SET007 Axioms: SET007+384.ax
%------------------------------------------------------------------------------
% File : SET007+384 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Product of Family of Universal Algebras
% 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 : pralg_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 82 ( 0 unt; 0 def)
% Number of atoms : 675 ( 87 equ)
% Maximal formula atoms : 30 ( 8 avg)
% Number of connectives : 708 ( 115 ~; 0 |; 364 &)
% ( 22 <=>; 207 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 40 ( 39 usr; 0 prp; 1-3 aty)
% Number of functors : 61 ( 61 usr; 3 con; 0-4 aty)
% Number of variables : 252 ( 245 !; 7 ?)
% 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(fc1_pralg_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ v1_xboole_0(k7_pralg_1(A))
& v1_relat_1(k7_pralg_1(A))
& v1_funct_1(k7_pralg_1(A))
& v1_unialg_1(k7_pralg_1(A),k1_tarski(k1_xboole_0))
& v2_unialg_1(k7_pralg_1(A),k1_tarski(k1_xboole_0)) ) ) ).
fof(rc1_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_finseq_1(B,A)
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) ) ) ).
fof(rc2_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k3_finseq_2(A))
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) ) ) ).
fof(rc3_pralg_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_pralg_1(A) ) ).
fof(cc1_pralg_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_pralg_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_pralg_1(A) ) ) ).
fof(rc4_pralg_1,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_pralg_1(B) ) ).
fof(rc5_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_pralg_1(B)
& v2_pralg_1(B)
& v3_pralg_1(B) ) ) ).
fof(fc2_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_pralg_1(B)
& m1_pboole(B,A) )
=> ( v1_relat_1(k12_pralg_1(A,B))
& v2_relat_1(k12_pralg_1(A,B))
& ~ v3_relat_1(k12_pralg_1(A,B))
& v1_funct_1(k12_pralg_1(A,B)) ) ) ).
fof(fc3_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A) )
=> ( ~ v1_xboole_0(k4_card_3(B))
& v1_fraenkel(k4_card_3(B)) ) ) ).
fof(rc6_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A) )
=> ? [C] :
( m1_pralg_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funcop_1(C)
& v4_pralg_1(C) ) ) ).
fof(t1_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k4_finseq_2(C,A) = k4_finseq_2(D,B)
=> C = D ) ) ) ) ) ).
fof(d1_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m2_finseq_1(D,A)
=> ( D = k1_pralg_1(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(D))
=> k1_funct_1(D,E) = k1_mcart_1(k1_funct_1(C,E)) ) ) ) ) ) ) ) ) ).
fof(d2_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m2_finseq_1(D,B)
=> ( D = k2_pralg_1(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(D))
=> k1_funct_1(D,E) = k2_mcart_1(k1_funct_1(C,E)) ) ) ) ) ) ) ) ) ).
fof(d3_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_funct_1(C)
& v1_unialg_1(C,A)
& v2_unialg_1(C,A)
& m2_relset_1(C,k3_finseq_2(A),A) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_funct_1(D)
& v1_unialg_1(D,B)
& v2_unialg_1(D,B)
& m2_relset_1(D,k3_finseq_2(B),B) )
=> ( k2_unialg_1(A,C) = k2_unialg_1(B,D)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& v1_funct_1(E)
& v1_unialg_1(E,k2_zfmisc_1(A,B))
& v2_unialg_1(E,k2_zfmisc_1(A,B))
& m2_relset_1(E,k3_finseq_2(k2_zfmisc_1(A,B)),k2_zfmisc_1(A,B)) )
=> ( E = k3_pralg_1(A,B,C,D)
<=> ( k1_relat_1(E) = k4_finseq_2(k2_unialg_1(A,C),k2_zfmisc_1(A,B))
& ! [F] :
( m2_finseq_1(F,k2_zfmisc_1(A,B))
=> ( r2_hidden(F,k1_relat_1(E))
