## SET007 Axioms: SET007+818.ax

```%------------------------------------------------------------------------------
% File     : SET007+818 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Recursive Definitions. Part II
% 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    : recdef_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   48 (  12 unit)
%            Number of atoms       :  274 ( 155 equality)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives :  244 (  18 ~  ;   0  |; 139  &)
%                                         (  12 <=>;  75 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   1 propositional; 0-5 arity)
%            Number of functors    :  108 (  67 constant; 0-5 arity)
%            Number of variables   :  209 (   0 singleton; 135 !;  74 ?)
%            Maximal term depth    :    4 (   1 average)
% 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_recdef_2,axiom,(
! [A] :
( ? [B,C,D] : A = k3_mcart_1(B,C,D)
=> ! [B] :
( B = k1_recdef_2(A)
<=> ! [C,D,E] :
( A = k3_mcart_1(C,D,E)
=> B = C ) ) ) )).

fof(d2_recdef_2,axiom,(
! [A] :
( ? [B,C,D] : A = k3_mcart_1(B,C,D)
=> ! [B] :
( B = k2_recdef_2(A)
<=> ! [C,D,E] :
( A = k3_mcart_1(C,D,E)
=> B = D ) ) ) )).

fof(d3_recdef_2,axiom,(
! [A] :
( ? [B,C,D] : A = k3_mcart_1(B,C,D)
=> ! [B] :
( B = k3_recdef_2(A)
<=> ! [C,D,E] :
( A = k3_mcart_1(C,D,E)
=> B = E ) ) ) )).

fof(t1_recdef_2,axiom,(
! [A] :
( ? [B,C,D] : A = k3_mcart_1(B,C,D)
=> A = k3_mcart_1(k1_recdef_2(A),k2_recdef_2(A),k3_recdef_2(A)) ) )).

fof(t2_recdef_2,axiom,(
! [A,B,C,D] :
( r2_hidden(A,k3_zfmisc_1(B,C,D))
=> ( r2_hidden(k1_recdef_2(A),B)
& r2_hidden(k2_recdef_2(A),C)
& r2_hidden(k3_recdef_2(A),D) ) ) )).

fof(t3_recdef_2,axiom,(
! [A,B,C,D] :
( r2_hidden(A,k3_zfmisc_1(B,C,D))
=> A = k3_mcart_1(k1_recdef_2(A),k2_recdef_2(A),k3_recdef_2(A)) ) )).

fof(d4_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E] : A = k4_mcart_1(B,C,D,E)
=> ! [B] :
( B = k4_recdef_2(A)
<=> ! [C,D,E,F] :
( A = k4_mcart_1(C,D,E,F)
=> B = C ) ) ) )).

fof(d5_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E] : A = k4_mcart_1(B,C,D,E)
=> ! [B] :
( B = k5_recdef_2(A)
<=> ! [C,D,E,F] :
( A = k4_mcart_1(C,D,E,F)
=> B = D ) ) ) )).

fof(d6_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E] : A = k4_mcart_1(B,C,D,E)
=> ! [B] :
( B = k6_recdef_2(A)
<=> ! [C,D,E,F] :
( A = k4_mcart_1(C,D,E,F)
=> B = E ) ) ) )).

fof(d7_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E] : A = k4_mcart_1(B,C,D,E)
=> ! [B] :
( B = k7_recdef_2(A)
<=> ! [C,D,E,F] :
( A = k4_mcart_1(C,D,E,F)
=> B = F ) ) ) )).

fof(t4_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E] : A = k4_mcart_1(B,C,D,E)
=> A = k4_mcart_1(k4_recdef_2(A),k5_recdef_2(A),k6_recdef_2(A),k7_recdef_2(A)) ) )).

fof(t5_recdef_2,axiom,(
! [A,B,C,D,E] :
( r2_hidden(A,k4_zfmisc_1(B,C,D,E))
=> ( r2_hidden(k4_recdef_2(A),B)
& r2_hidden(k5_recdef_2(A),C)
& r2_hidden(k6_recdef_2(A),D)
& r2_hidden(k7_recdef_2(A),E) ) ) )).

