SET007 Axioms: SET007+796.ax
%------------------------------------------------------------------------------
% File : SET007+796 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Magnitude Relation Properties of Radix-2^k SD 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_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 0 unt; 0 def)
% Number of atoms : 222 ( 44 equ)
% Maximal formula atoms : 15 ( 6 avg)
% Number of connectives : 199 ( 12 ~; 4 |; 39 &)
% ( 4 <=>; 140 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 2-3 aty)
% Number of functors : 29 ( 29 usr; 7 con; 0-4 aty)
% Number of variables : 112 ( 112 !; 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_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,A)
=> r2_hidden(k6_xcmplx_0(k1_radix_1(A),np__1),k3_radix_1(A)) ) ) ).
fof(t2_radix_5,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))
=> ( r1_xreal_0(A,np__1)
| r2_hidden(k5_binarith(A,np__1),k2_finseq_1(B)) ) ) ) ) ).
fof(t3_radix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,A)
=> r1_xreal_0(np__4,k1_radix_1(A)) ) ) ).
fof(t4_radix_5,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(t5_radix_5,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)
=> ( r2_hidden(A,k2_finseq_1(C))
=> k4_radix_1(A,B,C,k10_radix_1(B,C,np__0)) = np__0 ) ) ) ) ).
fof(t6_radix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> k8_radix_1(A,B,k10_radix_1(B,A,np__0)) = np__0 ) ) ) ).
fof(t7_radix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r2_hidden(np__1,k2_finseq_1(B))
& r1_xreal_0(np__2,A) )
=> k4_radix_1(np__1,A,B,k10_radix_1(A,B,np__1)) = np__1 ) ) ) ).
fof(t8_radix_5,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)
=> ( ( r2_hidden(A,k2_finseq_1(C))
& r1_xreal_0(np__2,B) )
=> ( r1_xreal_0(A,np__1)
| k4_radix_1(A,B,C,k10_radix_1(B,C,np__1)) = np__0 ) ) ) ) ) ).
fof(t9_radix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(np__2,B) )
=> k8_radix_1(A,B,k10_radix_1(B,A,np__1)) = np__1 ) ) ) ).
fof(t10_radix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,A)
=> k11_radix_1(k1_radix_1(A)) = np__1 ) ) ).
fof(t11_radix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,A)
=> k12_radix_1(k1_radix_1(A),A) = np__0 ) ) ).
fof(t12_radix_5,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_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_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_radix_1(E,B,A,C) = k4_radix_1(E,B,A,D) ) )
=> k8_radix_1(A,B,C) = k8_radix_1(A,B,D) ) ) ) ) ) ) ).
fof(t13_radix_5,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_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_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> r1_xreal_0(k4_radix_1(E,B,A,D),k4_radix_1(E,B,A,C)) ) )
=> r1_xreal_0(k8_radix_1(A,B,D),k8_radix_1(A,B,C)) ) ) ) ) ) ) ).
fof(t14_radix_5,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)
=> ( r1_xreal_0(np__2,B)
=> ! [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)))
=> ! [F] :
( m2_finseq_2(F,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
=> ( ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(G,k2_finseq_1(A))
& ~ ( k4_radix_1(G,B,A,C) = k4_radix_1(G,B,A,E)
& k4_radix_1(G,B,A,D) = k4_radix_1(G,B,A,F) )
& ~ ( k4_radix_1(G,B,A,D) = k4_radix_1(G,B,A,E)
& k4_radix_1(G,B,A,C) = k4_radix_1(G,B,A,F) ) ) )
=> k2_xcmplx_0(k8_radix_1(A,B,E),k8_radix_1(A,B,F)) = k2_xcmplx_0(k8_radix_1(A,B,C),k8_radix_1(A,B,D)) ) ) ) ) ) ) ) ) ) ).
fof(t15_radix_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(np__2,B) )
=> ! [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)))
=> ( ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k2_finseq_1(A))
& ~ ( k4_radix_1(F,B,A,C) = k4_radix_1(F,B,A,E)
& k4_radix_1(F,B,A,D) = np__0 )
& ~ ( k4_radix_1(F,B,A,D) = k4_radix_1(F,B,A,E)
& k4_radix_1(F,B,A,C) = np__0 ) ) )
=> k2_xcmplx_0(k8_radix_1(A,B,E),k8_radix_1(A,B,k10_radix_1(B,A,np__0))) = k2_xcmplx_0(k8_radix_1(A,B,C),k8_radix_1(A,B,D)) ) ) ) ) ) ) ) ).
fof(d1_radix_5,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,C)
=> ( ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(B,A)
| k1_radix_5(A,B,C) = k2_xcmplx_0(k4_xcmplx_0(k1_radix_1(C)),np__1) ) )
& ( ~ ( r1_xreal_0(np__1,A)
& ~ r1_xreal_0(B,A) )
=> k1_radix_5(A,B,C) = np__0 ) ) ) ) ) ) ).
