SET007 Axioms: SET007+539.ax
%------------------------------------------------------------------------------
% File : SET007+539 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Binary Arithmetics. Binary Sequences
% 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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 39 ( 5 unt; 0 def)
% Number of atoms : 179 ( 72 equ)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 170 ( 30 ~; 4 |; 39 &)
% ( 4 <=>; 93 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 36 ( 36 usr; 8 con; 0-5 aty)
% Number of variables : 68 ( 68 !; 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_binari_3,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))
=> ~ r1_xreal_0(k3_series_1(np__2,A),k9_binarith(A,B)) ) ) ).
fof(t2_binari_3,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))
=> ( k9_binarith(A,B) = k9_binarith(A,C)
=> B = C ) ) ) ) ).
fof(t3_binari_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( k3_finseq_5(A) = k3_finseq_5(B)
=> A = B ) ) ) ).
fof(t4_binari_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k5_euclid(k1_nat_1(A,np__1)) = k8_finseq_1(k1_numbers,k5_euclid(A),k13_binarith(k1_numbers,np__0)) ) ).
fof(t5_binari_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r2_hidden(k5_euclid(A),k13_finseq_1(k6_margrel1)) ) ).
fof(t6_binari_3,axiom,
! [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)
=> k6_binarith(A,B) = k4_finseqop(k1_numbers,A,np__1) ) ) ) ).
fof(t7_binari_3,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)
=> k9_binarith(A,B) = np__0 ) ) ) ).
fof(t8_binari_3,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)
=> k9_binarith(A,k6_binarith(A,B)) = k5_real_1(k3_series_1(np__2,A),np__1) ) ) ) ).
fof(t9_binari_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k4_finseq_5(k1_numbers,k5_euclid(A)) = k5_euclid(A) ) ).
fof(t10_binari_3,axiom,
! [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_finseq_5(k6_margrel1,k6_binarith(A,B)) = k6_binarith(A,B) ) ) ) ).
fof(t11_binari_3,axiom,
k2_binari_2(np__1) = k13_binarith(k6_margrel1,k8_margrel1) ).
fof(t12_binari_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> k9_binarith(A,k2_binari_2(A)) = np__1 ) ).
fof(t13_binari_3,axiom,
! [A] :
( m1_subset_1(A,k6_margrel1)
=> ! [B] :
( m1_subset_1(B,k6_margrel1)
=> ( ~ ( k3_binarith(A,B) = k8_margrel1
& A != k8_margrel1
& B != k8_margrel1 )
& ( ( A = k8_margrel1
| B = k8_margrel1 )
=> k3_binarith(A,B) = k8_margrel1 )
& ( k3_binarith(A,B) = k7_margrel1
=> ( A = k7_margrel1
& B = k7_margrel1 ) )
& ( ( A = k7_margrel1
& B = k7_margrel1 )
=> k3_binarith(A,B) = k7_margrel1 ) ) ) ) ).
fof(t14_binari_3,axiom,
! [A] :
( m1_subset_1(A,k6_margrel1)
=> ! [B] :
( m1_subset_1(B,k6_margrel1)
=> ( k11_binarith(np__1,k13_binarith(k6_margrel1,A),k13_binarith(k6_margrel1,B)) = k8_margrel1
<=> ( A = k8_margrel1
& B = k8_margrel1 ) ) ) ) ).
fof(t15_binari_3,axiom,
k6_binarith(np__1,k13_binarith(k6_margrel1,k7_margrel1)) = k13_binarith(k6_margrel1,k8_margrel1) ).
fof(t16_binari_3,axiom,
k6_binarith(np__1,k13_binarith(k6_margrel1,k8_margrel1)) = k13_binarith(k6_margrel1,k7_margrel1) ).
fof(t17_binari_3,axiom,
k10_binarith(np__1,k13_binarith(k6_margrel1,k7_margrel1),k13_binarith(k6_margrel1,k7_margrel1)) = k13_binarith(k6_margrel1,k7_margrel1) ).
fof(t18_binari_3,axiom,
( k10_binarith(np__1,k13_binarith(k6_margrel1,k7_margrel1),k13_binarith(k6_margrel1,k8_margrel1)) = k13_binarith(k6_margrel1,k8_margrel1)
& k10_binarith(np__1,k13_binarith(k6_margrel1,k8_margrel1),k13_binarith(k6_margrel1,k7_margrel1)) = k13_binarith(k6_margrel1,k8_margrel1) ) ).
fof(t19_binari_3,axiom,
k10_binarith(np__1,k13_binarith(k6_margrel1,k8_margrel1),k13_binarith(k6_margrel1,k8_margrel1)) = k13_binarith(k6_margrel1,k7_margrel1) ).
