SET007 Axioms: SET007+179.ax
%------------------------------------------------------------------------------
% File : SET007+179 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Properties of Restrictions of Finite Sequences
% 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 : finseq_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 105 ( 5 unt; 0 def)
% Number of atoms : 562 ( 85 equ)
% Maximal formula atoms : 11 ( 5 avg)
% Number of connectives : 538 ( 81 ~; 2 |; 156 &)
% ( 2 <=>; 297 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-3 aty)
% Number of functors : 37 ( 37 usr; 5 con; 0-4 aty)
% Number of variables : 293 ( 289 !; 4 ?)
% 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(cc1_finseq_5,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_realset1(A) ) ).
fof(fc1_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(k5_finseq_1(A))
& v1_relat_1(k5_finseq_1(A))
& v1_funct_1(k5_finseq_1(A))
& v1_realset1(k5_finseq_1(A))
& v1_finset_1(k5_finseq_1(A))
& v1_finseq_1(k5_finseq_1(A)) ) ).
fof(fc2_finseq_5,axiom,
! [A,B] :
( v1_relat_1(k10_finseq_1(A,B))
& v1_funct_1(k10_finseq_1(A,B))
& ~ v1_realset1(k10_finseq_1(A,B))
& v1_finset_1(k10_finseq_1(A,B))
& v1_finseq_1(k10_finseq_1(A,B)) ) ).
fof(rc1_finseq_5,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A) ) ).
fof(rc2_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_finseq_1(B,A)
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B) ) ) ).
fof(fc3_finseq_5,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& v2_funct_1(C)
& m1_finseq_1(C,B) )
=> ( v1_relat_1(k16_finseq_1(B,C,A))
& v1_funct_1(k16_finseq_1(B,C,A))
& v2_funct_1(k16_finseq_1(B,C,A))
& v1_finset_1(k16_finseq_1(B,C,A))
& v1_finseq_1(k16_finseq_1(B,C,A)) ) ) ).
fof(fc4_finseq_5,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& v2_funct_1(C)
& m1_finseq_1(C,B) )
=> ( v1_relat_1(k1_rfinseq(B,C,A))
& v1_funct_1(k1_rfinseq(B,C,A))
& v2_funct_1(k1_rfinseq(B,C,A))
& v1_finset_1(k1_rfinseq(B,C,A))
& v1_finseq_1(k1_rfinseq(B,C,A)) ) ) ).
fof(fc5_finseq_5,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& v2_funct_1(C)
& m1_finseq_1(C,A) )
=> ( v1_relat_1(k1_finseq_5(A,C,B))
& v1_funct_1(k1_finseq_5(A,C,B))
& v2_funct_1(k1_finseq_5(A,C,B))
& v1_finset_1(k1_finseq_5(A,C,B))
& v1_finseq_1(k1_finseq_5(A,C,B)) ) ) ).
fof(fc6_finseq_5,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_finseq_1(C,A) )
=> ( ~ v1_xboole_0(k2_finseq_5(A,C,B))
& v1_relat_1(k2_finseq_5(A,C,B))
& v1_funct_1(k2_finseq_5(A,C,B))
& v1_finset_1(k2_finseq_5(A,C,B))
& v1_finseq_1(k2_finseq_5(A,C,B)) ) ) ).
fof(fc7_finseq_5,axiom,
! [A] :
( ( v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_xboole_0(k3_finseq_5(A))
& v1_relat_1(k3_finseq_5(A))
& v1_funct_1(k3_finseq_5(A))
& v2_funct_1(k3_finseq_5(A))
& v1_realset1(k3_finseq_5(A))
& v1_finset_1(k3_finseq_5(A))
& v1_finseq_1(k3_finseq_5(A)) ) ) ).
fof(fc8_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(k3_finseq_5(A))
& v1_funct_1(k3_finseq_5(A))
& v2_funct_1(k3_finseq_5(A))
& v1_finset_1(k3_finseq_5(A))
& v1_finseq_1(k3_finseq_5(A)) ) ) ).
