SET007 Axioms: SET007+765.ax
%------------------------------------------------------------------------------
% File : SET007+765 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Notions and Properties of Orthoposets
% 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 : oposet_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 152 ( 35 unt; 0 def)
% Number of atoms : 800 ( 21 equ)
% Maximal formula atoms : 26 ( 5 avg)
% Number of connectives : 784 ( 136 ~; 0 |; 458 &)
% ( 24 <=>; 166 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 69 ( 67 usr; 1 prp; 0-3 aty)
% Number of functors : 20 ( 20 usr; 4 con; 0-4 aty)
% Number of variables : 154 ( 131 !; 23 ?)
% 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_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& v1_oposet_1(A) ) ).
fof(rc2_oposet_1,axiom,
! [A] :
? [B] :
( m1_relset_1(B,A,A)
& v1_relat_1(B)
& v2_relat_2(B)
& v5_relat_2(B)
& v8_relat_2(B) ) ).
fof(fc1_oposet_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> ( ~ v3_struct_0(g1_oposet_1(A,B,C))
& v1_oposet_1(g1_oposet_1(A,B,C)) ) ) ).
fof(rc3_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v1_oposet_1(A) ) ).
fof(rc4_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ? [B] :
( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
& v2_oposet_1(B,u1_struct_0(A)) ) ) ).
fof(fc2_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1) ) ).
fof(fc3_oposet_1,axiom,
( ~ v3_struct_0(k4_oposet_1)
& v1_oposet_1(k4_oposet_1) ) ).
fof(fc4_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1)
& v3_oposet_1(k3_oposet_1) ) ).
fof(rc5_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v3_oposet_1(A) ) ).
fof(fc5_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v2_orders_2(k3_oposet_1)
& v3_orders_2(k3_oposet_1)
& v4_orders_2(k3_oposet_1)
& v1_lattice3(k3_oposet_1)
& v2_lattice3(k3_oposet_1)
& v3_lattice3(k3_oposet_1)
& v1_yellow_0(k3_oposet_1)
& v2_yellow_0(k3_oposet_1)
& v3_yellow_0(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1)
& v3_oposet_1(k3_oposet_1) ) ).
fof(rc6_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v1_oposet_1(A) ) ).
fof(cc1_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_necklace(A) )
=> ( ~ v3_struct_0(A)
& v5_oposet_1(A) ) ) ) ).
fof(fc6_oposet_1,axiom,
( ~ v3_struct_0(k4_oposet_1)
& v3_necklace(k4_oposet_1)
& v1_oposet_1(k4_oposet_1)
& v5_oposet_1(k4_oposet_1) ) ).
fof(rc7_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v3_necklace(A)
& v1_oposet_1(A)
& v5_oposet_1(A) ) ).
fof(cc2_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v1_necklace(A) )
=> ( ~ v3_struct_0(A)
& v6_oposet_1(A) ) ) ) ).
fof(rc8_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v1_necklace(A)
& v1_oposet_1(A)
& v6_oposet_1(A) ) ).
fof(cc3_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_orders_2(A) )
=> ( ~ v3_struct_0(A)
& v7_oposet_1(A) ) ) ) ).
fof(fc7_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v2_orders_2(k3_oposet_1)
& v3_orders_2(k3_oposet_1)
& v4_orders_2(k3_oposet_1)
& v1_lattice3(k3_oposet_1)
& v2_lattice3(k3_oposet_1)
& v3_lattice3(k3_oposet_1)
& v1_yellow_0(k3_oposet_1)
& v2_yellow_0(k3_oposet_1)
& v3_yellow_0(k3_oposet_1)
& v1_necklace(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1)
& v3_oposet_1(k3_oposet_1)
& v6_oposet_1(k3_oposet_1)
& v7_oposet_1(k3_oposet_1) ) ).
fof(rc9_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v4_orders_2(A)
& v1_necklace(A)
& v1_oposet_1(A)
& v6_oposet_1(A)
& v7_oposet_1(A) ) ).
fof(cc4_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v8_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v2_necklace(A) ) ) ) ).
