SET007 Axioms: SET007+512.ax
%------------------------------------------------------------------------------
% File : SET007+512 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Equations in 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 : equation [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 3 unt; 0 def)
% Number of atoms : 500 ( 22 equ)
% Maximal formula atoms : 20 ( 8 avg)
% Number of connectives : 535 ( 95 ~; 1 |; 215 &)
% ( 6 <=>; 218 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 43 ( 41 usr; 1 prp; 0-4 aty)
% Number of functors : 41 ( 41 usr; 2 con; 0-6 aty)
% Number of variables : 240 ( 231 !; 9 ?)
% 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_equation,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& l3_msualg_1(B,A)
& m1_msualg_2(C,A,B) )
=> ~ v1_xboole_0(k2_equation(A,B,C)) ) ).
fof(rc1_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ? [B] :
( l3_msualg_1(B,A)
& v4_msualg_1(B,A)
& v5_msualg_1(B,A)
& v2_msafree(B,A)
& v1_msualg_6(B,A) ) ) ).
fof(fc2_equation,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A)
& v2_relat_1(C)
& v1_pre_circ(C,u1_struct_0(A))
& m4_pboole(C,u1_struct_0(A),u4_msualg_1(A,B)) )
=> ( v4_msualg_1(k12_msualg_2(A,B,C),A)
& v5_msualg_1(k12_msualg_2(A,B,C),A)
& v3_msafree2(k12_msualg_2(A,B,C),A)
& v1_msualg_6(k12_msualg_2(A,B,C),A) ) ) ).
fof(rc2_equation,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ? [C] :
( m1_msualg_2(C,A,B)
& v4_msualg_1(C,A)
& v5_msualg_1(C,A)
& v3_msafree2(C,A)
& v1_msualg_6(C,A) ) ) ).
fof(rc3_equation,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v1_msualg_6(B,A)
& l3_msualg_1(B,A) )
=> ? [C] :
( m1_msualg_2(C,A,B)
& v1_msualg_6(C,A) ) ) ).
fof(fc3_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ( v4_msualg_1(k3_equation(A),A)
& v5_msualg_1(k3_equation(A),A)
& v2_msafree(k3_equation(A),A)
& v1_msualg_6(k3_equation(A),A) ) ) ).
fof(fc4_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ( v1_relat_1(k4_equation(A))
& v2_relat_1(k4_equation(A))
& ~ v3_relat_1(k4_equation(A))
& v1_funct_1(k4_equation(A)) ) ) ).
fof(t1_equation,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,C)
& m2_relset_1(E,B,C) )
=> ( k2_relat_1(k7_funct_2(A,B,C,D,E)) = C
=> k2_relat_1(E) = C ) ) ) ) ) ).
fof(t2_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ! [F] :
( m3_pboole(F,A,C,D)
=> ( v2_msualg_3(k3_msualg_3(A,B,C,D,E,F),A,B,D)
=> v2_msualg_3(F,A,C,D) ) ) ) ) ) ) ).
fof(t3_equation,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(E,k1_funct_2(A,k1_funct_2(B,C)))
& r2_hidden(D,B) )
=> ( k1_relat_1(k1_funct_1(k10_funct_6(E),D)) = A
& r1_tarski(k2_relat_1(k1_funct_1(k10_funct_6(E),D)),C) ) ) ) ) ) ).
fof(t4_equation,axiom,
$true ).
fof(t5_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pzfmisc1(A,B,C)
=> ! [D] :
( ( v1_funcop_1(D)
& m1_pboole(D,A) )
=> ( ( r6_pboole(A,k1_extens_1(A,D),B)
& r2_pboole(A,k2_extens_1(A,D),C) )
=> m3_pboole(D,A,B,C) ) ) ) ) ) ).
fof(t6_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ! [E] :
( m4_pboole(E,A,B)
=> ! [F] :
( m4_pboole(F,A,B)
=> ! [G] :
( m4_pboole(G,A,E)
=> ( r6_pboole(A,F,G)
=> r6_pboole(A,k1_msafree(A,E,C,G,k1_msafree(A,B,C,E,D)),k1_msafree(A,B,C,F,D)) ) ) ) ) ) ) ) ).
fof(t7_equation,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m4_pboole(D,A,C)
=> ! [E] :
( m3_pboole(E,A,D,B)
=> ? [F] :
( m3_pboole(F,A,C,B)
& r6_pboole(A,k1_msafree(A,C,B,D,F),E) ) ) ) ) ) ).
fof(d1_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funcop_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m1_pboole(D,A)
=> ( D = k1_equation(A,B,C)
<=> ! [E] :
( r2_hidden(E,A)
=> k1_funct_1(D,E) = k10_relat_1(k1_funct_1(C,E),k1_funct_1(B,E)) ) ) ) ) ) ).
