SET007 Axioms: SET007+643.ax
%------------------------------------------------------------------------------
% File : SET007+643 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : High-Speed Algorithms for RSA Cryptograms
% 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_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 0 unt; 0 def)
% Number of atoms : 219 ( 40 equ)
% Maximal formula atoms : 17 ( 6 avg)
% Number of connectives : 191 ( 7 ~; 3 |; 38 &)
% ( 4 <=>; 139 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 11 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-3 aty)
% Number of functors : 44 ( 44 usr; 6 con; 0-6 aty)
% Number of variables : 143 ( 143 !; 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_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k4_nat_1(A,np__1) = np__0 ) ).
fof(t2_radix_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k6_int_1(k2_xcmplx_0(k6_int_1(A,C),k6_int_1(B,C)),C) = k6_int_1(k2_xcmplx_0(A,k6_int_1(B,C)),C) ) ) ) ) ).
fof(t3_radix_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k6_int_1(k3_xcmplx_0(A,B),C) = k6_int_1(k3_xcmplx_0(A,k6_int_1(B,C)),C) ) ) ) ) ).
fof(t4_radix_2,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(B,np__0)
| k3_nat_1(k4_nat_1(A,k2_wsierp_1(B,C)),k2_wsierp_1(B,k5_binarith(C,np__1))) = k4_nat_1(k3_nat_1(A,k2_wsierp_1(B,k5_binarith(C,np__1))),B) ) ) ) ) ) ).
fof(t5_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k2_finseq_1(B))
=> r2_hidden(k1_nat_1(A,np__1),k2_finseq_1(k1_nat_1(B,np__1))) ) ) ) ).
fof(t6_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k1_radix_1(A),np__0) ) ).
fof(t7_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k3_radix_1(A),k4_finseq_2(np__1,k3_radix_1(A)))
=> k8_radix_1(np__1,A,B) = k4_radix_1(np__1,A,np__1,B) ) ) ).
fof(t8_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_int_1(B)
=> k2_xcmplx_0(k12_radix_1(B,A),k3_xcmplx_0(k11_radix_1(B),k1_radix_1(A))) = B ) ) ).
fof(t9_radix_2,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(A),k4_finseq_2(k1_nat_1(B,np__1),k3_radix_1(A)))
=> ! [D] :
( m2_finseq_2(D,k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A)))
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(B))
=> k1_funct_1(C,E) = k1_funct_1(D,E) ) )
=> k7_wsierp_1(k7_radix_1(k1_nat_1(B,np__1),A,C)) = k7_wsierp_1(k4_wsierp_1(k1_numbers,k6_wsierp_1,k7_radix_1(B,A,D),k13_binarith(k6_wsierp_1,k6_radix_1(k1_nat_1(B,np__1),A,k1_nat_1(B,np__1),C)))) ) ) ) ) ) ).
fof(t10_radix_2,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(A),k4_finseq_2(k1_nat_1(B,np__1),k3_radix_1(A)))
=> ! [D] :
( m2_finseq_2(D,k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A)))
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(B))
=> k1_funct_1(C,E) = k1_funct_1(D,E) ) )
=> k2_xcmplx_0(k8_radix_1(B,A,D),k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),B),k4_radix_1(k1_nat_1(B,np__1),A,k1_nat_1(B,np__1),C))) = k8_radix_1(k1_nat_1(B,np__1),A,C) ) ) ) ) ) ).
fof(t11_radix_2,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_finseq_2(C,k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A)))
=> ! [D] :
( m2_finseq_2(D,k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A)))
=> ( r1_xreal_0(np__2,A)
=> k2_xcmplx_0(k8_radix_1(B,A,k14_radix_1(B,A,C,D)),k3_xcmplx_0(k11_radix_1(k2_xcmplx_0(k4_radix_1(B,A,B,C),k4_radix_1(B,A,B,D))),k2_wsierp_1(k1_radix_1(A),B))) = k2_xcmplx_0(k8_radix_1(B,A,C),k8_radix_1(B,A,D)) ) ) ) ) ) ) ).
fof(d1_radix_2,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,k5_numbers,k4_finseq_2(C,k5_numbers))
=> k1_radix_2(A,B,C,D) = k2_nat_1(k2_wsierp_1(k1_radix_1(B),k5_binarith(A,np__1)),k3_wsierp_1(D,A)) ) ) ) ) ).
fof(d2_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [D] :
( m2_finseq_2(D,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( D = k2_radix_2(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_finseq_4(k5_numbers,k5_numbers,D,E) = k1_radix_2(E,B,A,C) ) ) ) ) ) ) ) ).
fof(d3_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> k3_radix_2(A,B,C) = k9_wsierp_1(k2_radix_2(A,B,C)) ) ) ) ).
fof(d4_radix_2,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)
=> k4_radix_2(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(d5_radix_2,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,k5_numbers,k4_finseq_2(B,k5_numbers))
=> ( D = k5_radix_2(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(B))
=> k3_wsierp_1(D,E) = k4_radix_2(E,A,C) ) ) ) ) ) ) ) ).
fof(t12_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [D] :
( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
=> ( C = D
=> k2_radix_2(A,B,C) = k7_radix_1(A,B,D) ) ) ) ) ) ).
fof(t13_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [D] :
( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
=> ( C = D
=> k3_radix_2(A,B,C) = k8_radix_1(A,B,D) ) ) ) ) ) ).
fof(t14_radix_2,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)
=> k5_radix_2(C,B,A) = k10_radix_1(C,B,A) ) ) ) ).
fof(t15_radix_2,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_radix_1(A,B,C)
=> B = k3_radix_2(A,C,k5_radix_2(C,A,B)) ) ) ) ) ) ).