=> k1_funct_1(E,F) = k4_tarski(k1_funct_1(C,k1_pralg_1(A,B,F)),k1_funct_1(D,k2_pralg_1(A,B,F))) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_pralg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r1_unialg_2(A,B)
=> ! [C] :
( m2_finseq_1(C,k4_partfun1(k13_finseq_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))))
=> ( C = k4_pralg_1(A,B)
<=> ( k3_finseq_1(C) = k3_finseq_1(u1_unialg_1(A))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(C))
=> ! [E] :
( ( ~ v1_xboole_0(E)
& v1_funct_1(E)
& v1_unialg_1(E,u1_struct_0(A))
& v2_unialg_1(E,u1_struct_0(A))
& m2_relset_1(E,k3_finseq_2(u1_struct_0(A)),u1_struct_0(A)) )
=> ! [F] :
( ( ~ v1_xboole_0(F)
& v1_funct_1(F)
& v1_unialg_1(F,u1_struct_0(B))
& v2_unialg_1(F,u1_struct_0(B))
& m2_relset_1(F,k3_finseq_2(u1_struct_0(B)),u1_struct_0(B)) )
=> ( ( E = k1_funct_1(u1_unialg_1(A),D)
& F = k1_funct_1(u1_unialg_1(B),D) )
=> k1_funct_1(C,D) = k3_pralg_1(u1_struct_0(A),u1_struct_0(B),E,F) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t2_pralg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r1_unialg_2(A,B)
=> ( ~ v3_struct_0(g1_unialg_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k4_pralg_1(A,B)))
& v3_unialg_1(g1_unialg_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k4_pralg_1(A,B)))
& v6_unialg_1(g1_unialg_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k4_pralg_1(A,B)))
& v7_unialg_1(g1_unialg_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k4_pralg_1(A,B)))
& v8_unialg_1(g1_unialg_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k4_pralg_1(A,B)))
& l1_unialg_1(g1_unialg_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k4_pralg_1(A,B))) ) ) ) ) ).
fof(d5_pralg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r1_unialg_2(A,B)
=> k5_pralg_1(A,B) = g1_unialg_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),k4_pralg_1(A,B)) ) ) ) ).
fof(d6_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,B),k2_zfmisc_1(B,A))
& m2_relset_1(C,k2_zfmisc_1(A,B),k2_zfmisc_1(B,A)) )
=> ( C = k6_pralg_1(A,B)
<=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> k8_funct_2(k2_zfmisc_1(A,B),k2_zfmisc_1(B,A),C,D) = k1_domain_1(B,A,k3_domain_1(A,B,D),k2_domain_1(A,B,D)) ) ) ) ) ) ).
fof(t3_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> k9_funcop_1(k2_zfmisc_1(A,B),B,A,k6_pralg_1(A,B)) = k2_zfmisc_1(B,A) ) ) ).
fof(t4_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> v2_funct_1(k6_pralg_1(A,B)) ) ) ).
fof(t5_pralg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r1_unialg_2(A,B)
=> ( v1_funct_1(k6_pralg_1(u1_struct_0(A),u1_struct_0(B)))
& v1_funct_2(k6_pralg_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(k5_pralg_1(A,B)),u1_struct_0(k5_pralg_1(B,A)))
& m2_relset_1(k6_pralg_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(k5_pralg_1(A,B)),u1_struct_0(k5_pralg_1(B,A))) ) ) ) ) ).
fof(t6_pralg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r1_unialg_2(A,B)
=> ! [C] :
( m2_unialg_2(C,u1_struct_0(A),k1_unialg_2(A))
=> ! [D] :
( m2_unialg_2(D,u1_struct_0(B),k1_unialg_2(B))
=> ! [E] :
( m2_unialg_2(E,u1_struct_0(k5_pralg_1(A,B)),k1_unialg_2(k5_pralg_1(A,B)))
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( C = k1_funct_1(u1_unialg_1(A),F)
& D = k1_funct_1(u1_unialg_1(B),F)
& E = k1_funct_1(u1_unialg_1(k5_pralg_1(A,B)),F)
& r2_hidden(F,k4_finseq_1(u1_unialg_1(A))) )
=> ( k2_unialg_1(u1_struct_0(k5_pralg_1(A,B)),E) = k2_unialg_1(u1_struct_0(A),C)
& k2_unialg_1(u1_struct_0(k5_pralg_1(A,B)),E) = k2_unialg_1(u1_struct_0(B),D)
& E = k3_pralg_1(u1_struct_0(A),u1_struct_0(B),C,D) ) ) ) ) ) ) ) ) ) ).
