SET007 Axioms: SET007+40.ax
%------------------------------------------------------------------------------
% File : SET007+40 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Introduction to Arithmetic of Real Numbers
% 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 : xreal_0 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 48 ( 0 unt; 0 def)
% Number of atoms : 295 ( 4 equ)
% Maximal formula atoms : 25 ( 6 avg)
% Number of connectives : 339 ( 92 ~; 1 |; 194 &)
% ( 6 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 11 ( 10 usr; 0 prp; 1-2 aty)
% Number of functors : 12 ( 12 usr; 3 con; 0-2 aty)
% Number of variables : 79 ( 71 !; 8 ?)
% 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_xreal_0,axiom,
! [A] :
( v4_ordinal2(A)
=> v1_xreal_0(A) ) ).
fof(cc2_xreal_0,axiom,
! [A] :
( v1_xreal_0(A)
=> v1_xcmplx_0(A) ) ).
fof(rc1_xreal_0,axiom,
? [A] :
( v1_xcmplx_0(A)
& v1_xreal_0(A) ) ).
fof(fc1_xreal_0,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k4_xcmplx_0(A))
& v1_xreal_0(k4_xcmplx_0(A)) ) ) ).
fof(fc2_xreal_0,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v1_xcmplx_0(k5_xcmplx_0(A))
& v1_xreal_0(k5_xcmplx_0(A)) ) ) ).
fof(fc3_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ).
fof(fc4_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ).
fof(fc5_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ).
fof(fc6_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_xcmplx_0(k7_xcmplx_0(A,B))
& v1_xreal_0(k7_xcmplx_0(A,B)) ) ) ).
fof(cc3_xreal_0,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ( ~ v1_xboole_0(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A) ) ) ).
fof(cc4_xreal_0,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ( v1_xcmplx_0(A)
& v1_xreal_0(A)
& v2_xreal_0(A) ) ) ).
fof(cc5_xreal_0,axiom,
! [A] :
( ( v1_xreal_0(A)
& v3_xreal_0(A) )
=> ( ~ v1_xboole_0(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v2_xreal_0(A) ) ) ).
fof(cc6_xreal_0,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_xreal_0(A)
& ~ v2_xreal_0(A) )
=> ( v1_xcmplx_0(A)
& v1_xreal_0(A)
& v3_xreal_0(A) ) ) ).
fof(cc7_xreal_0,axiom,
! [A] :
( ( v1_xboole_0(A)
& v1_xreal_0(A) )
=> ( v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v2_xreal_0(A)
& ~ v3_xreal_0(A) ) ) ).
fof(cc8_xreal_0,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v2_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ( v1_xboole_0(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A) ) ) ).
fof(rc2_xreal_0,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& v2_xreal_0(A)
& ~ v3_xreal_0(A) ) ).
fof(rc3_xreal_0,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v2_xreal_0(A)
& v3_xreal_0(A) ) ).
fof(rc4_xreal_0,axiom,
? [A] :
( v1_xboole_0(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v2_xreal_0(A)
& ~ v3_xreal_0(A) ) ).
fof(fc7_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ).
fof(fc8_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ).
fof(fc9_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( ~ v1_xboole_0(k2_xcmplx_0(A,B))
& v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v2_xreal_0(k2_xcmplx_0(A,B))
& ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ).
fof(fc10_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( ~ v1_xboole_0(k2_xcmplx_0(B,A))
& v1_xcmplx_0(k2_xcmplx_0(B,A))
& v1_xreal_0(k2_xcmplx_0(B,A))
& v2_xreal_0(k2_xcmplx_0(B,A))
& ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ).
fof(fc11_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( ~ v1_xboole_0(k2_xcmplx_0(A,B))
& v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& ~ v2_xreal_0(k2_xcmplx_0(A,B))
& v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ).
fof(fc12_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( ~ v1_xboole_0(k2_xcmplx_0(B,A))
& v1_xcmplx_0(k2_xcmplx_0(B,A))
& v1_xreal_0(k2_xcmplx_0(B,A))
& ~ v2_xreal_0(k2_xcmplx_0(B,A))
& v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ).
fof(fc13_xreal_0,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v2_xreal_0(A) )
=> ( v1_xcmplx_0(k4_xcmplx_0(A))
& v1_xreal_0(k4_xcmplx_0(A))
& ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ).
fof(fc14_xreal_0,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ( v1_xcmplx_0(k4_xcmplx_0(A))
& v1_xreal_0(k4_xcmplx_0(A))
& ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ).
