SET007 Axioms: SET007+622.ax
%------------------------------------------------------------------------------
% File : SET007+622 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Radix-2^k Signed-Digit Number and its Adder Algorithm
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 5 unt; 0 def)
% Number of atoms : 264 ( 47 equ)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 222 ( 18 ~; 1 |; 50 &)
% ( 6 <=>; 147 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 41 ( 41 usr; 6 con; 0-5 aty)
% Number of variables : 149 ( 148 !; 1 ?)
% 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_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ~ v1_xboole_0(k2_radix_1(A)) ) ).
fof(t1_radix_1,axiom,
$true ).
fof(t2_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k4_nat_1(A,B) = k6_xcmplx_0(B,np__1)
=> k4_nat_1(k1_nat_1(A,np__1),B) = np__0 ) ) ) ).
fof(t3_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& ~ r1_xreal_0(k6_xcmplx_0(A,np__1),k4_nat_1(B,A))
& k4_nat_1(k1_nat_1(B,np__1),A) != k1_nat_1(k4_nat_1(B,A),np__1) ) ) ) ).
fof(t4_radix_1,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)
=> ( A != np__0
=> k4_nat_1(k4_nat_1(B,k2_nat_1(A,C)),C) = k4_nat_1(B,C) ) ) ) ) ).
fof(t5_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& k4_nat_1(k1_nat_1(B,np__1),A) != np__0
& k4_nat_1(k1_nat_1(B,np__1),A) != k1_nat_1(k4_nat_1(B,A),np__1) ) ) ) ).
fof(t6_radix_1,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)
=> ~ ( A != np__0
& B != np__0
& r1_xreal_0(A,k3_nat_1(k4_nat_1(C,k2_wsierp_1(A,B)),k2_wsierp_1(A,k5_binarith(B,np__1)))) ) ) ) ) ).
fof(t7_radix_1,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(A,B)
=> r1_nat_1(k2_wsierp_1(C,A),k2_wsierp_1(C,B)) ) ) ) ) ).
fof(t8_radix_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_xreal_0(np__0,B)
=> ( r1_xreal_0(A,np__0)
| k6_int_1(k6_int_1(C,k3_xcmplx_0(A,B)),B) = k6_int_1(C,B) ) ) ) ) ) ).
fof(d1_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_radix_1(A) = k4_power(np__2,A) ) ).
fof(t9_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_radix_1(A) != np__0 ) ).
fof(t10_radix_1,axiom,
! [A] :
( r2_hidden(A,k2_radix_1(np__0))
<=> A = np__0 ) ).
fof(t11_radix_1,axiom,
k2_radix_1(np__0) = k1_tarski(np__0) ).
fof(t12_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k2_radix_1(A),k2_radix_1(k1_nat_1(A,np__1))) ) ).
fof(t13_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( r2_hidden(B,k2_radix_1(A))
=> v1_int_1(B) ) ) ).
fof(t14_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k2_radix_1(A),k6_wsierp_1) ) ).
fof(t15_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> ( r2_hidden(B,k2_radix_1(A))
=> ( r1_xreal_0(B,k6_xcmplx_0(k1_radix_1(A),np__1))
& r1_xreal_0(k2_xcmplx_0(k4_xcmplx_0(k1_radix_1(A)),np__1),B) ) ) ) ) ).
fof(t16_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r2_hidden(np__0,k2_radix_1(A)) ) ).
fof(t17_radix_1,axiom,
$true ).
fof(t18_radix_1,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(C),k4_finseq_2(B,k3_radix_1(C)))
=> ( r2_hidden(A,k2_finseq_1(B))
=> m2_subset_1(k1_funct_1(D,A),k6_wsierp_1,k3_radix_1(C)) ) ) ) ) ) ).
fof(d3_radix_1,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))
=> k4_radix_1(A,B,C,D) = k1_funct_1(D,A) )
& ( A = np__0
=> k4_radix_1(A,B,C,D) = np__0 ) ) ) ) ) ) ).
fof(d4_radix_1,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)))
=> k5_radix_1(A,B,C,D) = k4_radix_1(A,B,C,D) ) ) ) ) ).
fof(t19_radix_1,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(C),k4_finseq_2(B,k3_radix_1(C)))
=> ( r2_hidden(A,k2_finseq_1(B))
=> m2_subset_1(k4_radix_1(A,C,B,D),k6_wsierp_1,k3_radix_1(C)) ) ) ) ) ) ).
