SET007 Axioms: SET007+755.ax
%------------------------------------------------------------------------------
% File : SET007+755 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : High Speed Adder Algorithm with Radix-2^k Sub Signed-Digit Number
% 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_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 13 ( 0 unt; 0 def)
% Number of atoms : 82 ( 11 equ)
% Maximal formula atoms : 9 ( 6 avg)
% Number of connectives : 70 ( 1 ~; 0 |; 18 &)
% ( 1 <=>; 50 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 10 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-3 aty)
% Number of functors : 27 ( 27 usr; 7 con; 0-5 aty)
% Number of variables : 46 ( 46 !; 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_radix_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,A)
& r1_xreal_0(k1_radix_1(A),np__2) ) ) ).
fof(t2_radix_4,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__3,C)
=> k5_radix_3(k2_xcmplx_0(k5_radix_3(A,C),k5_radix_3(B,C)),C) = np__0 ) ) ) ) ).
fof(t3_radix_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(np__2,A)
=> k7_radix_3(k1_nat_1(C,np__1),A,k1_nat_1(C,np__1),k10_radix_3(C,A,k10_radix_1(A,C,B))) = k5_radix_3(k4_radix_1(C,A,C,k10_radix_1(A,C,B)),A) ) ) ) ) ).
fof(t4_radix_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,A)
& r1_radix_1(np__1,B,A) )
=> k7_radix_3(k1_nat_1(np__1,np__1),A,k1_nat_1(np__1,np__1),k10_radix_3(np__1,A,k10_radix_1(A,np__1,B))) = k5_radix_3(B,A) ) ) ) ).
fof(t5_radix_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(np__1,C)
& r1_xreal_0(np__3,A)
& r1_radix_1(k1_nat_1(C,np__1),B,A) )
=> k7_radix_3(k1_nat_1(C,np__1),A,k1_nat_1(C,np__1),k10_radix_3(C,A,k10_radix_1(A,C,k4_nat_1(B,k2_wsierp_1(k1_radix_1(A),C))))) = k5_radix_3(k4_radix_1(C,A,C,k10_radix_1(A,C,B)),A) ) ) ) ) ).
fof(t6_radix_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,A)
& r1_radix_1(np__1,B,A) )
=> k7_radix_3(np__1,A,k1_nat_1(np__1,np__1),k10_radix_3(np__1,A,k10_radix_1(A,np__1,B))) = k6_xcmplx_0(B,k3_xcmplx_0(k5_radix_3(B,A),k1_radix_1(A))) ) ) ) ).
fof(t7_radix_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(np__1,C)
& r1_xreal_0(np__2,A)
& r1_radix_1(k1_nat_1(C,np__1),B,A) )
=> k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),C),k7_radix_3(k1_nat_1(C,np__1),A,k1_nat_1(k1_nat_1(C,np__1),np__1),k10_radix_3(k1_nat_1(C,np__1),A,k10_radix_1(A,k1_nat_1(C,np__1),B)))) = k2_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),C),k4_radix_1(k1_nat_1(C,np__1),A,k1_nat_1(C,np__1),k10_radix_1(A,k1_nat_1(C,np__1),B))),k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),k1_nat_1(C,np__1)),k5_radix_3(k4_radix_1(k1_nat_1(C,np__1),A,k1_nat_1(C,np__1),k10_radix_1(A,k1_nat_1(C,np__1),B)),A))),k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),C),k5_radix_3(k4_radix_1(C,A,k1_nat_1(C,np__1),k10_radix_1(A,k1_nat_1(C,np__1),B)),A))) ) ) ) ) ).
fof(d1_radix_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)
=> ! [D] :
( m2_finseq_2(D,k4_radix_3(C),k4_finseq_2(B,k4_radix_3(C)))
=> ! [E] :
( m2_finseq_2(E,k4_radix_3(C),k4_finseq_2(B,k4_radix_3(C)))
=> ( ( r2_hidden(A,k2_finseq_1(B))
& r1_xreal_0(np__2,C) )
=> k1_radix_4(A,B,C,D,E) = k2_xcmplx_0(k6_radix_3(k2_xcmplx_0(k7_radix_3(A,C,B,D),k7_radix_3(A,C,B,E)),C),k5_radix_3(k2_xcmplx_0(k7_radix_3(k5_binarith(A,np__1),C,B,D),k7_radix_3(k5_binarith(A,np__1),C,B,E)),C)) ) ) ) ) ) ) ).
fof(d2_radix_4,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,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
=> ! [E] :
( m2_finseq_2(E,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
=> ( E = k2_radix_4(A,B,C,D)
<=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k2_finseq_1(A))
=> k7_radix_3(F,B,A,E) = k1_radix_4(F,A,B,C,D) ) ) ) ) ) ) ) ) ).
fof(t8_radix_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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r2_hidden(E,k2_finseq_1(A))
& r1_xreal_0(np__2,B) )
=> k1_radix_4(E,k1_nat_1(k1_nat_1(A,np__1),np__1),B,k10_radix_3(k1_nat_1(A,np__1),B,k10_radix_1(B,k1_nat_1(A,np__1),C)),k10_radix_3(k1_nat_1(A,np__1),B,k10_radix_1(B,k1_nat_1(A,np__1),D))) = k1_radix_4(E,k1_nat_1(A,np__1),B,k10_radix_3(A,B,k10_radix_1(B,A,k4_nat_1(C,k2_wsierp_1(k1_radix_1(B),A)))),k10_radix_3(A,B,k10_radix_1(B,A,k4_nat_1(D,k2_wsierp_1(k1_radix_1(B),A))))) ) ) ) ) ) ) ).
fof(t9_radix_4,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__3,B)
& r1_radix_1(A,C,B)
& r1_radix_1(A,D,B) )
=> k1_nat_1(C,D) = k14_radix_3(k1_nat_1(A,np__1),B,k2_radix_4(k1_nat_1(A,np__1),B,k10_radix_3(A,B,k10_radix_1(B,A,C)),k10_radix_3(A,B,k10_radix_1(B,A,D)))) ) ) ) ) ) ) ).
fof(dt_k1_radix_4,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(B,k4_radix_3(C)))
& m1_subset_1(E,k4_finseq_2(B,k4_radix_3(C))) )
=> m2_subset_1(k1_radix_4(A,B,C,D,E),k6_wsierp_1,k4_radix_3(C)) ) ).
fof(dt_k2_radix_4,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,k4_radix_3(B)))
& m1_subset_1(D,k4_finseq_2(A,k4_radix_3(B))) )
=> m2_finseq_2(k2_radix_4(A,B,C,D),k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B))) ) ).
%------------------------------------------------------------------------------