SET007 Axioms: SET007+820.ax
%------------------------------------------------------------------------------
% File : SET007+820 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Fundamental Theorem of Arithmetic
% 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 : nat_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 111 ( 2 unt; 0 def)
% Number of atoms : 655 ( 81 equ)
% Maximal formula atoms : 13 ( 5 avg)
% Number of connectives : 631 ( 87 ~; 4 |; 311 &)
% ( 10 <=>; 219 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 41 ( 40 usr; 0 prp; 1-3 aty)
% Number of functors : 55 ( 55 usr; 6 con; 0-3 aty)
% Number of variables : 247 ( 244 !; 3 ?)
% 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_nat_3,axiom,
! [A] :
( v1_xboole_0(A)
=> ( v1_xboole_0(k1_card_1(A))
& v1_relat_1(k1_card_1(A))
& v1_funct_1(k1_card_1(A))
& v2_funct_1(k1_card_1(A))
& v1_finset_1(k1_card_1(A))
& v1_card_1(k1_card_1(A))
& v1_membered(k1_card_1(A))
& v2_membered(k1_card_1(A))
& v3_membered(k1_card_1(A))
& v4_membered(k1_card_1(A))
& v5_membered(k1_card_1(A))
& v1_polynom1(k1_card_1(A)) ) ) ).
fof(cc1_nat_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v7_seqm_3(A) )
=> ( v1_relat_1(A)
& v1_seq_1(A) ) ) ).
fof(rc1_nat_3,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_seq_1(A)
& v7_seqm_3(A)
& v3_facirc_1(A) ) ).
fof(fc2_nat_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A)
& v4_ordinal2(B) )
=> ( ~ v1_xboole_0(k2_newton(A,B))
& v4_ordinal2(k2_newton(A,B))
& v1_xreal_0(k2_newton(A,B))
& v2_xreal_0(k2_newton(A,B))
& ~ v3_xreal_0(k2_newton(A,B))
& v1_int_1(k2_newton(A,B))
& v1_xcmplx_0(k2_newton(A,B)) ) ) ).
fof(cc2_nat_3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_int_2(A)
=> ( ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xreal_0(A)
& v2_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_int_1(A)
& v1_xcmplx_0(A)
& v1_int_2(A) ) ) ) ).
fof(fc3_nat_3,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_seq_1(A)
& v4_ordinal2(B) )
=> ( v1_relat_1(k1_nat_3(A,B))
& v1_funct_1(k1_nat_3(A,B))
& v1_finseq_1(k1_nat_3(A,B))
& v1_seq_1(k1_nat_3(A,B)) ) ) ).
fof(fc4_nat_3,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v7_seqm_3(A)
& v4_ordinal2(B) )
=> ( v1_relat_1(k1_nat_3(A,B))
& v1_funct_1(k1_nat_3(A,B))
& v1_finseq_1(k1_nat_3(A,B))
& v1_seq_1(k1_nat_3(A,B))
& v7_seqm_3(k1_nat_3(A,B))
& v3_facirc_1(k1_nat_3(A,B)) ) ) ).
fof(rc2_nat_3,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B)
& v7_seqm_3(B)
& v1_polynom1(B)
& v3_facirc_1(B) ) ).
fof(fc5_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& m1_pboole(B,A)
& v4_ordinal2(C) )
=> ( v1_relat_1(k4_nat_3(A,B,C))
& v1_funct_1(k4_nat_3(A,B,C))
& v1_seq_1(k4_nat_3(A,B,C)) ) ) ).
fof(fc6_nat_3,axiom,
! [A,B,C] :
( ( v7_seqm_3(B)
& m1_pboole(B,A)
& v4_ordinal2(C) )
=> ( v1_relat_1(k4_nat_3(A,B,C))
& v1_funct_1(k4_nat_3(A,B,C))
& v1_seq_1(k4_nat_3(A,B,C))
& v7_seqm_3(k4_nat_3(A,B,C))
& v3_facirc_1(k4_nat_3(A,B,C)) ) ) ).