fof(fc8_oposet_1,axiom,
( ~ v3_struct_0(k4_oposet_1)
& v2_necklace(k4_oposet_1)
& v3_necklace(k4_oposet_1)
& v1_oposet_1(k4_oposet_1)
& v5_oposet_1(k4_oposet_1)
& v8_oposet_1(k4_oposet_1) ) ).
fof(rc10_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v2_necklace(A)
& v1_oposet_1(A)
& v8_oposet_1(A) ) ).
fof(cc5_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_orders_2(A) )
=> ( ~ v3_struct_0(A)
& v9_oposet_1(A) ) ) ) ).
fof(rc11_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_necklace(A)
& v1_oposet_1(A)
& v6_oposet_1(A)
& v7_oposet_1(A)
& v9_oposet_1(A) ) ).
fof(fc9_oposet_1,axiom,
( ~ v3_struct_0(k4_oposet_1)
& v3_orders_2(k4_oposet_1)
& v2_necklace(k4_oposet_1)
& v3_necklace(k4_oposet_1)
& v1_oposet_1(k4_oposet_1)
& v5_oposet_1(k4_oposet_1)
& v8_oposet_1(k4_oposet_1)
& v9_oposet_1(k4_oposet_1) ) ).
fof(rc12_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v3_orders_2(A)
& v2_necklace(A)
& v3_necklace(A)
& v1_oposet_1(A)
& v5_oposet_1(A)
& v8_oposet_1(A)
& v9_oposet_1(A) ) ).
fof(cc6_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v6_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v1_necklace(A)
& v6_oposet_1(A) ) ) ) ).
fof(cc7_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v7_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v4_orders_2(A)
& v7_oposet_1(A) ) ) ) ).
fof(cc8_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v9_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v3_orders_2(A)
& v9_oposet_1(A) ) ) ) ).
fof(cc9_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_necklace(A)
& v9_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v3_orders_2(A)
& v9_oposet_1(A) ) ) ) ).
fof(cc10_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_necklace(A)
& v3_necklace(A) )
=> ( ~ v3_struct_0(A)
& v2_necklace(A)
& v8_oposet_1(A) ) ) ) ).
fof(cc11_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v11_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v10_oposet_1(A) ) ) ) ).
fof(fc10_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v2_orders_2(k3_oposet_1)
& v3_orders_2(k3_oposet_1)
& v4_orders_2(k3_oposet_1)
& v1_lattice3(k3_oposet_1)
& v2_lattice3(k3_oposet_1)
& v3_lattice3(k3_oposet_1)
& v1_yellow_0(k3_oposet_1)
& v2_yellow_0(k3_oposet_1)
& v3_yellow_0(k3_oposet_1)
& v1_necklace(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1)
& v3_oposet_1(k3_oposet_1)
& v6_oposet_1(k3_oposet_1)
& v7_oposet_1(k3_oposet_1)
& v9_oposet_1(k3_oposet_1)
& v10_oposet_1(k3_oposet_1)
& v11_oposet_1(k3_oposet_1) ) ).
fof(rc13_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v1_oposet_1(A)
& v3_oposet_1(A)
& v10_oposet_1(A)
& v11_oposet_1(A)
& v12_oposet_1(A) ) ).
fof(fc11_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v2_orders_2(k3_oposet_1)
& v3_orders_2(k3_oposet_1)
& v4_orders_2(k3_oposet_1)
& v1_lattice3(k3_oposet_1)
& v2_lattice3(k3_oposet_1)
& v3_lattice3(k3_oposet_1)
& v1_yellow_0(k3_oposet_1)
& v2_yellow_0(k3_oposet_1)
& v3_yellow_0(k3_oposet_1)
& v1_necklace(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1)
& v3_oposet_1(k3_oposet_1)
& v6_oposet_1(k3_oposet_1)
& v7_oposet_1(k3_oposet_1)
& v9_oposet_1(k3_oposet_1)
& v10_oposet_1(k3_oposet_1)
& v11_oposet_1(k3_oposet_1)
& v12_oposet_1(k3_oposet_1) ) ).
fof(cc12_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v13_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v14_oposet_1(A) ) ) ) ).
