SET007 Axioms: SET007+778.ax
%------------------------------------------------------------------------------
% File : SET007+778 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Construction of Gr"obner bases.
% Version : [Urb08] axioms.
% English : Construction of Gr"obner bases. S-Polynomials and Standard
% Representations
% 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 : groeb_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 69 ( 3 unt; 0 def)
% Number of atoms : 1354 ( 42 equ)
% Maximal formula atoms : 48 ( 19 avg)
% Number of connectives : 1390 ( 105 ~; 7 |; 951 &)
% ( 13 <=>; 314 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 15 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 65 ( 63 usr; 1 prp; 0-7 aty)
% Number of functors : 45 ( 45 usr; 3 con; 0-5 aty)
% Number of variables : 343 ( 335 !; 8 ?)
% 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_groeb_2,axiom,
! [A,B] :
( ( v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& ~ v3_realset2(A)
& l1_rlvect_1(A)
& v8_rlvect_1(B,A)
& m1_subset_1(B,u1_struct_0(A)) )
=> v8_rlvect_1(k5_rlvect_1(A,B),A) ) ).
fof(fc2_groeb_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v5_rlvect_1(B)
& v7_vectsp_1(B)
& v1_binom(B)
& v1_algstr_1(B)
& l3_vectsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_relat_1(k5_polynom7(A,B,C,D))
& v1_funct_1(k5_polynom7(A,B,C,D))
& v1_funct_2(k5_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(k5_polynom7(A,B,C,D),k14_polynom1(A),B)
& v3_polynom7(k5_polynom7(A,B,C,D),A,B) ) ) ).
fof(fc3_groeb_2,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(A)
& v5_rlvect_1(B)
& v7_vectsp_1(B)
& v1_binom(B)
& v1_algstr_1(B)
& ~ v3_realset2(B)
& v2_vectsp_2(B)
& l3_vectsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& v1_polynom7(C,A,B)
& m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
& v8_rlvect_1(D,B)
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_relat_1(k5_polynom7(A,B,C,D))
& v1_funct_1(k5_polynom7(A,B,C,D))
& v1_funct_2(k5_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(k5_polynom7(A,B,C,D),k14_polynom1(A),B)
& v1_polynom7(k5_polynom7(A,B,C,D),A,B) ) ) ).
fof(fc4_groeb_2,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(A)
& v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m1_relset_1(B,k14_polynom1(A),k14_polynom1(A))
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C)
& v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> v1_finset_1(k4_groeb_2(A,B,C,D)) ) ).
fof(t1_groeb_2,axiom,
$true ).
fof(t2_groeb_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(D,C)
=> k16_finseq_1(u1_struct_0(A),k16_finseq_1(u1_struct_0(A),B,C),D) = k16_finseq_1(u1_struct_0(A),B,D) ) ) ) ) ) ).
fof(t3_groeb_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> ( r1_xreal_0(D,C)
| k1_funct_1(B,D) = k1_rlvect_1(A) ) ) )
=> k9_rlvect_1(A,B) = k9_rlvect_1(A,k16_finseq_1(u1_struct_0(A),B,C)) ) ) ) ) ).
fof(t4_groeb_2,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_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k9_rlvect_1(A,k2_finseq_7(u1_struct_0(A),B,C,D)) = k9_rlvect_1(A,B) ) ) ) ) ).
fof(d1_groeb_2,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) )
=> ( r3_polynom1(A,C,B)
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( D = k1_groeb_2(A,B,C)
<=> r6_pboole(A,k9_polynom1(A,C,D),B) ) ) ) ) ) ).
fof(d2_groeb_2,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) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( D = k2_groeb_2(A,B,C)
<=> ! [E] : k8_polynom1(D,E) = k4_square_1(k8_polynom1(B,E),k8_polynom1(C,E)) ) ) ) ) ).