fof(fc7_nat_3,axiom,
! [A,B] :
( ( v1_seq_1(B)
& m1_pboole(B,A) )
=> ( v1_xboole_0(k11_polynom1(k4_nat_3(A,B,np__0)))
& v1_relat_1(k11_polynom1(k4_nat_3(A,B,np__0)))
& v1_funct_1(k11_polynom1(k4_nat_3(A,B,np__0)))
& v2_funct_1(k11_polynom1(k4_nat_3(A,B,np__0)))
& v1_finset_1(k11_polynom1(k4_nat_3(A,B,np__0)))
& v1_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
& v2_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
& v3_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
& v4_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
& v5_membered(k11_polynom1(k4_nat_3(A,B,np__0)))
& v1_polynom1(k11_polynom1(k4_nat_3(A,B,np__0))) ) ) ).
fof(fc8_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& v1_polynom1(B)
& m1_pboole(B,A)
& v4_ordinal2(C) )
=> ( v1_relat_1(k4_nat_3(A,B,C))
& v1_funct_1(k4_nat_3(A,B,C))
& v1_seq_1(k4_nat_3(A,B,C))
& v1_polynom1(k4_nat_3(A,B,C)) ) ) ).
fof(fc9_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& m1_pboole(B,A)
& v1_seq_1(C)
& m1_pboole(C,A) )
=> ( v1_relat_1(k5_nat_3(A,B,C))
& v1_funct_1(k5_nat_3(A,B,C))
& v1_seq_1(k5_nat_3(A,B,C)) ) ) ).
fof(fc10_nat_3,axiom,
! [A,B,C] :
( ( v7_seqm_3(B)
& m1_pboole(B,A)
& v7_seqm_3(C)
& m1_pboole(C,A) )
=> ( v1_relat_1(k5_nat_3(A,B,C))
& v1_funct_1(k5_nat_3(A,B,C))
& v1_seq_1(k5_nat_3(A,B,C))
& v7_seqm_3(k5_nat_3(A,B,C))
& v3_facirc_1(k5_nat_3(A,B,C)) ) ) ).
fof(fc11_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& v1_polynom1(B)
& m1_pboole(B,A)
& v1_seq_1(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ( v1_relat_1(k5_nat_3(A,B,C))
& v1_funct_1(k5_nat_3(A,B,C))
& v1_seq_1(k5_nat_3(A,B,C))
& v1_polynom1(k5_nat_3(A,B,C)) ) ) ).
fof(fc12_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& m1_pboole(B,A)
& v1_seq_1(C)
& m1_pboole(C,A) )
=> ( v1_relat_1(k6_nat_3(A,B,C))
& v1_funct_1(k6_nat_3(A,B,C))
& v1_seq_1(k6_nat_3(A,B,C)) ) ) ).
fof(fc13_nat_3,axiom,
! [A,B,C] :
( ( v7_seqm_3(B)
& m1_pboole(B,A)
& v7_seqm_3(C)
& m1_pboole(C,A) )
=> ( v1_relat_1(k6_nat_3(A,B,C))
& v1_funct_1(k6_nat_3(A,B,C))
& v1_seq_1(k6_nat_3(A,B,C))
& v7_seqm_3(k6_nat_3(A,B,C))
& v3_facirc_1(k6_nat_3(A,B,C)) ) ) ).
fof(fc14_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& v1_polynom1(B)
& m1_pboole(B,A)
& v1_seq_1(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ( v1_relat_1(k6_nat_3(A,B,C))
& v1_funct_1(k6_nat_3(A,B,C))
& v1_seq_1(k6_nat_3(A,B,C))
& v1_polynom1(k6_nat_3(A,B,C)) ) ) ).
fof(fc15_nat_3,axiom,
! [A,B,C] :
( ( v7_seqm_3(B)
& m1_pboole(B,A)
& ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> ( v1_relat_1(k9_nat_3(A,B,C))
& v1_funct_1(k9_nat_3(A,B,C))
& v1_seq_1(k9_nat_3(A,B,C))
& v7_seqm_3(k9_nat_3(A,B,C))
& v3_facirc_1(k9_nat_3(A,B,C)) ) ) ).