fof(fc9_finseq_5,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,A) )
=> ( ~ v1_xboole_0(k5_finseq_5(A,B,D,C))
& v1_relat_1(k5_finseq_5(A,B,D,C))
& v1_funct_1(k5_finseq_5(A,B,D,C))
& v1_finset_1(k5_finseq_5(A,B,D,C))
& v1_finseq_1(k5_finseq_5(A,B,D,C)) ) ) ).
fof(t1_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> m2_subset_1(k2_xcmplx_0(k6_xcmplx_0(B,A),np__1),k1_numbers,k5_numbers) ) ) ) ).
fof(t2_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k2_finseq_1(B))
=> r2_hidden(k2_xcmplx_0(k6_xcmplx_0(B,A),np__1),k2_finseq_1(B)) ) ) ) ).
fof(t3_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( k10_relat_1(A,k1_tarski(C)) = k1_tarski(B)
=> ( r2_hidden(B,k1_relat_1(A))
& r2_hidden(C,k2_relat_1(A))
& k1_funct_1(A,B) = C ) ) ) ).
fof(t4_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
<=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> k10_relat_1(A,k1_tarski(k1_funct_1(A,B))) = k1_tarski(B) ) ) ) ).
fof(t5_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( ( v2_funct_1(A)
& r2_hidden(B,k2_relat_1(A))
& r2_hidden(C,k2_relat_1(A))
& k10_relat_1(A,k1_tarski(B)) = k10_relat_1(A,k1_tarski(C)) )
=> B = C ) ) ).
fof(t6_finseq_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( r2_hidden(np__1,k4_finseq_1(A))
& r2_hidden(k3_finseq_1(A),k4_finseq_1(A)) ) ) ).
fof(t7_finseq_5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& k1_nat_1(B,np__1) = k3_finseq_1(A) ) ) ).
fof(t8_finseq_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k3_finseq_1(k7_finseq_1(k9_finseq_1(A),B)) = k1_nat_1(np__1,k3_finseq_1(B)) ) ).
fof(t9_finseq_5,axiom,
$true ).
fof(t10_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( ( r2_hidden(B,k2_relat_1(A))
& r2_hidden(C,k2_relat_1(A))
& k5_finseq_4(A,B) = k5_finseq_4(A,C) )
=> B = C ) ) ).
fof(t11_finseq_5,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) )
=> ( ( r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(B))
& C = k7_relat_1(B,k2_finseq_1(A)) )
=> k7_relat_1(B,k2_finseq_1(k1_nat_1(A,np__1))) = k7_finseq_1(C,k9_finseq_1(k1_funct_1(B,k1_nat_1(A,np__1)))) ) ) ) ) ).
fof(t12_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_finseq_1(B) )
=> ( r2_hidden(A,k4_finseq_1(B))
=> k5_finseq_4(B,k1_funct_1(B,A)) = A ) ) ) ).
fof(t13_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( ( k4_finseq_1(B) = k4_finseq_1(C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> k4_finseq_4(k5_numbers,A,B,D) = k4_finseq_4(k5_numbers,A,C,D) ) ) )
=> B = C ) ) ) ) ).
fof(t14_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( ( k3_finseq_1(B) = k3_finseq_1(C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(B)) )
=> k4_finseq_4(k5_numbers,A,B,D) = k4_finseq_4(k5_numbers,A,C,D) ) ) )
=> B = C ) ) ) ) ).
fof(t15_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( k3_finseq_1(B) = np__1
=> B = k12_finseq_1(A,k4_finseq_4(k5_numbers,A,B,np__1)) ) ) ) ).
fof(t16_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k4_finseq_4(k5_numbers,A,k8_finseq_1(A,k12_finseq_1(A,B),C),np__1) = B ) ) ) ).
fof(t17_finseq_5,axiom,
$true ).
fof(t18_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> r1_xreal_0(k3_finseq_1(k16_finseq_1(B,C,A)),k3_finseq_1(C)) ) ) ).
fof(t19_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> r1_xreal_0(k3_finseq_1(k16_finseq_1(B,C,A)),A) ) ) ).
fof(t20_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> r1_tarski(k4_finseq_1(k16_finseq_1(B,C,A)),k4_finseq_1(C)) ) ) ).
fof(t21_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> r1_tarski(k2_relat_1(k16_finseq_1(B,C,A)),k2_relat_1(C)) ) ) ) ).
