SET007 Axioms: SET007+466.ax
%------------------------------------------------------------------------------
% File : SET007+466 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : More on Products of Many Sorted 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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 53 ( 4 unt; 0 def)
% Number of atoms : 507 ( 65 equ)
% Maximal formula atoms : 22 ( 9 avg)
% Number of connectives : 592 ( 138 ~; 3 |; 198 &)
% ( 9 <=>; 244 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 12 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-4 aty)
% Number of functors : 50 ( 50 usr; 2 con; 0-6 aty)
% Number of variables : 268 ( 256 !; 12 ?)
% 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_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B) )
=> ( v4_msualg_1(k15_pralg_2(A,B,C),B)
& v5_msualg_1(k15_pralg_2(A,B,C),B) ) ) ).
fof(fc2_pralg_3,axiom,
! [A] :
( v1_setfam_1(A)
=> ( v1_relat_1(k6_relat_1(A))
& v2_relat_1(k6_relat_1(A))
& v1_relat_2(k6_relat_1(A))
& v3_relat_2(k6_relat_1(A))
& v4_relat_2(k6_relat_1(A))
& v8_relat_2(k6_relat_1(A))
& v1_funct_1(k6_relat_1(A)) ) ) ).
fof(fc3_pralg_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,u1_msualg_1(A))
& m2_pralg_2(D,B,A) )
=> ( v1_relat_1(k2_pralg_3(A,k15_pralg_2(B,A,D),C))
& v1_funct_1(k2_pralg_3(A,k15_pralg_2(B,A,D),C)) ) ) ).
fof(t1_pralg_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( r2_hidden(A,k4_card_3(B))
=> r2_hidden(k7_relat_1(A,C),k4_card_3(k7_relat_1(B,C))) ) ) ) ).
fof(t2_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(A)) )
=> ! [F] :
( m2_pralg_2(F,E,B)
=> ( k7_relat_1(C,E) = F
=> k10_pralg_2(E,B,D,F) = k7_relat_1(k10_pralg_2(A,B,D,C),E) ) ) ) ) ) ) ) ).
fof(t3_pralg_3,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,B,B)
& m2_relset_1(C,B,B) )
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(B),k8_eqrel_1(B,C))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(B),k8_eqrel_1(B,C))
=> ( ( r2_hidden(A,D)
& r2_hidden(A,E) )
=> D = E ) ) ) ) ) ).
fof(t4_pralg_3,axiom,
$true ).
fof(t5_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_fraenkel(C)
& v1_pralg_2(C) )
=> ( C = k2_relat_1(B)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( r2_hidden(D,k1_relat_1(B))
& r2_hidden(E,k1_pralg_2(C)) )
=> k1_funct_1(k1_funct_1(B,D),E) = k1_funct_1(k1_funct_1(k10_funct_6(B),E),D) ) ) ) ) ) ) ).
fof(d1_pralg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> k2_pralg_3(A,B,C) = k1_funct_1(k5_msualg_1(A,C,B),k1_xboole_0) ) ) ) ).
fof(t6_pralg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ( k1_msualg_1(A,C) = k1_xboole_0
=> ( k4_msualg_1(A,C,B) = k1_xboole_0
| r2_hidden(k2_pralg_3(A,B,C),k4_msualg_1(A,C,B)) ) ) ) ) ) ).
fof(t10_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_pralg_2(D,A,B)
=> ! [E] :
( m1_subset_1(E,u1_msualg_1(B))
=> ( k1_msualg_1(B,E) = k1_xboole_0
=> k1_funct_1(k2_pralg_3(B,k15_pralg_2(A,B,D),E),C) = k2_pralg_3(B,k6_pralg_2(A,B,D,C),E) ) ) ) ) ) ) ).
fof(t11_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( k1_msualg_1(B,D) = k1_xboole_0
& k1_relat_1(E) = A
& ! [F] :
( m1_subset_1(F,A)
=> k1_funct_1(E,F) = k2_pralg_3(B,k6_pralg_2(A,B,C,F),D) ) )
=> E = k2_pralg_3(B,k15_pralg_2(A,B,C),D) ) ) ) ) ) ) ).
fof(t12_pralg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( l3_msualg_1(C,A)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(A))
=> ! [E] :
( m1_subset_1(E,k3_msualg_1(A,D,B))
=> ( ( E = k1_xboole_0
& k1_msualg_1(A,D) = k1_xboole_0 )
=> ( k3_msualg_1(A,D,B) = k1_xboole_0
| k3_msualg_1(A,D,C) = k1_xboole_0
| ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> k5_msualg_3(A,B,C,D,F,E) = k1_xboole_0 ) ) ) ) ) ) ) ) ).