fof(d3_groeb_2,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_groeb_2(A,B,C)
<=> ! [D] :
( k8_polynom1(B,D) = np__0
| k8_polynom1(C,D) = np__0 ) ) ) ) ).
fof(t5_groeb_2,axiom,
$true ).
fof(t6_groeb_2,axiom,
$true ).
fof(t7_groeb_2,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) )
=> ( r3_polynom1(A,B,k2_groeb_2(A,B,C))
& r3_polynom1(A,C,k2_groeb_2(A,B,C)) ) ) ) ).
fof(t8_groeb_2,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) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( ( r3_polynom1(A,B,D)
& r3_polynom1(A,C,D) )
=> r3_polynom1(A,k2_groeb_2(A,B,C),D) ) ) ) ) ).
fof(t9_groeb_2,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_groeb_2(A,B,C)
<=> r6_pboole(A,k2_groeb_2(A,B,C),k9_polynom1(A,B,C)) ) ) ) ).
fof(t10_groeb_2,axiom,
! [A,B] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> r6_pboole(A,k1_groeb_2(A,B,B),k16_polynom1(A)) ) ).
fof(t11_groeb_2,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) )
=> ( r3_polynom1(A,C,B)
<=> r6_pboole(A,k2_groeb_2(A,B,C),B) ) ) ) ).
fof(t12_groeb_2,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) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( r3_polynom1(A,B,k2_groeb_2(A,C,D))
=> r3_polynom1(A,k2_groeb_2(A,C,B),k2_groeb_2(A,C,D)) ) ) ) ) ).
fof(t13_groeb_2,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) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( r3_polynom1(A,k2_groeb_2(A,C,B),k2_groeb_2(A,C,D))
=> r3_polynom1(A,k2_groeb_2(A,B,D),k2_groeb_2(A,C,D)) ) ) ) ) ).
fof(t14_groeb_2,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) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( r3_polynom1(A,k2_groeb_2(A,B,D),k2_groeb_2(A,C,D))
=> r3_polynom1(A,B,k2_groeb_2(A,C,D)) ) ) ) ) ).
fof(t15_groeb_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k14_polynom1(A))) )
=> ? [D] :
( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A)
& r2_hidden(D,C)
& ! [E] :
( ( v7_seqm_3(E)
& v1_polynom1(E)
& m1_pboole(E,A) )
=> ( r2_hidden(E,C)
=> r1_termord(A,B,D,E) ) ) ) ) ) ) ).
fof(t16_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v5_rlvect_1(B)
& v4_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) )
=> ! [E] :
( ( v7_seqm_3(E)
& v1_polynom1(E)
& m1_pboole(E,A) )
=> k1_polyred(A,E,B,k22_polynom1(A,B,C,D)) = k22_polynom1(A,B,k1_polyred(A,E,B,C),k1_polyred(A,E,B,D)) ) ) ) ) ) ).
fof(t17_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l1_rlvect_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> k1_polyred(A,D,B,k24_polynom1(A,B,C)) = k24_polynom1(A,B,k1_polyred(A,D,B,C)) ) ) ) ) ).
fof(t18_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_vectsp_1(B)
& v1_algstr_1(B)
& v3_algstr_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> k1_polyred(A,D,B,k5_polynom7(A,B,C,E)) = k5_polynom7(A,B,k1_polyred(A,D,B,C),E) ) ) ) ) ) ).
fof(t19_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) )
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> k5_polynom7(A,B,k22_polynom1(A,B,C,D),E) = k22_polynom1(A,B,k5_polynom7(A,B,C,E),k5_polynom7(A,B,D,E)) ) ) ) ) ) ).
fof(t20_groeb_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k24_polynom1(A,B,k4_polynom7(A,B,C)) = k4_polynom7(A,B,k5_rlvect_1(B,C)) ) ) ).