fof(t7_pralg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( r1_unialg_2(A,B)
=> r1_unialg_2(k5_pralg_1(A,B),A) ) ) ) ).
fof(t8_pralg_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v6_unialg_1(C)
& v7_unialg_1(C)
& v8_unialg_1(C)
& l1_unialg_1(C) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v6_unialg_1(D)
& v7_unialg_1(D)
& v8_unialg_1(D)
& l1_unialg_1(D) )
=> ( ( m3_unialg_2(A,B)
& m3_unialg_2(C,D)
& r1_unialg_2(B,D) )
=> m3_unialg_2(k5_pralg_1(A,C),k5_pralg_1(B,D)) ) ) ) ) ) ).
fof(d7_pralg_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k3_finseq_2(k1_tarski(k1_xboole_0)),k1_tarski(k1_xboole_0)) )
=> ( B = k7_pralg_1(A)
<=> ( k1_relat_1(B) = k1_tarski(k2_finseq_2(A,k1_xboole_0))
& k2_relat_1(B) = k1_tarski(k1_xboole_0) ) ) ) ) ).
fof(t9_pralg_1,axiom,
! [A] :
( v4_ordinal2(A)
=> k2_unialg_1(k1_tarski(k1_xboole_0),k7_pralg_1(A)) = A ) ).
fof(d8_pralg_1,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k4_partfun1(k13_finseq_1(k1_tarski(k1_xboole_0)),k1_tarski(k1_xboole_0)))
=> ( B = k8_pralg_1(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))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( D = k1_funct_1(A,C)
=> k1_funct_1(B,C) = k7_pralg_1(D) ) ) ) ) ) ) ) ) ).
fof(t10_pralg_1,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ( v4_unialg_1(k8_pralg_1(A),k1_tarski(k1_xboole_0))
& v5_unialg_1(k8_pralg_1(A),k1_tarski(k1_xboole_0))
& v2_relat_1(k8_pralg_1(A)) ) ) ).
fof(t11_pralg_1,axiom,
! [A] :
( m2_finseq_1(A,k5_numbers)
=> ( A != k1_xboole_0
=> ( ~ v3_struct_0(g1_unialg_1(k1_tarski(k1_xboole_0),k8_pralg_1(A)))
& v3_unialg_1(g1_unialg_1(k1_tarski(k1_xboole_0),k8_pralg_1(A)))
& v6_unialg_1(g1_unialg_1(k1_tarski(k1_xboole_0),k8_pralg_1(A)))
& v7_unialg_1(g1_unialg_1(k1_tarski(k1_xboole_0),k8_pralg_1(A)))
& v8_unialg_1(g1_unialg_1(k1_tarski(k1_xboole_0),k8_pralg_1(A)))
& l1_unialg_1(g1_unialg_1(k1_tarski(k1_xboole_0),k8_pralg_1(A))) ) ) ) ).
fof(d9_pralg_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_finseq_1(A,k5_numbers) )
=> k9_pralg_1(A) = g1_unialg_1(k1_tarski(k1_xboole_0),k8_pralg_1(A)) ) ).
fof(d10_pralg_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_pralg_1(A)
<=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> ( ~ v3_struct_0(k1_funct_1(A,B))
& v6_unialg_1(k1_funct_1(A,B))
& v7_unialg_1(k1_funct_1(A,B))
& v8_unialg_1(k1_funct_1(A,B))
& l1_unialg_1(k1_funct_1(A,B)) ) ) ) ) ).
fof(d11_pralg_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v2_pralg_1(A)
<=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> l1_struct_0(k1_funct_1(A,B)) ) ) ) ).
fof(d12_pralg_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v3_pralg_1(A)
<=> ! [B,C] :
( ( r2_hidden(B,k1_relat_1(A))
& r2_hidden(C,k1_relat_1(A)) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v6_unialg_1(D)
& v7_unialg_1(D)
& v8_unialg_1(D)
& l1_unialg_1(D) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& v6_unialg_1(E)
& v7_unialg_1(E)
& v8_unialg_1(E)
& l1_unialg_1(E) )
=> ( ( D = k1_funct_1(A,B)
& E = k1_funct_1(A,C) )
=> k3_unialg_1(D) = k3_unialg_1(E) ) ) ) ) ) ) ).