fof(d2_radix_5,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(A,k3_radix_1(C)))
=> ( D = k2_radix_5(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_radix_1(E,C,A,D) = k1_radix_5(E,B,C) ) ) ) ) ) ) ) ).
fof(d3_radix_5,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,C)
=> ( ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(B,A)
| k3_radix_5(A,B,C) = k6_xcmplx_0(k1_radix_1(C),np__1) ) )
& ( ~ ( r1_xreal_0(np__1,A)
& ~ r1_xreal_0(B,A) )
=> k3_radix_5(A,B,C) = np__0 ) ) ) ) ) ) ).
fof(d4_radix_5,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(A,k3_radix_1(C)))
=> ( D = k4_radix_5(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_radix_1(E,C,A,D) = k3_radix_5(E,B,C) ) ) ) ) ) ) ) ).
fof(d5_radix_5,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,C)
=> ( ( A = B
=> k5_radix_5(A,B,C) = np__1 )
& ( A != B
=> k5_radix_5(A,B,C) = np__0 ) ) ) ) ) ) ).
fof(d6_radix_5,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(A,k3_radix_1(C)))
=> ( D = k6_radix_5(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_radix_1(E,C,A,D) = k5_radix_5(E,B,C) ) ) ) ) ) ) ) ).
fof(d7_radix_5,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,C)
=> ( ( A = B
=> k7_radix_5(A,B,C) = k6_xcmplx_0(k1_radix_1(C),np__1) )
& ( A != B
=> k7_radix_5(A,B,C) = np__0 ) ) ) ) ) ) ).
fof(d8_radix_5,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(A,k3_radix_1(C)))
=> ( D = k8_radix_5(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_finseq_1(A))
=> k4_radix_1(E,C,A,D) = k7_radix_5(E,B,C) ) ) ) ) ) ) ) ).
fof(t16_radix_5,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,A)
& r1_xreal_0(np__2,C)
& r2_hidden(B,k2_finseq_1(A)) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k2_xcmplx_0(k4_radix_1(D,C,A,k4_radix_5(A,B,C)),k4_radix_1(D,C,A,k2_radix_5(A,B,C))) = np__0 ) ) ) ) ) ) ).
fof(t17_radix_5,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)
=> ( ( r2_hidden(B,k2_finseq_1(A))
& r1_xreal_0(np__2,C) )
=> k2_xcmplx_0(k8_radix_1(A,C,k4_radix_5(A,B,C)),k8_radix_1(A,C,k2_radix_5(A,B,C))) = k8_radix_1(A,C,k10_radix_1(C,A,np__0)) ) ) ) ) ) ).
fof(t18_radix_5,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)
=> ( ( r2_hidden(B,k2_finseq_1(A))
& r1_xreal_0(np__2,C) )
=> k8_radix_1(A,C,k6_radix_5(A,B,C)) = k2_xcmplx_0(k8_radix_1(A,C,k4_radix_5(A,B,C)),k8_radix_1(A,C,k10_radix_1(C,A,np__1))) ) ) ) ) ) ).
fof(t19_radix_5,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)
=> ( ( r2_hidden(B,k2_finseq_1(A))
& r1_xreal_0(np__2,C) )
=> k8_radix_1(k1_nat_1(A,np__1),C,k6_radix_5(k1_nat_1(A,np__1),k1_nat_1(B,np__1),C)) = k2_xcmplx_0(k8_radix_1(k1_nat_1(A,np__1),C,k6_radix_5(k1_nat_1(A,np__1),B,C)),k8_radix_1(k1_nat_1(A,np__1),C,k8_radix_5(k1_nat_1(A,np__1),B,C))) ) ) ) ) ).
fof(dt_k1_radix_5,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(k1_radix_5(A,B,C),k6_wsierp_1,k3_radix_1(C)) ) ).
fof(dt_k2_radix_5,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(k2_radix_5(A,B,C),k3_radix_1(C),k4_finseq_2(A,k3_radix_1(C))) ) ).
fof(dt_k3_radix_5,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(k3_radix_5(A,B,C),k6_wsierp_1,k3_radix_1(C)) ) ).
fof(dt_k4_radix_5,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(k4_radix_5(A,B,C),k3_radix_1(C),k4_finseq_2(A,k3_radix_1(C))) ) ).
fof(dt_k5_radix_5,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(k5_radix_5(A,B,C),k6_wsierp_1,k3_radix_1(C)) ) ).
fof(dt_k6_radix_5,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(k6_radix_5(A,B,C),k3_radix_1(C),k4_finseq_2(A,k3_radix_1(C))) ) ).
fof(dt_k7_radix_5,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(k7_radix_5(A,B,C),k6_wsierp_1,k3_radix_1(C)) ) ).
fof(dt_k8_radix_5,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(k8_radix_5(A,B,C),k3_radix_1(C),k4_finseq_2(A,k3_radix_1(C))) ) ).
%------------------------------------------------------------------------------