SET007 Axioms: SET007+702.ax
%------------------------------------------------------------------------------
% File : SET007+702 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Polynomials with Coefficients in a Ring of Polynomials
% 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 : polynom6 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 25 ( 0 unt; 0 def)
% Number of atoms : 297 ( 28 equ)
% Maximal formula atoms : 31 ( 11 avg)
% Number of connectives : 312 ( 40 ~; 1 |; 148 &)
% ( 6 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 16 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 43 ( 42 usr; 0 prp; 1-3 aty)
% Number of functors : 30 ( 30 usr; 4 con; 0-5 aty)
% Number of variables : 125 ( 119 !; 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(fc1_polynom6,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B)
& ~ v1_xboole_0(B) )
=> ( v3_ordinal1(k14_ordinal2(A,B))
& ~ v1_xboole_0(k14_ordinal2(A,B)) ) ) ).
fof(fc2_polynom6,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v3_ordinal1(B)
& ~ v1_xboole_0(B) )
=> ( v3_ordinal1(k14_ordinal2(B,A))
& ~ v1_xboole_0(k14_ordinal2(B,A)) ) ) ).
fof(fc3_polynom6,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ( ~ v3_struct_0(k30_polynom1(A,B))
& v3_rlvect_1(k30_polynom1(A,B))
& v4_rlvect_1(k30_polynom1(A,B))
& v5_rlvect_1(k30_polynom1(A,B))
& v6_rlvect_1(k30_polynom1(A,B))
& v2_group_1(k30_polynom1(A,B))
& v4_group_1(k30_polynom1(A,B))
& v3_vectsp_1(k30_polynom1(A,B))
& v4_vectsp_1(k30_polynom1(A,B))
& v5_vectsp_1(k30_polynom1(A,B))
& v6_vectsp_1(k30_polynom1(A,B))
& v7_vectsp_1(k30_polynom1(A,B))
& v8_vectsp_1(k30_polynom1(A,B))
& ~ v3_realset2(k30_polynom1(A,B))
& v1_algstr_1(k30_polynom1(A,B))
& v2_algstr_1(k30_polynom1(A,B))
& v3_algstr_1(k30_polynom1(A,B))
& v4_algstr_1(k30_polynom1(A,B))
& v5_algstr_1(k30_polynom1(A,B))
& v6_algstr_1(k30_polynom1(A,B)) ) ) ).
fof(t1_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v3_ordinal1(E)
& D = k14_ordinal2(A,E)
& r2_hidden(E,B) ) )
=> k14_ordinal2(A,B) = k2_xboole_0(A,C) ) ) ) ).
fof(t2_polynom6,axiom,
! [A] :
( v3_ordinal1(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_polynom1(A,B,C)
& ! [D] :
( v3_ordinal1(D)
=> ~ ( r2_hidden(D,A)
& ~ r1_xreal_0(k8_polynom1(C,D),k8_polynom1(B,D))
& ! [E] :
( v3_ordinal1(E)
=> ( r2_hidden(E,D)
=> k8_polynom1(B,E) = k8_polynom1(C,E) ) ) ) ) ) ) ) ) ).
fof(d1_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,B,k14_polynom1(B))
=> ! [E] :
( m1_polynom1(E,k14_ordinal2(A,B),k14_polynom1(k14_ordinal2(A,B)))
=> ( E = k1_polynom6(A,B,C,D)
<=> ! [F] :
( v3_ordinal1(F)
=> ( ( r2_hidden(F,A)
=> k8_polynom1(E,F) = k8_polynom1(C,F) )
& ( r2_hidden(F,k4_xboole_0(k14_ordinal2(A,B),A))
=> k8_polynom1(E,F) = k8_polynom1(D,k5_ordinal3(F,A)) ) ) ) ) ) ) ) ) ) ).
fof(t3_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,B,k14_polynom1(B))
=> ( B = k1_xboole_0
=> k1_polynom6(A,B,C,D) = C ) ) ) ) ) ).