fof(d13_pralg_1,axiom,
! [A,B] :
( ( v2_pralg_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pboole(C,A)
=> ( C = k12_pralg_1(A,B)
<=> ! [D] :
~ ( r2_hidden(D,A)
& ! [E] :
( l1_struct_0(E)
=> ~ ( E = k1_funct_1(B,D)
& k1_funct_1(C,D) = u1_struct_0(E) ) ) ) ) ) ) ).
fof(d14_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_pralg_1(B)
& v3_pralg_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m2_finseq_1(C,k5_numbers)
=> ( C = k13_pralg_1(A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> C = k3_unialg_1(k11_pralg_1(A,B,D)) ) ) ) ) ) ).
fof(d15_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_funcop_1(C)
& m1_pboole(C,A) )
=> ( m1_pralg_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ( ~ v1_xboole_0(k1_funct_1(C,D))
& v1_funct_1(k1_funct_1(C,D))
& v1_unialg_1(k1_funct_1(C,D),k1_funct_1(B,D))
& v2_unialg_1(k1_funct_1(C,D),k1_funct_1(B,D))
& m2_relset_1(k1_funct_1(C,D),k3_finseq_2(k1_funct_1(B,D)),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(d16_pralg_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v4_pralg_1(A)
<=> ! [B,C] :
( ( r2_hidden(B,k1_relat_1(A))
& r2_hidden(C,k1_relat_1(A)) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( k1_funct_1(A,B) = D
& k1_funct_1(A,C) = E )
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( ~ v1_xboole_0(H)
=> ! [I] :
( ~ v1_xboole_0(I)
=> ( ( k1_relat_1(D) = k4_finseq_2(F,H)
& k1_relat_1(E) = k4_finseq_2(G,I) )
=> ! [J] :
( ( ~ v1_xboole_0(J)
& v1_funct_1(J)
& v1_unialg_1(J,H)
& v2_unialg_1(J,H)
& m2_relset_1(J,k3_finseq_2(H),H) )
=> ! [K] :
( ( ~ v1_xboole_0(K)
& v1_funct_1(K)
& v1_unialg_1(K,I)
& v2_unialg_1(K,I)
& m2_relset_1(K,k3_finseq_2(I),I) )
=> ( ( D = J
& E = K )
=> k2_unialg_1(H,J) = k2_unialg_1(I,K) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pralg_1(C,A,B)
=> ( v4_pralg_1(C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> k2_unialg_1(k1_funct_1(B,D),k14_pralg_1(A,B,C,D)) = k2_unialg_1(k1_funct_1(B,E),k14_pralg_1(A,B,C,E)) ) ) ) ) ) ) ).
fof(d17_pralg_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k15_pralg_1(A,B)
<=> ( k1_relat_1(C) = k1_relat_1(A)
& ! [D] :
( r2_hidden(D,k1_relat_1(A))
=> k1_funct_1(C,D) = k1_funct_1(k1_funct_1(A,D),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(d18_pralg_1,axiom,
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( D = k16_pralg_1(A,B,C)
<=> ! [E] :
( r2_hidden(E,A)
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( F = k1_funct_1(B,E)
=> k1_funct_1(D,E) = k1_funct_1(F,k1_funct_1(C,E)) ) ) ) ) ) ) ) ).
fof(d19_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v4_pralg_1(C)
& m1_pralg_1(C,A,B) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( D = k20_pralg_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> D = k2_unialg_1(k1_funct_1(B,E),k14_pralg_1(A,B,C,E)) ) ) ) ) ) ) ).
fof(d20_pralg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( C = k21_pralg_1(A,B)
<=> ! [D] :
( r2_hidden(D,A)
=> k1_funct_1(C,D) = k1_subset_1(k1_funct_1(B,D)) ) ) ) ) ).