fof(fc16_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& v1_polynom1(B)
& m1_pboole(B,A)
& ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> ( v1_relat_1(k9_nat_3(A,B,C))
& v1_funct_1(k9_nat_3(A,B,C))
& v1_polynom1(k9_nat_3(A,B,C)) ) ) ).
fof(fc17_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_relat_1(k11_nat_3(A))
& v1_funct_1(k11_nat_3(A))
& v1_seq_1(k11_nat_3(A))
& v7_seqm_3(k11_nat_3(A))
& v3_facirc_1(k11_nat_3(A)) ) ) ).
fof(fc18_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ( v1_relat_1(k11_nat_3(A))
& v1_funct_1(k11_nat_3(A))
& v1_seq_1(k11_nat_3(A))
& v7_seqm_3(k11_nat_3(A))
& v1_polynom1(k11_nat_3(A))
& v3_facirc_1(k11_nat_3(A)) ) ) ).
fof(fc19_nat_3,axiom,
( v1_xboole_0(k11_polynom1(k11_nat_3(np__1)))
& v1_relat_1(k11_polynom1(k11_nat_3(np__1)))
& v1_funct_1(k11_polynom1(k11_nat_3(np__1)))
& v2_funct_1(k11_polynom1(k11_nat_3(np__1)))
& v1_finset_1(k11_polynom1(k11_nat_3(np__1)))
& v1_membered(k11_polynom1(k11_nat_3(np__1)))
& v2_membered(k11_polynom1(k11_nat_3(np__1)))
& v3_membered(k11_polynom1(k11_nat_3(np__1)))
& v4_membered(k11_polynom1(k11_nat_3(np__1)))
& v5_membered(k11_polynom1(k11_nat_3(np__1)))
& v1_polynom1(k11_polynom1(k11_nat_3(np__1))) ) ).
fof(fc20_nat_3,axiom,
! [A,B] :
( ( v1_int_2(A)
& m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ( ~ v1_xboole_0(k11_polynom1(k11_nat_3(k3_newton(A,B))))
& v1_finset_1(k11_polynom1(k11_nat_3(k3_newton(A,B))))
& v1_realset1(k11_polynom1(k11_nat_3(k3_newton(A,B)))) ) ) ).
fof(fc21_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m1_subset_1(A,k5_numbers) )
=> ( ~ v1_xboole_0(k11_polynom1(k11_nat_3(A)))
& v1_finset_1(k11_polynom1(k11_nat_3(A)))
& v1_realset1(k11_polynom1(k11_nat_3(A))) ) ) ).
fof(fc22_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ( v1_relat_1(k12_nat_3(A))
& v1_funct_1(k12_nat_3(A))
& v1_seq_1(k12_nat_3(A))
& v7_seqm_3(k12_nat_3(A))
& v1_polynom1(k12_nat_3(A))
& v3_facirc_1(k12_nat_3(A)) ) ) ).
fof(t1_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ( ( r1_nat_1(A,C)
& r1_nat_1(B,D) )
=> r1_nat_1(k3_xcmplx_0(A,B),k3_xcmplx_0(C,D)) ) ) ) ) ) ).
fof(t2_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,np__1)
=> r1_xreal_0(B,k2_newton(A,B)) ) ) ) ).
fof(t3_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( A != np__0
=> r1_nat_1(B,k2_newton(B,A)) ) ) ) ).
fof(t4_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ( r1_nat_1(k2_newton(C,B),D)
=> ( r1_xreal_0(B,A)
| r1_nat_1(k2_newton(C,k2_xcmplx_0(A,np__1)),D) ) ) ) ) ) ) ).
fof(t5_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( ( v1_int_2(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r1_nat_1(C,k2_newton(A,B))
=> r1_nat_1(C,A) ) ) ) ) ).
fof(t6_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( ( v1_int_2(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r1_nat_1(C,k3_newton(B,A))
=> C = B ) ) ) ) ).
fof(t7_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( m1_trees_4(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k2_relat_1(B))
=> r1_nat_1(A,k10_wsierp_1(B)) ) ) ) ).
fof(t8_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_trees_4(B,k5_numbers,k12_newton)
=> ( r1_nat_1(A,k10_wsierp_1(B))
=> r2_hidden(A,k2_relat_1(B)) ) ) ) ).