fof(t6_recdef_2,axiom,(
! [A,B,C,D,E] :
( r2_hidden(A,k4_zfmisc_1(B,C,D,E))
=> A = k4_mcart_1(k4_recdef_2(A),k5_recdef_2(A),k6_recdef_2(A),k7_recdef_2(A)) ) )).

fof(d8_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E,F] : A = k1_mcart_2(B,C,D,E,F)
=> ! [B] :
( B = k8_recdef_2(A)
<=> ! [C,D,E,F,G] :
( A = k1_mcart_2(C,D,E,F,G)
=> B = C ) ) ) )).

fof(d9_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E,F] : A = k1_mcart_2(B,C,D,E,F)
=> ! [B] :
( B = k9_recdef_2(A)
<=> ! [C,D,E,F,G] :
( A = k1_mcart_2(C,D,E,F,G)
=> B = D ) ) ) )).

fof(d10_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E,F] : A = k1_mcart_2(B,C,D,E,F)
=> ! [B] :
( B = k10_recdef_2(A)
<=> ! [C,D,E,F,G] :
( A = k1_mcart_2(C,D,E,F,G)
=> B = E ) ) ) )).

fof(d11_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E,F] : A = k1_mcart_2(B,C,D,E,F)
=> ! [B] :
( B = k11_recdef_2(A)
<=> ! [C,D,E,F,G] :
( A = k1_mcart_2(C,D,E,F,G)
=> B = F ) ) ) )).

fof(d12_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E,F] : A = k1_mcart_2(B,C,D,E,F)
=> ! [B] :
( B = k12_recdef_2(A)
<=> ! [C,D,E,F,G] :
( A = k1_mcart_2(C,D,E,F,G)
=> B = G ) ) ) )).

fof(t7_recdef_2,axiom,(
! [A] :
( ? [B,C,D,E,F] : A = k1_mcart_2(B,C,D,E,F)
=> A = k1_mcart_2(k8_recdef_2(A),k9_recdef_2(A),k10_recdef_2(A),k11_recdef_2(A),k12_recdef_2(A)) ) )).

fof(t8_recdef_2,axiom,(
! [A,B,C,D,E,F] :
( r2_hidden(A,k2_mcart_2(B,C,D,E,F))
=> ( r2_hidden(k8_recdef_2(A),B)
& r2_hidden(k9_recdef_2(A),C)
& r2_hidden(k10_recdef_2(A),D)
& r2_hidden(k11_recdef_2(A),E)
& r2_hidden(k12_recdef_2(A),F) ) ) )).

fof(t9_recdef_2,axiom,(
! [A,B,C,D,E,F] :
( r2_hidden(A,k2_mcart_2(B,C,D,E,F))
=> A = k1_mcart_2(k8_recdef_2(A),k9_recdef_2(A),k10_recdef_2(A),k11_recdef_2(A),k12_recdef_2(A)) ) )).

fof(s1_recdef_2,axiom,
( ( ! [A] :
( r2_hidden(A,f1_s1_recdef_2)
=> ( ~ ( p1_s1_recdef_2(A)
& p2_s1_recdef_2(A) )
& ~ ( p1_s1_recdef_2(A)
& p3_s1_recdef_2(A) )
& ~ ( p2_s1_recdef_2(A)
& p3_s1_recdef_2(A) ) ) )
& ! [A] :
~ ( r2_hidden(A,f1_s1_recdef_2)
& ~ p1_s1_recdef_2(A)
& ~ p2_s1_recdef_2(A)
& ~ p3_s1_recdef_2(A) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s1_recdef_2
& ! [B] :
( r2_hidden(B,f1_s1_recdef_2)
=> ( ( p1_s1_recdef_2(B)
=> k1_funct_1(A,B) = f2_s1_recdef_2(B) )
& ( p2_s1_recdef_2(B)
=> k1_funct_1(A,B) = f3_s1_recdef_2(B) )
& ( p3_s1_recdef_2(B)
=> k1_funct_1(A,B) = f4_s1_recdef_2(B) ) ) ) ) )).