fof(t13_pralg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_msualg_1(A))
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( ( v5_msualg_1(D,A)
& l3_msualg_1(D,A) )
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
=> ! [F] :
( m1_subset_1(F,k3_msualg_1(A,B,C))
=> r2_hidden(F,k4_card_3(k2_funct_6(k5_relat_1(k1_msualg_1(A,B),E)))) ) ) ) ) ) ) ).
fof(t14_pralg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_msualg_1(A))
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( ( v5_msualg_1(D,A)
& l3_msualg_1(D,A) )
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
=> ! [F] :
( m1_subset_1(F,k3_msualg_1(A,B,C))
=> ! [G] :
( r2_hidden(G,k4_finseq_1(k1_msualg_1(A,B)))
=> k1_funct_1(k6_msualg_3(A,C,D,B,E,F),G) = k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D),E,k4_finseq_4(k5_numbers,u1_struct_0(A),k1_msualg_1(A,B),G)),k1_funct_1(F,G)) ) ) ) ) ) ) ) ).
fof(t16_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( m1_subset_1(E,k3_msualg_1(B,D,k15_pralg_2(A,B,C)))
=> ! [F] :
( r2_hidden(F,k4_finseq_1(k1_msualg_1(B,D)))
=> r2_hidden(k1_funct_1(E,F),k4_card_3(k10_pralg_2(A,B,k4_finseq_4(k5_numbers,u1_struct_0(B),k1_msualg_1(B,D),F),C))) ) ) ) ) ) ) ).
fof(t17_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( r2_hidden(F,k4_finseq_1(k1_msualg_1(B,D)))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ( G = k1_funct_1(k1_msualg_1(B,D),F)
=> ! [H] :
( m1_subset_1(H,k3_msualg_1(B,D,k15_pralg_2(A,B,C)))
=> ! [I] :
( ( v1_relat_1(I)
& v1_funct_1(I) )
=> ( I = k1_funct_1(H,F)
=> r2_hidden(k1_funct_1(I,E),k1_funct_1(u4_msualg_1(B,k6_pralg_2(A,B,C,E)),G)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( m1_subset_1(E,k3_msualg_1(B,D,k15_pralg_2(A,B,C)))
=> ( k1_msualg_1(B,D) != k1_xboole_0
=> r2_hidden(k10_funct_6(E),k4_card_3(k2_funct_6(k13_pralg_2(A,B,C,D)))) ) ) ) ) ) ) ).
fof(t19_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( m1_subset_1(E,k3_msualg_1(B,D,k15_pralg_2(A,B,C)))
=> ( k1_msualg_1(B,D) != k1_xboole_0
=> r2_hidden(E,k1_relat_1(k2_pralg_2(k3_pralg_2(k13_pralg_2(A,B,C,D))))) ) ) ) ) ) ) ).
fof(t20_pralg_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( m1_subset_1(E,k3_msualg_1(B,D,k15_pralg_2(A,B,C)))
=> r2_hidden(k8_funct_2(k3_msualg_1(B,D,k15_pralg_2(A,B,C)),k4_msualg_1(B,D,k15_pralg_2(A,B,C)),k5_msualg_1(B,D,k15_pralg_2(A,B,C)),E),k4_card_3(k10_pralg_2(A,B,k2_msualg_1(B,D),C))) ) ) ) ) ).
fof(t21_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,u1_msualg_1(B))
=> ( k1_msualg_1(B,E) != k1_xboole_0
=> ! [F] :
( ( v5_msualg_1(F,B)
& l3_msualg_1(F,B) )
=> ! [G] :
( m1_subset_1(G,k3_msualg_1(B,E,k15_pralg_2(A,B,C)))
=> m1_subset_1(k1_funct_1(k10_funct_6(G),D),k3_msualg_1(B,E,k6_pralg_2(A,B,C,D))) ) ) ) ) ) ) ) ) ).
fof(t22_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,u1_msualg_1(B))
=> ! [F] :
( m1_subset_1(F,k3_msualg_1(B,E,k15_pralg_2(A,B,C)))
=> ! [G] :
( r2_hidden(G,k4_finseq_1(k1_msualg_1(B,E)))
=> ! [H] :
( ( v1_relat_1(H)
& v1_funct_1(H) )
=> ( H = k1_funct_1(F,G)
=> k1_funct_1(k1_funct_1(k10_funct_6(F),D),G) = k1_funct_1(H,D) ) ) ) ) ) ) ) ) ) ).