fof(d6_radix_2,axiom,
! [A] :
( v1_int_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)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_finseq_2(F,k3_radix_1(D),k4_finseq_2(E,k3_radix_1(D)))
=> k6_radix_2(A,B,C,D,E,F) = k6_int_1(k3_xcmplx_0(A,k4_radix_1(C,D,E,F)),B) ) ) ) ) ) ) ).
fof(d7_radix_2,axiom,
! [A] :
( v1_int_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)
=> ! [E] :
( m2_finseq_2(E,k3_radix_1(B),k4_finseq_2(D,k3_radix_1(B)))
=> ( r1_xreal_0(np__1,D)
=> ! [F] :
( m2_finseq_2(F,k6_wsierp_1,k4_finseq_2(D,k6_wsierp_1))
=> ( F = k7_radix_2(A,B,C,D,E)
<=> ( k1_funct_1(F,np__1) = k6_radix_2(A,C,D,B,D,E)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,G)
& r1_xreal_0(G,k6_xcmplx_0(D,np__1))
& ! [H] :
( v1_int_1(H)
=> ! [I] :
( v1_int_1(I)
=> ~ ( H = k1_funct_1(F,G)
& I = k1_funct_1(F,k1_nat_1(G,np__1))
& I = k6_int_1(k2_xcmplx_0(k3_xcmplx_0(k1_radix_1(B),H),k6_radix_2(A,C,k5_binarith(D,G),B,D,E)),C) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t16_radix_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ! [B] :
( v1_int_1(B)
=> ! [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)
=> ( r1_radix_1(A,C,E)
=> ( r1_xreal_0(D,np__0)
| ! [F] :
( m2_finseq_2(F,k3_radix_1(E),k4_finseq_2(A,k3_radix_1(E)))
=> ( F = k10_radix_1(E,A,C)
=> k1_funct_1(k7_radix_2(B,E,D,A,F),A) = k6_int_1(k3_xcmplx_0(B,C),D) ) ) ) ) ) ) ) ) ) ) ).
fof(d8_radix_2,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_finseq_2(E,k5_numbers,k4_finseq_2(A,k5_numbers))
=> k8_radix_2(A,B,C,D,E) = k4_nat_1(k2_wsierp_1(D,k4_finseq_4(k5_numbers,k5_numbers,E,C)),B) ) ) ) ) ) ).
fof(d9_radix_2,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_finseq_2(E,k5_numbers,k4_finseq_2(D,k5_numbers))
=> ( r1_xreal_0(np__1,D)
=> ! [F] :
( m2_finseq_2(F,k5_numbers,k4_finseq_2(D,k5_numbers))
=> ( F = k9_radix_2(A,B,C,D,E)
<=> ( k3_wsierp_1(F,np__1) = k8_radix_2(D,B,D,C,E)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,G)
& r1_xreal_0(G,k6_xcmplx_0(D,np__1))
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ~ ( H = k3_wsierp_1(F,G)
& I = k3_wsierp_1(F,k1_nat_1(G,np__1))
& I = k4_nat_1(k2_nat_1(k4_nat_1(k2_wsierp_1(H,k1_radix_1(A)),B),k8_radix_2(D,B,k5_binarith(D,G),C,E)),B) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_radix_2,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)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_radix_1(A,E,C)
=> ( r1_xreal_0(D,np__0)
| ! [F] :
( m2_finseq_2(F,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( F = k5_radix_2(C,A,E)
=> k3_wsierp_1(k9_radix_2(C,D,B,A,F),A) = k4_nat_1(k2_wsierp_1(B,E),D) ) ) ) ) ) ) ) ) ) ) ).
fof(dt_k1_radix_2,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,k5_numbers)) )
=> m2_subset_1(k1_radix_2(A,B,C,D),k1_numbers,k5_numbers) ) ).
fof(dt_k2_radix_2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k4_finseq_2(A,k5_numbers)) )
=> m2_finseq_2(k2_radix_2(A,B,C),k5_numbers,k4_finseq_2(A,k5_numbers)) ) ).
fof(dt_k3_radix_2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k4_finseq_2(A,k5_numbers)) )
=> m2_subset_1(k3_radix_2(A,B,C),k1_numbers,k5_numbers) ) ).
fof(dt_k4_radix_2,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(k4_radix_2(A,B,C),k1_numbers,k5_numbers) ) ).
fof(dt_k5_radix_2,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(k5_radix_2(A,B,C),k5_numbers,k4_finseq_2(B,k5_numbers)) ) ).
fof(dt_k6_radix_2,axiom,
! [A,B,C,D,E,F] :
( ( v1_int_1(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers)
& m1_subset_1(E,k5_numbers)
& m1_subset_1(F,k4_finseq_2(E,k3_radix_1(D))) )
=> v1_int_1(k6_radix_2(A,B,C,D,E,F)) ) ).
fof(dt_k7_radix_2,axiom,
! [A,B,C,D,E] :
( ( v1_int_1(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers)
& m1_subset_1(E,k4_finseq_2(D,k3_radix_1(B))) )
=> m2_finseq_2(k7_radix_2(A,B,C,D,E),k6_wsierp_1,k4_finseq_2(D,k6_wsierp_1)) ) ).
fof(dt_k8_radix_2,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,k5_numbers)
& m1_subset_1(E,k4_finseq_2(A,k5_numbers)) )
=> m2_subset_1(k8_radix_2(A,B,C,D,E),k1_numbers,k5_numbers) ) ).
fof(dt_k9_radix_2,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,k5_numbers)
& m1_subset_1(E,k4_finseq_2(D,k5_numbers)) )
=> m2_finseq_2(k9_radix_2(A,B,C,D,E),k5_numbers,k4_finseq_2(D,k5_numbers)) ) ).
%------------------------------------------------------------------------------