fof(t21_groeb_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( ( r2_hidden(E,D)
& r2_hidden(F,D) )
=> ( E = F
| ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
& v3_polynom7(G,A,C)
& m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k14_polynom1(A),u1_struct_0(C))
& v3_polynom7(H,A,C)
& m2_relset_1(H,k14_polynom1(A),u1_struct_0(C)) )
=> ( k5_termord(A,B,C,k29_polynom1(A,C,G,E)) = k5_termord(A,B,C,k29_polynom1(A,C,H,F))
=> r1_rewrite1(k3_polyred(A,B,C,D),k25_polynom1(A,C,k29_polynom1(A,C,G,E),k29_polynom1(A,C,H,F)),k26_polynom1(A,C)) ) ) ) ) ) ) )
=> ( r2_hidden(k26_polynom1(A,C),D)
| r1_groeb_1(A,B,C,D) ) ) ) ) ) ) ).
fof(d4_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> k3_groeb_2(A,B,C,D,E) = k25_polynom1(A,C,k5_polynom7(A,C,k1_polyred(A,k1_groeb_2(A,k2_groeb_2(A,k3_termord(A,B,C,D),k3_termord(A,B,C,E)),k3_termord(A,B,C,D)),C,D),k4_termord(A,B,C,E)),k5_polynom7(A,C,k1_polyred(A,k1_groeb_2(A,k2_groeb_2(A,k3_termord(A,B,C,D),k3_termord(A,B,C,E)),k3_termord(A,B,C,E)),C,E),k4_termord(A,B,C,D))) ) ) ) ) ) ).
fof(t22_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( ( r2_hidden(E,D)
& r2_hidden(F,D) )
=> r2_hidden(k3_groeb_2(A,B,C,E,F),k7_ideal_1(k30_polynom1(A,C),D)) ) ) ) ) ) ) ) ).
fof(t23_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( D = E
=> k3_groeb_2(A,B,C,D,E) = k26_polynom1(A,C) ) ) ) ) ) ) ).
fof(t24_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v3_polynom7(D,A,C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v3_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> k3_groeb_2(A,B,C,D,E) = k26_polynom1(A,C) ) ) ) ) ) ).
fof(t25_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ( k3_groeb_2(A,B,C,D,k26_polynom1(A,C)) = k26_polynom1(A,C)
& k3_groeb_2(A,B,C,k26_polynom1(A,C),D) = k26_polynom1(A,C) ) ) ) ) ) ).
fof(t26_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( k3_groeb_2(A,B,C,D,E) = k26_polynom1(A,C)
| r2_termord(A,B,k3_termord(A,B,C,k3_groeb_2(A,B,C,D,E)),k2_groeb_2(A,k3_termord(A,B,C,D),k3_termord(A,B,C,E))) ) ) ) ) ) ) ).
fof(t27_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& v1_polynom7(D,A,C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& v1_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r3_polynom1(A,k3_termord(A,B,C,E),k3_termord(A,B,C,D))
=> r8_polyred(A,B,C,k5_polynom7(A,C,D,k4_termord(A,B,C,E)),E,k3_groeb_2(A,B,C,D,E)) ) ) ) ) ) ) ).
fof(t28_groeb_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( r1_groeb_1(A,B,C,D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(G,k14_polynom1(A),C)
& m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
=> ( ( r2_hidden(E,D)
& r2_hidden(F,D)
& r4_rewrite1(k3_polyred(A,B,C,D),k3_groeb_2(A,B,C,E,F),G) )
=> G = k26_polynom1(A,C) ) ) ) ) ) ) ) ) ) ).
fof(t29_groeb_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(G,k14_polynom1(A),C)
& m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
=> ( ( r2_hidden(E,D)
& r2_hidden(F,D)
& r4_rewrite1(k3_polyred(A,B,C,D),k3_groeb_2(A,B,C,E,F),G) )
=> G = k26_polynom1(A,C) ) ) ) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( ( r2_hidden(E,D)
& r2_hidden(F,D) )
=> r1_rewrite1(k3_polyred(A,B,C,D),k3_groeb_2(A,B,C,E,F),k26_polynom1(A,C)) ) ) ) ) ) ) ) ) ).
