SET007 Axioms: SET007+97.ax
%------------------------------------------------------------------------------
% File : SET007+97 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Pigeon Hole Principle
% 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_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 102 ( 15 unt; 0 def)
% Number of atoms : 536 ( 85 equ)
% Maximal formula atoms : 15 ( 5 avg)
% Number of connectives : 473 ( 39 ~; 0 |; 223 &)
% ( 20 <=>; 191 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 6 con; 0-4 aty)
% Number of variables : 226 ( 222 !; 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(d1_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r1_finseq_4(A,B)
<=> k10_relat_1(A,k9_relat_1(A,k1_tarski(B))) = k1_tarski(B) ) ) ).
fof(t1_finseq_4,axiom,
$true ).
fof(t2_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r1_finseq_4(A,B)
=> r2_hidden(B,k1_relat_1(A)) ) ) ).
fof(t3_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r1_finseq_4(A,B)
<=> ( r2_hidden(B,k1_relat_1(A))
& k10_relat_1(A,k1_tarski(k1_funct_1(A,B))) = k1_tarski(B) ) ) ) ).
fof(t4_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r1_finseq_4(A,B)
<=> ( r2_hidden(B,k1_relat_1(A))
& ! [C] :
~ ( r2_hidden(C,k1_relat_1(A))
& B != C
& k1_funct_1(A,B) = k1_funct_1(A,C) ) ) ) ) ).
fof(t5_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> r1_finseq_4(A,B) )
<=> v2_funct_1(A) ) ) ).
fof(d2_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
<=> ? [C] :
( v1_finset_1(C)
& C = k10_relat_1(A,k1_tarski(B))
& k4_card_1(C) = np__1 ) ) ) ).
fof(t6_finseq_4,axiom,
$true ).
fof(t7_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
=> r2_hidden(B,k2_relat_1(A)) ) ) ).
fof(t8_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
<=> ? [C] : k1_tarski(C) = k10_relat_1(A,k1_tarski(B)) ) ) ).
fof(t9_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
<=> ? [C] :
( r2_hidden(C,k1_relat_1(A))
& B = k1_funct_1(A,C)
& ! [D] :
~ ( r2_hidden(D,k1_relat_1(A))
& D != C
& k1_funct_1(A,D) = B ) ) ) ) ).
fof(t10_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_funct_1(A)
<=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> r2_finseq_4(A,B) ) ) ) ).
fof(t11_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r1_finseq_4(A,B)
<=> ( r2_hidden(B,k1_relat_1(A))
& r2_finseq_4(A,k1_funct_1(A,B)) ) ) ) ).
fof(d3_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
=> ! [C] :
( C = k1_finseq_4(A,B)
<=> ( r2_hidden(C,k1_relat_1(A))
& k1_funct_1(A,C) = B ) ) ) ) ).
fof(t12_finseq_4,axiom,
$true ).
fof(t13_finseq_4,axiom,
$true ).
fof(t14_finseq_4,axiom,
$true ).
fof(t15_finseq_4,axiom,
$true ).
fof(t16_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
=> k9_relat_1(A,k1_tarski(k1_finseq_4(A,B))) = k1_tarski(B) ) ) ).
fof(t17_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
=> k10_relat_1(A,k1_tarski(B)) = k1_tarski(k1_finseq_4(A,B)) ) ) ).
fof(t18_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v2_funct_1(A)
& r2_hidden(B,k2_relat_1(A)) )
=> k1_funct_1(k2_funct_1(A),B) = k1_finseq_4(A,B) ) ) ).
fof(t19_finseq_4,axiom,
$true ).
fof(t20_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r1_finseq_4(A,B)
=> k1_finseq_4(A,k1_funct_1(A,B)) = B ) ) ).
fof(t21_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
=> r1_finseq_4(A,k1_finseq_4(A,B)) ) ) ).
fof(d4_finseq_4,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,B) )
=> ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> k4_finseq_4(A,B,C,D) = k1_funct_1(C,D) ) ) ).