fof(t23_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ( k1_msualg_1(B,D) != k1_xboole_0
=> ! [E] :
( m1_subset_1(E,k3_msualg_1(B,D,k15_pralg_2(A,B,C)))
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ( G = k8_funct_2(k3_msualg_1(B,D,k15_pralg_2(A,B,C)),k4_msualg_1(B,D,k15_pralg_2(A,B,C)),k5_msualg_1(B,D,k15_pralg_2(A,B,C)),E)
=> k1_funct_1(G,F) = k1_funct_1(k5_msualg_1(B,D,k6_pralg_2(A,B,C,F)),k1_funct_1(k10_funct_6(E),F)) ) ) ) ) ) ) ) ) ) ).
fof(d2_pralg_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k3_pralg_3(A,B)
<=> ( k1_relat_1(C) = k4_card_3(A)
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(D,k1_relat_1(C))
=> k1_funct_1(C,D) = k1_funct_1(D,B) ) ) ) ) ) ) ).
fof(d3_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m3_pboole(E,u1_struct_0(B),u4_msualg_1(B,k15_pralg_2(A,B,C)),u4_msualg_1(B,k6_pralg_2(A,B,C,D)))
=> ( E = k4_pralg_3(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> k1_msualg_3(u1_struct_0(B),u4_msualg_1(B,k15_pralg_2(A,B,C)),u4_msualg_1(B,k6_pralg_2(A,B,C,D)),E,F) = k3_pralg_3(k10_pralg_2(A,B,F,C),D) ) ) ) ) ) ) ) ).
fof(t24_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( m1_subset_1(E,k3_msualg_1(B,D,k15_pralg_2(A,B,C)))
=> ~ ( k3_msualg_1(B,D,k15_pralg_2(A,B,C)) != k1_xboole_0
& k1_msualg_1(B,D) != k1_xboole_0
& ~ ! [F] :
( m1_subset_1(F,A)
=> k6_msualg_3(B,k15_pralg_2(A,B,C),k6_pralg_2(A,B,C,F),D,k4_pralg_3(A,B,C,F),E) = k1_funct_1(k10_funct_6(E),F) ) ) ) ) ) ) ) ).
fof(t25_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_pralg_2(D,A,B)
=> r1_msualg_3(B,k15_pralg_2(A,B,D),k6_pralg_2(A,B,D,C),k4_pralg_3(A,B,D,C)) ) ) ) ) ).
fof(t28_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_pralg_2(D,A,B)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( ( v5_msualg_1(F,B)
& l3_msualg_1(F,B) )
=> ! [G] :
( ( v1_funcop_1(G)
& m1_pboole(G,A) )
=> ( ! [H] :
( m1_subset_1(H,A)
=> ? [I] :
( m3_pboole(I,u1_struct_0(B),u4_msualg_1(B,F),u4_msualg_1(B,k6_pralg_2(A,B,D,H)))
& I = k1_funct_1(G,H)
& r1_msualg_3(B,F,k6_pralg_2(A,B,D,H),I) ) )
=> ! [H] :
( m3_pboole(H,u1_struct_0(B),u4_msualg_1(B,F),u4_msualg_1(B,k6_pralg_2(A,B,D,C)))
=> ( H = k1_funct_1(G,C)
=> ! [I] :
( r2_hidden(I,k1_funct_1(u4_msualg_1(B,F),E))
=> ! [J] :
( ( v1_relat_1(J)
& v1_funct_1(J) )
=> ( J = k1_funct_1(k10_funct_6(k1_funct_1(k10_funct_6(G),E)),I)
=> k1_funct_1(J,C) = k1_funct_1(k1_msualg_3(u1_struct_0(B),u4_msualg_1(B,F),u4_msualg_1(B,k6_pralg_2(A,B,D,C)),H,E),I) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t29_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v5_msualg_1(E,B)
& l3_msualg_1(E,B) )
=> ! [F] :
( ( v1_funcop_1(F)
& m1_pboole(F,A) )
=> ( ! [G] :
( m1_subset_1(G,A)
=> ? [H] :
( m3_pboole(H,u1_struct_0(B),u4_msualg_1(B,E),u4_msualg_1(B,k6_pralg_2(A,B,C,G)))
& H = k1_funct_1(F,G)
& r1_msualg_3(B,E,k6_pralg_2(A,B,C,G),H) ) )
=> ! [G] :
( r2_hidden(G,k1_funct_1(u4_msualg_1(B,E),D))
=> r2_hidden(k1_funct_1(k10_funct_6(k1_funct_1(k10_funct_6(F),D)),G),k4_card_3(k10_pralg_2(A,B,D,C))) ) ) ) ) ) ) ) ) ).