fof(t20_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_wsierp_1,k4_finseq_2(np__1,k6_wsierp_1))
=> ( k4_finseq_4(k5_numbers,k6_wsierp_1,B,np__1) = A
=> B = k13_binarith(k1_numbers,A) ) ) ) ).
fof(d5_radix_1,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)))
=> k6_radix_1(A,B,C,D) = k3_xcmplx_0(k2_wsierp_1(k1_radix_1(B),k5_binarith(A,np__1)),k5_radix_1(A,B,C,D)) ) ) ) ) ).
fof(d6_radix_1,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,k6_wsierp_1,k4_finseq_2(A,k6_wsierp_1))
=> ( D = k7_radix_1(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) = k6_radix_1(E,B,A,C) ) ) ) ) ) ) ) ).
fof(d7_radix_1,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)))
=> k8_radix_1(A,B,C) = k7_wsierp_1(k7_radix_1(A,B,C)) ) ) ) ).
fof(d8_radix_1,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)
=> k9_radix_1(A,B,C) = k3_nat_1(k4_nat_1(C,k2_wsierp_1(k1_radix_1(B),A)),k2_wsierp_1(k1_radix_1(B),k5_binarith(A,np__1))) ) ) ) ).
fof(d9_radix_1,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(B,k3_radix_1(A)))
=> ( D = k10_radix_1(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(B))
=> k4_radix_1(E,A,B,D) = k9_radix_1(E,A,C) ) ) ) ) ) ) ) ).
fof(d10_radix_1,axiom,
! [A] :
( v1_int_1(A)
=> ( ( ~ r1_xreal_0(A,np__2)
=> k11_radix_1(A) = np__1 )
& ( ~ r1_xreal_0(k4_xcmplx_0(np__2),A)
=> k11_radix_1(A) = k4_xcmplx_0(np__1) )
& ( ( r1_xreal_0(A,np__2)
& r1_xreal_0(k4_xcmplx_0(np__2),A) )
=> k11_radix_1(A) = np__0 ) ) ) ).
fof(t21_radix_1,axiom,
k11_radix_1(np__0) = np__0 ).
fof(d11_radix_1,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k12_radix_1(A,B) = k6_xcmplx_0(A,k3_xcmplx_0(k11_radix_1(A),k1_radix_1(B))) ) ) ).
fof(t22_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k12_radix_1(np__0,A) = np__0 ) ).
fof(t23_radix_1,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)) )
=> ( r1_xreal_0(k2_xcmplx_0(k4_xcmplx_0(k1_radix_1(A)),np__2),k12_radix_1(k2_xcmplx_0(B,C),A))
& r1_xreal_0(k12_radix_1(k2_xcmplx_0(B,C),A),k6_xcmplx_0(k1_radix_1(A),np__2)) ) ) ) ) ) ).
fof(d12_radix_1,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_radix_1(A,B,C)
<=> ~ r1_xreal_0(k2_wsierp_1(k1_radix_1(C),A),B) ) ) ) ) ).
fof(t24_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_radix_1(np__1,A,B)
=> k4_radix_1(np__1,B,np__1,k10_radix_1(B,np__1,A)) = A ) ) ) ).
fof(t25_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_radix_1(B,C,A)
=> C = k8_radix_1(B,A,k10_radix_1(A,B,C)) ) ) ) ) ) ).
fof(t26_radix_1,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(np__2,A)
& r1_radix_1(np__1,B,A)
& r1_radix_1(np__1,C,A) )
=> k11_radix_1(k2_xcmplx_0(k4_radix_1(np__1,A,np__1,k10_radix_1(A,np__1,B)),k4_radix_1(np__1,A,np__1,k10_radix_1(A,np__1,C)))) = k11_radix_1(k1_nat_1(B,C)) ) ) ) ) ).
fof(t27_radix_1,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_radix_1(k1_nat_1(B,np__1),A,C)
=> k4_radix_1(k1_nat_1(B,np__1),C,k1_nat_1(B,np__1),k10_radix_1(C,k1_nat_1(B,np__1),A)) = k3_nat_1(A,k2_wsierp_1(k1_radix_1(C),B)) ) ) ) ) ).
fof(d13_radix_1,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)))
=> ! [E] :
( m2_finseq_2(E,k3_radix_1(A),k4_finseq_2(C,k3_radix_1(A)))
=> ( ( r2_hidden(B,k2_finseq_1(C))
& r1_xreal_0(np__2,A) )
=> k13_radix_1(A,B,C,D,E) = k2_xcmplx_0(k12_radix_1(k2_xcmplx_0(k4_radix_1(B,A,C,D),k4_radix_1(B,A,C,E)),A),k11_radix_1(k2_xcmplx_0(k4_radix_1(k5_binarith(B,np__1),A,C,D),k4_radix_1(k5_binarith(B,np__1),A,C,E)))) ) ) ) ) ) ) ).