fof(t22_finseq_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,A)
=> k4_finseq_4(A,B,C,D) = k8_funct_2(A,B,C,D) ) ) ) ) ).
fof(t23_finseq_4,axiom,
$true ).
fof(t24_finseq_4,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B)) )
=> k4_finseq_4(k5_numbers,A,B,C) = k1_funct_1(B,C) ) ) ) ).
fof(t25_finseq_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k4_finseq_4(k5_numbers,A,k12_finseq_1(A,B),np__1) = B ) ) ).
fof(t26_finseq_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( k4_finseq_4(k5_numbers,A,k2_finseq_4(A,B,C),np__1) = B
& k4_finseq_4(k5_numbers,A,k2_finseq_4(A,B,C),np__2) = C ) ) ) ) ).
fof(t27_finseq_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( k4_finseq_4(k5_numbers,A,k3_finseq_4(A,B,C,D),np__1) = B
& k4_finseq_4(k5_numbers,A,k3_finseq_4(A,B,C,D),np__2) = C
& k4_finseq_4(k5_numbers,A,k3_finseq_4(A,B,C,D),np__3) = D ) ) ) ) ) ).
fof(d5_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] : k5_finseq_4(A,B) = k1_funct_1(k14_finseq_1(k10_relat_1(A,k1_tarski(B))),np__1) ) ).
fof(t28_finseq_4,axiom,
$true ).
fof(t29_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> k1_funct_1(A,k5_finseq_4(A,B)) = B ) ) ).
fof(t30_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> r2_hidden(k5_finseq_4(A,B),k4_finseq_1(A)) ) ) ).
fof(t31_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ( r1_xreal_0(np__1,k5_finseq_4(A,B))
& r1_xreal_0(k5_finseq_4(A,B),k3_finseq_1(A)) ) ) ) ).
fof(t32_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ( m2_subset_1(k6_xcmplx_0(k5_finseq_4(A,B),np__1),k1_numbers,k5_numbers)
& m2_subset_1(k6_xcmplx_0(k3_finseq_1(A),k5_finseq_4(A,B)),k1_numbers,k5_numbers) ) ) ) ).
fof(t33_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> r2_hidden(k5_finseq_4(A,B),k10_relat_1(A,k1_tarski(B))) ) ) ).
fof(t34_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(A))
& ~ r1_xreal_0(k5_finseq_4(A,B),C)
& k1_funct_1(A,C) = B ) ) ) ).
fof(t35_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
=> k1_finseq_4(A,B) = k5_finseq_4(A,B) ) ) ).
fof(t36_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(A))
& C != k5_finseq_4(A,B)
& k1_funct_1(A,C) = B ) ) ) ) ).
fof(t37_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( r2_hidden(B,k2_relat_1(A))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(A))
& C != k5_finseq_4(A,B)
& k1_funct_1(A,C) = B ) ) )
=> r2_finseq_4(A,B) ) ) ).
fof(t38_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
<=> ( r2_hidden(B,k2_relat_1(A))
& k1_tarski(k5_finseq_4(A,B)) = k10_relat_1(A,k1_tarski(B)) ) ) ) ).
fof(t39_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v2_funct_1(A)
& r2_hidden(B,k2_relat_1(A)) )
=> k1_tarski(k5_finseq_4(A,B)) = k10_relat_1(A,k1_tarski(B)) ) ) ).
fof(t40_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
<=> k3_finseq_1(k1_finseq_3(A,k1_tarski(B))) = k6_xcmplx_0(k3_finseq_1(A),np__1) ) ) ).
fof(t41_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(k1_finseq_3(A,k1_tarski(B))))
=> ( ( ~ r1_xreal_0(k5_finseq_4(A,B),C)
=> k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,C) )
& ( r1_xreal_0(k5_finseq_4(A,B),C)
=> k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,k1_nat_1(C,np__1)) ) ) ) ) ) ) ).
