SET007 Axioms: SET007+714.ax
%------------------------------------------------------------------------------
% File : SET007+714 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Half Open Intervals in 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 : rcomp_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 33 ( 1 unt; 0 def)
% Number of atoms : 173 ( 22 equ)
% Maximal formula atoms : 9 ( 5 avg)
% Number of connectives : 161 ( 21 ~; 0 |; 29 &)
% ( 5 <=>; 106 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-2 aty)
% Number of functors : 16 ( 16 usr; 2 con; 0-3 aty)
% Number of variables : 97 ( 95 !; 2 ?)
% 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_rcomp_2,axiom,
$true ).
fof(t2_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( ~ r1_xreal_0(B,A)
& ~ r1_xreal_0(B,C) )
<=> ~ r1_xreal_0(B,k2_square_1(A,C)) ) ) ) ) ).
fof(t3_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(A,k1_rcomp_2(B,C))
<=> ( r1_xreal_0(B,A)
& ~ r1_xreal_0(C,A) ) ) ) ) ) ).
fof(t4_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(A,k2_rcomp_2(B,C))
<=> ( ~ r1_xreal_0(A,B)
& r1_xreal_0(A,C) ) ) ) ) ) ).
fof(t5_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> k1_rcomp_2(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k1_tarski(A)) ) ) ) ).
fof(t6_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> k2_rcomp_2(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k1_tarski(B)) ) ) ) ).
fof(t7_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_rcomp_2(A,A) = k1_xboole_0 ) ).
fof(t8_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_rcomp_2(A,A) = k1_xboole_0 ) ).
fof(t9_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k1_rcomp_2(B,A) = k1_xboole_0 ) ) ) ).
fof(t10_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k2_rcomp_2(B,A) = k1_xboole_0 ) ) ) ).
fof(t11_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(B,C) )
=> k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_2(B,C)) = k1_rcomp_2(A,C) ) ) ) ) ).
fof(t12_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(B,C) )
=> k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(B,C)) = k2_rcomp_2(A,C) ) ) ) ) ).
fof(t13_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(A,C)
& r1_xreal_0(B,D)
& r1_xreal_0(C,D) )
=> k1_rcomp_1(A,D) = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_1(B,C)),k2_rcomp_2(C,D)) ) ) ) ) ) ).
fof(t14_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(B,A)
& ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(D,B)
& ~ r1_xreal_0(D,C)
& k2_rcomp_1(A,D) != k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_1(B,C)),k1_rcomp_2(C,D)) ) ) ) ) ) ).
fof(t15_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k5_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_2(C,D)) = k1_rcomp_2(k2_square_1(A,C),k1_square_1(B,D)) ) ) ) ) ).
fof(t16_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> k5_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(C,D)) = k2_rcomp_2(k2_square_1(A,C),k1_square_1(B,D)) ) ) ) ) ).
fof(t17_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_tarski(k2_rcomp_1(A,B),k1_rcomp_2(A,B))
& r1_tarski(k2_rcomp_1(A,B),k2_rcomp_2(A,B))
& r1_tarski(k1_rcomp_2(A,B),k1_rcomp_1(A,B))
& r1_tarski(k2_rcomp_2(A,B),k1_rcomp_1(A,B)) ) ) ) ).
fof(t18_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r2_hidden(A,k1_rcomp_2(B,C))
& r2_hidden(D,k1_rcomp_2(B,C)) )
=> r1_tarski(k1_rcomp_1(A,D),k1_rcomp_2(B,C)) ) ) ) ) ) ).
fof(t19_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r2_hidden(A,k2_rcomp_2(B,C))
& r2_hidden(D,k2_rcomp_2(B,C)) )
=> r1_tarski(k1_rcomp_1(A,D),k2_rcomp_2(B,C)) ) ) ) ) ) ).
fof(t20_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(B,C) )
=> k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k2_rcomp_2(B,C)) = k1_rcomp_1(A,C) ) ) ) ) ).
fof(t21_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(B,C) )
=> k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_1(B,C)) = k1_rcomp_1(A,C) ) ) ) ) ).
fof(t22_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ~ r1_xboole_0(k1_rcomp_2(A,B),k1_rcomp_2(C,D))
=> r1_xreal_0(C,B) ) ) ) ) ) ).
fof(t23_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ~ r1_xboole_0(k2_rcomp_2(A,B),k2_rcomp_2(C,D))
=> r1_xreal_0(C,B) ) ) ) ) ) ).
fof(t24_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ~ r1_xboole_0(k1_rcomp_2(A,B),k1_rcomp_2(C,D))
=> k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_2(C,D)) = k1_rcomp_2(k1_square_1(A,C),k2_square_1(B,D)) ) ) ) ) ) ).
fof(t25_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ~ r1_xboole_0(k2_rcomp_2(A,B),k2_rcomp_2(C,D))
=> k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(C,D)) = k2_rcomp_2(k1_square_1(A,C),k2_square_1(B,D)) ) ) ) ) ) ).
fof(t26_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ~ r1_xboole_0(k1_rcomp_2(A,B),k1_rcomp_2(C,D))
=> k6_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_2(C,D)) = k4_subset_1(k1_numbers,k1_rcomp_2(A,C),k1_rcomp_2(D,B)) ) ) ) ) ) ).
fof(t27_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ~ r1_xboole_0(k2_rcomp_2(A,B),k2_rcomp_2(C,D))
=> k6_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(C,D)) = k4_subset_1(k1_numbers,k2_rcomp_2(A,C),k2_rcomp_2(D,B)) ) ) ) ) ) ).
fof(dt_k1_rcomp_2,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k2_rcomp_2,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> m1_subset_1(k2_rcomp_2(A,B),k1_zfmisc_1(k1_numbers)) ) ).
fof(d1_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k1_rcomp_2(A,B) = a_2_0_rcomp_2(A,B) ) ) ).
fof(d2_rcomp_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_rcomp_2(A,B) = a_2_1_rcomp_2(A,B) ) ) ).
fof(fraenkel_a_2_0_rcomp_2,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( r2_hidden(A,a_2_0_rcomp_2(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = D
& r1_xreal_0(B,D)
& ~ r1_xreal_0(C,D) ) ) ) ).
fof(fraenkel_a_2_1_rcomp_2,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( r2_hidden(A,a_2_1_rcomp_2(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = D
& ~ r1_xreal_0(D,B)
& r1_xreal_0(D,C) ) ) ) ).
%------------------------------------------------------------------------------