SET007 Axioms: SET007+760.ax
%------------------------------------------------------------------------------
% File : SET007+760 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Integers by Binary Arithmetics and Addition of Integers
% 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 : binari_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 42 ( 0 unt; 0 def)
% Number of atoms : 243 ( 43 equ)
% Maximal formula atoms : 15 ( 5 avg)
% Number of connectives : 245 ( 44 ~; 9 |; 64 &)
% ( 1 <=>; 127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 8 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 34 ( 34 usr; 8 con; 0-4 aty)
% Number of variables : 100 ( 98 !; 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_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> r1_xreal_0(k1_nat_1(A,np__1),k2_nat_1(A,np__2)) ) ) ).
fof(t2_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_xreal_0(A,k3_series_1(np__2,A)) ) ).
fof(t3_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k7_euclid(A,k5_euclid(A),k5_euclid(A)) = k5_euclid(A) ) ).
fof(t4_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(C,A)
& r1_xreal_0(A,B)
& C != A
& ~ ( r1_xreal_0(k1_nat_1(C,np__1),A)
& r1_xreal_0(A,B) ) ) ) ) ) ).
fof(t5_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( ( B = k5_euclid(A)
& C = k5_euclid(A) )
=> k7_binarith(A,B,C) = k5_euclid(A) ) ) ) ) ).
fof(t6_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( ( B = k5_euclid(A)
& C = k5_euclid(A) )
=> k10_binarith(A,B,C) = k5_euclid(A) ) ) ) ) ).
fof(t7_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( B = k5_euclid(A)
=> k4_binari_2(A,B) = np__0 ) ) ) ).
fof(t8_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k1_nat_1(A,B),k5_real_1(C,np__1))
=> ( ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(C,B) ) ) ) ) ) ).
fof(t9_binari_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_xreal_0(A,k2_xcmplx_0(B,C))
=> ( r1_xreal_0(np__0,B)
| r1_xreal_0(np__0,C)
| ( ~ r1_xreal_0(B,A)
& ~ r1_xreal_0(C,A) ) ) ) ) ) ) ).
fof(t10_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k1_nat_1(B,C),k5_real_1(k3_series_1(np__2,A),np__1))
=> k11_binarith(A,k1_binari_3(A,B),k1_binari_3(A,C)) = k7_margrel1 ) ) ) ) ).
fof(t11_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k1_nat_1(B,C),k5_real_1(k3_series_1(np__2,A),np__1))
=> k9_binarith(A,k10_binarith(A,k1_binari_3(A,B),k1_binari_3(A,C))) = k1_nat_1(B,C) ) ) ) ) ).
fof(t12_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( k4_finseq_4(k5_numbers,k6_margrel1,B,A) = k8_margrel1
=> r1_xreal_0(k3_series_1(np__2,k5_binarith(A,np__1)),k9_binarith(A,B)) ) ) ) ).
fof(t13_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k1_nat_1(B,C),k5_real_1(k3_series_1(np__2,k5_binarith(A,np__1)),np__1))
=> k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,k1_binari_3(A,B),k1_binari_3(A,C)),A) = k7_margrel1 ) ) ) ) ).
fof(t14_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r1_xreal_0(k1_nat_1(A,B),k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1))
=> k4_binari_2(C,k10_binarith(C,k1_binari_3(C,A),k1_binari_3(C,B))) = k1_nat_1(A,B) ) ) ) ) ).
fof(t15_binari_4,axiom,
! [A] :
( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
=> ( A = k13_binarith(k6_margrel1,k8_margrel1)
=> k4_binari_2(np__1,A) = k1_real_1(np__1) ) ) ).
fof(t16_binari_4,axiom,
! [A] :
( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
=> ( A = k13_binarith(k6_margrel1,k7_margrel1)
=> k4_binari_2(np__1,A) = np__0 ) ) ).
fof(t17_binari_4,axiom,
! [A] :
( v2_margrel1(A)
=> k1_binarith(k8_margrel1,A) = k8_margrel1 ) ).
fof(t18_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ( r1_xreal_0(np__0,k5_real_1(k3_series_1(np__2,k5_binarith(A,np__1)),np__1))
& r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(A,np__1))),np__0) ) ) ).
fof(t19_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( ( B = k5_euclid(A)
& C = k5_euclid(A) )
=> r1_binarith(A,B,C) ) ) ) ) ).
fof(t20_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( v1_int_1(B)
=> k6_int_1(k3_xcmplx_0(B,A),A) = np__0 ) ) ).
fof(d1_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k1_binari_4(A,B)
<=> ( r1_xreal_0(B,k3_series_1(np__2,C))
& r1_xreal_0(A,C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(B,k3_series_1(np__2,D))
& r1_xreal_0(A,D) )
=> r1_xreal_0(C,D) ) ) ) ) ) ) ) ).
fof(t21_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> r1_xreal_0(k1_binari_4(C,B),k1_binari_4(C,A)) ) ) ) ) ).
fof(t22_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> r1_xreal_0(k1_binari_4(B,C),k1_binari_4(A,C)) ) ) ) ) ).
fof(t23_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> k1_binari_4(A,np__1) = A ) ) ).
fof(t24_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,k3_series_1(np__2,B))
=> k1_binari_4(B,A) = B ) ) ) ).
fof(t25_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,k3_series_1(np__2,B))
& r1_xreal_0(k1_binari_4(B,A),B) ) ) ) ).