fof(d1_nat_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_seq_1(A) )
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( C = k1_nat_3(A,B)
<=> ( k3_finseq_1(C) = k3_finseq_1(A)
& ! [D] :
( r2_hidden(D,k4_finseq_1(C))
=> k1_funct_1(C,D) = k3_newton(k1_seq_1(A,D),B) ) ) ) ) ) ) ).
fof(t9_nat_3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k2_nat_3(A,np__0) = k1_newton(k3_finseq_1(A),np__1) ) ).
fof(t10_nat_3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k2_nat_3(A,np__1) = A ) ).
fof(t11_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> k2_nat_3(k6_finseq_1(k1_numbers),A) = k6_finseq_1(k1_numbers) ) ).
fof(t12_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k2_nat_3(k13_binarith(k1_numbers,B),A) = k13_binarith(k1_numbers,k3_newton(B,A)) ) ) ).
fof(t13_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> k2_nat_3(k8_finseq_1(k1_numbers,C,k13_binarith(k1_numbers,B)),A) = k8_finseq_1(k1_numbers,k2_nat_3(C,A),k2_nat_3(k13_binarith(k1_numbers,B),A)) ) ) ) ).
fof(t14_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> k16_rvsum_1(k2_nat_3(B,k2_xcmplx_0(A,np__1))) = k11_binop_2(k16_rvsum_1(k2_nat_3(B,A)),k16_rvsum_1(B)) ) ) ).
fof(t15_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> k16_rvsum_1(k2_nat_3(B,A)) = k3_newton(k16_rvsum_1(B),A) ) ) ).
fof(d2_nat_3,axiom,
! [A,B] :
( ( v1_seq_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( m1_pboole(D,A)
=> ( D = k4_nat_3(A,B,C)
<=> ! [E] : k1_funct_1(D,E) = k3_xcmplx_0(C,k1_seq_1(B,E)) ) ) ) ) ).
fof(t16_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B,C] :
( ( v1_seq_1(C)
& m1_pboole(C,B) )
=> ( A != np__0
=> k11_polynom1(C) = k11_polynom1(k4_nat_3(B,C,A)) ) ) ) ).
fof(d3_nat_3,axiom,
! [A,B] :
( ( v1_seq_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_seq_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m1_pboole(D,A)
=> ( D = k5_nat_3(A,B,C)
<=> ! [E] :
( ( r1_xreal_0(k1_seq_1(B,E),k1_seq_1(C,E))
=> k1_funct_1(D,E) = k1_seq_1(B,E) )
& ( ~ r1_xreal_0(k1_seq_1(B,E),k1_seq_1(C,E))
=> k1_funct_1(D,E) = k1_seq_1(C,E) ) ) ) ) ) ) ).
fof(t17_nat_3,axiom,
! [A,B] :
( ( v1_seq_1(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_seq_1(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> r1_tarski(k11_polynom1(k5_nat_3(A,B,C)),k2_xboole_0(k11_polynom1(B),k11_polynom1(C))) ) ) ).
fof(d4_nat_3,axiom,
! [A,B] :
( ( v1_seq_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_seq_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m1_pboole(D,A)
=> ( D = k6_nat_3(A,B,C)
<=> ! [E] :
( ( r1_xreal_0(k1_seq_1(B,E),k1_seq_1(C,E))
=> k1_funct_1(D,E) = k1_seq_1(C,E) )
& ( ~ r1_xreal_0(k1_seq_1(B,E),k1_seq_1(C,E))
=> k1_funct_1(D,E) = k1_seq_1(B,E) ) ) ) ) ) ) ).
fof(t18_nat_3,axiom,
! [A,B] :
( ( v1_seq_1(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v1_seq_1(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> r1_tarski(k11_polynom1(k6_nat_3(A,B,C)),k2_xboole_0(k11_polynom1(B),k11_polynom1(C))) ) ) ).
fof(d5_nat_3,axiom,
! [A,B] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> ! [C] :
( v4_ordinal2(C)
=> ( C = k7_nat_3(A,B)
<=> ? [D] :
( m1_trees_4(D,k1_numbers,k5_numbers)
& C = k10_wsierp_1(D)
& D = k5_relat_1(k1_uproots(k1_polynom2(A,B)),B) ) ) ) ) ).