fof(d21_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v4_pralg_1(C)
& m1_pralg_1(C,A,B) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_funct_1(D)
& v1_unialg_1(D,k4_card_3(B))
& v2_unialg_1(D,k4_card_3(B))
& m2_relset_1(D,k3_finseq_2(k4_card_3(B)),k4_card_3(B)) )
=> ( D = k22_pralg_1(A,B,C)
<=> ( k1_relat_1(D) = k4_finseq_2(k20_pralg_1(A,B,C),k4_card_3(B))
& ! [E] :
( m2_finseq_2(E,k4_card_3(B),k3_finseq_2(k4_card_3(B)))
=> ( r2_hidden(E,k1_relat_1(D))
=> ( ( k4_finseq_1(E) = k1_xboole_0
=> k1_funct_1(D,E) = k16_pralg_1(A,C,k21_pralg_1(A,B)) )
& ( k4_finseq_1(E) != k1_xboole_0
=> ! [F] :
( ~ v1_xboole_0(F)
=> ! [G] :
( m1_pboole(G,k2_zfmisc_1(A,F))
=> ( ( F = k4_finseq_1(E)
& G = k18_pralg_1(k4_finseq_1(E),A,k17_pralg_1(A,B,E)) )
=> k1_funct_1(D,E) = k16_pralg_1(A,C,k19_pralg_1(A,F,G)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d22_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_pralg_1(B)
& v3_pralg_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( v4_ordinal2(C)
=> ( r2_hidden(C,k4_finseq_1(k13_pralg_1(A,B)))
=> ! [D] :
( ( v4_pralg_1(D)
& m1_pralg_1(D,A,k12_pralg_1(A,B)) )
=> ( D = k23_pralg_1(A,B,C)
<=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m2_unialg_2(F,u1_struct_0(k11_pralg_1(A,B,E)),k1_unialg_2(k11_pralg_1(A,B,E)))
=> ( k1_funct_1(u1_unialg_1(k11_pralg_1(A,B,E)),C) = F
=> k14_pralg_1(A,k12_pralg_1(A,B),D,E) = F ) ) ) ) ) ) ) ) ) ).
fof(d23_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_pralg_1(B)
& v3_pralg_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m2_finseq_1(C,k4_partfun1(k13_finseq_1(k4_card_3(k12_pralg_1(A,B))),k4_card_3(k12_pralg_1(A,B))))
=> ( C = k24_pralg_1(A,B)
<=> ( k3_finseq_1(C) = k3_finseq_1(k13_pralg_1(A,B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(C))
=> k1_funct_1(C,D) = k22_pralg_1(A,k12_pralg_1(A,B),k23_pralg_1(A,B,D)) ) ) ) ) ) ) ) ).
fof(d24_pralg_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_pralg_1(B)
& v3_pralg_1(B)
& m1_pboole(B,A) )
=> k25_pralg_1(A,B) = g1_unialg_1(k4_card_3(k12_pralg_1(A,B)),k24_pralg_1(A,B)) ) ) ).
fof(dt_m1_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pralg_1(C,A,B)
=> ( v1_funcop_1(C)
& m1_pboole(C,A) ) ) ) ).
fof(existence_m1_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A) )
=> ? [C] : m1_pralg_1(C,A,B) ) ).
fof(dt_k1_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_finseq_1(C,k2_zfmisc_1(A,B)) )
=> m2_finseq_1(k1_pralg_1(A,B,C),A) ) ).
fof(redefinition_k1_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_finseq_1(C,k2_zfmisc_1(A,B)) )
=> k1_pralg_1(A,B,C) = k15_mcart_1(C) ) ).
fof(dt_k2_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_finseq_1(C,k2_zfmisc_1(A,B)) )
=> m2_finseq_1(k2_pralg_1(A,B,C),B) ) ).
fof(redefinition_k2_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_finseq_1(C,k2_zfmisc_1(A,B)) )
=> k2_pralg_1(A,B,C) = k16_mcart_1(C) ) ).
fof(dt_k3_pralg_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(C)
& v1_unialg_1(C,A)
& v2_unialg_1(C,A)
& m1_relset_1(C,k3_finseq_2(A),A)
& ~ v1_xboole_0(D)
& v1_funct_1(D)
& v1_unialg_1(D,B)
& v2_unialg_1(D,B)
& m1_relset_1(D,k3_finseq_2(B),B) )
=> ( ~ v1_xboole_0(k3_pralg_1(A,B,C,D))
& v1_funct_1(k3_pralg_1(A,B,C,D))
& v1_unialg_1(k3_pralg_1(A,B,C,D),k2_zfmisc_1(A,B))
& v2_unialg_1(k3_pralg_1(A,B,C,D),k2_zfmisc_1(A,B))
& m2_relset_1(k3_pralg_1(A,B,C,D),k3_finseq_2(k2_zfmisc_1(A,B)),k2_zfmisc_1(A,B)) ) ) ).
