SET007 Axioms: SET007+271.ax
%------------------------------------------------------------------------------
% File : SET007+271 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Ordered Rings - Part I
% 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 : o_ring_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 23 ( 0 unt; 0 def)
% Number of atoms : 189 ( 33 equ)
% Maximal formula atoms : 17 ( 8 avg)
% Number of connectives : 227 ( 61 ~; 2 |; 85 &)
% ( 13 <=>; 66 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-3 aty)
% Number of functors : 12 ( 12 usr; 4 con; 0-3 aty)
% Number of variables : 69 ( 62 !; 7 ?)
% 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(d1_o_ring_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(C,k3_finseq_1(B))
=> ( np__0 = C
| k1_o_ring_1(A,B,C) = k1_funct_1(B,C) ) ) ) ) ) ).
fof(d2_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_o_ring_1(A,B) = k1_group_1(A,B,B) ) ) ).
fof(d3_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_o_ring_1(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B = k2_o_ring_1(A,C) ) ) ) ) ).
fof(d4_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( v2_o_ring_1(B,A)
<=> ( k3_finseq_1(B) != np__0
& v1_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,np__1),A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( C != np__0
& ~ r1_xreal_0(k3_finseq_1(B),C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( v1_o_ring_1(D,A)
& k1_o_ring_1(u1_struct_0(A),B,k1_nat_1(C,np__1)) = k2_rlvect_1(A,k1_o_ring_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ) ) ) ).
fof(d5_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v3_o_ring_1(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_struct_0(A))
& v2_o_ring_1(C,A)
& B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) ).
fof(d6_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( v4_o_ring_1(B,A)
<=> ( k3_finseq_1(B) != np__0
& v1_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,np__1),A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( C != np__0
& ~ r1_xreal_0(k3_finseq_1(B),C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( v1_o_ring_1(D,A)
& k1_o_ring_1(u1_struct_0(A),B,k1_nat_1(C,np__1)) = k1_group_1(A,k1_o_ring_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ) ) ) ).
fof(d7_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v5_o_ring_1(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_struct_0(A))
& v4_o_ring_1(C,A)
& B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) ).
fof(d8_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( v6_o_ring_1(B,A)
<=> ( k3_finseq_1(B) != np__0
& v5_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,np__1),A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( C != np__0
& ~ r1_xreal_0(k3_finseq_1(B),C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( v5_o_ring_1(D,A)
& k1_o_ring_1(u1_struct_0(A),B,k1_nat_1(C,np__1)) = k2_rlvect_1(A,k1_o_ring_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ) ) ) ).
fof(d9_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v7_o_ring_1(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_struct_0(A))
& v6_o_ring_1(C,A)
& B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) ).
fof(d10_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( v8_o_ring_1(B,A)
<=> ( k3_finseq_1(B) != np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( C != np__0
& r1_xreal_0(C,k3_finseq_1(B))
& ~ v5_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,C),A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( k1_o_ring_1(u1_struct_0(A),B,C) = k1_group_1(A,k1_o_ring_1(u1_struct_0(A),B,D),k1_o_ring_1(u1_struct_0(A),B,E))
& D != np__0
& ~ r1_xreal_0(C,D)
& E != np__0
& ~ r1_xreal_0(C,E) ) ) ) ) ) ) ) ) ) ).
fof(d11_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v9_o_ring_1(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_struct_0(A))
& v8_o_ring_1(C,A)
& B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) ).
fof(d12_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( v10_o_ring_1(B,A)
<=> ( k3_finseq_1(B) != np__0
& v9_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,np__1),A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( C != np__0
& ~ r1_xreal_0(k3_finseq_1(B),C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( v9_o_ring_1(D,A)
& k1_o_ring_1(u1_struct_0(A),B,k1_nat_1(C,np__1)) = k2_rlvect_1(A,k1_o_ring_1(u1_struct_0(A),B,C),D) ) ) ) ) ) ) ) ) ).
fof(d13_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v11_o_ring_1(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_struct_0(A))
& v10_o_ring_1(C,A)
& B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) ).
fof(d14_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( v12_o_ring_1(B,A)
<=> ( k3_finseq_1(B) != np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( C != np__0
& r1_xreal_0(C,k3_finseq_1(B))
& ~ v9_o_ring_1(k1_o_ring_1(u1_struct_0(A),B,C),A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( ( k1_o_ring_1(u1_struct_0(A),B,C) = k1_group_1(A,k1_o_ring_1(u1_struct_0(A),B,D),k1_o_ring_1(u1_struct_0(A),B,E))
| k1_o_ring_1(u1_struct_0(A),B,C) = k2_rlvect_1(A,k1_o_ring_1(u1_struct_0(A),B,D),k1_o_ring_1(u1_struct_0(A),B,E)) )
& D != np__0
& ~ r1_xreal_0(C,D)
& E != np__0
& ~ r1_xreal_0(C,E) ) ) ) ) ) ) ) ) ) ).
fof(d15_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v13_o_ring_1(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_struct_0(A))
& v12_o_ring_1(C,A)
& B = k1_o_ring_1(u1_struct_0(A),C,k3_finseq_1(C)) ) ) ) ) ).
fof(t1_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_o_ring_1(B,A)
=> ( v3_o_ring_1(B,A)
& v5_o_ring_1(B,A)
& v7_o_ring_1(B,A)
& v9_o_ring_1(B,A)
& v11_o_ring_1(B,A)
& v13_o_ring_1(B,A) ) ) ) ) ).
fof(t2_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v3_o_ring_1(B,A)
=> ( v7_o_ring_1(B,A)
& v11_o_ring_1(B,A)
& v13_o_ring_1(B,A) ) ) ) ) ).
fof(t3_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v5_o_ring_1(B,A)
=> ( v7_o_ring_1(B,A)
& v9_o_ring_1(B,A)
& v11_o_ring_1(B,A)
& v13_o_ring_1(B,A) ) ) ) ) ).
fof(t4_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v7_o_ring_1(B,A)
=> ( v11_o_ring_1(B,A)
& v13_o_ring_1(B,A) ) ) ) ) ).
fof(t5_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v9_o_ring_1(B,A)
=> ( v11_o_ring_1(B,A)
& v13_o_ring_1(B,A) ) ) ) ) ).
fof(t6_o_ring_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v11_o_ring_1(B,A)
=> v13_o_ring_1(B,A) ) ) ) ).
fof(dt_k1_o_ring_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,A)
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k1_o_ring_1(A,B,C),A) ) ).
fof(dt_k2_o_ring_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k2_o_ring_1(A,B),u1_struct_0(A)) ) ).
%------------------------------------------------------------------------------