fof(t30_groeb_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( ( r2_hidden(E,D)
& r2_hidden(F,D) )
=> r1_rewrite1(k3_polyred(A,B,C,D),k3_groeb_2(A,B,C,E,F),k26_polynom1(A,C)) ) ) )
=> ( r2_hidden(k26_polynom1(A,C),D)
| r1_groeb_1(A,B,C,D) ) ) ) ) ) ) ).
fof(t31_groeb_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,D)
=> ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v3_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) ) )
=> ( r2_hidden(k26_polynom1(A,C),D)
| r1_groeb_1(A,B,C,D) ) ) ) ) ) ) ).
fof(t32_groeb_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_ideal_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> m2_ideal_1(k16_finseq_1(u1_struct_0(A),C,D),A,B) ) ) ) ) ).
fof(t33_groeb_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_ideal_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> m2_ideal_1(k1_rfinseq(u1_struct_0(A),C,D),A,B) ) ) ) ) ).
fof(t34_groeb_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m2_ideal_1(D,A,B)
=> ( r1_tarski(B,C)
=> m2_ideal_1(D,A,C) ) ) ) ) ) ).
fof(d6_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,B)))) )
=> ! [E] :
( m2_ideal_1(E,k30_polynom1(A,B),D)
=> ( r2_groeb_2(A,B,C,D,E)
<=> ( k9_rlvect_1(k30_polynom1(A,B),E) = C
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k4_finseq_1(E))
& ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(G,A,B)
& m2_relset_1(G,k14_polynom1(A),u1_struct_0(B)) )
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(H,k14_polynom1(A),B)
& m2_relset_1(H,k14_polynom1(A),u1_struct_0(B)) )
=> ~ ( r2_hidden(H,D)
& k4_finseq_4(k5_numbers,u1_struct_0(k30_polynom1(A,B)),E,F) = k28_polynom1(A,B,G,H) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t36_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(D,k14_polynom1(A),B)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,B)))) )
=> ! [F] :
( m2_ideal_1(F,k30_polynom1(A,B),E)
=> ! [G] :
( m2_ideal_1(G,k30_polynom1(A,B),E)
=> ( ( r2_groeb_2(A,B,C,E,F)
& r2_groeb_2(A,B,D,E,G) )
=> r2_groeb_2(A,B,k22_polynom1(A,B,C,D),E,k5_groeb_2(A,B,E,F,G)) ) ) ) ) ) ) ) ) ).
fof(d7_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [F] :
( m2_ideal_1(F,k30_polynom1(A,C),E)
=> ! [G] :
( ( v7_seqm_3(G)
& v1_polynom1(G)
& m1_pboole(G,A) )
=> ( r3_groeb_2(A,B,C,D,E,F,G)
<=> ( k9_rlvect_1(k30_polynom1(A,C),F) = D
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(H,k4_finseq_1(F))
& ! [I] :
( ( v1_funct_1(I)
& v1_funct_2(I,k14_polynom1(A),u1_struct_0(C))
& v1_polynom7(I,A,C)
& v3_polynom7(I,A,C)
& m2_relset_1(I,k14_polynom1(A),u1_struct_0(C)) )
=> ! [J] :
( ( v1_funct_1(J)
& v1_funct_2(J,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(J,k14_polynom1(A),C)
& v1_polynom7(J,A,C)
& m2_relset_1(J,k14_polynom1(A),u1_struct_0(C)) )
=> ~ ( r2_hidden(J,E)
& k4_finseq_4(k5_numbers,u1_struct_0(k30_polynom1(A,C)),F,H) = k28_polynom1(A,C,I,J)
& r1_termord(A,B,k3_termord(A,B,C,k28_polynom1(A,C,I,J)),G) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d8_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [F] :
( m2_ideal_1(F,k30_polynom1(A,C),E)
=> ( r4_groeb_2(A,B,C,D,E,F)
<=> r3_groeb_2(A,B,C,D,E,F,k3_termord(A,B,C,D)) ) ) ) ) ) ) ) ).
