SET007 Axioms: SET007+677.ax
%------------------------------------------------------------------------------
% File : SET007+677 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Binomial Theorem for Algebraic Structures
% 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 : binom [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 2 unt; 0 def)
% Number of atoms : 402 ( 56 equ)
% Maximal formula atoms : 17 ( 7 avg)
% Number of connectives : 400 ( 55 ~; 0 |; 190 &)
% ( 5 <=>; 150 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-3 aty)
% Number of functors : 43 ( 43 usr; 13 con; 0-6 aty)
% Number of variables : 155 ( 149 !; 6 ?)
% 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(rc1_binom,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v2_algstr_1(A) ) ).
fof(rc2_binom,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v3_algstr_1(A) ) ).
fof(rc3_binom,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v1_binom(A) ) ).
fof(cc1_binom,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_algstr_1(A)
& v3_algstr_1(A) )
=> ( ~ v3_struct_0(A)
& v1_binom(A) ) ) ) ).
fof(cc2_binom,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v1_binom(A) )
=> ( ~ v3_struct_0(A)
& v2_algstr_1(A)
& v3_algstr_1(A) ) ) ) ).
fof(cc3_binom,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v3_algstr_1(A) )
=> ( ~ v3_struct_0(A)
& v2_algstr_1(A) ) ) ) ).
fof(cc4_binom,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v2_algstr_1(A) )
=> ( ~ v3_struct_0(A)
& v3_algstr_1(A) ) ) ) ).
fof(cc5_binom,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A) )
=> ( ~ v3_struct_0(A)
& v3_algstr_1(A) ) ) ) ).
fof(rc4_binom,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v1_binom(A) ) ).
fof(d1_binom,axiom,
$true ).
fof(d2_binom,axiom,
$true ).
fof(d3_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v1_binom(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( k2_rlvect_1(A,B,C) = k2_rlvect_1(A,B,D)
=> C = D )
& ( k2_rlvect_1(A,C,B) = k2_rlvect_1(A,D,B)
=> C = D ) ) ) ) ) ) ) ).
fof(t1_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) ) ) ).
fof(t2_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) ) ) ).
fof(t3_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k9_rlvect_1(A,k12_finseq_1(u1_struct_0(A),B)) = B ) ) ).
fof(t4_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k9_rlvect_1(A,k6_polynom1(A,C,B)) = k1_group_1(A,B,k9_rlvect_1(A,C)) ) ) ) ).
fof(t5_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k9_rlvect_1(A,k7_polynom1(A,C,B)) = k1_group_1(A,k9_rlvect_1(A,C),B) ) ) ) ).
fof(t6_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> k9_rlvect_1(A,k7_polynom1(A,C,B)) = k9_rlvect_1(A,k6_polynom1(A,C,B)) ) ) ) ).
fof(d4_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ( k4_relset_1(k5_numbers,u1_struct_0(A),B) = k4_relset_1(k5_numbers,u1_struct_0(A),C)
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(A))
=> ( D = k1_binom(A,B,C)
<=> ( k4_relset_1(k5_numbers,u1_struct_0(A),D) = k4_relset_1(k5_numbers,u1_struct_0(A),B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,E)
& r1_xreal_0(E,k3_finseq_1(D)) )
=> k4_finseq_4(k5_numbers,u1_struct_0(A),D,E) = k2_rlvect_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),B,E),k4_finseq_4(k5_numbers,u1_struct_0(A),C,E)) ) ) ) ) ) ) ) ) ) ).
fof(t7_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ( k4_relset_1(k5_numbers,u1_struct_0(A),B) = k4_relset_1(k5_numbers,u1_struct_0(A),C)
=> k9_rlvect_1(A,k1_binom(A,B,C)) = k4_rlvect_1(A,k9_rlvect_1(A,B),k9_rlvect_1(A,C)) ) ) ) ) ).
fof(d5_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_binom(A,B,C) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,C) ) ) ) ).
fof(t8_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k2_binom(A,B,np__0) = k2_group_1(A)
& k2_binom(A,B,np__1) = B ) ) ) ).
fof(t9_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_binom(A,B,k1_nat_1(C,np__1)) = k1_group_1(A,k2_binom(A,B,C),B) ) ) ) ).
fof(t10_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_binom(A,k10_group_1(A,B,C),D) = k10_group_1(A,k2_binom(A,B,D),k2_binom(A,C,D)) ) ) ) ) ).
fof(t11_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_binom(A,B,k1_nat_1(C,D)) = k1_group_1(A,k2_binom(A,B,C),k2_binom(A,B,D)) ) ) ) ) ).
fof(t12_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_binom(A,k2_binom(A,B,C),D) = k2_binom(A,B,k2_nat_1(C,D)) ) ) ) ) ).
fof(d6_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(B,k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A)) )
=> ( B = k3_binom(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,np__0,C) = k1_rlvect_1(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,k1_nat_1(D,np__1),C) = k2_rlvect_1(A,C,k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,D,C)) ) ) ) ) ) ) ).
fof(d7_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A))
& m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A)) )
=> ( B = k4_binom(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,np__0) = k1_rlvect_1(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,k1_nat_1(D,np__1)) = k2_rlvect_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,D),C) ) ) ) ) ) ) ).
