SET007 Axioms: SET007+23.ax
%------------------------------------------------------------------------------
% File : SET007+23 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Binary Operations
% 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 : binop_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 55 ( 14 unt; 0 def)
% Number of atoms : 369 ( 32 equ)
% Maximal formula atoms : 13 ( 6 avg)
% Number of connectives : 334 ( 20 ~; 0 |; 130 &)
% ( 31 <=>; 153 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-3 aty)
% Number of functors : 11 ( 11 usr; 3 con; 0-6 aty)
% Number of variables : 174 ( 170 !; 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_binop_1,axiom,
! [A] :
( m1_relset_1(A,k2_zfmisc_1(k1_xboole_0,k1_xboole_0),k1_xboole_0)
=> ( ( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(k1_xboole_0,k1_xboole_0),k1_xboole_0) )
=> ( v1_funct_1(A)
& v1_xboole_0(A)
& v1_funct_2(A,k2_zfmisc_1(k1_xboole_0,k1_xboole_0),k1_xboole_0)
& v1_binop_1(A,k1_xboole_0)
& v2_binop_1(A,k1_xboole_0) ) ) ) ).
fof(d1_binop_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] : k1_binop_1(A,B,C) = k1_funct_1(A,k4_tarski(B,C)) ) ).
fof(t1_binop_1,axiom,
$true ).
fof(t2_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,B),C)
& m2_relset_1(E,k2_zfmisc_1(A,B),C) )
=> ( ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,B)
=> k2_binop_1(A,B,C,D,F,G) = k2_binop_1(A,B,C,E,F,G) ) )
=> D = E ) ) ) ) ) ) ).
fof(d2_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> k1_binop_1(B,C,D) = k1_binop_1(B,D,C) ) ) ) ) ).
fof(d3_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v2_binop_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> k1_binop_1(B,C,k1_binop_1(B,D,E)) = k1_binop_1(B,k1_binop_1(B,C,D),E) ) ) ) ) ) ).
fof(d4_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v3_binop_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> k1_binop_1(B,C,C) = C ) ) ) ).
fof(d5_binop_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r1_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> k1_binop_1(C,B,D) = D ) ) ) ) ).
fof(d6_binop_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r2_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> k1_binop_1(C,D,B) = D ) ) ) ) ).
fof(d7_binop_1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r3_binop_1(A,B,C)
<=> ( r1_binop_1(A,B,C)
& r2_binop_1(A,B,C) ) ) ) ) ).
fof(t3_binop_1,axiom,
$true ).
fof(t4_binop_1,axiom,
$true ).
fof(t5_binop_1,axiom,
$true ).
fof(t6_binop_1,axiom,
$true ).
fof(t7_binop_1,axiom,
$true ).
fof(t8_binop_1,axiom,
$true ).
fof(t9_binop_1,axiom,
$true ).
fof(t10_binop_1,axiom,
$true ).
fof(t11_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r3_binop_1(A,C,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ( k1_binop_1(B,C,D) = D
& k1_binop_1(B,D,C) = D ) ) ) ) ) ).
fof(t12_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( v1_binop_1(B,A)
=> ( r3_binop_1(A,C,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> k1_binop_1(B,C,D) = D ) ) ) ) ) ).
fof(t13_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( v1_binop_1(B,A)
=> ( r3_binop_1(A,C,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> k1_binop_1(B,D,C) = D ) ) ) ) ) ).
fof(t14_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( v1_binop_1(B,A)
=> ( r3_binop_1(A,C,B)
<=> r1_binop_1(A,C,B) ) ) ) ) ).
fof(t15_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( v1_binop_1(B,A)
=> ( r3_binop_1(A,C,B)
<=> r2_binop_1(A,C,B) ) ) ) ) ).
fof(t16_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( v1_binop_1(B,A)
=> ( r1_binop_1(A,C,B)
<=> r2_binop_1(A,C,B) ) ) ) ) ).
fof(t17_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r1_binop_1(A,C,B)
& r2_binop_1(A,D,B) )
=> C = D ) ) ) ) ).
fof(t18_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r3_binop_1(A,C,B)
& r3_binop_1(A,D,B) )
=> C = D ) ) ) ) ).
fof(d8_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( ? [C] :
( m1_subset_1(C,A)
& r3_binop_1(A,C,B) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( C = k3_binop_1(A,B)
<=> r3_binop_1(A,C,B) ) ) ) ) ).
fof(d9_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r4_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> k1_binop_1(B,D,k1_binop_1(C,E,F)) = k1_binop_1(C,k1_binop_1(B,D,E),k1_binop_1(B,D,F)) ) ) ) ) ) ) ).
fof(d10_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r5_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> k1_binop_1(B,k1_binop_1(C,D,E),F) = k1_binop_1(C,k1_binop_1(B,D,F),k1_binop_1(B,E,F)) ) ) ) ) ) ) ).
fof(d11_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r6_binop_1(A,B,C)
<=> ( r4_binop_1(A,B,C)
& r5_binop_1(A,B,C) ) ) ) ) ).
fof(t19_binop_1,axiom,
$true ).
fof(t20_binop_1,axiom,
$true ).
fof(t21_binop_1,axiom,
$true ).
fof(t22_binop_1,axiom,
$true ).