fof(s2_recdef_2,axiom,
( ( ! [A] :
( r2_hidden(A,f1_s2_recdef_2)
=> ( ~ ( p1_s2_recdef_2(A)
& p2_s2_recdef_2(A) )
& ~ ( p1_s2_recdef_2(A)
& p3_s2_recdef_2(A) )
& ~ ( p1_s2_recdef_2(A)
& p4_s2_recdef_2(A) )
& ~ ( p2_s2_recdef_2(A)
& p3_s2_recdef_2(A) )
& ~ ( p2_s2_recdef_2(A)
& p4_s2_recdef_2(A) )
& ~ ( p3_s2_recdef_2(A)
& p4_s2_recdef_2(A) ) ) )
& ! [A] :
~ ( r2_hidden(A,f1_s2_recdef_2)
& ~ p1_s2_recdef_2(A)
& ~ p2_s2_recdef_2(A)
& ~ p3_s2_recdef_2(A)
& ~ p4_s2_recdef_2(A) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s2_recdef_2
& ! [B] :
( r2_hidden(B,f1_s2_recdef_2)
=> ( ( p1_s2_recdef_2(B)
=> k1_funct_1(A,B) = f2_s2_recdef_2(B) )
& ( p2_s2_recdef_2(B)
=> k1_funct_1(A,B) = f3_s2_recdef_2(B) )
& ( p3_s2_recdef_2(B)
=> k1_funct_1(A,B) = f4_s2_recdef_2(B) )
& ( p4_s2_recdef_2(B)
=> k1_funct_1(A,B) = f5_s2_recdef_2(B) ) ) ) ) )).

fof(s3_recdef_2,axiom,
( ! [A] :
( m1_subset_1(A,k5_numbers)
=> ! [B] :
( m1_subset_1(B,f1_s3_recdef_2)
=> ! [C] :
( m1_subset_1(C,f2_s3_recdef_2)
=> ? [D] :
( m1_subset_1(D,f1_s3_recdef_2)
& ? [E] :
( m1_subset_1(E,f2_s3_recdef_2)
& p1_s3_recdef_2(A,B,C,D,E) ) ) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,f1_s3_recdef_2)
& m2_relset_1(A,k5_numbers,f1_s3_recdef_2)
& ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,f2_s3_recdef_2)
& m2_relset_1(B,k5_numbers,f2_s3_recdef_2)
& k8_funct_2(k5_numbers,f1_s3_recdef_2,A,np__0) = f3_s3_recdef_2
& k8_funct_2(k5_numbers,f2_s3_recdef_2,B,np__0) = f4_s3_recdef_2
& ! [C] :
( m1_subset_1(C,k5_numbers)
=> p1_s3_recdef_2(C,k8_funct_2(k5_numbers,f1_s3_recdef_2,A,C),k8_funct_2(k5_numbers,f2_s3_recdef_2,B,C),k8_funct_2(k5_numbers,f1_s3_recdef_2,A,k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,f2_s3_recdef_2,B,k1_nat_1(C,np__1))) ) ) ) )).

fof(s4_recdef_2,axiom,(
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = k5_numbers
& k1_funct_1(A,np__0) = f1_s4_recdef_2
& k1_funct_1(A,np__1) = f2_s4_recdef_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_funct_1(A,k1_nat_1(B,np__2)) = f3_s4_recdef_2(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) )).

fof(s5_recdef_2,axiom,(
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,f1_s5_recdef_2)
& m2_relset_1(A,k5_numbers,f1_s5_recdef_2)
& k8_funct_2(k5_numbers,f1_s5_recdef_2,A,np__0) = f2_s5_recdef_2
& k8_funct_2(k5_numbers,f1_s5_recdef_2,A,np__1) = f3_s5_recdef_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s5_recdef_2,A,k1_nat_1(B,np__2)) = f4_s5_recdef_2(B,k8_funct_2(k5_numbers,f1_s5_recdef_2,A,B),k8_funct_2(k5_numbers,f1_s5_recdef_2,A,k1_nat_1(B,np__1))) ) ) )).