fof(t8_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m3_pboole(E,A,B,C)
=> m4_pboole(k14_pboole(A,D,E),A,C) ) ) ) ) ).
fof(t9_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m3_pboole(E,A,B,C)
=> m4_pboole(k1_equation(A,D,E),A,B) ) ) ) ) ).
fof(t10_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ( v2_msualg_3(D,A,B,C)
=> r6_pboole(A,k14_pboole(A,B,D),C) ) ) ) ) ).
fof(t11_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m3_pboole(D,A,B,C)
=> ( r1_pzfmisc1(A,B,C)
=> r6_pboole(A,k1_equation(A,C,D),B) ) ) ) ) ).
fof(t12_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v1_funcop_1(C)
& m1_pboole(C,A) )
=> ( r2_pboole(A,B,k2_extens_1(A,C))
=> r6_pboole(A,k14_pboole(A,k1_equation(A,B,C),C),B) ) ) ) ).
fof(t13_equation,axiom,
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pboole(C,A)
=> r2_pboole(A,k14_pboole(A,C,B),k2_extens_1(A,B)) ) ) ).
fof(t14_equation,axiom,
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> r6_pboole(A,k14_pboole(A,k1_extens_1(A,B),B),k2_extens_1(A,B)) ) ).
fof(t15_equation,axiom,
! [A,B] :
( ( v1_funcop_1(B)
& m1_pboole(B,A) )
=> r6_pboole(A,k1_equation(A,k2_extens_1(A,B),B),k1_extens_1(A,B)) ) ).
fof(t16_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> r6_pboole(A,k14_pboole(A,B,k2_msualg_3(A,B)),B) ) ).
fof(t17_equation,axiom,
! [A,B] :
( m1_pboole(B,A)
=> r6_pboole(A,k1_equation(A,B,k2_msualg_3(A,B)),B) ) ).
fof(t18_equation,axiom,
$true ).
fof(t19_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> m1_msualg_2(B,A,g3_msualg_1(A,u4_msualg_1(A,B),u5_msualg_1(A,B))) ) ) ).
fof(t20_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_msualg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(A))
=> ! [E] :
( r2_hidden(E,k3_msualg_1(A,D,C))
=> r2_hidden(E,k3_msualg_1(A,D,B)) ) ) ) ) ) ).
fof(t21_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_msualg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(A))
=> ! [E] :
( r2_hidden(E,k3_msualg_1(A,D,C))
=> k1_funct_1(k5_msualg_1(A,D,C),E) = k1_funct_1(k5_msualg_1(A,D,B),E) ) ) ) ) ) ).
fof(t22_equation,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_msualg_2(D,B,k15_pralg_2(A,B,C))
=> ! [E] :
( m1_subset_1(E,u1_msualg_1(B))
=> ! [F] :
( r2_hidden(F,k3_msualg_1(B,E,D))
=> ( v1_relat_1(k1_funct_1(k5_msualg_1(B,E,D),F))
& v1_funct_1(k1_funct_1(k5_msualg_1(B,E,D),F))
& v1_relat_1(k1_funct_1(k5_msualg_1(B,E,k15_pralg_2(A,B,C)),F))
& v1_funct_1(k1_funct_1(k5_msualg_1(B,E,k15_pralg_2(A,B,C)),F)) ) ) ) ) ) ) ).
fof(d2_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_msualg_2(C,A,B)
=> ! [D] :
( D = k2_equation(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( v4_msualg_1(F,A)
& m1_msualg_2(F,A,B)
& E = F
& m1_msualg_2(C,A,F) ) ) ) ) ) ) ).
fof(t23_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( l3_msualg_1(C,A)
=> ! [D] :
( m1_msualg_2(D,A,C)
=> ! [E] :
( m4_pboole(E,u1_struct_0(A),u4_msualg_1(A,C))
=> ( r6_pboole(u1_struct_0(A),E,u4_msualg_1(A,D))
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B))
=> ! [G] :
( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,B))
=> ( r6_pboole(u1_struct_0(A),G,k1_msafree(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B),E,F))
=> ! [H] :
( m1_subset_1(H,u1_msualg_1(A))
=> ! [I] :
( m1_subset_1(I,k3_msualg_1(A,H,C))
=> ! [J] :
( m1_subset_1(J,k3_msualg_1(A,H,D))
=> ( I = J
=> ( k3_msualg_1(A,H,D) = k1_xboole_0
| k5_msualg_3(A,C,B,H,F,I) = k5_msualg_3(A,D,B,H,G,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v1_msualg_6(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( ( v1_msualg_6(D,A)
& m1_msualg_2(D,A,C) )
=> ! [E] :
( m4_pboole(E,u1_struct_0(A),u4_msualg_1(A,C))
=> ( r6_pboole(u1_struct_0(A),E,u4_msualg_1(A,D))
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B))
=> ( r1_msualg_3(A,C,B,F)
=> ! [G] :
( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,B))
=> ( r6_pboole(u1_struct_0(A),G,k1_msafree(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,B),E,F))
=> r1_msualg_3(A,D,B,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t25_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v4_msualg_1(C,A)
& v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m1_msafree(D,A,B)
=> ! [E] :
( ( v2_relat_1(E)
& m1_msafree(E,A,C) )
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ( ( r2_pboole(u1_struct_0(A),E,k14_pboole(u1_struct_0(A),D,F))
& r1_msualg_3(A,B,C,F) )
=> r2_msualg_3(A,B,C,F) ) ) ) ) ) ) ) ).