fof(t30_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( ( v5_msualg_1(D,B)
& l3_msualg_1(D,B) )
=> ! [E] :
( ( v1_funcop_1(E)
& m1_pboole(E,A) )
=> ~ ( ! [F] :
( m1_subset_1(F,A)
=> ? [G] :
( m3_pboole(G,u1_struct_0(B),u4_msualg_1(B,D),u4_msualg_1(B,k6_pralg_2(A,B,C,F)))
& G = k1_funct_1(E,F)
& r1_msualg_3(B,D,k6_pralg_2(A,B,C,F),G) ) )
& ! [F] :
( m3_pboole(F,u1_struct_0(B),u4_msualg_1(B,D),u4_msualg_1(B,k15_pralg_2(A,B,C)))
=> ~ ( r1_msualg_3(B,D,k15_pralg_2(A,B,C),F)
& ! [G] :
( m1_subset_1(G,A)
=> k3_msualg_3(u1_struct_0(B),u4_msualg_1(B,D),u4_msualg_1(B,k15_pralg_2(A,B,C)),u4_msualg_1(B,k6_pralg_2(A,B,C,G)),F,k4_pralg_3(A,B,C,G)) = k1_funct_1(E,G) ) ) ) ) ) ) ) ) ) ).
fof(d4_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C) )
=> ! [D] :
( m1_pboole(D,A)
=> ( m1_pralg_3(D,A,B,C)
<=> ! [E] :
( r2_hidden(E,A)
=> m2_pralg_2(k1_funct_1(D,E),k1_funct_1(B,E),C) ) ) ) ) ) ) ).
fof(d5_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ! [E] :
( m1_pralg_3(E,k8_eqrel_1(A,D),k1_pralg_3(k8_eqrel_1(A,D)),B)
=> ( E = k5_pralg_3(A,B,C,D)
<=> ! [F] :
( r2_hidden(F,k8_eqrel_1(A,D))
=> k1_funct_1(E,F) = k7_relat_1(C,F) ) ) ) ) ) ) ) ).
fof(d6_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m1_pralg_3(D,A,C,B)
=> ! [E] :
( m2_pralg_2(E,A,B)
=> ( E = k6_pralg_3(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,A)
=> ~ ( r2_hidden(F,A)
& ! [G] :
( ~ v1_xboole_0(G)
=> ! [H] :
( m2_pralg_2(H,G,B)
=> ~ ( G = k1_funct_1(C,F)
& H = k1_funct_1(D,F)
& k6_pralg_2(A,B,E,F) = k15_pralg_2(G,B,H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t31_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( ( v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> r6_msualg_3(B,k15_pralg_2(A,B,C),k15_pralg_2(k8_eqrel_1(A,D),B,k6_pralg_3(k8_eqrel_1(A,D),B,k1_pralg_3(k8_eqrel_1(A,D)),k5_pralg_3(A,B,C,D)))) ) ) ) ) ).
fof(dt_m1_pralg_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A)
& ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C) )
=> ! [D] :
( m1_pralg_3(D,A,B,C)
=> m1_pboole(D,A) ) ) ).
fof(existence_m1_pralg_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A)
& ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C) )
=> ? [D] : m1_pralg_3(D,A,B,C) ) ).
fof(dt_k1_pralg_3,axiom,
! [A] : m1_pboole(k1_pralg_3(A),A) ).
fof(redefinition_k1_pralg_3,axiom,
! [A] : k1_pralg_3(A) = k6_relat_1(A) ).
fof(dt_k2_pralg_3,axiom,
$true ).
fof(dt_k3_pralg_3,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k3_pralg_3(A,B))
& v1_funct_1(k3_pralg_3(A,B)) ) ) ).
fof(dt_k4_pralg_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B)
& m1_subset_1(D,A) )
=> m3_pboole(k4_pralg_3(A,B,C,D),u1_struct_0(B),u4_msualg_1(B,k15_pralg_2(A,B,C)),u4_msualg_1(B,k6_pralg_2(A,B,C,D))) ) ).
fof(dt_k5_pralg_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B)
& v3_relat_2(D)
& v8_relat_2(D)
& v1_partfun1(D,A,A)
& m1_relset_1(D,A,A) )
=> m1_pralg_3(k5_pralg_3(A,B,C,D),k8_eqrel_1(A,D),k1_pralg_3(k8_eqrel_1(A,D)),B) ) ).
