SET007 Axioms: SET007+754.ax
%------------------------------------------------------------------------------
% File : SET007+754 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Improvement of Radix-2^k Signed-Digit Number for High Speed Circuit
% 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 : radix_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 54 ( 4 unt; 0 def)
% Number of atoms : 233 ( 29 equ)
% Maximal formula atoms : 11 ( 4 avg)
% Number of connectives : 188 ( 9 ~; 1 |; 50 &)
% ( 4 <=>; 124 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 11 ( 9 usr; 1 prp; 0-3 aty)
% Number of functors : 40 ( 40 usr; 6 con; 0-4 aty)
% Number of variables : 127 ( 125 !; 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(fc1_radix_3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ~ v1_xboole_0(k1_radix_3(A)) ) ).
fof(fc2_radix_3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ~ v1_xboole_0(k2_radix_3(A)) ) ).
fof(t1_radix_3,axiom,
$true ).
fof(t2_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ( r2_hidden(B,k2_radix_3(A))
=> ( r1_xreal_0(k6_xcmplx_0(k4_xcmplx_0(k1_radix_1(k5_binarith(A,np__1))),np__1),B)
& r1_xreal_0(B,k1_radix_1(k5_binarith(A,np__1))) ) ) ) ) ).
fof(t3_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k1_radix_3(A),k2_radix_3(A)) ) ).
fof(t4_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k1_radix_3(A),k1_radix_3(k1_nat_1(A,np__1))) ) ).
fof(t5_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,A)
=> r1_tarski(k2_radix_3(A),k3_radix_1(A)) ) ) ).
fof(t6_radix_3,axiom,
r2_hidden(np__0,k1_radix_3(np__0)) ).
fof(t7_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r2_hidden(np__0,k1_radix_3(A)) ) ).
fof(t8_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r2_hidden(np__0,k2_radix_3(A)) ) ).
fof(t9_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( r2_hidden(B,k2_radix_3(A))
=> v1_int_1(B) ) ) ).
fof(t10_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k2_radix_3(A),k6_wsierp_1) ) ).
fof(t11_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k1_radix_3(A),k6_wsierp_1) ) ).
fof(t12_radix_3,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)
=> ! [D] :
( m2_finseq_2(D,k4_radix_3(C),k4_finseq_2(B,k4_radix_3(C)))
=> ( r2_hidden(A,k2_finseq_1(B))
=> m2_subset_1(k1_funct_1(D,A),k6_wsierp_1,k4_radix_3(C)) ) ) ) ) ) ).
fof(d3_radix_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(k1_radix_1(k5_binarith(B,np__1)),A)
=> k5_radix_3(A,B) = np__1 )
& ( ~ r1_xreal_0(k4_xcmplx_0(k1_radix_1(k5_binarith(B,np__1))),A)
=> k5_radix_3(A,B) = k4_xcmplx_0(np__1) )
& ( r1_xreal_0(k4_xcmplx_0(k1_radix_1(k5_binarith(B,np__1))),A)
=> ( r1_xreal_0(k1_radix_1(k5_binarith(B,np__1)),A)
| k5_radix_3(A,B) = np__0 ) ) ) ) ) ).
fof(d4_radix_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k6_radix_3(A,B) = k6_xcmplx_0(A,k3_xcmplx_0(k1_radix_1(B),k5_radix_3(A,B))) ) ) ).
fof(t13_radix_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,B)
=> ( r1_xreal_0(k4_xcmplx_0(np__1),k5_radix_3(A,B))
& r1_xreal_0(k5_radix_3(A,B),np__1) ) ) ) ) ).
fof(t14_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ( ( r1_xreal_0(np__2,A)
& r2_hidden(B,k3_radix_1(A)) )
=> ( r1_xreal_0(k4_xcmplx_0(k1_radix_1(k5_binarith(A,np__1))),k6_radix_3(B,A))
& r1_xreal_0(k6_radix_3(B,A),k6_xcmplx_0(k1_radix_1(k5_binarith(A,np__1)),np__1)) ) ) ) ) ).
fof(t15_radix_3,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,B)
=> r2_hidden(k5_radix_3(A,B),k3_radix_3(B)) ) ) ) ).
fof(t16_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r1_xreal_0(np__2,A)
& r2_hidden(B,k3_radix_1(A))
& r2_hidden(C,k3_radix_1(A)) )
=> r2_hidden(k2_xcmplx_0(k6_radix_3(B,A),k5_radix_3(C,A)),k4_radix_3(A)) ) ) ) ) ).
fof(t17_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,A)
=> k5_radix_3(np__0,A) = np__0 ) ) ).