fof(t23_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r6_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ( k1_binop_1(B,D,k1_binop_1(C,E,F)) = k1_binop_1(C,k1_binop_1(B,D,E),k1_binop_1(B,D,F))
& k1_binop_1(B,k1_binop_1(C,D,E),F) = k1_binop_1(C,k1_binop_1(B,D,F),k1_binop_1(B,E,F)) ) ) ) ) ) ) ) ).
fof(t24_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(C,A)
=> ( r6_binop_1(A,C,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> k2_binop_1(A,A,A,C,D,k2_binop_1(A,A,A,B,E,F)) = k2_binop_1(A,A,A,B,k2_binop_1(A,A,A,C,D,E),k2_binop_1(A,A,A,C,D,F)) ) ) ) ) ) ) ) ) ).
fof(t25_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(C,A)
=> ( r6_binop_1(A,C,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> k2_binop_1(A,A,A,C,k2_binop_1(A,A,A,B,D,E),F) = k2_binop_1(A,A,A,B,k2_binop_1(A,A,A,C,D,F),k2_binop_1(A,A,A,C,E,F)) ) ) ) ) ) ) ) ) ).
fof(t26_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(C,A)
=> ( r6_binop_1(A,C,B)
<=> r4_binop_1(A,C,B) ) ) ) ) ) ).
fof(t27_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(C,A)
=> ( r6_binop_1(A,C,B)
<=> r5_binop_1(A,C,B) ) ) ) ) ) ).
fof(t28_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(C,A)
=> ( r5_binop_1(A,C,B)
<=> r4_binop_1(A,C,B) ) ) ) ) ) ).
fof(d12_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r7_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> k1_funct_1(B,k1_binop_1(C,D,E)) = k1_binop_1(C,k1_funct_1(B,D),k1_funct_1(B,E)) ) ) ) ) ) ).
fof(d13_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_binop_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> k2_binop_1(A,A,A,B,C,D) = k2_binop_1(A,A,A,B,D,C) ) ) ) ) ) ).
fof(d14_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v2_binop_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> k2_binop_1(A,A,A,B,C,k2_binop_1(A,A,A,B,D,E)) = k2_binop_1(A,A,A,B,k2_binop_1(A,A,A,B,C,D),E) ) ) ) ) ) ) ).
fof(d15_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v3_binop_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> k2_binop_1(A,A,A,B,C,C) = C ) ) ) ) ).
fof(d16_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r1_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> k2_binop_1(A,A,A,C,B,D) = D ) ) ) ) ) ).
fof(d17_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r2_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> k2_binop_1(A,A,A,C,D,B) = D ) ) ) ) ) ).
fof(d18_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r4_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> k2_binop_1(A,A,A,B,D,k2_binop_1(A,A,A,C,E,F)) = k2_binop_1(A,A,A,C,k2_binop_1(A,A,A,B,D,E),k2_binop_1(A,A,A,B,D,F)) ) ) ) ) ) ) ) ).
fof(d19_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r5_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> k2_binop_1(A,A,A,B,k2_binop_1(A,A,A,C,D,E),F) = k2_binop_1(A,A,A,C,k2_binop_1(A,A,A,B,D,F),k2_binop_1(A,A,A,B,E,F)) ) ) ) ) ) ) ) ).
fof(d20_binop_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r7_binop_1(A,B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> k8_funct_2(A,A,B,k2_binop_1(A,A,A,C,D,E)) = k2_binop_1(A,A,A,C,k8_funct_2(A,A,B,D),k8_funct_2(A,A,B,E)) ) ) ) ) ) ) ).
fof(s1_binop_1,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_binop_1)
=> ! [B] :
( m1_subset_1(B,f1_s1_binop_1)
=> ? [C] :
( m1_subset_1(C,f1_s1_binop_1)
& p1_s1_binop_1(A,B,C) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(f1_s1_binop_1,f1_s1_binop_1),f1_s1_binop_1)
& m2_relset_1(A,k2_zfmisc_1(f1_s1_binop_1,f1_s1_binop_1),f1_s1_binop_1)
& ! [B] :
( m1_subset_1(B,f1_s1_binop_1)
=> ! [C] :
( m1_subset_1(C,f1_s1_binop_1)
=> p1_s1_binop_1(B,C,k2_binop_1(f1_s1_binop_1,f1_s1_binop_1,f1_s1_binop_1,A,B,C)) ) ) ) ) ).
fof(s2_binop_1,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(f1_s2_binop_1,f1_s2_binop_1),f1_s2_binop_1)
& m2_relset_1(A,k2_zfmisc_1(f1_s2_binop_1,f1_s2_binop_1),f1_s2_binop_1)
& ! [B] :
( m1_subset_1(B,f1_s2_binop_1)
=> ! [C] :
( m1_subset_1(C,f1_s2_binop_1)
=> k2_binop_1(f1_s2_binop_1,f1_s2_binop_1,f1_s2_binop_1,A,B,C) = f2_s2_binop_1(B,C) ) ) ) ).
fof(dt_k1_binop_1,axiom,
$true ).
fof(dt_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C)
& m1_subset_1(E,A)
& m1_subset_1(F,B) )
=> m1_subset_1(k2_binop_1(A,B,C,D,E,F),C) ) ).
fof(redefinition_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C)
& m1_subset_1(E,A)
& m1_subset_1(F,B) )
=> k2_binop_1(A,B,C,D,E,F) = k1_binop_1(D,E,F) ) ).
fof(dt_k3_binop_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> m1_subset_1(k3_binop_1(A,B),A) ) ).
%------------------------------------------------------------------------------