fof(t19_nat_3,axiom,
! [A,B] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ( r1_xboole_0(k1_polynom2(A,B),k1_polynom2(A,C))
=> k8_nat_3(A,k9_polynom1(A,B,C)) = k24_binop_2(k8_nat_3(A,B),k8_nat_3(A,C)) ) ) ) ).
fof(d6_nat_3,axiom,
! [A,B] :
( ( v1_seq_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> ! [D] :
( m1_pboole(D,A)
=> ( D = k9_nat_3(A,B,C)
<=> ( k11_polynom1(D) = k11_polynom1(B)
& ! [E] : k1_funct_1(D,E) = k3_newton(k1_seq_1(B,E),C) ) ) ) ) ) ).
fof(t20_nat_3,axiom,
! [A] : k8_nat_3(A,k16_polynom1(A)) = np__1 ).
fof(d7_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( B != np__1
& A != np__0
& ~ ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k10_nat_3(A,B)
<=> ( r1_nat_1(k2_newton(B,C),A)
& ~ r1_nat_1(k2_newton(B,k23_binop_2(C,np__1)),A) ) ) ) ) ) ) ).
fof(t21_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( A != np__1
=> k10_nat_3(np__1,A) = np__0 ) ) ).
fof(t22_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__1)
=> k10_nat_3(A,A) = np__1 ) ) ).
fof(t23_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( A != np__0
& ~ r1_xreal_0(B,A)
& B != np__1
& k10_nat_3(A,B) != np__0 ) ) ) ).
fof(t24_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ~ ( A != np__1
& A != B
& k10_nat_3(B,A) != np__0 ) ) ) ).
fof(t25_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,np__1)
=> k10_nat_3(k2_newton(A,B),A) = B ) ) ) ).
fof(t26_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_nat_1(A,k2_newton(A,k10_nat_3(B,A)))
=> ( A = np__1
| B = np__0
| r1_nat_1(A,B) ) ) ) ) ).
fof(t27_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( A != np__1
=> ( ( B != np__0
& k10_nat_3(B,A) = np__0 )
<=> ~ r1_nat_1(A,B) ) ) ) ) ).
fof(t28_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> k10_nat_3(k3_xcmplx_0(B,C),A) = k23_binop_2(k10_nat_3(B,A),k10_nat_3(C,A)) ) ) ) ).
fof(t29_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> k2_wsierp_1(A,k10_nat_3(k3_xcmplx_0(B,C),A)) = k24_binop_2(k2_wsierp_1(A,k10_nat_3(B,A)),k2_wsierp_1(A,k10_nat_3(C,A))) ) ) ) ).
fof(t30_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> ( r1_nat_1(C,B)
=> r1_xreal_0(k10_nat_3(C,A),k10_nat_3(B,A)) ) ) ) ) ).
fof(t31_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> ( r1_nat_1(C,B)
=> k10_nat_3(k3_nat_1(B,C),A) = k5_binarith(k10_nat_3(B,A),k10_nat_3(C,A)) ) ) ) ) ).
fof(t32_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> k10_nat_3(k2_newton(C,A),B) = k3_xcmplx_0(A,k10_nat_3(C,B)) ) ) ) ).
fof(d8_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( m1_pboole(B,k12_newton)
=> ( B = k11_nat_3(A)
<=> ! [C] :
( ( v1_int_2(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> k1_funct_1(B,C) = k10_nat_3(A,C) ) ) ) ) ).
fof(t33_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( r2_hidden(B,k1_relat_1(k11_nat_3(A)))
=> ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) ) ) ) ).
fof(t34_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( r2_hidden(B,k11_polynom1(k11_nat_3(A)))
=> ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) ) ) ) ).
fof(t35_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( ~ r1_xreal_0(A,B)
& B != np__0
& k1_funct_1(k11_nat_3(B),A) != np__0 ) ) ) ).
fof(t36_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r2_hidden(A,k11_polynom1(k11_nat_3(B)))
=> r1_nat_1(A,B) ) ) ) ).