fof(t4_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,B,k14_polynom1(B))
=> ( A = k1_xboole_0
=> k1_polynom6(A,B,C,D) = D ) ) ) ) ) ).
fof(t5_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,B,k14_polynom1(B))
=> ( r6_pboole(k14_ordinal2(A,B),k1_polynom6(A,B,C,D),k16_polynom1(k14_ordinal2(A,B)))
<=> ( r6_pboole(A,C,k16_polynom1(A))
& r6_pboole(B,D,k16_polynom1(B)) ) ) ) ) ) ) ).
fof(t6_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,k14_ordinal2(A,B),k14_polynom1(k14_ordinal2(A,B)))
=> ? [D] :
( m1_polynom1(D,A,k14_polynom1(A))
& ? [E] :
( m1_polynom1(E,B,k14_polynom1(B))
& r6_pboole(k14_ordinal2(A,B),C,k1_polynom6(A,B,D,E)) ) ) ) ) ) ).
fof(t7_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,A,k14_polynom1(A))
=> ! [E] :
( m1_polynom1(E,B,k14_polynom1(B))
=> ! [F] :
( m1_polynom1(F,B,k14_polynom1(B))
=> ( r6_pboole(k14_ordinal2(A,B),k1_polynom6(A,B,C,E),k1_polynom6(A,B,D,F))
=> ( r6_pboole(A,C,D)
& r6_pboole(B,E,F) ) ) ) ) ) ) ) ) ).
fof(t8_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v7_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] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(B)) )
=> k28_polynom1(A,B,k23_polynom1(A,B,C,D),E) = k23_polynom1(A,B,k28_polynom1(A,B,C,E),k28_polynom1(A,B,D,E)) ) ) ) ) ) ).
fof(d2_polynom6,axiom,
! [A] :
( ( v3_ordinal1(A)
& ~ v1_xboole_0(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& ~ v1_xboole_0(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v2_group_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(k30_polynom1(B,C)))
& v2_polynom1(D,k14_polynom1(A),k30_polynom1(B,C))
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(k30_polynom1(B,C))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(k14_ordinal2(A,B)),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(k14_ordinal2(A,B)),C)
& m2_relset_1(E,k14_polynom1(k14_ordinal2(A,B)),u1_struct_0(C)) )
=> ( E = k2_polynom6(A,B,C,D)
<=> ! [F] :
( m1_polynom1(F,k14_ordinal2(A,B),k14_polynom1(k14_ordinal2(A,B)))
=> ? [G] :
( m1_polynom1(G,A,k14_polynom1(A))
& ? [H] :
( m1_polynom1(H,B,k14_polynom1(B))
& ? [I] :
( v1_funct_1(I)
& v1_funct_2(I,k14_polynom1(B),u1_struct_0(C))
& v2_polynom1(I,k14_polynom1(B),C)
& m2_relset_1(I,k14_polynom1(B),u1_struct_0(C))
& I = k15_polynom1(A,k30_polynom1(B,C),D,G)
& r6_pboole(k14_ordinal2(A,B),F,k1_polynom6(A,B,G,H))
& k15_polynom1(k14_ordinal2(A,B),C,E,F) = k15_polynom1(B,C,I,H) ) ) ) ) ) ) ) ) ) ) ).
fof(t9_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,A,k14_polynom1(A))
=> ! [E] :
( m1_polynom1(E,B,k14_polynom1(B))
=> ! [F] :
( m1_polynom1(F,B,k14_polynom1(B))
=> ( ( r3_polynom1(A,C,D)
& r3_polynom1(B,E,F) )
=> r3_polynom1(k14_ordinal2(A,B),k1_polynom6(A,B,C,E),k1_polynom6(A,B,D,F)) ) ) ) ) ) ) ) ).