fof(fc15_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ).
fof(fc16_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(B,A))
& v1_xreal_0(k6_xcmplx_0(B,A))
& ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ).
fof(fc17_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( ~ v1_xboole_0(k6_xcmplx_0(A,B))
& v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v2_xreal_0(k6_xcmplx_0(A,B))
& ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ).
fof(fc18_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( ~ v1_xboole_0(k6_xcmplx_0(B,A))
& v1_xcmplx_0(k6_xcmplx_0(B,A))
& v1_xreal_0(k6_xcmplx_0(B,A))
& ~ v2_xreal_0(k6_xcmplx_0(B,A))
& v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ).
fof(fc19_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( ~ v1_xboole_0(k6_xcmplx_0(A,B))
& v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& ~ v2_xreal_0(k6_xcmplx_0(A,B))
& v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ).
fof(fc20_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( ~ v1_xboole_0(k6_xcmplx_0(B,A))
& v1_xcmplx_0(k6_xcmplx_0(B,A))
& v1_xreal_0(k6_xcmplx_0(B,A))
& v2_xreal_0(k6_xcmplx_0(B,A))
& ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ).
fof(fc21_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ).
fof(fc22_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(B,A))
& v1_xreal_0(k3_xcmplx_0(B,A))
& ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ).
fof(fc23_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ).
fof(fc24_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ).
fof(fc25_xreal_0,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v2_xreal_0(A) )
=> ( v1_xcmplx_0(k5_xcmplx_0(A))
& v1_xreal_0(k5_xcmplx_0(A))
& ~ v2_xreal_0(k5_xcmplx_0(A)) ) ) ).
fof(fc26_xreal_0,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ( v1_xcmplx_0(k5_xcmplx_0(A))
& v1_xreal_0(k5_xcmplx_0(A))
& ~ v3_xreal_0(k5_xcmplx_0(A)) ) ) ).
fof(fc27_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( v1_xcmplx_0(k7_xcmplx_0(A,B))
& v1_xreal_0(k7_xcmplx_0(A,B))
& ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) ).
fof(fc28_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( v1_xcmplx_0(k7_xcmplx_0(B,A))
& v1_xreal_0(k7_xcmplx_0(B,A))
& ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) ).
fof(fc29_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ( v1_xcmplx_0(k7_xcmplx_0(A,B))
& v1_xreal_0(k7_xcmplx_0(A,B))
& ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ).
fof(fc30_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& ~ v2_xreal_0(A)
& v1_xreal_0(B)
& ~ v2_xreal_0(B) )
=> ( v1_xcmplx_0(k7_xcmplx_0(A,B))
& v1_xreal_0(k7_xcmplx_0(A,B))
& ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ).
fof(d1_xreal_0,axiom,
! [A] :
( v1_xreal_0(A)
<=> r2_hidden(A,k1_numbers) ) ).
fof(d2_xreal_0,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_arytm_2) )
=> ( r1_xreal_0(A,B)
<=> ? [C] :
( m1_subset_1(C,k2_arytm_2)
& ? [D] :
( m1_subset_1(D,k2_arytm_2)
& A = C
& B = D
& r1_arytm_2(C,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)) )
=> ( r1_xreal_0(A,B)
<=> ? [C] :
( m1_subset_1(C,k2_arytm_2)
& ? [D] :
( m1_subset_1(D,k2_arytm_2)
& A = k4_tarski(np__0,C)
& B = k4_tarski(np__0,D)
& r1_arytm_2(D,C) ) ) ) )
& ~ ( ~ ( r2_hidden(A,k2_arytm_2)
& r2_hidden(B,k2_arytm_2) )
& ~ ( 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)) )
& ~ ( r1_xreal_0(A,B)
<=> ( r2_hidden(B,k2_arytm_2)
& r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) ) ) ) ) ) ) ).
fof(d3_xreal_0,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v2_xreal_0(A)
<=> ~ r1_xreal_0(A,np__0) ) ) ).
fof(d4_xreal_0,axiom,
! [A] :
( v1_xreal_0(A)
=> ( v3_xreal_0(A)
<=> ~ r1_xreal_0(np__0,A) ) ) ).
fof(reflexivity_r1_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> r1_xreal_0(A,A) ) ).
fof(connectedness_r1_xreal_0,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( r1_xreal_0(A,B)
| r1_xreal_0(B,A) ) ) ).
%------------------------------------------------------------------------------