SET007 Axioms: SET007+30.ax
%------------------------------------------------------------------------------
% File : SET007+30 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Non-Negative Real Numbers. 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 : arytm_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 32 ( 1 unt; 0 def)
% Number of atoms : 161 ( 38 equ)
% Maximal formula atoms : 9 ( 5 avg)
% Number of connectives : 145 ( 16 ~; 4 |; 18 &)
% ( 3 <=>; 104 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-2 aty)
% Number of functors : 7 ( 7 usr; 2 con; 0-2 aty)
% Number of variables : 80 ( 80 !; 0 ?)
% 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_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( k7_arytm_2(A,B) = B
=> A = k1_xboole_0 ) ) ) ).
fof(t2_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ~ ( k8_arytm_2(A,B) = k1_xboole_0
& A != k1_xboole_0
& B != k1_xboole_0 ) ) ) ).
fof(t3_arytm_1,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)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(B,C) )
=> r1_arytm_2(A,C) ) ) ) ) ).
fof(t4_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(B,A) )
=> A = B ) ) ) ).
fof(t5_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& B = k1_xboole_0 )
=> A = k1_xboole_0 ) ) ) ).
fof(t6_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( A = k1_xboole_0
=> r1_arytm_2(A,B) ) ) ) ).
fof(t7_arytm_1,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)
=> ( r1_arytm_2(A,B)
<=> r1_arytm_2(k7_arytm_2(A,C),k7_arytm_2(B,C)) ) ) ) ) ).
fof(t8_arytm_1,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)
=> ( r1_arytm_2(A,B)
=> r1_arytm_2(k8_arytm_2(A,C),k8_arytm_2(B,C)) ) ) ) ) ).
fof(d1_arytm_1,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)
=> ( ( r1_arytm_2(B,A)
=> ( C = k1_arytm_1(A,B)
<=> k7_arytm_2(C,B) = A ) )
& ( ~ r1_arytm_2(B,A)
=> ( C = k1_arytm_1(A,B)
<=> C = k1_xboole_0 ) ) ) ) ) ) ).
fof(t9_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ~ ( ~ r1_arytm_2(A,B)
& k1_arytm_1(A,B) = k1_xboole_0 ) ) ) ).
fof(t10_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r1_arytm_2(A,B)
& k1_arytm_1(B,A) = k1_xboole_0 )
=> A = B ) ) ) ).
fof(t11_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> r1_arytm_2(k1_arytm_1(A,B),A) ) ) ).
fof(t12_arytm_1,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)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(A,C) )
=> k7_arytm_2(B,k1_arytm_1(C,A)) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) ).
fof(t13_arytm_1,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)
=> ( r1_arytm_2(A,B)
=> k7_arytm_2(C,k1_arytm_1(B,A)) = k1_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) ).
fof(t14_arytm_1,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)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(C,A) )
=> k7_arytm_2(k1_arytm_1(B,A),C) = k1_arytm_1(B,k1_arytm_1(A,C)) ) ) ) ) ).
fof(t15_arytm_1,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)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(A,C) )
=> k7_arytm_2(k1_arytm_1(C,A),B) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) ).
fof(t16_arytm_1,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)
=> ( r1_arytm_2(A,B)
=> r1_arytm_2(k1_arytm_1(C,B),k1_arytm_1(C,A)) ) ) ) ) ).
fof(t17_arytm_1,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)
=> ( r1_arytm_2(A,B)
=> r1_arytm_2(k1_arytm_1(A,C),k1_arytm_1(B,C)) ) ) ) ) ).
fof(d2_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r1_arytm_2(B,A)
=> k2_arytm_1(A,B) = k1_arytm_1(A,B) )
& ( ~ r1_arytm_2(B,A)
=> k2_arytm_1(A,B) = k4_tarski(k1_xboole_0,k1_arytm_1(B,A)) ) ) ) ) ).
fof(t18_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> k2_arytm_1(A,A) = k1_xboole_0 ) ).
fof(t19_arytm_1,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( A = k1_xboole_0
=> ( B = k1_xboole_0
| k2_arytm_1(A,B) = k4_tarski(k1_xboole_0,B) ) ) ) ) ).
fof(t20_arytm_1,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)
=> ( r1_arytm_2(A,B)
=> k7_arytm_2(C,k1_arytm_1(B,A)) = k2_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) ).
fof(t21_arytm_1,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)
=> ( ~ r1_arytm_2(A,B)
=> k2_arytm_1(C,k1_arytm_1(A,B)) = k2_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) ).
fof(t22_arytm_1,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)
=> ( r1_arytm_2(A,B)
=> ( r1_arytm_2(A,C)
| k2_arytm_1(B,k1_arytm_1(A,C)) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) ) ).
fof(t23_arytm_1,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)
=> ~ ( ~ r1_arytm_2(A,B)
& ~ r1_arytm_2(A,C)
& k2_arytm_1(B,k1_arytm_1(A,C)) != k2_arytm_1(C,k1_arytm_1(A,B)) ) ) ) ) ).
fof(t24_arytm_1,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)
=> ( r1_arytm_2(A,B)
=> k2_arytm_1(B,k7_arytm_2(A,C)) = k2_arytm_1(k1_arytm_1(B,A),C) ) ) ) ) ).
fof(t25_arytm_1,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)
=> ( ( r1_arytm_2(A,B)
& r1_arytm_2(C,B) )
=> k2_arytm_1(k1_arytm_1(B,C),A) = k2_arytm_1(k1_arytm_1(B,A),C) ) ) ) ) ).
fof(t26_arytm_1,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)
=> ( r1_arytm_2(A,B)
=> k8_arytm_2(C,k1_arytm_1(B,A)) = k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) ) ) ) ) ).
fof(t27_arytm_1,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)
=> ~ ( ~ r1_arytm_2(A,B)
& C != k1_xboole_0
& k4_tarski(k1_xboole_0,k8_arytm_2(C,k1_arytm_1(A,B))) != k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) ) ) ) ) ).
fof(t28_arytm_1,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)
=> ( r1_arytm_2(B,A)
=> ( k1_arytm_1(A,B) = k1_xboole_0
| C = k1_xboole_0
| k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) = k4_tarski(k1_xboole_0,k8_arytm_2(C,k1_arytm_1(A,B))) ) ) ) ) ) ).
fof(dt_k1_arytm_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_arytm_2)
& m1_subset_1(B,k2_arytm_2) )
=> m1_subset_1(k1_arytm_1(A,B),k2_arytm_2) ) ).
fof(dt_k2_arytm_1,axiom,
$true ).
%------------------------------------------------------------------------------