fof(cc13_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v14_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v13_oposet_1(A) ) ) ) ).
fof(cc14_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v14_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v7_oposet_1(A)
& v9_oposet_1(A) ) ) ) ).
fof(cc15_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A) )
=> ( ~ v3_struct_0(A)
& v13_oposet_1(A)
& v14_oposet_1(A) ) ) ) ).
fof(rc14_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_oposet_1(A)
& v3_oposet_1(A)
& v7_oposet_1(A)
& v9_oposet_1(A)
& v13_oposet_1(A)
& v14_oposet_1(A)
& v15_oposet_1(A) ) ).
fof(fc12_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v2_orders_2(k3_oposet_1)
& v3_orders_2(k3_oposet_1)
& v4_orders_2(k3_oposet_1)
& v1_lattice3(k3_oposet_1)
& v2_lattice3(k3_oposet_1)
& v3_lattice3(k3_oposet_1)
& v1_yellow_0(k3_oposet_1)
& v2_yellow_0(k3_oposet_1)
& v3_yellow_0(k3_oposet_1)
& v1_necklace(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1)
& v3_oposet_1(k3_oposet_1)
& v6_oposet_1(k3_oposet_1)
& v7_oposet_1(k3_oposet_1)
& v9_oposet_1(k3_oposet_1)
& v10_oposet_1(k3_oposet_1)
& v11_oposet_1(k3_oposet_1)
& v12_oposet_1(k3_oposet_1)
& v13_oposet_1(k3_oposet_1)
& v14_oposet_1(k3_oposet_1)
& v15_oposet_1(k3_oposet_1) ) ).
fof(cc16_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v17_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v16_oposet_1(A) ) ) ) ).
fof(cc17_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v16_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v5_oposet_1(A) ) ) ) ).
fof(cc18_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_necklace(A)
& v16_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v5_oposet_1(A)
& v16_oposet_1(A)
& v17_oposet_1(A) ) ) ) ).
fof(cc19_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v17_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v3_necklace(A)
& v5_oposet_1(A) ) ) ) ).
fof(fc13_oposet_1,axiom,
( ~ v3_struct_0(k4_oposet_1)
& v3_orders_2(k4_oposet_1)
& v2_necklace(k4_oposet_1)
& v3_necklace(k4_oposet_1)
& v1_oposet_1(k4_oposet_1)
& v5_oposet_1(k4_oposet_1)
& v8_oposet_1(k4_oposet_1)
& v9_oposet_1(k4_oposet_1)
& v16_oposet_1(k4_oposet_1)
& v17_oposet_1(k4_oposet_1) ) ).
fof(rc15_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v3_necklace(A)
& v1_oposet_1(A)
& v5_oposet_1(A)
& v16_oposet_1(A)
& v17_oposet_1(A) ) ).
fof(cc20_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( ( ~ v3_struct_0(A)
& v14_oposet_1(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v7_oposet_1(A)
& v9_oposet_1(A)
& v13_oposet_1(A)
& v14_oposet_1(A) ) ) ) ).
fof(fc14_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v2_orders_2(k3_oposet_1)
& v3_orders_2(k3_oposet_1)
& v4_orders_2(k3_oposet_1)
& v1_lattice3(k3_oposet_1)
& v2_lattice3(k3_oposet_1)
& v3_lattice3(k3_oposet_1)
& v1_yellow_0(k3_oposet_1)
& v2_yellow_0(k3_oposet_1)
& v3_yellow_0(k3_oposet_1)
& v1_necklace(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1)
& v3_oposet_1(k3_oposet_1)
& v6_oposet_1(k3_oposet_1)
& v7_oposet_1(k3_oposet_1)
& v9_oposet_1(k3_oposet_1)
& v10_oposet_1(k3_oposet_1)
& v11_oposet_1(k3_oposet_1)
& v12_oposet_1(k3_oposet_1)
& v13_oposet_1(k3_oposet_1)
& v14_oposet_1(k3_oposet_1)
& v15_oposet_1(k3_oposet_1)
& v19_oposet_1(k3_oposet_1) ) ).