fof(d9_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> ( r5_groeb_2(A,B,C,D,E,F)
<=> ? [G] :
( m2_ideal_1(G,k30_polynom1(A,C),E)
& r3_groeb_2(A,B,C,D,E,G,F) ) ) ) ) ) ) ) ) ).
fof(d10_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( r6_groeb_2(A,B,C,D,E)
<=> ? [F] :
( m2_ideal_1(F,k30_polynom1(A,C),E)
& r4_groeb_2(A,B,C,D,E,F) ) ) ) ) ) ) ) ).
fof(t37_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [F] :
( m2_ideal_1(F,k30_polynom1(A,C),E)
=> ! [G] :
( ( v7_seqm_3(G)
& v1_polynom1(G)
& m1_pboole(G,A) )
=> ( r3_groeb_2(A,B,C,D,E,F,G)
=> r2_groeb_2(A,C,D,E,F) ) ) ) ) ) ) ) ) ).
fof(t38_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [G] :
( m2_ideal_1(G,k30_polynom1(A,C),F)
=> ! [H] :
( m2_ideal_1(H,k30_polynom1(A,C),F)
=> ! [I] :
( ( v7_seqm_3(I)
& v1_polynom1(I)
& m1_pboole(I,A) )
=> ( ( r3_groeb_2(A,B,C,D,F,G,I)
& r3_groeb_2(A,B,C,E,F,H,I) )
=> r3_groeb_2(A,B,C,k22_polynom1(A,C,D,E),F,k5_groeb_2(A,C,F,G,H),I) ) ) ) ) ) ) ) ) ) ) ).
fof(t39_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [G] :
( m2_ideal_1(G,k30_polynom1(A,C),F)
=> ! [H] :
( m2_ideal_1(H,k30_polynom1(A,C),F)
=> ! [I] :
( ( v7_seqm_3(I)
& v1_polynom1(I)
& m1_pboole(I,A) )
=> ! [J] :
( m2_subset_1(J,k1_numbers,k5_numbers)
=> ( ( r3_groeb_2(A,B,C,D,F,G,I)
& H = k16_finseq_1(u1_struct_0(k30_polynom1(A,C)),G,J)
& E = k9_rlvect_1(k30_polynom1(A,C),k1_rfinseq(u1_struct_0(k30_polynom1(A,C)),G,J)) )
=> r3_groeb_2(A,B,C,k25_polynom1(A,C,D,E),F,H,I) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t40_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [G] :
( m2_ideal_1(G,k30_polynom1(A,C),F)
=> ! [H] :
( m2_ideal_1(H,k30_polynom1(A,C),F)
=> ! [I] :
( ( v7_seqm_3(I)
& v1_polynom1(I)
& m1_pboole(I,A) )
=> ! [J] :
( m2_subset_1(J,k1_numbers,k5_numbers)
=> ( ( r3_groeb_2(A,B,C,D,F,G,I)
& H = k1_rfinseq(u1_struct_0(k30_polynom1(A,C)),G,J)
& E = k9_rlvect_1(k30_polynom1(A,C),k16_finseq_1(u1_struct_0(k30_polynom1(A,C)),G,J))
& r1_xreal_0(J,k3_finseq_1(G)) )
=> r3_groeb_2(A,B,C,k25_polynom1(A,C,D,E),F,H,I) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t41_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& v1_polynom7(D,A,C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [F] :
( m2_ideal_1(F,k30_polynom1(A,C),E)
=> ~ ( r2_groeb_2(A,C,D,E,F)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k14_polynom1(A),u1_struct_0(C))
& v1_polynom7(H,A,C)
& v3_polynom7(H,A,C)
& m2_relset_1(H,k14_polynom1(A),u1_struct_0(C)) )
=> ! [I] :
( ( v1_funct_1(I)
& v1_funct_2(I,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(I,k14_polynom1(A),C)
& v1_polynom7(I,A,C)
& m2_relset_1(I,k14_polynom1(A),u1_struct_0(C)) )
=> ~ ( r2_hidden(G,k4_finseq_1(F))
& r2_hidden(I,E)
& k1_funct_1(F,G) = k28_polynom1(A,C,H,I)
& r1_termord(A,B,k3_termord(A,B,C,D),k3_termord(A,B,C,k28_polynom1(A,C,H,I))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t42_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& v1_polynom7(D,A,C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [F] :