fof(t22_finseq_5,axiom,
$true ).
fof(t23_finseq_5,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ( ~ v1_xboole_0(B)
=> k16_finseq_1(A,B,np__1) = k9_finseq_1(k4_finseq_4(k5_numbers,A,B,np__1)) ) ) ).
fof(t24_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( k1_nat_1(A,np__1) = k3_finseq_1(C)
=> C = k8_finseq_1(B,k16_finseq_1(B,C,A),k12_finseq_1(B,k4_finseq_4(k5_numbers,B,C,k3_finseq_1(C)))) ) ) ) ) ).
fof(t25_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( r1_xreal_0(A,k3_finseq_1(C))
=> k16_finseq_1(B,k8_finseq_1(B,C,D),A) = k16_finseq_1(B,C,A) ) ) ) ) ).
fof(t26_finseq_5,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k16_finseq_1(A,k8_finseq_1(A,B,C),k3_finseq_1(B)) = B ) ) ).
fof(t27_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C,D] :
( m2_finseq_1(D,C)
=> ( r2_hidden(B,k2_relat_1(D))
=> k7_finseq_1(k6_finseq_4(D,B),k12_finseq_1(A,B)) = k16_finseq_1(C,D,k5_finseq_4(D,B)) ) ) ) ) ).
fof(t28_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> r1_xreal_0(k3_finseq_1(k1_rfinseq(B,C,A)),k3_finseq_1(C)) ) ) ) ).
fof(t29_finseq_5,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] :
( m2_finseq_1(D,C)
=> ( r2_hidden(A,k4_finseq_1(k1_rfinseq(C,D,B)))
=> r2_hidden(k1_nat_1(B,A),k4_finseq_1(D)) ) ) ) ) ) ).
fof(t30_finseq_5,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] :
( m2_finseq_1(D,C)
=> ( r2_hidden(A,k4_finseq_1(k1_rfinseq(C,D,B)))
=> k4_finseq_4(k5_numbers,C,k1_rfinseq(C,D,B),A) = k4_finseq_4(k5_numbers,C,D,k1_nat_1(B,A)) ) ) ) ) ) ).
fof(t31_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> k1_rfinseq(A,B,np__0) = B ) ) ).
fof(t32_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( ~ v1_xboole_0(B)
=> B = k8_finseq_1(A,k12_finseq_1(A,k4_finseq_4(k5_numbers,A,B,np__1)),k1_rfinseq(A,B,np__1)) ) ) ) ).
fof(t33_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( k1_nat_1(A,np__1) = k3_finseq_1(C)
=> k1_rfinseq(B,C,A) = k12_finseq_1(B,k4_finseq_4(k5_numbers,B,C,k3_finseq_1(C))) ) ) ) ) ).
fof(t34_finseq_5,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] :
( m2_finseq_1(D,C)
=> ( ( k1_nat_1(A,np__1) = B
& r2_hidden(B,k4_finseq_1(D)) )
=> k8_finseq_1(C,k12_finseq_1(C,k4_finseq_4(k5_numbers,C,D,B)),k1_rfinseq(C,D,B)) = k1_rfinseq(C,D,A) ) ) ) ) ) ).
fof(t35_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> ( r1_xreal_0(k3_finseq_1(C),A)
=> v1_xboole_0(k1_rfinseq(B,C,A)) ) ) ) ).
fof(t36_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> r1_tarski(k2_relat_1(k1_rfinseq(B,C,A)),k2_relat_1(C)) ) ) ) ).
fof(t37_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( v2_funct_1(C)
=> r1_xboole_0(k2_relat_1(k16_finseq_1(B,C,A)),k2_relat_1(k1_rfinseq(B,C,A))) ) ) ) ) ).
fof(t38_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r2_hidden(B,k2_relat_1(C))
=> k7_finseq_4(C,B) = k1_rfinseq(A,C,k5_finseq_4(C,B)) ) ) ) ) ).
fof(t39_finseq_5,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)
=> k1_rfinseq(B,k8_finseq_1(B,C,D),k1_nat_1(k3_finseq_1(C),A)) = k1_rfinseq(B,D,A) ) ) ) ) ).