fof(s6_recdef_2,axiom,
( ( k1_relat_1(f3_s6_recdef_2) = k5_numbers
& k1_funct_1(f3_s6_recdef_2,np__0) = f1_s6_recdef_2
& k1_funct_1(f3_s6_recdef_2,np__1) = f2_s6_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(f3_s6_recdef_2,k1_nat_1(A,np__2)) = f5_s6_recdef_2(A,k1_funct_1(f3_s6_recdef_2,A),k1_funct_1(f3_s6_recdef_2,k1_nat_1(A,np__1))) )
& k1_relat_1(f4_s6_recdef_2) = k5_numbers
& k1_funct_1(f4_s6_recdef_2,np__0) = f1_s6_recdef_2
& k1_funct_1(f4_s6_recdef_2,np__1) = f2_s6_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(f4_s6_recdef_2,k1_nat_1(A,np__2)) = f5_s6_recdef_2(A,k1_funct_1(f4_s6_recdef_2,A),k1_funct_1(f4_s6_recdef_2,k1_nat_1(A,np__1))) ) )
=> f3_s6_recdef_2 = f4_s6_recdef_2 )).

fof(s7_recdef_2,axiom,
( ( k8_funct_2(k5_numbers,f1_s7_recdef_2,f4_s7_recdef_2,np__0) = f2_s7_recdef_2
& k8_funct_2(k5_numbers,f1_s7_recdef_2,f4_s7_recdef_2,np__1) = f3_s7_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s7_recdef_2,f4_s7_recdef_2,k1_nat_1(A,np__2)) = f6_s7_recdef_2(A,k8_funct_2(k5_numbers,f1_s7_recdef_2,f4_s7_recdef_2,A),k8_funct_2(k5_numbers,f1_s7_recdef_2,f4_s7_recdef_2,k1_nat_1(A,np__1))) )
& k8_funct_2(k5_numbers,f1_s7_recdef_2,f5_s7_recdef_2,np__0) = f2_s7_recdef_2
& k8_funct_2(k5_numbers,f1_s7_recdef_2,f5_s7_recdef_2,np__1) = f3_s7_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s7_recdef_2,f5_s7_recdef_2,k1_nat_1(A,np__2)) = f6_s7_recdef_2(A,k8_funct_2(k5_numbers,f1_s7_recdef_2,f5_s7_recdef_2,A),k8_funct_2(k5_numbers,f1_s7_recdef_2,f5_s7_recdef_2,k1_nat_1(A,np__1))) ) )
=> f4_s7_recdef_2 = f5_s7_recdef_2 )).

fof(s8_recdef_2,axiom,(
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = k5_numbers
& k1_funct_1(A,np__0) = f1_s8_recdef_2
& k1_funct_1(A,np__1) = f2_s8_recdef_2
& k1_funct_1(A,np__2) = f3_s8_recdef_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_funct_1(A,k1_nat_1(B,np__3)) = f4_s8_recdef_2(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1)),k1_funct_1(A,k1_nat_1(B,np__2))) ) ) )).

fof(s9_recdef_2,axiom,(
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,f1_s9_recdef_2)
& m2_relset_1(A,k5_numbers,f1_s9_recdef_2)
& k8_funct_2(k5_numbers,f1_s9_recdef_2,A,np__0) = f2_s9_recdef_2
& k8_funct_2(k5_numbers,f1_s9_recdef_2,A,np__1) = f3_s9_recdef_2
& k8_funct_2(k5_numbers,f1_s9_recdef_2,A,np__2) = f4_s9_recdef_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s9_recdef_2,A,k1_nat_1(B,np__3)) = f5_s9_recdef_2(B,k8_funct_2(k5_numbers,f1_s9_recdef_2,A,B),k8_funct_2(k5_numbers,f1_s9_recdef_2,A,k1_nat_1(B,np__1)),k8_funct_2(k5_numbers,f1_s9_recdef_2,A,k1_nat_1(B,np__2))) ) ) )).