fof(t37_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers)
& r1_nat_1(B,A) )
=> ( v1_xboole_0(A)
| r2_hidden(B,k11_polynom1(k11_nat_3(A))) ) ) ) ) ).
fof(t38_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ~ ( r1_nat_1(A,B)
& k8_polynom1(k11_nat_3(B),A) = np__0 ) ) ) ).
fof(t39_nat_3,axiom,
r6_pboole(k12_newton,k11_nat_3(np__1),k16_polynom1(k12_newton)) ).
fof(t40_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> k8_polynom1(k11_nat_3(k3_newton(B,A)),B) = A ) ) ).
fof(t41_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> k8_polynom1(k11_nat_3(A),A) = np__1 ) ).
fof(t42_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_int_2(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ( A != np__0
=> k1_polynom2(k12_newton,k11_nat_3(k3_newton(B,A))) = k1_tarski(B) ) ) ) ).
fof(t43_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> k1_polynom2(k12_newton,k11_nat_3(A)) = k1_tarski(A) ) ).
fof(t44_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ( r2_int_2(A,B)
=> r1_xboole_0(k1_polynom2(k12_newton,k11_nat_3(A)),k1_polynom2(k12_newton,k11_nat_3(B))) ) ) ) ).
fof(t45_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> r1_tarski(k1_polynom2(k12_newton,k11_nat_3(A)),k1_polynom2(k12_newton,k11_nat_3(k3_xcmplx_0(A,B)))) ) ) ).
fof(t46_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> k1_polynom2(k12_newton,k11_nat_3(k24_binop_2(A,B))) = k4_subset_1(k12_newton,k1_polynom2(k12_newton,k11_nat_3(A)),k1_polynom2(k12_newton,k11_nat_3(B))) ) ) ).
fof(t47_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ( r2_int_2(A,B)
=> k4_card_1(k1_polynom2(k12_newton,k11_nat_3(k24_binop_2(A,B)))) = k23_binop_2(k4_card_1(k1_polynom2(k12_newton,k11_nat_3(A))),k4_card_1(k1_polynom2(k12_newton,k11_nat_3(B)))) ) ) ) ).
fof(t48_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> k1_polynom2(k12_newton,k11_nat_3(A)) = k1_polynom2(k12_newton,k11_nat_3(k2_newton(A,B))) ) ) ).
fof(t49_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> r6_pboole(k12_newton,k11_nat_3(k3_xcmplx_0(A,B)),k9_polynom1(k12_newton,k11_nat_3(A),k11_nat_3(B))) ) ) ).
fof(t50_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ( r1_nat_1(A,B)
=> r6_pboole(k12_newton,k11_nat_3(k3_nat_1(B,A)),k10_polynom1(k12_newton,k11_nat_3(B),k11_nat_3(A))) ) ) ) ).
fof(t51_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> r6_pboole(k12_newton,k11_nat_3(k2_newton(B,A)),k4_nat_3(k12_newton,k11_nat_3(B),A)) ) ) ).
fof(t52_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ( k1_polynom2(k12_newton,k11_nat_3(A)) = k1_xboole_0
=> A = np__1 ) ) ).
fof(t53_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> r6_pboole(k12_newton,k11_nat_3(k6_nat_1(B,A)),k5_nat_3(k12_newton,k11_nat_3(B),k11_nat_3(A))) ) ) ).
fof(t54_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> r6_pboole(k12_newton,k11_nat_3(k5_nat_1(B,A)),k6_nat_3(k12_newton,k11_nat_3(B),k11_nat_3(A))) ) ) ).
fof(d9_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ! [B] :
( m1_pboole(B,k12_newton)
=> ( B = k12_nat_3(A)
<=> ( k11_polynom1(B) = k1_polynom2(k12_newton,k11_nat_3(A))
& ! [C] :
( v4_ordinal2(C)
=> ( r2_hidden(C,k1_polynom2(k12_newton,k11_nat_3(A)))
=> k1_funct_1(B,C) = k2_newton(C,k10_nat_3(A,C)) ) ) ) ) ) ) ).
fof(t55_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ( k10_nat_3(B,A) = np__0
=> k8_polynom1(k12_nat_3(B),A) = np__0 ) ) ) ).