fof(t42_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v2_funct_1(A)
& r2_hidden(B,k2_relat_1(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(k1_finseq_3(A,k1_tarski(B))))
=> ( ~ ( k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,C)
& r1_xreal_0(k5_finseq_4(A,B),C) )
& ( ~ r1_xreal_0(k5_finseq_4(A,B),C)
=> k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,C) )
& ( k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,k1_nat_1(C,np__1))
=> r1_xreal_0(k5_finseq_4(A,B),C) )
& ( r1_xreal_0(k5_finseq_4(A,B),C)
=> k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,k1_nat_1(C,np__1)) ) ) ) ) ) ) ).
fof(d6_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( C = k6_finseq_4(A,B)
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& D = k6_xcmplx_0(k5_finseq_4(A,B),np__1)
& C = k7_relat_1(A,k2_finseq_1(D)) ) ) ) ) ) ).
fof(t43_finseq_4,axiom,
$true ).
fof(t44_finseq_4,axiom,
$true ).
fof(t45_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k2_relat_1(A))
& C = k6_xcmplx_0(k5_finseq_4(A,B),np__1) )
=> k7_relat_1(A,k2_finseq_1(C)) = k6_finseq_4(A,B) ) ) ) ).
fof(t46_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> k3_finseq_1(k6_finseq_4(A,B)) = k6_xcmplx_0(k5_finseq_4(A,B),np__1) ) ) ).
fof(t47_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k2_relat_1(A))
& C = k6_xcmplx_0(k5_finseq_4(A,B),np__1) )
=> k4_finseq_1(k6_finseq_4(A,B)) = k2_finseq_1(C) ) ) ) ).
fof(t48_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k2_relat_1(A))
& r2_hidden(C,k4_finseq_1(k6_finseq_4(A,B))) )
=> k1_funct_1(A,C) = k1_funct_1(k6_finseq_4(A,B),C) ) ) ) ).
fof(t49_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
~ ( r2_hidden(B,k2_relat_1(A))
& r2_hidden(B,k2_relat_1(k6_finseq_4(A,B))) ) ) ).
fof(t50_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> r1_xboole_0(k2_relat_1(k6_finseq_4(A,B)),k1_tarski(B)) ) ) ).
fof(t51_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> r1_tarski(k2_relat_1(k6_finseq_4(A,B)),k2_relat_1(A)) ) ) ).
fof(t52_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ( k5_finseq_4(A,B) = np__1
<=> k6_finseq_4(A,B) = k1_xboole_0 ) ) ) ).
fof(t53_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( ~ v1_xboole_0(C)
=> ( ( r2_hidden(B,k2_relat_1(A))
& m2_finseq_1(A,C) )
=> m2_finseq_1(k6_finseq_4(A,B),C) ) ) ) ).
fof(d7_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( C = k7_finseq_4(A,B)
<=> ( k3_finseq_1(C) = k6_xcmplx_0(k3_finseq_1(A),k5_finseq_4(A,B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(C))
=> k1_funct_1(C,D) = k1_funct_1(A,k1_nat_1(D,k5_finseq_4(A,B))) ) ) ) ) ) ) ) ).
fof(t54_finseq_4,axiom,
$true ).
fof(t55_finseq_4,axiom,
$true ).
fof(t56_finseq_4,axiom,
$true ).
fof(t57_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k2_relat_1(A))
& C = k6_xcmplx_0(k3_finseq_1(A),k5_finseq_4(A,B)) )
=> k4_finseq_1(k7_finseq_4(A,B)) = k2_finseq_1(C) ) ) ) ).
fof(t58_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k2_relat_1(A))
& r2_hidden(C,k4_finseq_1(k7_finseq_4(A,B))) )
=> r2_hidden(k1_nat_1(C,k5_finseq_4(A,B)),k4_finseq_1(A)) ) ) ) ).
fof(t59_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> r1_tarski(k2_relat_1(k7_finseq_4(A,B)),k2_relat_1(A)) ) ) ).