fof(t10_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,k14_ordinal2(A,B)) )
=> ! [D] :
( m1_polynom1(D,A,k14_polynom1(A))
=> ! [E] :
( m1_polynom1(E,B,k14_polynom1(B))
=> ~ ( r3_polynom1(k14_ordinal2(A,B),C,k1_polynom6(A,B,D,E))
& ! [F] :
( m1_polynom1(F,A,k14_polynom1(A))
=> ! [G] :
( m1_polynom1(G,B,k14_polynom1(B))
=> ~ ( r3_polynom1(A,F,D)
& r3_polynom1(B,G,E)
& r6_pboole(k14_ordinal2(A,B),C,k1_polynom6(A,B,F,G)) ) ) ) ) ) ) ) ) ) ).
fof(t11_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,A,k14_polynom1(A))
=> ! [E] :
( m1_polynom1(E,B,k14_polynom1(B))
=> ! [F] :
( m1_polynom1(F,B,k14_polynom1(B))
=> ( r1_polynom1(k14_ordinal2(A,B),k1_polynom6(A,B,C,E),k1_polynom6(A,B,D,F))
<=> ( r1_polynom1(A,C,D)
| ( r6_pboole(A,C,D)
& r1_polynom1(B,E,F) ) ) ) ) ) ) ) ) ) ).
fof(t12_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,B,k14_polynom1(B))
=> ! [E] :
( m2_finseq_1(E,k3_finseq_2(k14_polynom1(k14_ordinal2(A,B))))
=> ( ( k4_finseq_1(E) = k4_finseq_1(k20_polynom1(A,C))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k4_finseq_1(k20_polynom1(A,C)))
& ! [G] :
( m1_polynom1(G,A,k14_polynom1(A))
=> ! [H] :
( m2_finseq_1(H,k14_polynom1(k14_ordinal2(A,B)))
=> ~ ( H = k1_matrlin(k14_polynom1(k14_ordinal2(A,B)),k5_numbers,k3_finseq_2(k14_polynom1(k14_ordinal2(A,B))),E,F)
& k4_finseq_4(k5_numbers,k13_polynom1(A),k20_polynom1(A,C),F) = G
& k3_finseq_1(H) = k3_finseq_1(k20_polynom1(B,D))
& ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(I,k4_finseq_1(H))
& ! [J] :
( m1_polynom1(J,B,k14_polynom1(B))
=> ~ ( k4_finseq_4(k5_numbers,k13_polynom1(B),k20_polynom1(B,D),I) = J
& k4_finseq_4(k5_numbers,k14_polynom1(k14_ordinal2(A,B)),H,I) = k1_polynom6(A,B,G,J) ) ) ) ) ) ) ) ) ) )
=> k20_polynom1(k14_ordinal2(A,B),k1_polynom6(A,B,C,D)) = k15_dtconstr(k14_polynom1(k14_ordinal2(A,B)),E) ) ) ) ) ) ) ).
fof(t13_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,A,k14_polynom1(A))
=> ! [E] :
( m1_polynom1(E,A,k14_polynom1(A))
=> ! [F] :
( m1_polynom1(F,B,k14_polynom1(B))
=> ! [G] :
( m1_polynom1(G,B,k14_polynom1(B))
=> ! [H] :
( m1_polynom1(H,B,k14_polynom1(B))
=> ( ( r6_pboole(A,E,k10_polynom1(A,D,C))
& r6_pboole(B,H,k10_polynom1(B,G,F)) )
=> r6_pboole(k14_ordinal2(A,B),k10_polynom1(k14_ordinal2(A,B),k1_polynom6(A,B,D,G),k1_polynom6(A,B,C,F)),k1_polynom6(A,B,E,H)) ) ) ) ) ) ) ) ) ) ).