fof(t56_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ( k10_nat_3(B,A) != np__0
=> k8_polynom1(k12_nat_3(B),A) = k2_wsierp_1(A,k10_nat_3(B,A)) ) ) ) ).
fof(t57_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ( k1_polynom2(k12_newton,k12_nat_3(A)) = k1_xboole_0
=> A = np__1 ) ) ).
fof(t58_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ( r2_int_2(A,B)
=> r6_pboole(k12_newton,k12_nat_3(k24_binop_2(A,B)),k9_polynom1(k12_newton,k12_nat_3(A),k12_nat_3(B))) ) ) ) ).
fof(t59_nat_3,axiom,
! [A] :
( ( v1_int_2(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> k8_polynom1(k12_nat_3(k3_newton(A,B)),A) = k3_newton(A,B) ) ) ).
fof(t60_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> r6_pboole(k12_newton,k12_nat_3(k2_newton(A,B)),k9_nat_3(k12_newton,k12_nat_3(A),B)) ) ) ).
fof(t61_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> k8_nat_3(k12_newton,k12_nat_3(A)) = A ) ).
fof(dt_k1_nat_3,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A)
& v1_seq_1(A)
& v4_ordinal2(B) )
=> ( v1_relat_1(k1_nat_3(A,B))
& v1_funct_1(k1_nat_3(A,B))
& v1_finseq_1(k1_nat_3(A,B)) ) ) ).
fof(dt_k2_nat_3,axiom,
! [A,B] :
( ( m1_finseq_1(A,k1_numbers)
& v4_ordinal2(B) )
=> m2_finseq_1(k2_nat_3(A,B),k1_numbers) ) ).
fof(redefinition_k2_nat_3,axiom,
! [A,B] :
( ( m1_finseq_1(A,k1_numbers)
& v4_ordinal2(B) )
=> k2_nat_3(A,B) = k1_nat_3(A,B) ) ).
fof(dt_k3_nat_3,axiom,
! [A,B] :
( ( m1_finseq_1(A,k5_numbers)
& v4_ordinal2(B) )
=> m1_trees_4(k3_nat_3(A,B),k1_numbers,k5_numbers) ) ).
fof(redefinition_k3_nat_3,axiom,
! [A,B] :
( ( m1_finseq_1(A,k5_numbers)
& v4_ordinal2(B) )
=> k3_nat_3(A,B) = k1_nat_3(A,B) ) ).
fof(dt_k4_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& m1_pboole(B,A)
& v4_ordinal2(C) )
=> m1_pboole(k4_nat_3(A,B,C),A) ) ).
fof(dt_k5_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& m1_pboole(B,A)
& v1_seq_1(C)
& m1_pboole(C,A) )
=> m1_pboole(k5_nat_3(A,B,C),A) ) ).
fof(dt_k6_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& m1_pboole(B,A)
& v1_seq_1(C)
& m1_pboole(C,A) )
=> m1_pboole(k6_nat_3(A,B,C),A) ) ).
fof(dt_k7_nat_3,axiom,
! [A,B] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> v4_ordinal2(k7_nat_3(A,B)) ) ).
fof(dt_k8_nat_3,axiom,
! [A,B] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> m2_subset_1(k8_nat_3(A,B),k1_numbers,k5_numbers) ) ).
fof(redefinition_k8_nat_3,axiom,
! [A,B] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> k8_nat_3(A,B) = k7_nat_3(A,B) ) ).
fof(dt_k9_nat_3,axiom,
! [A,B,C] :
( ( v1_seq_1(B)
& m1_pboole(B,A)
& ~ v1_xboole_0(C)
& v4_ordinal2(C) )
=> m1_pboole(k9_nat_3(A,B,C),A) ) ).
fof(dt_k10_nat_3,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> m2_subset_1(k10_nat_3(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k11_nat_3,axiom,
! [A] :
( v4_ordinal2(A)
=> m1_pboole(k11_nat_3(A),k12_newton) ) ).
fof(dt_k12_nat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> m1_pboole(k12_nat_3(A),k12_newton) ) ).
%------------------------------------------------------------------------------