fof(t60_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
<=> ( r2_hidden(B,k2_relat_1(A))
& ~ r2_hidden(B,k2_relat_1(k7_finseq_4(A,B))) ) ) ) ).
fof(t61_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
~ ( r2_hidden(B,k2_relat_1(A))
& v2_funct_1(A)
& r2_hidden(B,k2_relat_1(k7_finseq_4(A,B))) ) ) ).
fof(t62_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
<=> ( r2_hidden(B,k2_relat_1(A))
& r1_xboole_0(k2_relat_1(k7_finseq_4(A,B)),k1_tarski(B)) ) ) ) ).
fof(t63_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( r2_hidden(B,k2_relat_1(A))
& v2_funct_1(A) )
=> r1_xboole_0(k2_relat_1(k7_finseq_4(A,B)),k1_tarski(B)) ) ) ).
fof(t64_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ( k5_finseq_4(A,B) = k3_finseq_1(A)
<=> k7_finseq_4(A,B) = k1_xboole_0 ) ) ) ).
fof(t65_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( ~ v1_xboole_0(C)
=> ( ( r2_hidden(B,k2_relat_1(A))
& m2_finseq_1(A,C) )
=> m2_finseq_1(k7_finseq_4(A,B),C) ) ) ) ).
fof(t66_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> A = k7_finseq_1(k7_finseq_1(k6_finseq_4(A,B),k9_finseq_1(B)),k7_finseq_4(A,B)) ) ) ).
fof(t67_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( r2_hidden(B,k2_relat_1(A))
& v2_funct_1(A) )
=> v2_funct_1(k6_finseq_4(A,B)) ) ) ).
fof(t68_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( r2_hidden(B,k2_relat_1(A))
& v2_funct_1(A) )
=> v2_funct_1(k7_finseq_4(A,B)) ) ) ).
fof(t69_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_finseq_4(A,B)
<=> ( r2_hidden(B,k2_relat_1(A))
& k1_finseq_3(A,k1_tarski(B)) = k7_finseq_1(k6_finseq_4(A,B),k7_finseq_4(A,B)) ) ) ) ).
fof(t70_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( r2_hidden(B,k2_relat_1(A))
& v2_funct_1(A) )
=> k1_finseq_3(A,k1_tarski(B)) = k7_finseq_1(k6_finseq_4(A,B),k7_finseq_4(A,B)) ) ) ).
fof(t71_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( r2_hidden(B,k2_relat_1(A))
& v2_funct_1(k1_finseq_3(A,k1_tarski(B)))
& k1_finseq_3(A,k1_tarski(B)) = k7_finseq_1(k6_finseq_4(A,B),k7_finseq_4(A,B)) )
=> v2_funct_1(A) ) ) ).
fof(t72_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( r2_hidden(B,k2_relat_1(A))
& v2_funct_1(A) )
=> r1_xboole_0(k2_relat_1(k6_finseq_4(A,B)),k2_relat_1(k7_finseq_4(A,B))) ) ) ).
fof(t73_finseq_4,axiom,
! [A] :
~ ( v1_finset_1(A)
& ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ~ ( k2_relat_1(B) = A
& v2_funct_1(B) ) ) ) ).
fof(t74_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( ( r1_tarski(k2_relat_1(A),k4_finseq_1(A))
& v2_funct_1(A) )
=> k2_relat_1(A) = k4_finseq_1(A) ) ) ).
fof(t75_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( k2_relat_1(A) = k4_finseq_1(A)
=> v2_funct_1(A) ) ) ).
fof(t76_finseq_4,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) )
=> ( ( k2_relat_1(A) = k2_relat_1(B)
& k3_finseq_1(A) = k3_finseq_1(B)
& v2_funct_1(B) )
=> v2_funct_1(A) ) ) ) ).
fof(t77_finseq_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v2_funct_1(A)
<=> k4_card_1(k2_relat_1(A)) = k3_finseq_1(A) ) ) ).