fof(dt_k4_pralg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> m2_finseq_1(k4_pralg_1(A,B),k4_partfun1(k13_finseq_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)))) ) ).
fof(dt_k5_pralg_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_unialg_1(A)
& v7_unialg_1(A)
& v8_unialg_1(A)
& l1_unialg_1(A)
& ~ v3_struct_0(B)
& v6_unialg_1(B)
& v7_unialg_1(B)
& v8_unialg_1(B)
& l1_unialg_1(B) )
=> ( ~ v3_struct_0(k5_pralg_1(A,B))
& v3_unialg_1(k5_pralg_1(A,B))
& v6_unialg_1(k5_pralg_1(A,B))
& v7_unialg_1(k5_pralg_1(A,B))
& v8_unialg_1(k5_pralg_1(A,B))
& l1_unialg_1(k5_pralg_1(A,B)) ) ) ).
fof(dt_k6_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ( v1_funct_1(k6_pralg_1(A,B))
& v1_funct_2(k6_pralg_1(A,B),k2_zfmisc_1(A,B),k2_zfmisc_1(B,A))
& m2_relset_1(k6_pralg_1(A,B),k2_zfmisc_1(A,B),k2_zfmisc_1(B,A)) ) ) ).
fof(dt_k7_pralg_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_funct_1(k7_pralg_1(A))
& m2_relset_1(k7_pralg_1(A),k3_finseq_2(k1_tarski(k1_xboole_0)),k1_tarski(k1_xboole_0)) ) ) ).
fof(dt_k8_pralg_1,axiom,
! [A] :
( m1_finseq_1(A,k5_numbers)
=> m2_finseq_1(k8_pralg_1(A),k4_partfun1(k13_finseq_1(k1_tarski(k1_xboole_0)),k1_tarski(k1_xboole_0))) ) ).
fof(dt_k9_pralg_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(A,k5_numbers) )
=> ( ~ v3_struct_0(k9_pralg_1(A))
& v3_unialg_1(k9_pralg_1(A))
& v6_unialg_1(k9_pralg_1(A))
& v7_unialg_1(k9_pralg_1(A))
& v8_unialg_1(k9_pralg_1(A))
& l1_unialg_1(k9_pralg_1(A)) ) ) ).
fof(dt_k10_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_pralg_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> l1_struct_0(k10_pralg_1(A,B,C)) ) ).
fof(redefinition_k10_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_pralg_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> k10_pralg_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k11_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_pralg_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> ( ~ v3_struct_0(k11_pralg_1(A,B,C))
& v6_unialg_1(k11_pralg_1(A,B,C))
& v7_unialg_1(k11_pralg_1(A,B,C))
& v8_unialg_1(k11_pralg_1(A,B,C))
& l1_unialg_1(k11_pralg_1(A,B,C)) ) ) ).
fof(redefinition_k11_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_pralg_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> k11_pralg_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k12_pralg_1,axiom,
! [A,B] :
( ( v2_pralg_1(B)
& m1_pboole(B,A) )
=> m1_pboole(k12_pralg_1(A,B),A) ) ).
fof(dt_k13_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_pralg_1(B)
& v3_pralg_1(B)
& m1_pboole(B,A) )
=> m2_finseq_1(k13_pralg_1(A,B),k5_numbers) ) ).
fof(dt_k14_pralg_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& m1_pralg_1(C,A,B)
& m1_subset_1(D,A) )
=> ( ~ v1_xboole_0(k14_pralg_1(A,B,C,D))
& v1_funct_1(k14_pralg_1(A,B,C,D))
& v1_unialg_1(k14_pralg_1(A,B,C,D),k1_funct_1(B,D))
& v2_unialg_1(k14_pralg_1(A,B,C,D),k1_funct_1(B,D))
& m2_relset_1(k14_pralg_1(A,B,C,D),k3_finseq_2(k1_funct_1(B,D)),k1_funct_1(B,D)) ) ) ).