fof(t14_polynom6,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_polynom1(C,A,k14_polynom1(A))
=> ! [D] :
( m1_polynom1(D,B,k14_polynom1(B))
=> ! [E] :
( m2_finseq_1(E,k3_finseq_2(k4_finseq_2(np__2,k14_polynom1(k14_ordinal2(A,B)))))
=> ( ( k4_finseq_1(E) = k4_finseq_1(k21_polynom1(A,C))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k4_finseq_1(k21_polynom1(A,C)))
& ! [G] :
( m1_polynom1(G,A,k14_polynom1(A))
=> ! [H] :
( m1_polynom1(H,A,k14_polynom1(A))
=> ! [I] :
( m2_finseq_1(I,k4_finseq_2(np__2,k14_polynom1(k14_ordinal2(A,B))))
=> ~ ( I = k1_matrlin(k4_finseq_2(np__2,k14_polynom1(k14_ordinal2(A,B))),k5_numbers,k3_finseq_2(k4_finseq_2(np__2,k14_polynom1(k14_ordinal2(A,B)))),E,F)
& k1_matrlin(k14_polynom1(A),k5_numbers,k4_finseq_2(np__2,k14_polynom1(A)),k21_polynom1(A,C),F) = k2_polynom3(k14_polynom1(A),G,H)
& k3_finseq_1(I) = k3_finseq_1(k21_polynom1(B,D))
& ! [J] :
( m2_subset_1(J,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(J,k4_finseq_1(I))
& ! [K] :
( m1_polynom1(K,B,k14_polynom1(B))
=> ! [L] :
( m1_polynom1(L,B,k14_polynom1(B))
=> ~ ( k1_matrlin(k14_polynom1(B),k5_numbers,k4_finseq_2(np__2,k14_polynom1(B)),k21_polynom1(B,D),J) = k2_polynom3(k14_polynom1(B),K,L)
& k1_matrlin(k14_polynom1(k14_ordinal2(A,B)),k5_numbers,k4_finseq_2(np__2,k14_polynom1(k14_ordinal2(A,B))),I,J) = k2_polynom3(k14_polynom1(k14_ordinal2(A,B)),k1_polynom6(A,B,G,K),k1_polynom6(A,B,H,L)) ) ) ) ) ) ) ) ) ) ) ) )
=> k21_polynom1(k14_ordinal2(A,B),k1_polynom6(A,B,C,D)) = k15_dtconstr(k4_finseq_2(np__2,k14_polynom1(k14_ordinal2(A,B))),E) ) ) ) ) ) ) ).
fof(t15_polynom6,axiom,
! [A] :
( ( v3_ordinal1(A)
& ~ v1_xboole_0(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& ~ v1_xboole_0(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> r2_quofield(k30_polynom1(A,k30_polynom1(B,C)),k30_polynom1(k14_ordinal2(A,B),C)) ) ) ) ).
fof(symmetry_r1_polynom6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ( r1_polynom6(A,B)
=> r1_polynom6(B,A) ) ) ).
fof(reflexivity_r1_polynom6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> r1_polynom6(A,A) ) ).
fof(redefinition_r1_polynom6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ( r1_polynom6(A,B)
<=> r2_quofield(A,B) ) ) ).
fof(dt_k1_polynom6,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(A)
& v3_ordinal1(B)
& m1_subset_1(C,k14_polynom1(A))
& m1_subset_1(D,k14_polynom1(B)) )
=> m1_polynom1(k1_polynom6(A,B,C,D),k14_ordinal2(A,B),k14_polynom1(k14_ordinal2(A,B))) ) ).
fof(dt_k2_polynom6,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(A)
& ~ v1_xboole_0(A)
& v3_ordinal1(B)
& ~ v1_xboole_0(B)
& ~ v3_struct_0(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v2_group_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C)
& v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(k30_polynom1(B,C)))
& v2_polynom1(D,k14_polynom1(A),k30_polynom1(B,C))
& m1_relset_1(D,k14_polynom1(A),u1_struct_0(k30_polynom1(B,C))) )
=> ( v1_funct_1(k2_polynom6(A,B,C,D))
& v1_funct_2(k2_polynom6(A,B,C,D),k14_polynom1(k14_ordinal2(A,B)),u1_struct_0(C))
& v2_polynom1(k2_polynom6(A,B,C,D),k14_polynom1(k14_ordinal2(A,B)),C)
& m2_relset_1(k2_polynom6(A,B,C,D),k14_polynom1(k14_ordinal2(A,B)),u1_struct_0(C)) ) ) ).
%------------------------------------------------------------------------------