fof(d14_radix_1,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,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
=> ! [E] :
( m2_finseq_2(E,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
=> ( E = k14_radix_1(A,B,C,D)
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k2_finseq_1(A))
=> k4_radix_1(F,B,A,E) = k13_radix_1(B,F,A,C,D) ) ) ) ) ) ) ) ) ).
fof(t28_radix_1,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(np__2,A)
& r1_radix_1(np__1,B,A)
& r1_radix_1(np__1,C,A) )
=> k8_radix_1(np__1,A,k14_radix_1(np__1,A,k10_radix_1(A,np__1,B),k10_radix_1(A,np__1,C))) = k12_radix_1(k1_nat_1(B,C),A) ) ) ) ) ).
fof(t29_radix_1,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,B)
& r1_radix_1(A,C,B)
& r1_radix_1(A,D,B) )
=> k1_nat_1(C,D) = k2_xcmplx_0(k8_radix_1(A,B,k14_radix_1(A,B,k10_radix_1(B,A,C),k10_radix_1(B,A,D))),k3_xcmplx_0(k2_wsierp_1(k1_radix_1(B),A),k11_radix_1(k2_xcmplx_0(k4_radix_1(A,B,A,k10_radix_1(B,A,C)),k4_radix_1(A,B,A,k10_radix_1(B,A,D)))))) ) ) ) ) ) ) ).
fof(dt_k1_radix_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m2_subset_1(k1_radix_1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k2_radix_1,axiom,
$true ).
fof(dt_k3_radix_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v1_xboole_0(k3_radix_1(A))
& m1_subset_1(k3_radix_1(A),k1_zfmisc_1(k6_wsierp_1)) ) ) ).
fof(redefinition_k3_radix_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> k3_radix_1(A) = k2_radix_1(A) ) ).
fof(dt_k4_radix_1,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(k4_radix_1(A,B,C,D)) ) ).
fof(dt_k5_radix_1,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(k5_radix_1(A,B,C,D),k1_numbers,k6_wsierp_1) ) ).
fof(dt_k6_radix_1,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(k6_radix_1(A,B,C,D),k1_numbers,k6_wsierp_1) ) ).
fof(dt_k7_radix_1,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(k7_radix_1(A,B,C),k6_wsierp_1,k4_finseq_2(A,k6_wsierp_1)) ) ).
fof(dt_k8_radix_1,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))) )
=> v1_int_1(k8_radix_1(A,B,C)) ) ).
fof(dt_k9_radix_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers) )
=> m2_subset_1(k9_radix_1(A,B,C),k6_wsierp_1,k3_radix_1(B)) ) ).
fof(dt_k10_radix_1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers) )
=> m2_finseq_2(k10_radix_1(A,B,C),k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A))) ) ).
fof(dt_k11_radix_1,axiom,
! [A] :
( v1_int_1(A)
=> v1_int_1(k11_radix_1(A)) ) ).
fof(dt_k12_radix_1,axiom,
! [A,B] :
( ( v1_int_1(A)
& m1_subset_1(B,k5_numbers) )
=> v1_int_1(k12_radix_1(A,B)) ) ).
fof(dt_k13_radix_1,axiom,
! [A,B,C,D,E] :
( ( 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(A)))
& m1_subset_1(E,k4_finseq_2(C,k3_radix_1(A))) )
=> m2_subset_1(k13_radix_1(A,B,C,D,E),k6_wsierp_1,k3_radix_1(A)) ) ).
fof(dt_k14_radix_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k4_finseq_2(A,k3_radix_1(B)))
& m1_subset_1(D,k4_finseq_2(A,k3_radix_1(B))) )
=> m2_finseq_2(k14_radix_1(A,B,C,D),k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B))) ) ).
fof(d2_radix_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_radix_1(A) = a_1_0_radix_1(A) ) ).
fof(fraenkel_a_1_0_radix_1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_0_radix_1(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k6_wsierp_1)
& A = C
& r1_xreal_0(C,k6_xcmplx_0(k1_radix_1(B),np__1))
& r1_xreal_0(k2_xcmplx_0(k4_xcmplx_0(k1_radix_1(B)),np__1),C) ) ) ) ).
%------------------------------------------------------------------------------