fof(t40_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k1_rfinseq(A,k8_finseq_1(A,B,C),k3_finseq_1(B)) = C ) ) ) ).
fof(t41_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r2_hidden(B,k2_relat_1(C))
=> k4_finseq_4(k5_numbers,A,C,k5_finseq_4(C,B)) = B ) ) ) ) ).
fof(t42_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( r2_hidden(A,k4_finseq_1(C))
=> r1_xreal_0(k5_finseq_4(C,k4_finseq_4(k5_numbers,B,C,A)),A) ) ) ) ) ).
fof(t43_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( r2_hidden(C,k2_relat_1(k16_finseq_1(B,D,A)))
=> k5_finseq_4(k16_finseq_1(B,D,A),C) = k5_finseq_4(D,C) ) ) ) ) ) ).
fof(t44_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ( ( r2_hidden(A,k4_finseq_1(C))
& v2_funct_1(C) )
=> k5_finseq_4(C,k4_finseq_4(k5_numbers,B,C,A)) = A ) ) ) ) ).
fof(d1_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] : k1_finseq_5(A,B,C) = k16_finseq_1(A,B,k5_finseq_4(B,C)) ) ) ).
fof(t45_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r2_hidden(B,k2_relat_1(C))
=> k3_finseq_1(k1_finseq_5(A,C,B)) = k5_finseq_4(C,B) ) ) ) ) ).
fof(t46_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( ( r2_hidden(C,k2_relat_1(D))
& r2_hidden(A,k2_finseq_1(k5_finseq_4(D,C))) )
=> k4_finseq_4(k5_numbers,B,k1_finseq_5(B,D,C),A) = k4_finseq_4(k5_numbers,B,D,A) ) ) ) ) ) ).
fof(t47_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r2_hidden(B,k2_relat_1(C))
=> k4_finseq_4(k5_numbers,A,k1_finseq_5(A,C,B),np__1) = k4_finseq_4(k5_numbers,A,C,np__1) ) ) ) ) ).
fof(t48_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r2_hidden(B,k2_relat_1(C))
=> k4_finseq_4(k5_numbers,A,k1_finseq_5(A,C,B),k5_finseq_4(C,B)) = B ) ) ) ) ).
fof(t49_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( ( r2_hidden(D,k2_relat_1(C))
& r2_hidden(B,k2_relat_1(C))
& r1_xreal_0(k5_finseq_4(C,D),k5_finseq_4(C,B)) )
=> r2_hidden(D,k2_relat_1(k1_finseq_5(A,C,B))) ) ) ) ) ).
fof(t50_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ~ ( r2_hidden(B,k2_relat_1(C))
& v1_xboole_0(k1_finseq_5(A,C,B)) ) ) ) ) ).
fof(t51_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> r1_tarski(k2_relat_1(k1_finseq_5(A,C,B)),k2_relat_1(C)) ) ) ) ).
fof(d2_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> k2_finseq_5(A,B,C) = k8_finseq_1(A,k12_finseq_1(A,C),k1_rfinseq(A,B,k5_finseq_4(B,C))) ) ) ) ).
fof(t52_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ~ ( r2_hidden(B,k2_relat_1(C))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( k1_nat_1(D,np__1) = k5_finseq_4(C,B)
& k2_finseq_5(A,C,B) = k1_rfinseq(A,C,D) ) ) ) ) ) ) ).
fof(t53_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r2_hidden(B,k2_relat_1(C))
=> k3_finseq_1(k2_finseq_5(A,C,B)) = k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(C),k5_finseq_4(C,B)),np__1) ) ) ) ) ).
fof(t54_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( ( r2_hidden(C,k2_relat_1(D))
& r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(k2_finseq_5(B,D,C))) )
=> r2_hidden(k1_nat_1(A,k5_finseq_4(D,C)),k4_finseq_1(D)) ) ) ) ) ) ).
fof(t55_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( ( r2_hidden(C,k2_relat_1(D))
& r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(k2_finseq_5(B,D,C))) )
=> k4_finseq_4(k5_numbers,B,k2_finseq_5(B,D,C),k1_nat_1(A,np__1)) = k4_finseq_4(k5_numbers,B,D,k1_nat_1(A,k5_finseq_4(D,C))) ) ) ) ) ) ).