fof(t78_finseq_4,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( ( k4_card_1(A) = k4_card_1(B)
& v2_funct_1(C) )
=> k2_relat_1(C) = B ) ) ) ) ).
fof(t79_finseq_4,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( ( k4_card_1(A) = k4_card_1(B)
& k2_relat_1(C) = B )
=> v2_funct_1(C) ) ) ) ) ).
fof(t80_finseq_4,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,A)
& m2_relset_1(C,B,A) )
=> ~ ( r2_hidden(k1_card_1(A),k1_card_1(B))
& A != k1_xboole_0
& ! [D,E] :
~ ( r2_hidden(D,B)
& r2_hidden(E,B)
& D != E
& k1_funct_1(C,D) = k1_funct_1(C,E) ) ) ) ).
fof(t81_finseq_4,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ~ ( r2_hidden(k1_card_1(A),k1_card_1(B))
& ! [D] :
~ ( r2_hidden(D,B)
& ! [E] :
~ ( r2_hidden(E,A)
& k1_funct_1(C,E) = D ) ) ) ) ).
fof(t82_finseq_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> k4_finseq_4(k5_numbers,A,k8_finseq_1(A,B,k12_finseq_1(A,C)),k1_nat_1(k3_finseq_1(B),np__1)) = C ) ) ) ).
fof(t83_finseq_4,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)
=> ( r2_hidden(A,k4_finseq_1(C))
=> k4_finseq_4(k5_numbers,B,k8_finseq_1(B,C,D),A) = k4_finseq_4(k5_numbers,B,C,A) ) ) ) ) ) ).
fof(t84_finseq_4,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)
=> ( r2_hidden(A,k4_finseq_1(D))
=> k4_finseq_4(k5_numbers,B,k8_finseq_1(B,C,D),k1_nat_1(k3_finseq_1(C),A)) = k4_finseq_4(k5_numbers,B,D,A) ) ) ) ) ) ).
fof(t85_finseq_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C,D] :
( m2_finseq_1(D,C)
=> ( r2_hidden(B,k4_finseq_1(k16_finseq_1(C,D,A)))
=> k4_finseq_4(k5_numbers,C,k16_finseq_1(C,D,A),B) = k4_finseq_4(k5_numbers,C,D,B) ) ) ) ) ).
fof(t86_finseq_4,axiom,
! [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)
=> ( ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(D,k2_finseq_1(C)) )
=> ( r2_hidden(D,k4_finseq_1(B))
& k4_finseq_4(k5_numbers,A,k16_finseq_1(A,B,C),D) = k4_finseq_4(k5_numbers,A,B,D) ) ) ) ) ) ).
fof(dt_k1_finseq_4,axiom,
$true ).
fof(dt_k2_finseq_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m2_finseq_1(k2_finseq_4(A,B,C),A) ) ).
fof(redefinition_k2_finseq_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k2_finseq_4(A,B,C) = k10_finseq_1(B,C) ) ).
fof(dt_k3_finseq_4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,A) )
=> m2_finseq_1(k3_finseq_4(A,B,C,D),A) ) ).
fof(redefinition_k3_finseq_4,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,A) )
=> k3_finseq_4(A,B,C,D) = k11_finseq_1(B,C,D) ) ).
fof(dt_k4_finseq_4,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& m1_relset_1(C,A,B) )
=> m1_subset_1(k4_finseq_4(A,B,C,D),B) ) ).
fof(dt_k5_finseq_4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> m2_subset_1(k5_finseq_4(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k6_finseq_4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(k6_finseq_4(A,B))
& v1_funct_1(k6_finseq_4(A,B))
& v1_finseq_1(k6_finseq_4(A,B)) ) ) ).
fof(dt_k7_finseq_4,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(k7_finseq_4(A,B))
& v1_funct_1(k7_finseq_4(A,B))
& v1_finseq_1(k7_finseq_4(A,B)) ) ) ).
%------------------------------------------------------------------------------