fof(s10_recdef_2,axiom,
( ( k1_relat_1(f4_s10_recdef_2) = k5_numbers
& k1_funct_1(f4_s10_recdef_2,np__0) = f1_s10_recdef_2
& k1_funct_1(f4_s10_recdef_2,np__1) = f2_s10_recdef_2
& k1_funct_1(f4_s10_recdef_2,np__2) = f3_s10_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(f4_s10_recdef_2,k1_nat_1(A,np__3)) = f6_s10_recdef_2(A,k1_funct_1(f4_s10_recdef_2,A),k1_funct_1(f4_s10_recdef_2,k1_nat_1(A,np__1)),k1_funct_1(f4_s10_recdef_2,k1_nat_1(A,np__2))) )
& k1_relat_1(f5_s10_recdef_2) = k5_numbers
& k1_funct_1(f5_s10_recdef_2,np__0) = f1_s10_recdef_2
& k1_funct_1(f5_s10_recdef_2,np__1) = f2_s10_recdef_2
& k1_funct_1(f5_s10_recdef_2,np__2) = f3_s10_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(f5_s10_recdef_2,k1_nat_1(A,np__3)) = f6_s10_recdef_2(A,k1_funct_1(f5_s10_recdef_2,A),k1_funct_1(f5_s10_recdef_2,k1_nat_1(A,np__1)),k1_funct_1(f5_s10_recdef_2,k1_nat_1(A,np__2))) ) )
=> f4_s10_recdef_2 = f5_s10_recdef_2 )).

fof(s11_recdef_2,axiom,
( ( k8_funct_2(k5_numbers,f1_s11_recdef_2,f5_s11_recdef_2,np__0) = f2_s11_recdef_2
& k8_funct_2(k5_numbers,f1_s11_recdef_2,f5_s11_recdef_2,np__1) = f3_s11_recdef_2
& k8_funct_2(k5_numbers,f1_s11_recdef_2,f5_s11_recdef_2,np__2) = f4_s11_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s11_recdef_2,f5_s11_recdef_2,k1_nat_1(A,np__3)) = f7_s11_recdef_2(A,k8_funct_2(k5_numbers,f1_s11_recdef_2,f5_s11_recdef_2,A),k8_funct_2(k5_numbers,f1_s11_recdef_2,f5_s11_recdef_2,k1_nat_1(A,np__1)),k8_funct_2(k5_numbers,f1_s11_recdef_2,f5_s11_recdef_2,k1_nat_1(A,np__2))) )
& k8_funct_2(k5_numbers,f1_s11_recdef_2,f6_s11_recdef_2,np__0) = f2_s11_recdef_2
& k8_funct_2(k5_numbers,f1_s11_recdef_2,f6_s11_recdef_2,np__1) = f3_s11_recdef_2
& k8_funct_2(k5_numbers,f1_s11_recdef_2,f6_s11_recdef_2,np__2) = f4_s11_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s11_recdef_2,f6_s11_recdef_2,k1_nat_1(A,np__3)) = f7_s11_recdef_2(A,k8_funct_2(k5_numbers,f1_s11_recdef_2,f6_s11_recdef_2,A),k8_funct_2(k5_numbers,f1_s11_recdef_2,f6_s11_recdef_2,k1_nat_1(A,np__1)),k8_funct_2(k5_numbers,f1_s11_recdef_2,f6_s11_recdef_2,k1_nat_1(A,np__2))) ) )
=> f5_s11_recdef_2 = f6_s11_recdef_2 )).

fof(s12_recdef_2,axiom,(
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = k5_numbers
& k1_funct_1(A,np__0) = f1_s12_recdef_2
& k1_funct_1(A,np__1) = f2_s12_recdef_2
& k1_funct_1(A,np__2) = f3_s12_recdef_2
& k1_funct_1(A,np__3) = f4_s12_recdef_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_funct_1(A,k1_nat_1(B,np__4)) = f5_s12_recdef_2(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1)),k1_funct_1(A,k1_nat_1(B,np__2)),k1_funct_1(A,k1_nat_1(B,np__3))) ) ) )).

fof(s13_recdef_2,axiom,(
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,f1_s13_recdef_2)
& m2_relset_1(A,k5_numbers,f1_s13_recdef_2)
& k8_funct_2(k5_numbers,f1_s13_recdef_2,A,np__0) = f2_s13_recdef_2
& k8_funct_2(k5_numbers,f1_s13_recdef_2,A,np__1) = f3_s13_recdef_2
& k8_funct_2(k5_numbers,f1_s13_recdef_2,A,np__2) = f4_s13_recdef_2
& k8_funct_2(k5_numbers,f1_s13_recdef_2,A,np__3) = f5_s13_recdef_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s13_recdef_2,A,k1_nat_1(B,np__4)) = f6_s13_recdef_2(B,k8_funct_2(k5_numbers,f1_s13_recdef_2,A,B),k8_funct_2(k5_numbers,f1_s13_recdef_2,A,k1_nat_1(B,np__1)),k8_funct_2(k5_numbers,f1_s13_recdef_2,A,k1_nat_1(B,np__2)),k8_funct_2(k5_numbers,f1_s13_recdef_2,A,k1_nat_1(B,np__3))) ) ) )).