fof(t56_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k4_finseq_4(k5_numbers,A,k2_finseq_5(A,C,B),np__1) = B ) ) ) ).
fof(t57_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r2_hidden(B,k2_relat_1(C))
=> k4_finseq_4(k5_numbers,A,k2_finseq_5(A,C,B),k3_finseq_1(k2_finseq_5(A,C,B))) = k4_finseq_4(k5_numbers,A,C,k3_finseq_1(C)) ) ) ) ) ).
fof(t58_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( r2_hidden(B,k2_relat_1(C))
=> r1_tarski(k2_relat_1(k2_finseq_5(A,C,B)),k2_relat_1(C)) ) ) ) ) ).
fof(t59_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ( ( r2_hidden(B,k2_relat_1(C))
& v2_funct_1(C) )
=> v2_funct_1(k2_finseq_5(A,C,B)) ) ) ) ) ).
fof(d3_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( B = k3_finseq_5(A)
<=> ( k3_finseq_1(B) = k3_finseq_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(B))
=> k1_funct_1(B,C) = k1_funct_1(A,k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(A),C),np__1)) ) ) ) ) ) ) ).
fof(t60_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( k4_finseq_1(A) = k4_finseq_1(k3_finseq_5(A))
& k2_relat_1(A) = k2_relat_1(k3_finseq_5(A)) ) ) ).
fof(t61_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( r2_hidden(A,k4_finseq_1(B))
=> k1_funct_1(k3_finseq_5(B),A) = k1_funct_1(B,k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(B),A),np__1)) ) ) ) ).
fof(t62_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k4_finseq_1(A))
& k1_nat_1(B,C) = k1_nat_1(k3_finseq_1(A),np__1) )
=> r2_hidden(C,k4_finseq_1(k3_finseq_5(A))) ) ) ) ) ).
fof(t63_finseq_5,axiom,
! [A] : k3_finseq_5(k9_finseq_1(A)) = k9_finseq_1(A) ).
fof(t64_finseq_5,axiom,
! [A,B] : k3_finseq_5(k10_finseq_1(A,B)) = k10_finseq_1(B,A) ).
fof(t65_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( k1_funct_1(A,np__1) = k1_funct_1(k3_finseq_5(A),k3_finseq_1(A))
& k1_funct_1(A,k3_finseq_1(A)) = k1_funct_1(k3_finseq_5(A),np__1) ) ) ).
fof(t66_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] : k3_finseq_5(k7_finseq_1(A,k9_finseq_1(B))) = k7_finseq_1(k9_finseq_1(B),k3_finseq_5(A)) ) ).
fof(t67_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k3_finseq_5(k7_finseq_1(A,B)) = k7_finseq_1(k3_finseq_5(B),k3_finseq_5(A)) ) ) ).
fof(t68_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ( ~ v1_xboole_0(B)
=> ( k4_finseq_4(k5_numbers,A,B,np__1) = k4_finseq_4(k5_numbers,A,k4_finseq_5(A,B),k3_finseq_1(B))
& k4_finseq_4(k5_numbers,A,B,k3_finseq_1(B)) = k4_finseq_4(k5_numbers,A,k4_finseq_5(A,B),np__1) ) ) ) ) ).
fof(t69_finseq_5,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] :
( m2_finseq_1(D,C)
=> ( ( r2_hidden(A,k4_finseq_1(D))
& k1_nat_1(A,B) = k1_nat_1(k3_finseq_1(D),np__1) )
=> k4_finseq_4(k5_numbers,C,D,A) = k4_finseq_4(k5_numbers,C,k4_finseq_5(C,D),B) ) ) ) ) ) ).
fof(d4_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k5_finseq_5(A,B,C,D) = k8_finseq_1(A,k8_finseq_1(A,k16_finseq_1(A,B,D),k12_finseq_1(A,C)),k1_rfinseq(A,B,D)) ) ) ) ) ).
fof(t70_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k5_finseq_5(A,C,B,np__0) = k8_finseq_1(A,k12_finseq_1(A,B),C) ) ) ) ).