fof(t20_binari_3,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))
=> ( ( k4_finseq_4(k5_numbers,k6_margrel1,B,A) = k8_margrel1
& k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,k2_binari_2(A)),A) = k8_margrel1 )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_subset_1(D,k1_numbers,k5_numbers) )
=> ( r1_xreal_0(D,A)
=> ( D = np__1
| ( k4_finseq_4(k5_numbers,k6_margrel1,B,D) = k8_margrel1
& k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,k2_binari_2(A)),D) = k8_margrel1 ) ) ) ) ) ) ) ) ).
fof(t21_binari_3,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
& k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,k2_binari_2(A)),A) = k8_margrel1 )
=> k7_binarith(A,B,k2_binari_2(A)) = k6_binarith(A,k2_binari_2(A)) ) ) ) ).
fof(t22_binari_3,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))
=> ( ( C = k5_euclid(A)
& k4_finseq_4(k5_numbers,k6_margrel1,B,A) = k8_margrel1
& k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,k2_binari_2(A)),A) = k8_margrel1 )
=> B = k6_binarith(A,C) ) ) ) ) ).
fof(t23_binari_3,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)
=> k7_binarith(A,k6_binarith(A,B),k2_binari_2(A)) = k6_binarith(A,k2_binari_2(A)) ) ) ) ).
fof(t24_binari_3,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))
=> ( C = k5_euclid(A)
=> ( k11_binarith(A,B,k2_binari_2(A)) = k8_margrel1
<=> B = k6_binarith(A,C) ) ) ) ) ) ).
fof(t25_binari_3,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)
=> k10_binarith(A,k6_binarith(A,B),k2_binari_2(A)) = B ) ) ) ).
fof(d1_binari_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( C = k1_binari_3(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k4_finseq_4(k5_numbers,k6_margrel1,C,D) = k2_cqc_lang(k6_margrel1,k4_nat_1(k3_nat_1(B,k3_series_1(np__2,k5_binarith(D,np__1))),np__2),np__0,k7_margrel1,k8_margrel1) ) ) ) ) ) ) ).
fof(t26_binari_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_binari_3(A,np__0) = k5_euclid(A) ) ).
fof(t27_binari_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_series_1(np__2,A),B)
=> k1_funct_1(k1_binari_3(k1_nat_1(A,np__1),B),k1_nat_1(A,np__1)) = k7_margrel1 ) ) ) ).
fof(t28_binari_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_series_1(np__2,A),B)
=> k1_binari_3(k1_nat_1(A,np__1),B) = k12_binarith(A,np__1,k6_margrel1,k1_binari_3(A,B),k13_binarith(k6_margrel1,k7_margrel1)) ) ) ) ).
fof(t29_binari_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_binari_3(k1_nat_1(A,np__1),k3_series_1(np__2,A)) = k8_finseq_1(k1_numbers,k5_euclid(A),k13_binarith(k1_numbers,np__1)) ) ).
fof(t30_binari_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k3_series_1(np__2,A),B)
=> ( r1_xreal_0(k3_series_1(np__2,k1_nat_1(A,np__1)),B)
| k1_funct_1(k1_binari_3(k1_nat_1(A,np__1),B),k1_nat_1(A,np__1)) = k8_margrel1 ) ) ) ) ).
fof(t31_binari_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k3_series_1(np__2,A),B)
=> ( r1_xreal_0(k3_series_1(np__2,k1_nat_1(A,np__1)),B)
| k1_binari_3(k1_nat_1(A,np__1),B) = k12_binarith(A,np__1,k6_margrel1,k1_binari_3(A,k5_binarith(B,k3_series_1(np__2,A))),k13_binarith(k6_margrel1,k8_margrel1)) ) ) ) ) ).
fof(t32_binari_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_series_1(np__2,A),B)
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( C = k5_euclid(A)
=> ( k1_binari_3(A,B) = k6_binarith(A,C)
<=> B = k5_real_1(k3_series_1(np__2,A),np__1) ) ) ) ) ) ) ).
fof(t33_binari_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_series_1(np__2,A),k1_nat_1(B,np__1))
=> k11_binarith(A,k1_binari_3(A,B),k2_binari_2(A)) = k7_margrel1 ) ) ) ).
fof(t34_binari_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_series_1(np__2,A),k1_nat_1(B,np__1))
=> k1_binari_3(A,k1_nat_1(B,np__1)) = k10_binarith(A,k1_binari_3(A,B),k2_binari_2(A)) ) ) ) ).
fof(t35_binari_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_binari_3(k1_nat_1(A,np__1),B) = k7_finseq_1(k13_binarith(k1_numbers,k4_nat_1(B,np__2)),k1_binari_3(A,k3_nat_1(B,np__2))) ) ) ).
fof(t36_binari_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_series_1(np__2,A),B)
=> k9_binarith(A,k1_binari_3(A,B)) = B ) ) ) ).
fof(t37_binari_3,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))
=> k1_binari_3(A,k9_binarith(A,B)) = B ) ) ).
fof(dt_k1_binari_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_finseq_2(k1_binari_3(A,B),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) ).
%------------------------------------------------------------------------------