( m2_ideal_1(F,k30_polynom1(A,C),E)
=> ~ ( r4_groeb_2(A,B,C,D,E,F)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k14_polynom1(A),u1_struct_0(C))
& v1_polynom7(H,A,C)
& v3_polynom7(H,A,C)
& m2_relset_1(H,k14_polynom1(A),u1_struct_0(C)) )
=> ! [I] :
( ( v1_funct_1(I)
& v1_funct_2(I,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(I,k14_polynom1(A),C)
& v1_polynom7(I,A,C)
& m2_relset_1(I,k14_polynom1(A),u1_struct_0(C)) )
=> ~ ( r2_hidden(I,E)
& r2_hidden(G,k4_finseq_1(F))
& k4_finseq_4(k5_numbers,u1_struct_0(k30_polynom1(A,C)),F,G) = k28_polynom1(A,C,H,I)
& r6_pboole(A,k3_termord(A,B,C,D),k3_termord(A,B,C,k28_polynom1(A,C,H,I))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t43_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( r1_rewrite1(k3_polyred(A,B,C,E),D,k26_polynom1(A,C))
=> r6_groeb_2(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t44_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& v1_polynom7(D,A,C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( r6_groeb_2(A,B,C,D,E)
=> r10_polyred(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t45_groeb_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( r1_groeb_1(A,B,C,D)
<=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& v1_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
=> r6_groeb_2(A,B,C,E,D) ) ) ) ) ) ) ) ).
fof(dt_k1_groeb_2,axiom,
! [A,B,C] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A)
& v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ( v7_seqm_3(k1_groeb_2(A,B,C))
& v1_polynom1(k1_groeb_2(A,B,C))
& m1_pboole(k1_groeb_2(A,B,C),A) ) ) ).
fof(dt_k2_groeb_2,axiom,
! [A,B,C] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A)
& v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ( v7_seqm_3(k2_groeb_2(A,B,C))
& v1_polynom1(k2_groeb_2(A,B,C))
& m1_pboole(k2_groeb_2(A,B,C),A) ) ) ).
fof(commutativity_k2_groeb_2,axiom,
! [A,B,C] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A)
& v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> k2_groeb_2(A,B,C) = k2_groeb_2(A,C,B) ) ).
fof(idempotence_k2_groeb_2,axiom,
! [A,B,C] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A)
& v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> k2_groeb_2(A,B,B) = B ) ).
fof(dt_k3_groeb_2,axiom,
! [A,B,C,D,E] :
( ( v3_ordinal1(A)
& v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m1_relset_1(B,k14_polynom1(A),k14_polynom1(A))
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C)
& v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m1_relset_1(D,k14_polynom1(A),u1_struct_0(C))
& v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m1_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( v1_funct_1(k3_groeb_2(A,B,C,D,E))
& v1_funct_2(k3_groeb_2(A,B,C,D,E),k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(k3_groeb_2(A,B,C,D,E),k14_polynom1(A),C)
& m2_relset_1(k3_groeb_2(A,B,C,D,E),k14_polynom1(A),u1_struct_0(C)) ) ) ).
