## SET007 Axioms: SET007+36.ax

%------------------------------------------------------------------------------
% File     : SET007+36 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Introduction to Arithmetics
% 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    : arytm_0 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   43 (   6 unit)
%            Number of atoms       :  226 (  90 equality)
%            Maximal formula depth :   22 (   6 average)
%            Number of connectives :  206 (  23 ~  ;   4  |;  76  &)
%                                         (  12 <=>;  91 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   1 propositional; 0-2 arity)
%            Number of functors    :   18 (   6 constant; 0-5 arity)
%            Number of variables   :   95 (   0 singleton;  77 !;  18 ?)
%            Maximal term depth    :    3 (   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(t1_arytm_0,axiom,(
r1_tarski(k2_arytm_2,k1_numbers) )).

fof(t2_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ( A != k12_arytm_3
=> r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers) ) ) )).

fof(t3_arytm_0,axiom,(
! [A] :
~ ( r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers)
& A = k12_arytm_3 ) )).

fof(t4_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> r2_hidden(k2_arytm_1(A,B),k1_numbers) ) ) )).

fof(t5_arytm_0,axiom,(
r1_subset_1(k2_arytm_2,k2_zfmisc_1(k1_tarski(k12_arytm_3),k2_arytm_2)) )).

fof(t6_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( k2_arytm_1(A,B) = k12_arytm_3
=> A = B ) ) ) )).

fof(t7_arytm_0,axiom,(
! [A,B] : k13_arytm_3 != k4_tarski(A,B) )).

fof(t8_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( k8_arytm_2(A,B) = k8_arytm_2(A,C)
=> ( A = k12_arytm_3
| B = C ) ) ) ) ) )).

fof(d1_arytm_0,axiom,(
\$true )).

fof(d2_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_arytm_2) )
=> ( C = k1_arytm_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& A = D
& B = E
& C = k7_arytm_2(D,E) ) ) ) )
& ( ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
=> ( C = k1_arytm_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& A = D
& B = k4_tarski(np__0,E)
& C = k2_arytm_1(D,E) ) ) ) )
& ( ( r2_hidden(B,k2_arytm_2)
& r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
=> ( C = k1_arytm_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& A = k4_tarski(np__0,D)
& B = E
& C = k2_arytm_1(E,D) ) ) ) )
& ~ ( ~ ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_arytm_2) )
& ~ ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
& ~ ( r2_hidden(B,k2_arytm_2)
& r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
& ~ ( C = k1_arytm_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& A = k4_tarski(np__0,D)
& B = k4_tarski(np__0,E)
& C = k4_tarski(np__0,k7_arytm_2(D,E)) ) ) ) ) ) ) ) ) )).

fof(d3_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_arytm_2) )
=> ( C = k2_arytm_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& A = D
& B = E
& C = k8_arytm_2(D,E) ) ) ) )
& ( ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
=> ( A = np__0
| ( C = k2_arytm_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& A = D
& B = k4_tarski(np__0,E)
& C = k4_tarski(np__0,k8_arytm_2(D,E)) ) ) ) ) )
& ( ( r2_hidden(B,k2_arytm_2)
& r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
=> ( B = np__0
| ( C = k2_arytm_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& A = k4_tarski(np__0,D)
& B = E
& C = k4_tarski(np__0,k8_arytm_2(E,D)) ) ) ) ) )
& ( ( r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
& r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
=> ( C = k2_arytm_0(A,B)
<=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& A = k4_tarski(np__0,D)
& B = k4_tarski(np__0,E)
& C = k8_arytm_2(E,D) ) ) ) )
& ~ ( ~ ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_arytm_2) )
& ~ ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
& A != np__0 )
& ~ ( r2_hidden(B,k2_arytm_2)
& r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
& B != np__0 )
& ~ ( r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
& r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
& ~ ( C = k2_arytm_0(A,B)
<=> C = np__0 ) ) ) ) ) ) )).

fof(d4_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k3_arytm_0(A)
<=> k1_arytm_0(A,B) = np__0 ) ) ) )).

fof(d5_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( A != np__0
=> ( B = k4_arytm_0(A)
<=> k2_arytm_0(A,B) = k13_arytm_3 ) )
& ( A = np__0
=> ( B = k4_arytm_0(A)
<=> B = np__0 ) ) ) ) ) )).