fof(t26_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( ( v4_msualg_1(D,A)
& v5_msualg_1(D,A)
& v2_msafree(D,A)
& l3_msualg_1(D,A) )
=> ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C))
=> ( r2_msualg_3(A,B,C,E)
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,C))
=> ~ ( r1_msualg_3(A,D,C,F)
& ! [G] :
( m3_pboole(G,u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,B))
=> ~ ( r1_msualg_3(A,D,B,G)
& r6_pboole(u1_struct_0(A),F,k3_msualg_3(u1_struct_0(A),u4_msualg_1(A,D),u4_msualg_1(A,B),u4_msualg_1(A,C),G,E)) ) ) ) ) ) ) ) ) ) ) ).
fof(t27_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m2_pralg_2(D,B,A)
=> ~ ( ! [E] :
( m1_subset_1(E,B)
=> ? [F] :
( v4_msualg_1(F,A)
& v5_msualg_1(F,A)
& v3_msafree2(F,A)
& m1_msualg_2(F,A,C)
& F = k6_pralg_2(B,A,D,E) ) )
& ! [E] :
( ( v4_msualg_1(E,A)
& v5_msualg_1(E,A)
& v3_msafree2(E,A)
& m1_msualg_2(E,A,C) )
=> ~ ! [F] :
( m1_subset_1(F,B)
=> m1_msualg_2(k6_pralg_2(B,A,D,F),A,E) ) ) ) ) ) ) ) ).
fof(t28_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v4_msualg_1(C,A)
& v5_msualg_1(C,A)
& v3_msafree2(C,A)
& m1_msualg_2(C,A,B) )
=> ! [D] :
( ( v4_msualg_1(D,A)
& v5_msualg_1(D,A)
& v3_msafree2(D,A)
& m1_msualg_2(D,A,B) )
=> ? [E] :
( v4_msualg_1(E,A)
& v5_msualg_1(E,A)
& v3_msafree2(E,A)
& m1_msualg_2(E,A,B)
& m1_msualg_2(C,A,E)
& m1_msualg_2(D,A,E) ) ) ) ) ) ).
fof(t30_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_msafree(B,A)
& v1_msualg_6(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v1_msafree(C,A,B)
& m1_msafree(C,A,B) )
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
=> ( ( r2_pboole(u1_struct_0(A),D,C)
& v1_msualg_6(k12_msualg_2(A,B,D),A) )
=> v2_msafree(k12_msualg_2(A,B,D),A) ) ) ) ) ) ).
fof(d3_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> k3_equation(A) = k11_msafree(A,k2_pre_circ(u1_struct_0(A),k5_numbers)) ) ).
fof(d4_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> k4_equation(A) = k11_pboole(u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,k3_equation(A))) ) ).
fof(t31_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_funct_1(k4_equation(A),B))
=> r2_hidden(k1_mcart_1(C),k1_funct_1(u4_msualg_1(A,k3_equation(A)),B)) ) ) ) ).
fof(t32_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_funct_1(k4_equation(A),B))
=> r2_hidden(k2_mcart_1(C),k1_funct_1(u4_msualg_1(A,k3_equation(A)),B)) ) ) ) ).
fof(d5_equation,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))
=> ! [D] :
( m1_subset_1(D,k1_funct_1(k4_equation(A),C))
=> ( r1_equation(A,B,C,D)
<=> ! [E] :
( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B))
=> ( r1_msualg_3(A,k3_equation(A),B,E)
=> k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B),E,C),k1_mcart_1(D)) = k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k3_equation(A)),u4_msualg_1(A,B),E,C),k2_mcart_1(D)) ) ) ) ) ) ) ) ).
fof(d6_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m4_pboole(C,u1_struct_0(A),k4_equation(A))
=> ( r2_equation(A,B,C)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(k4_equation(A),D))
=> ( r2_hidden(E,k1_funct_1(C,D))
=> r1_equation(A,B,D,E) ) ) ) ) ) ) ) ).
fof(t33_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_funct_1(k4_equation(A),C))
=> ! [E] :
( ( v4_msualg_1(E,A)
& v5_msualg_1(E,A)
& m1_msualg_2(E,A,B) )
=> ( r1_equation(A,B,C,D)
=> r1_equation(A,E,C,D) ) ) ) ) ) ) ).