fof(dt_k4_groeb_2,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(A)
& v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m1_relset_1(B,k14_polynom1(A),k14_polynom1(A))
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> m1_subset_1(k4_groeb_2(A,B,C,D),k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) ).
fof(dt_k5_groeb_2,axiom,
! [A,B,C,D,E] :
( ( v3_ordinal1(A)
& ~ v3_struct_0(B)
& v2_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,B))))
& m2_ideal_1(D,k30_polynom1(A,B),C)
& m2_ideal_1(E,k30_polynom1(A,B),C) )
=> m2_ideal_1(k5_groeb_2(A,B,C,D,E),k30_polynom1(A,B),C) ) ).
fof(redefinition_k5_groeb_2,axiom,
! [A,B,C,D,E] :
( ( v3_ordinal1(A)
& ~ v3_struct_0(B)
& v2_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,B))))
& m2_ideal_1(D,k30_polynom1(A,B),C)
& m2_ideal_1(E,k30_polynom1(A,B),C) )
=> k5_groeb_2(A,B,C,D,E) = k7_finseq_1(D,E) ) ).
fof(d5_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> k4_groeb_2(A,B,C,D) = a_4_0_groeb_2(A,B,C,D) ) ) ) ) ).
fof(t35_groeb_2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,B)))) )
=> ! [E] :
( m2_ideal_1(E,k30_polynom1(A,B),D)
=> ( r2_groeb_2(A,B,C,D,E)
=> r1_tarski(k12_polynom1(k14_polynom1(A),B,C),k3_tarski(a_4_1_groeb_2(A,B,D,E))) ) ) ) ) ) ) ).
fof(fraenkel_a_4_0_groeb_2,axiom,
! [A,B,C,D,E] :
( ( v3_ordinal1(B)
& v1_partfun1(C,k14_polynom1(B),k14_polynom1(B))
& v1_relat_2(C)
& v4_relat_2(C)
& v6_relat_2(C)
& v8_relat_2(C)
& m2_relset_1(C,k14_polynom1(B),k14_polynom1(B))
& v2_group_1(D)
& v4_group_1(D)
& v7_group_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_vectsp_1(D)
& v9_vectsp_1(D)
& ~ v3_realset2(D)
& l3_vectsp_1(D)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(B,D)))) )
=> ( r2_hidden(A,a_4_0_groeb_2(B,C,D,E))
<=> ? [F,G] :
( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(B),u1_struct_0(D))
& v2_polynom1(F,k14_polynom1(B),D)
& m2_relset_1(F,k14_polynom1(B),u1_struct_0(D))
& v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(B),u1_struct_0(D))
& v2_polynom1(G,k14_polynom1(B),D)
& m2_relset_1(G,k14_polynom1(B),u1_struct_0(D))
& A = k3_groeb_2(B,C,D,F,G)
& r2_hidden(F,E)
& r2_hidden(G,E) ) ) ) ).
fof(fraenkel_a_4_1_groeb_2,axiom,
! [A,B,C,D,E] :
( ( v3_ordinal1(B)
& ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C)
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(B,C))))
& m2_ideal_1(E,k30_polynom1(B,C),D) )
=> ( r2_hidden(A,a_4_1_groeb_2(B,C,D,E))
<=> ? [F,G] :
( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(B),u1_struct_0(C))
& v3_polynom7(F,B,C)
& m2_relset_1(F,k14_polynom1(B),u1_struct_0(C))
& v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(B),u1_struct_0(C))
& v2_polynom1(G,k14_polynom1(B),C)
& m2_relset_1(G,k14_polynom1(B),u1_struct_0(C))
& A = k12_polynom1(k14_polynom1(B),C,k28_polynom1(B,C,F,G))
& ? [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
& r2_hidden(H,k4_finseq_1(E))
& k4_finseq_4(k5_numbers,u1_struct_0(k30_polynom1(B,C)),E,H) = k28_polynom1(B,C,F,G) ) ) ) ) ).
%------------------------------------------------------------------------------