fof(dt_k6_pralg_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v2_relat_1(C)
& m1_pboole(C,A)
& m1_pralg_3(D,A,C,B) )
=> m2_pralg_2(k6_pralg_3(A,B,C,D),A,B) ) ).
fof(t7_pralg_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_funct_1(u4_msualg_1(A,B),C) != k1_xboole_0
=> k1_msualg_2(A,B,C) = a_3_0_pralg_3(A,B,C) ) ) ) ) ).
fof(t8_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ( k1_msualg_1(B,D) = k1_xboole_0
=> r2_hidden(k1_funct_1(k10_funct_6(k12_pralg_2(A,B,C)),D),k1_funct_2(A,k1_funct_2(k1_tarski(k1_xboole_0),k3_tarski(a_4_0_pralg_3(A,B,C,D))))) ) ) ) ) ) ).
fof(t9_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ( k1_msualg_1(B,D) = k1_xboole_0
=> r2_hidden(k2_pralg_3(B,k15_pralg_2(A,B,C),D),k1_funct_2(A,k3_tarski(a_4_0_pralg_3(A,B,C,D)))) ) ) ) ) ) ).
fof(t15_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( m1_subset_1(E,k3_msualg_1(B,D,k15_pralg_2(A,B,C)))
=> r2_hidden(E,k1_funct_2(k4_finseq_1(k1_msualg_1(B,D)),k1_funct_2(A,k3_tarski(a_3_1_pralg_3(A,B,C))))) ) ) ) ) ) ).
fof(t26_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_pralg_2(D,A,B)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( ( v5_msualg_1(F,B)
& l3_msualg_1(F,B) )
=> ! [G] :
( ( v1_funcop_1(G)
& m1_pboole(G,A) )
=> ( ! [H] :
( m1_subset_1(H,A)
=> ? [I] :
( m3_pboole(I,u1_struct_0(B),u4_msualg_1(B,F),u4_msualg_1(B,k6_pralg_2(A,B,D,H)))
& I = k1_funct_1(G,H)
& r1_msualg_3(B,F,k6_pralg_2(A,B,D,H),I) ) )
=> ( r2_hidden(G,k1_funct_2(A,k1_funct_2(u1_struct_0(B),a_3_2_pralg_3(A,B,G))))
& k1_funct_1(k1_funct_1(k10_funct_6(G),E),C) = k1_funct_1(k1_funct_1(G,C),E) ) ) ) ) ) ) ) ) ) ).
fof(t27_pralg_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v5_msualg_1(E,B)
& l3_msualg_1(E,B) )
=> ! [F] :
( ( v1_funcop_1(F)
& m1_pboole(F,A) )
=> ( ! [G] :
( m1_subset_1(G,A)
=> ? [H] :
( m3_pboole(H,u1_struct_0(B),u4_msualg_1(B,E),u4_msualg_1(B,k6_pralg_2(A,B,C,G)))
& H = k1_funct_1(F,G)
& r1_msualg_3(B,E,k6_pralg_2(A,B,C,G),H) ) )
=> r2_hidden(k1_funct_1(k10_funct_6(F),D),k1_funct_2(A,k1_funct_2(k1_funct_1(u4_msualg_1(B,E),D),k3_tarski(a_3_1_pralg_3(A,B,C))))) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_0_pralg_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& l3_msualg_1(C,B)
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_pralg_3(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_msualg_1(B))
& A = k2_pralg_3(B,C,E)
& k2_msualg_1(B,E) = D
& k1_msualg_1(B,E) = k1_xboole_0 ) ) ) ).
fof(fraenkel_a_4_0_pralg_3,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C)
& m2_pralg_2(D,B,C)
& m1_subset_1(E,u1_msualg_1(C)) )
=> ( r2_hidden(A,a_4_0_pralg_3(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,B)
& A = k4_msualg_1(C,E,k6_pralg_2(B,C,D,F)) ) ) ) ).
fof(fraenkel_a_3_1_pralg_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C)
& m2_pralg_2(D,B,C) )
=> ( r2_hidden(A,a_3_1_pralg_3(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,B)
& m1_subset_1(F,u1_struct_0(C))
& A = k1_funct_1(u4_msualg_1(C,k6_pralg_2(B,C,D,E)),F) ) ) ) ).
fof(fraenkel_a_3_2_pralg_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C)
& v1_funcop_1(D)
& m1_pboole(D,B) )
=> ( r2_hidden(A,a_3_2_pralg_3(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,u1_struct_0(C))
& m1_subset_1(F,B)
& A = k1_funct_1(k1_funct_1(D,F),E) ) ) ) ).
%------------------------------------------------------------------------------