fof(t34_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m4_pboole(C,u1_struct_0(A),k4_equation(A))
=> ! [D] :
( ( v4_msualg_1(D,A)
& v5_msualg_1(D,A)
& m1_msualg_2(D,A,B) )
=> ( r2_equation(A,B,C)
=> r2_equation(A,D,C) ) ) ) ) ) ).
fof(t35_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_funct_1(k4_equation(A),D))
=> ( ( r6_msualg_3(A,B,C)
& r1_equation(A,B,D,E) )
=> r1_equation(A,C,D,E) ) ) ) ) ) ) ).
fof(t36_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),k4_equation(A))
=> ( ( r6_msualg_3(A,B,C)
& r2_equation(A,B,D) )
=> r2_equation(A,C,D) ) ) ) ) ) ).
fof(t37_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_funct_1(k4_equation(A),C))
=> ! [E] :
( ( v3_msualg_4(E,A,B)
& v4_msualg_4(E,A,B)
& m1_msualg_4(E,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
=> ( r1_equation(A,B,C,D)
=> r1_equation(A,k14_msualg_4(A,B,E),C,D) ) ) ) ) ) ) ).
fof(t38_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m4_pboole(C,u1_struct_0(A),k4_equation(A))
=> ! [D] :
( ( v3_msualg_4(D,A,B)
& v4_msualg_4(D,A,B)
& m1_msualg_4(D,u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,B)) )
=> ( r2_equation(A,B,C)
=> r2_equation(A,k14_msualg_4(A,B,D),C) ) ) ) ) ) ).
fof(t39_equation,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,k1_funct_1(k4_equation(B),C))
=> ! [E] :
( m2_pralg_2(E,A,B)
=> ( ! [F] :
~ ( r2_hidden(F,A)
& ! [G] :
( l3_msualg_1(G,B)
=> ~ ( G = k1_funct_1(E,F)
& r1_equation(B,G,C,D) ) ) )
=> r1_equation(B,k15_pralg_2(A,B,E),C,D) ) ) ) ) ) ).
fof(t40_equation,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m4_pboole(C,u1_struct_0(B),k4_equation(B))
=> ! [D] :
( m2_pralg_2(D,A,B)
=> ( ! [E] :
~ ( r2_hidden(E,A)
& ! [F] :
( l3_msualg_1(F,B)
=> ~ ( F = k1_funct_1(D,E)
& r2_equation(B,F,C) ) ) )
=> r2_equation(B,k15_pralg_2(A,B,D),C) ) ) ) ) ).
fof(dt_k1_equation,axiom,
! [A,B,C] :
( ( m1_pboole(B,A)
& v1_funcop_1(C)
& m1_pboole(C,A) )
=> m1_pboole(k1_equation(A,B,C),A) ) ).
fof(dt_k2_equation,axiom,
$true ).
fof(dt_k3_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> l3_msualg_1(k3_equation(A),A) ) ).
fof(dt_k4_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> m1_pboole(k4_equation(A),u1_struct_0(A)) ) ).
fof(dt_k5_equation,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k1_funct_1(u4_msualg_1(A,k3_equation(A)),B))
& m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k3_equation(A)),B)) )
=> m1_subset_1(k5_equation(A,B,C,D),k1_funct_1(k4_equation(A),B)) ) ).
fof(redefinition_k5_equation,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k1_funct_1(u4_msualg_1(A,k3_equation(A)),B))
& m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k3_equation(A)),B)) )
=> k5_equation(A,B,C,D) = k4_tarski(C,D) ) ).
fof(t29_equation,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( C = a_2_0_equation(A,B)
=> ! [D] :
( m2_pralg_2(D,C,A)
=> ~ ( ! [E] :
( r2_hidden(E,C)
=> E = k1_funct_1(D,E) )
& ! [E] :
( ( v4_msualg_1(E,A)
& v5_msualg_1(E,A)
& m1_msualg_2(E,A,k15_pralg_2(C,A,D)) )
=> ! [F] :
( m3_pboole(F,u1_struct_0(A),u4_msualg_1(A,E),u4_msualg_1(A,B))
=> ~ r2_msualg_3(A,E,B,F) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_0_equation,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v5_msualg_1(C,B)
& l3_msualg_1(C,B) )
=> ( r2_hidden(A,a_2_0_equation(B,C))
<=> ? [D] :
( m1_subset_1(D,k14_msualg_2(B,C))
& A = D
& ? [E] :
( v4_msualg_1(E,B)
& v5_msualg_1(E,B)
& v3_msafree2(E,B)
& m1_msualg_2(E,B,C)
& E = D ) ) ) ) ).
%------------------------------------------------------------------------------