fof(d2_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ( ( ~ r1_xreal_0(np__0,B)
=> k2_binari_4(A,B) = k1_binari_3(A,k1_prepower(k2_xcmplx_0(k3_series_1(np__2,k1_binari_4(A,k1_prepower(B))),B))) )
& ( r1_xreal_0(np__0,B)
=> k2_binari_4(A,B) = k1_binari_3(A,k1_prepower(B)) ) ) ) ) ).
fof(t26_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_binari_4(A,np__0) = k5_euclid(A) ) ).
fof(t27_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( v1_int_1(B)
=> ( ( r1_xreal_0(B,k5_real_1(k3_series_1(np__2,k5_binarith(A,np__1)),np__1))
& r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(A,np__1))),B) )
=> k4_binari_2(A,k2_binari_4(A,B)) = B ) ) ) ).
fof(t28_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( k6_int_1(B,k3_series_1(np__2,A)) = k6_int_1(C,k3_series_1(np__2,A))
=> ( ( ~ ( r1_xreal_0(np__0,B)
& r1_xreal_0(np__0,C) )
& ~ ( ~ r1_xreal_0(np__0,B)
& ~ r1_xreal_0(np__0,C) ) )
| k2_binari_4(A,B) = k2_binari_4(A,C) ) ) ) ) ) ).
fof(t29_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_1(B,C,k3_series_1(np__2,A))
=> ( ( ~ ( r1_xreal_0(np__0,B)
& r1_xreal_0(np__0,C) )
& ~ ( ~ r1_xreal_0(np__0,B)
& ~ r1_xreal_0(np__0,C) ) )
| k2_binari_4(A,B) = k2_binari_4(A,C) ) ) ) ) ) ).
fof(t30_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k4_nat_1(B,k3_series_1(np__2,A)) = k4_nat_1(C,k3_series_1(np__2,A))
=> k1_binari_3(A,B) = k1_binari_3(A,C) ) ) ) ) ).
fof(t31_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_int_1(B,C,k3_series_1(np__2,A))
=> k1_binari_3(A,B) = k1_binari_3(A,C) ) ) ) ) ).
fof(t32_binari_4,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,A) )
=> k4_finseq_4(k5_numbers,k6_margrel1,k2_binari_4(k1_nat_1(A,np__1),B),C) = k4_finseq_4(k5_numbers,k6_margrel1,k2_binari_4(A,B),C) ) ) ) ) ).
fof(t33_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ? [C] :
( m1_subset_1(C,k6_margrel1)
& k2_binari_4(k1_nat_1(A,np__1),B) = k8_finseq_1(k6_margrel1,k2_binari_4(A,B),k13_binarith(k6_margrel1,C)) ) ) ) ).
fof(t34_binari_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( m1_subset_1(C,k6_margrel1)
& k1_binari_3(k1_nat_1(A,np__1),B) = k8_finseq_1(k6_margrel1,k1_binari_3(A,B),k13_binarith(k6_margrel1,C)) ) ) ) ).
fof(t35_binari_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( ( r1_xreal_0(k1_real_1(k3_series_1(np__2,C)),k2_xcmplx_0(A,B))
& r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),A)
& r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),B) )
=> ( r1_xreal_0(np__0,A)
| r1_xreal_0(np__0,B)
| k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(k1_nat_1(C,np__1),k2_binari_4(k1_nat_1(C,np__1),A),k2_binari_4(k1_nat_1(C,np__1),B)),k1_nat_1(C,np__1)) = k8_margrel1 ) ) ) ) ) ).
fof(t36_binari_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( ( r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),k2_xcmplx_0(A,B))
& r1_xreal_0(k2_xcmplx_0(A,B),k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1))
& r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) )
=> k4_binari_2(C,k10_binarith(C,k2_binari_4(C,A),k2_binari_4(C,B))) = k2_xcmplx_0(A,B) ) ) ) ) ).
fof(t37_binari_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( ( r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(k1_nat_1(C,np__1),np__1))),k2_xcmplx_0(A,B))
& r1_xreal_0(k2_xcmplx_0(A,B),k5_real_1(k3_series_1(np__2,k5_binarith(k1_nat_1(C,np__1),np__1)),np__1))
& r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),A)
& r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),B) )
=> ( r1_xreal_0(np__0,A)
| r1_xreal_0(np__0,B)
| k4_binari_2(k1_nat_1(C,np__1),k10_binarith(k1_nat_1(C,np__1),k2_binari_4(k1_nat_1(C,np__1),A),k2_binari_4(k1_nat_1(C,np__1),B))) = k2_xcmplx_0(A,B) ) ) ) ) ) ).
fof(t38_binari_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( ( r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),A)
& r1_xreal_0(A,k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1))
& r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),B)
& r1_xreal_0(B,k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1))
& r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),k2_xcmplx_0(A,B))
& r1_xreal_0(k2_xcmplx_0(A,B),k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1)) )
=> ( ( ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(np__0,B) )
& ~ ( ~ r1_xreal_0(np__0,A)
& r1_xreal_0(np__0,B) ) )
| k4_binari_2(C,k10_binarith(C,k2_binari_4(C,A),k2_binari_4(C,B))) = k2_xcmplx_0(A,B) ) ) ) ) ) ).
fof(dt_k1_binari_4,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k1_binari_4(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k2_binari_4,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_int_1(B) )
=> m2_finseq_2(k2_binari_4(A,B),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) ).
%------------------------------------------------------------------------------