fof(s14_recdef_2,axiom,
( ( k1_relat_1(f5_s14_recdef_2) = k5_numbers
& k1_funct_1(f5_s14_recdef_2,np__0) = f1_s14_recdef_2
& k1_funct_1(f5_s14_recdef_2,np__1) = f2_s14_recdef_2
& k1_funct_1(f5_s14_recdef_2,np__2) = f3_s14_recdef_2
& k1_funct_1(f5_s14_recdef_2,np__3) = f4_s14_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(f5_s14_recdef_2,k1_nat_1(A,np__4)) = f7_s14_recdef_2(A,k1_funct_1(f5_s14_recdef_2,A),k1_funct_1(f5_s14_recdef_2,k1_nat_1(A,np__1)),k1_funct_1(f5_s14_recdef_2,k1_nat_1(A,np__2)),k1_funct_1(f5_s14_recdef_2,k1_nat_1(A,np__3))) )
& k1_relat_1(f6_s14_recdef_2) = k5_numbers
& k1_funct_1(f6_s14_recdef_2,np__0) = f1_s14_recdef_2
& k1_funct_1(f6_s14_recdef_2,np__1) = f2_s14_recdef_2
& k1_funct_1(f6_s14_recdef_2,np__2) = f3_s14_recdef_2
& k1_funct_1(f6_s14_recdef_2,np__3) = f4_s14_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(f6_s14_recdef_2,k1_nat_1(A,np__4)) = f7_s14_recdef_2(A,k1_funct_1(f6_s14_recdef_2,A),k1_funct_1(f6_s14_recdef_2,k1_nat_1(A,np__1)),k1_funct_1(f6_s14_recdef_2,k1_nat_1(A,np__2)),k1_funct_1(f6_s14_recdef_2,k1_nat_1(A,np__3))) ) )
=> f5_s14_recdef_2 = f6_s14_recdef_2 )).

fof(s15_recdef_2,axiom,
( ( k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,np__0) = f2_s15_recdef_2
& k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,np__1) = f3_s15_recdef_2
& k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,np__2) = f4_s15_recdef_2
& k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,np__3) = f5_s15_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,k1_nat_1(A,np__4)) = f8_s15_recdef_2(A,k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,A),k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,k1_nat_1(A,np__1)),k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,k1_nat_1(A,np__2)),k8_funct_2(k5_numbers,f1_s15_recdef_2,f6_s15_recdef_2,k1_nat_1(A,np__3))) )
& k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,np__0) = f2_s15_recdef_2
& k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,np__1) = f3_s15_recdef_2
& k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,np__2) = f4_s15_recdef_2
& k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,np__3) = f5_s15_recdef_2
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,k1_nat_1(A,np__4)) = f8_s15_recdef_2(A,k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,A),k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,k1_nat_1(A,np__1)),k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,k1_nat_1(A,np__2)),k8_funct_2(k5_numbers,f1_s15_recdef_2,f7_s15_recdef_2,k1_nat_1(A,np__3))) ) )
=> f6_s15_recdef_2 = f7_s15_recdef_2 )).

fof(dt_k1_recdef_2,axiom,(
\$true )).

fof(dt_k2_recdef_2,axiom,(
\$true )).

fof(dt_k3_recdef_2,axiom,(
\$true )).

fof(dt_k4_recdef_2,axiom,(
\$true )).

fof(dt_k5_recdef_2,axiom,(
\$true )).

fof(dt_k6_recdef_2,axiom,(
\$true )).

fof(dt_k7_recdef_2,axiom,(
\$true )).

fof(dt_k8_recdef_2,axiom,(
\$true )).

fof(dt_k9_recdef_2,axiom,(
\$true )).

fof(dt_k10_recdef_2,axiom,(
\$true )).

fof(dt_k11_recdef_2,axiom,(
\$true )).

fof(dt_k12_recdef_2,axiom,(
\$true )).
%------------------------------------------------------------------------------
```