fof(d5_radix_3,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)
=> ! [D] :
( m2_finseq_2(D,k4_radix_3(B),k4_finseq_2(C,k4_radix_3(B)))
=> ( ( r2_hidden(A,k2_finseq_1(C))
=> k7_radix_3(A,B,C,D) = k1_funct_1(D,A) )
& ( A = np__0
=> k7_radix_3(A,B,C,D) = np__0 ) ) ) ) ) ) ).
fof(d6_radix_3,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)
=> ! [D] :
( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(C,k3_radix_1(B)))
=> ( ( r2_hidden(A,k2_finseq_1(C))
=> k8_radix_3(A,B,C,D) = k2_xcmplx_0(k6_radix_3(k4_radix_1(A,B,C,D),B),k5_radix_3(k4_radix_1(k5_binarith(A,np__1),B,C,D),B)) )
& ( A = k1_nat_1(C,np__1)
=> k8_radix_3(A,B,C,D) = k5_radix_3(k4_radix_1(k5_binarith(A,np__1),B,C,D),B) )
& ~ ( ~ r2_hidden(A,k2_finseq_1(C))
& A != k1_nat_1(C,np__1)
& k8_radix_3(A,B,C,D) != np__0 ) ) ) ) ) ) ).
fof(t18_radix_3,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)
=> ! [D] :
( m2_finseq_2(D,k3_radix_1(A),k4_finseq_2(C,k3_radix_1(A)))
=> ( ( r1_xreal_0(np__2,A)
& r2_hidden(B,k2_finseq_1(k1_nat_1(C,np__1))) )
=> m2_subset_1(k8_radix_3(B,A,C,D),k6_wsierp_1,k4_radix_3(A)) ) ) ) ) ) ).
fof(d7_radix_3,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)
=> ! [D] :
( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(C,k3_radix_1(B)))
=> ( ( r1_xreal_0(np__2,B)
& r2_hidden(A,k2_finseq_1(k1_nat_1(C,np__1))) )
=> k9_radix_3(A,B,C,D) = k8_radix_3(A,B,C,D) ) ) ) ) ) ).
fof(d8_radix_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,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
=> ! [D] :
( m2_finseq_2(D,k4_radix_3(B),k4_finseq_2(k1_nat_1(A,np__1),k4_radix_3(B)))
=> ( D = k10_radix_3(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(k1_nat_1(A,np__1)))
=> k7_radix_3(E,B,k1_nat_1(A,np__1),D) = k9_radix_3(E,B,A,C) ) ) ) ) ) ) ) ).
fof(t19_radix_3,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)
=> ! [D] :
( m2_finseq_2(D,k4_radix_3(C),k4_finseq_2(B,k4_radix_3(C)))
=> ( r2_hidden(A,k2_finseq_1(B))
=> m2_subset_1(k7_radix_3(A,C,B,D),k6_wsierp_1,k4_radix_3(C)) ) ) ) ) ) ).
fof(t20_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r1_xreal_0(np__2,A)
& r2_hidden(B,k3_radix_1(A))
& r2_hidden(C,k4_radix_3(A)) )
=> r2_hidden(k6_radix_3(k2_xcmplx_0(B,C),A),k3_radix_3(A)) ) ) ) ) ).
fof(d9_radix_3,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)
=> ! [D] :
( m2_finseq_2(D,k4_radix_3(B),k4_finseq_2(C,k4_radix_3(B)))
=> k11_radix_3(A,B,C,D) = k7_radix_3(A,B,C,D) ) ) ) ) ).
fof(d10_radix_3,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)
=> ! [D] :
( m2_finseq_2(D,k4_radix_3(B),k4_finseq_2(C,k4_radix_3(B)))
=> k12_radix_3(A,B,C,D) = k3_xcmplx_0(k2_wsierp_1(k1_radix_1(B),k5_binarith(A,np__1)),k11_radix_3(A,B,C,D)) ) ) ) ) ).
fof(d11_radix_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,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
=> ! [D] :
( m2_finseq_2(D,k6_wsierp_1,k4_finseq_2(A,k6_wsierp_1))
=> ( D = k13_radix_3(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_finseq_4(k5_numbers,k6_wsierp_1,D,E) = k12_radix_3(E,B,A,C) ) ) ) ) ) ) ) ).
fof(d12_radix_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,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
=> k14_radix_3(A,B,C) = k7_wsierp_1(k13_radix_3(A,B,C)) ) ) ) ).
