrix_1,axiom,
$true ).
fof(dt_k10_matrix_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> m1_matrix_1(k10_matrix_1(A,B),u1_struct_0(A),B,B) ) ).
fof(dt_k11_matrix_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> m1_matrix_1(k11_matrix_1(A,B),u1_struct_0(A),B,B) ) ).
fof(dt_k12_matrix_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers)
& m1_matrix_1(C,u1_struct_0(A),B,B) )
=> m1_matrix_1(k12_matrix_1(A,B,C),u1_struct_0(A),B,B) ) ).
fof(dt_k13_matrix_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers)
& m1_matrix_1(C,u1_struct_0(A),B,B)
& m1_matrix_1(D,u1_struct_0(A),B,B) )
=> m1_matrix_1(k13_matrix_1(A,B,C,D),u1_struct_0(A),B,B) ) ).
fof(dt_k14_matrix_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( ~ v3_struct_0(k14_matrix_1(A,B))
& v1_rlvect_1(k14_matrix_1(A,B))
& v3_rlvect_1(k14_matrix_1(A,B))
& v4_rlvect_1(k14_matrix_1(A,B))
& v5_rlvect_1(k14_matrix_1(A,B))
& v6_rlvect_1(k14_matrix_1(A,B))
& l1_rlvect_1(k14_matrix_1(A,B)) ) ) ).
%------------------------------------------------------------------------------