fof(d8_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_binom(A,B,C) = k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),k3_binom(A),C,B) ) ) ) ).
fof(d9_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k6_binom(A,B,C) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k4_binom(A),B,C) ) ) ) ).
fof(t13_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k5_binom(A,B,np__0) = k1_rlvect_1(A)
& k6_binom(A,B,np__0) = k1_rlvect_1(A) ) ) ) ).
fof(t14_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k5_binom(A,B,np__1) = B ) ) ).
fof(t15_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_binom(A,B,np__1) = B ) ) ).
fof(t16_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k5_binom(A,B,k1_nat_1(C,D)) = k2_rlvect_1(A,k5_binom(A,B,C),k5_binom(A,B,D)) ) ) ) ) ).
fof(t17_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k6_binom(A,B,k1_nat_1(C,D)) = k2_rlvect_1(A,k6_binom(A,B,C),k6_binom(A,B,D)) ) ) ) ) ).
fof(t18_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_binom(A,B,C) = k6_binom(A,B,C) ) ) ) ).
fof(t19_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_binom(A,B,C) = k6_binom(A,B,C) ) ) ) ).
fof(t20_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_group_1(A,k5_binom(A,B,D),C) = k5_binom(A,k1_group_1(A,B,C),D) ) ) ) ) ).
fof(t21_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_group_1(A,C,k5_binom(A,B,D)) = k6_binom(A,k1_group_1(A,C,B),D) ) ) ) ) ).
fof(t22_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_algstr_1(A)
& v1_binom(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_group_1(A,k6_binom(A,B,D),C) = k1_group_1(A,B,k5_binom(A,C,D)) ) ) ) ) ).
fof(d10_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(A))
=> ( E = k8_binom(A,B,C,D)
<=> ( k3_finseq_1(E) = k1_nat_1(D,np__1)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k4_relset_1(k5_numbers,u1_struct_0(A),E))
& H = k6_xcmplx_0(F,np__1)
& G = k6_xcmplx_0(D,H) )
=> k4_finseq_4(k5_numbers,u1_struct_0(A),E,F) = k1_group_1(A,k5_binom(A,k2_binom(A,B,G),k7_binom(H,D)),k2_binom(A,C,H)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_binom(A,B,C,np__0) = k12_finseq_1(u1_struct_0(A),k2_group_1(A)) ) ) ) ).
fof(t24_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_funct_1(k8_binom(A,B,C,D),np__1) = k2_binom(A,B,D) ) ) ) ) ).
fof(t25_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_funct_1(k8_binom(A,B,C,D),k1_nat_1(D,np__1)) = k2_binom(A,C,D) ) ) ) ) ).
fof(t26_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v1_algstr_1(A)
& v1_binom(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_binom(A,k4_rlvect_1(A,B,C),D) = k9_rlvect_1(A,k8_binom(A,B,C,D)) ) ) ) ) ).
fof(s1_binom,axiom,
( ( p1_s1_binom(f1_s1_binom)
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(f1_s1_binom,A)
& p1_s1_binom(A) )
=> p1_s1_binom(k1_nat_1(A,np__1)) ) ) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(f1_s1_binom,A)
=> p1_s1_binom(A) ) ) ) ).
fof(s2_binom,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(k5_numbers,f1_s2_binom),f2_s2_binom)
& m2_relset_1(A,k2_zfmisc_1(k5_numbers,f1_s2_binom),f2_s2_binom)
& ! [B] :
( m1_subset_1(B,f1_s2_binom)
=> ( k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,np__0,B) = f3_s2_binom
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,k1_nat_1(C,np__1),B) = k2_binop_1(f1_s2_binom,f2_s2_binom,f2_s2_binom,f4_s2_binom,B,k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,C,B)) ) ) ) ) ).
fof(s3_binom,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k2_zfmisc_1(f1_s3_binom,k5_numbers),f2_s3_binom)
& m2_relset_1(A,k2_zfmisc_1(f1_s3_binom,k5_numbers),f2_s3_binom)
& ! [B] :
( m1_subset_1(B,f1_s3_binom)
=> ( k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,np__0) = f3_s3_binom
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,k1_nat_1(C,np__1)) = k2_binop_1(f2_s3_binom,f1_s3_binom,f2_s3_binom,f4_s3_binom,k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,C),B) ) ) ) ) ).
fof(dt_k1_binom,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_finseq_1(B,u1_struct_0(A))
& m1_finseq_1(C,u1_struct_0(A)) )
=> m2_finseq_1(k1_binom(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k2_binom,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k2_binom(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k3_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v1_funct_1(k3_binom(A))
& v1_funct_2(k3_binom(A),k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(k3_binom(A),k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_k4_binom,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v1_funct_1(k4_binom(A))
& v1_funct_2(k4_binom(A),k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A))
& m2_relset_1(k4_binom(A),k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A)) ) ) ).
fof(dt_k5_binom,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k5_binom(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k6_binom,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k6_binom(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k7_binom,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k7_binom(A,B),k1_numbers,k5_numbers) ) ).
fof(redefinition_k7_binom,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k7_binom(A,B) = k7_newton(A,B) ) ).
fof(dt_k8_binom,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k5_numbers) )
=> m2_finseq_1(k8_binom(A,B,C,D),u1_struct_0(A)) ) ).
%------------------------------------------------------------------------------