fof(redefinition_k14_pralg_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& m1_pralg_1(C,A,B)
& m1_subset_1(D,A) )
=> k14_pralg_1(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k15_pralg_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k15_pralg_1(A,B))
& v1_funct_1(k15_pralg_1(A,B)) ) ) ).
fof(dt_k16_pralg_1,axiom,
! [A,B,C] :
( ( v1_funcop_1(B)
& m1_pboole(B,A)
& m1_pboole(C,A) )
=> m1_pboole(k16_pralg_1(A,B,C),A) ) ).
fof(redefinition_k16_pralg_1,axiom,
! [A,B,C] :
( ( v1_funcop_1(B)
& m1_pboole(B,A)
& m1_pboole(C,A) )
=> k16_pralg_1(A,B,C) = k15_pralg_1(B,C) ) ).
fof(dt_k17_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& m1_finseq_1(C,k4_card_3(B)) )
=> m1_pboole(k17_pralg_1(A,B,C),k2_zfmisc_1(k4_finseq_1(C),A)) ) ).
fof(redefinition_k17_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& m1_finseq_1(C,k4_card_3(B)) )
=> k17_pralg_1(A,B,C) = k4_funct_5(C) ) ).
fof(dt_k18_pralg_1,axiom,
! [A,B,C] :
( m1_pboole(C,k2_zfmisc_1(A,B))
=> m1_pboole(k18_pralg_1(A,B,C),k2_zfmisc_1(B,A)) ) ).
fof(redefinition_k18_pralg_1,axiom,
! [A,B,C] :
( m1_pboole(C,k2_zfmisc_1(A,B))
=> k18_pralg_1(A,B,C) = k2_funct_4(C) ) ).
fof(dt_k19_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_pboole(C,k2_zfmisc_1(A,B)) )
=> m1_pboole(k19_pralg_1(A,B,C),A) ) ).
fof(redefinition_k19_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_pboole(C,k2_zfmisc_1(A,B)) )
=> k19_pralg_1(A,B,C) = k3_funct_5(C) ) ).
fof(dt_k20_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& v4_pralg_1(C)
& m1_pralg_1(C,A,B) )
=> m2_subset_1(k20_pralg_1(A,B,C),k1_numbers,k5_numbers) ) ).
fof(dt_k21_pralg_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> m1_pboole(k21_pralg_1(A,B),A) ) ).
fof(dt_k22_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& v4_pralg_1(C)
& m1_pralg_1(C,A,B) )
=> ( ~ v1_xboole_0(k22_pralg_1(A,B,C))
& v1_funct_1(k22_pralg_1(A,B,C))
& v1_unialg_1(k22_pralg_1(A,B,C),k4_card_3(B))
& v2_unialg_1(k22_pralg_1(A,B,C),k4_card_3(B))
& m2_relset_1(k22_pralg_1(A,B,C),k3_finseq_2(k4_card_3(B)),k4_card_3(B)) ) ) ).
fof(dt_k23_pralg_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_pralg_1(B)
& v3_pralg_1(B)
& m1_pboole(B,A)
& v4_ordinal2(C) )
=> ( v4_pralg_1(k23_pralg_1(A,B,C))
& m1_pralg_1(k23_pralg_1(A,B,C),A,k12_pralg_1(A,B)) ) ) ).
fof(dt_k24_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_pralg_1(B)
& v3_pralg_1(B)
& m1_pboole(B,A) )
=> m2_finseq_1(k24_pralg_1(A,B),k4_partfun1(k13_finseq_1(k4_card_3(k12_pralg_1(A,B))),k4_card_3(k12_pralg_1(A,B)))) ) ).
fof(dt_k25_pralg_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_pralg_1(B)
& v3_pralg_1(B)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k25_pralg_1(A,B))
& v3_unialg_1(k25_pralg_1(A,B))
& v6_unialg_1(k25_pralg_1(A,B))
& v7_unialg_1(k25_pralg_1(A,B))
& v8_unialg_1(k25_pralg_1(A,B))
& l1_unialg_1(k25_pralg_1(A,B)) ) ) ).
%------------------------------------------------------------------------------