fof(t9_arytm_0,axiom,(
\$true )).

fof(t10_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ r2_hidden(k5_funct_4(k1_numbers,np__0,k13_arytm_3,A,B),k1_numbers) ) ) )).

fof(d6_arytm_0,axiom,(
\$true )).

fof(d7_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( B = np__0
=> k5_arytm_0(A,B) = A )
& ( B != np__0
=> k5_arytm_0(A,B) = k5_funct_4(k1_numbers,np__0,k13_arytm_3,A,B) ) ) ) ) )).

fof(t11_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k2_numbers)
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& A = k5_arytm_0(B,C) ) ) ) )).

fof(t12_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( k5_arytm_0(A,B) = k5_arytm_0(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) )).

fof(t13_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = np__0
=> k1_arytm_0(A,B) = A ) ) ) )).

fof(t14_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = np__0
=> k2_arytm_0(A,B) = np__0 ) ) ) )).

fof(t15_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k2_arytm_0(A,k2_arytm_0(B,C)) = k2_arytm_0(k2_arytm_0(A,B),C) ) ) ) )).

fof(t16_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k2_arytm_0(A,k1_arytm_0(B,C)) = k1_arytm_0(k2_arytm_0(A,B),k2_arytm_0(A,C)) ) ) ) )).

fof(t17_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k2_arytm_0(k3_arytm_0(A),B) = k3_arytm_0(k2_arytm_0(A,B)) ) ) )).

fof(t18_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> r2_hidden(k2_arytm_0(A,A),k2_arytm_2) ) )).

fof(t19_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( k1_arytm_0(k2_arytm_0(A,A),k2_arytm_0(B,B)) = np__0
=> A = np__0 ) ) ) )).

fof(t20_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( k2_arytm_0(A,B) = k13_arytm_3
& k2_arytm_0(A,C) = k13_arytm_3 )
=> ( A = np__0
| B = C ) ) ) ) ) )).

fof(t21_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k13_arytm_3
=> k2_arytm_0(A,B) = A ) ) ) )).

fof(t22_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B != np__0
=> k2_arytm_0(k2_arytm_0(A,B),k4_arytm_0(B)) = A ) ) ) )).

fof(t23_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( k2_arytm_0(A,B) = np__0
& A != np__0
& B != np__0 ) ) ) )).

fof(t24_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k4_arytm_0(k2_arytm_0(A,B)) = k2_arytm_0(k4_arytm_0(A),k4_arytm_0(B)) ) ) )).

fof(t25_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k1_arytm_0(A,k1_arytm_0(B,C)) = k1_arytm_0(k1_arytm_0(A,B),C) ) ) ) )).

fof(t26_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(k5_arytm_0(A,B),k1_numbers)
=> B = np__0 ) ) ) )).

fof(t27_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k3_arytm_0(k1_arytm_0(A,B)) = k1_arytm_0(k3_arytm_0(A),k3_arytm_0(B)) ) ) )).

fof(dt_k1_arytm_0,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k1_arytm_0(A,B),k1_numbers) ) )).

fof(commutativity_k1_arytm_0,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k1_arytm_0(A,B) = k1_arytm_0(B,A) ) )).

fof(dt_k2_arytm_0,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k2_arytm_0(A,B),k1_numbers) ) )).

fof(commutativity_k2_arytm_0,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> k2_arytm_0(A,B) = k2_arytm_0(B,A) ) )).

fof(dt_k3_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k3_arytm_0(A),k1_numbers) ) )).

fof(involutiveness_k3_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> k3_arytm_0(k3_arytm_0(A)) = A ) )).

fof(dt_k4_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(k4_arytm_0(A),k1_numbers) ) )).

fof(involutiveness_k4_arytm_0,axiom,(
! [A] :
( m1_subset_1(A,k1_numbers)
=> k4_arytm_0(k4_arytm_0(A)) = A ) )).

fof(dt_k5_arytm_0,axiom,(
! [A,B] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> m1_subset_1(k5_arytm_0(A,B),k2_numbers) ) )).
%------------------------------------------------------------------------------