fof(t71_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( r1_xreal_0(k3_finseq_1(D),A)
=> k5_finseq_5(B,D,C,A) = k8_finseq_1(B,D,k12_finseq_1(B,C)) ) ) ) ) ) ).
fof(t72_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> k3_finseq_1(k5_finseq_5(B,D,C,A)) = k1_nat_1(k3_finseq_1(D),np__1) ) ) ) ) ).
fof(t73_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> k2_relat_1(k5_finseq_5(B,D,C,A)) = k2_xboole_0(k1_tarski(C),k2_relat_1(D)) ) ) ) ) ).
fof(t74_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> r2_hidden(C,k2_relat_1(k5_finseq_5(B,D,C,A))) ) ) ) ) ).
fof(t75_finseq_5,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)
=> ! [E] :
( m2_finseq_1(E,C)
=> ( r2_hidden(A,k4_finseq_1(k16_finseq_1(C,E,B)))
=> k4_finseq_4(k5_numbers,C,k5_finseq_5(C,E,D,B),A) = k4_finseq_4(k5_numbers,C,E,A) ) ) ) ) ) ) ).
fof(t76_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( r1_xreal_0(A,k3_finseq_1(D))
=> k4_finseq_4(k5_numbers,B,k5_finseq_5(B,D,C,A),k1_nat_1(A,np__1)) = C ) ) ) ) ) ).
fof(t77_finseq_5,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)
=> ! [E] :
( m2_finseq_1(E,C)
=> ( ( r1_xreal_0(k1_nat_1(A,np__1),B)
& r1_xreal_0(B,k3_finseq_1(E)) )
=> k4_finseq_4(k5_numbers,C,k5_finseq_5(C,E,D,A),k1_nat_1(B,np__1)) = k4_finseq_4(k5_numbers,C,E,B) ) ) ) ) ) ) ).
fof(t78_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( r1_xreal_0(np__1,A)
=> ( v1_xboole_0(D)
| k4_finseq_4(k5_numbers,B,k5_finseq_5(B,D,C,A),np__1) = k4_finseq_4(k5_numbers,B,D,np__1) ) ) ) ) ) ) ).
fof(t79_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m2_finseq_1(D,B)
=> ( v2_funct_1(D)
=> ( r2_hidden(C,k2_relat_1(D))
| v2_funct_1(k5_finseq_5(B,D,C,A)) ) ) ) ) ) ) ).
fof(t80_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,D)
=> ( k16_finseq_1(A,k16_finseq_1(A,B,C),D) = k16_finseq_1(A,B,C)
& k16_finseq_1(A,k16_finseq_1(A,B,D),C) = k16_finseq_1(A,B,C) ) ) ) ) ) ) ).
fof(t81_finseq_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k16_finseq_1(B,k6_finseq_1(B),A) = k6_finseq_1(B) ) ) ).
fof(t82_finseq_5,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_finseq_5(A,k6_finseq_1(A)) = k6_finseq_1(A) ) ).
fof(s1_finseq_5,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& k3_finseq_1(A) = f1_s1_finseq_5
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k4_finseq_1(A))
=> k1_funct_1(A,B) = f2_s1_finseq_5(B) ) ) ) ).
fof(dt_k1_finseq_5,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A) )
=> m2_finseq_1(k1_finseq_5(A,B,C),A) ) ).
fof(dt_k2_finseq_5,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A)
& m1_subset_1(C,A) )
=> m2_finseq_1(k2_finseq_5(A,B,C),A) ) ).
fof(dt_k3_finseq_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(k3_finseq_5(A))
& v1_funct_1(k3_finseq_5(A))
& v1_finseq_1(k3_finseq_5(A)) ) ) ).
fof(dt_k4_finseq_5,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A) )
=> m2_finseq_1(k4_finseq_5(A,B),A) ) ).
fof(redefinition_k4_finseq_5,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A) )
=> k4_finseq_5(A,B) = k3_finseq_5(B) ) ).
fof(dt_k5_finseq_5,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,k5_numbers) )
=> m2_finseq_1(k5_finseq_5(A,B,C,D),A) ) ).
%------------------------------------------------------------------------------