fof(t21_radix_3,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(D,k2_finseq_1(A))
& r1_xreal_0(np__2,B) )
=> k7_radix_3(D,B,k1_nat_1(k1_nat_1(A,np__1),np__1),k10_radix_3(k1_nat_1(A,np__1),B,k10_radix_1(B,k1_nat_1(A,np__1),C))) = k7_radix_3(D,B,k1_nat_1(A,np__1),k10_radix_3(A,B,k10_radix_1(B,A,k4_nat_1(C,k2_wsierp_1(k1_radix_1(B),A))))) ) ) ) ) ) ).
fof(t22_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,B)
& r1_radix_1(A,C,B) )
=> C = k14_radix_3(k1_nat_1(A,np__1),B,k10_radix_3(A,B,k10_radix_1(B,A,C))) ) ) ) ) ) ).
fof(dt_k1_radix_3,axiom,
$true ).
fof(dt_k2_radix_3,axiom,
$true ).
fof(dt_k3_radix_3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k3_radix_3(A))
& m1_subset_1(k3_radix_3(A),k1_zfmisc_1(k6_wsierp_1)) ) ) ).
fof(redefinition_k3_radix_3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> k3_radix_3(A) = k1_radix_3(A) ) ).
fof(dt_k4_radix_3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k4_radix_3(A))
& m1_subset_1(k4_radix_3(A),k1_zfmisc_1(k6_wsierp_1)) ) ) ).
fof(redefinition_k4_radix_3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> k4_radix_3(A) = k2_radix_3(A) ) ).
fof(dt_k5_radix_3,axiom,
! [A,B] :
( ( v1_int_1(A)
& m1_subset_1(B,k5_numbers) )
=> v1_int_1(k5_radix_3(A,B)) ) ).
fof(dt_k6_radix_3,axiom,
! [A,B] :
( ( v1_int_1(A)
& m1_subset_1(B,k5_numbers) )
=> v1_int_1(k6_radix_3(A,B)) ) ).
fof(dt_k7_radix_3,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k4_finseq_2(C,k4_radix_3(B))) )
=> v1_int_1(k7_radix_3(A,B,C,D)) ) ).
fof(dt_k8_radix_3,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k4_finseq_2(C,k3_radix_1(B))) )
=> v1_int_1(k8_radix_3(A,B,C,D)) ) ).
fof(dt_k9_radix_3,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k4_finseq_2(C,k3_radix_1(B))) )
=> m2_subset_1(k9_radix_3(A,B,C,D),k6_wsierp_1,k4_radix_3(B)) ) ).
fof(dt_k10_radix_3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k4_finseq_2(A,k3_radix_1(B))) )
=> m2_finseq_2(k10_radix_3(A,B,C),k4_radix_3(B),k4_finseq_2(k1_nat_1(A,np__1),k4_radix_3(B))) ) ).
fof(dt_k11_radix_3,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k4_finseq_2(C,k4_radix_3(B))) )
=> m2_subset_1(k11_radix_3(A,B,C,D),k1_numbers,k6_wsierp_1) ) ).
fof(dt_k12_radix_3,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k4_finseq_2(C,k4_radix_3(B))) )
=> m2_subset_1(k12_radix_3(A,B,C,D),k1_numbers,k6_wsierp_1) ) ).
fof(dt_k13_radix_3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k4_finseq_2(A,k4_radix_3(B))) )
=> m2_finseq_2(k13_radix_3(A,B,C),k6_wsierp_1,k4_finseq_2(A,k6_wsierp_1)) ) ).
fof(dt_k14_radix_3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k4_finseq_2(A,k4_radix_3(B))) )
=> v1_int_1(k14_radix_3(A,B,C)) ) ).
fof(d1_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_radix_3(A) = a_1_0_radix_3(A) ) ).
fof(d2_radix_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_radix_3(A) = a_1_1_radix_3(A) ) ).
fof(fraenkel_a_1_0_radix_3,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_0_radix_3(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k6_wsierp_1)
& A = C
& r1_xreal_0(k4_xcmplx_0(k1_radix_1(k5_binarith(B,np__1))),C)
& r1_xreal_0(C,k6_xcmplx_0(k1_radix_1(k5_binarith(B,np__1)),np__1)) ) ) ) ).
fof(fraenkel_a_1_1_radix_3,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_1_radix_3(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k6_wsierp_1)
& A = C
& r1_xreal_0(k6_xcmplx_0(k4_xcmplx_0(k1_radix_1(k5_binarith(B,np__1))),np__1),C)
& r1_xreal_0(C,k1_radix_1(k5_binarith(B,np__1))) ) ) ) ).
%------------------------------------------------------------------------------