fof(rc16_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v7_oposet_1(A)
& v9_oposet_1(A)
& v13_oposet_1(A)
& v14_oposet_1(A)
& v15_oposet_1(A)
& v19_oposet_1(A) ) ).
fof(fc15_oposet_1,axiom,
( ~ v3_struct_0(k3_oposet_1)
& v2_orders_2(k3_oposet_1)
& v3_orders_2(k3_oposet_1)
& v4_orders_2(k3_oposet_1)
& v1_lattice3(k3_oposet_1)
& v2_lattice3(k3_oposet_1)
& v3_lattice3(k3_oposet_1)
& v1_yellow_0(k3_oposet_1)
& v2_yellow_0(k3_oposet_1)
& v3_yellow_0(k3_oposet_1)
& v1_necklace(k3_oposet_1)
& v3_realset2(k3_oposet_1)
& v1_oposet_1(k3_oposet_1)
& v3_oposet_1(k3_oposet_1)
& v6_oposet_1(k3_oposet_1)
& v7_oposet_1(k3_oposet_1)
& v9_oposet_1(k3_oposet_1)
& v10_oposet_1(k3_oposet_1)
& v11_oposet_1(k3_oposet_1)
& v12_oposet_1(k3_oposet_1)
& v13_oposet_1(k3_oposet_1)
& v14_oposet_1(k3_oposet_1)
& v15_oposet_1(k3_oposet_1)
& v19_oposet_1(k3_oposet_1)
& v20_oposet_1(k3_oposet_1)
& v21_oposet_1(k3_oposet_1) ) ).
fof(rc17_oposet_1,axiom,
? [A] :
( l1_oposet_1(A)
& ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v7_oposet_1(A)
& v9_oposet_1(A)
& v13_oposet_1(A)
& v14_oposet_1(A)
& v20_oposet_1(A)
& v21_oposet_1(A) ) ).
fof(d1_oposet_1,axiom,
! [A,B] : k1_oposet_1(A,B) = k1_xboole_0 ).
fof(d2_oposet_1,axiom,
! [A,B] : k2_oposet_1(A,B) = k2_zfmisc_1(A,B) ).
fof(t1_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k3_relat_1(k6_partfun1(A)) = A ) ).
fof(t2_oposet_1,axiom,
$true ).
fof(t3_oposet_1,axiom,
k7_vectsp_2 = k1_tarski(k4_tarski(k1_xboole_0,k1_xboole_0)) ).
fof(t4_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r3_orders_2(A,B,C)
=> ( k1_yellow_0(A,k2_struct_0(A,B,C)) = C
& k2_yellow_0(A,k2_struct_0(A,B,C)) = B ) ) ) ) ) ).
fof(t5_oposet_1,axiom,
$true ).
fof(t6_oposet_1,axiom,
! [A,B] : k3_relat_1(k1_oposet_1(A,B)) = k1_xboole_0 ).
fof(t7_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( r1_relat_2(B,A)
=> ( v1_relat_2(B)
& k3_relat_1(B) = A ) ) ) ) ).
fof(t8_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( r3_relat_2(B,A)
=> v3_relat_2(B) ) ) ) ).
fof(t9_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,A)
=> ( ( v3_relat_2(C)
& r1_tarski(k3_relat_1(C),B) )
=> r3_relat_2(C,B) ) ) ) ) ).
fof(t10_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,A)
=> ( ( v4_relat_2(C)
& r1_tarski(k3_relat_1(C),B) )
=> r4_relat_2(C,B) ) ) ) ) ).
fof(t11_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( r4_relat_2(B,A)
=> v4_relat_2(B) ) ) ) ).
fof(t12_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,A)
=> ( ( v8_relat_2(C)
& r1_tarski(k3_relat_1(C),B) )
=> r8_relat_2(C,B) ) ) ) ) ).
fof(t13_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( r8_relat_2(B,A)
=> v8_relat_2(B) ) ) ) ).
fof(t14_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,A)
=> ( ( v5_relat_2(C)
& r1_tarski(k3_relat_1(C),B) )
=> r5_relat_2(C,B) ) ) ) ) ).
fof(t15_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( r5_relat_2(B,A)
=> v5_relat_2(B) ) ) ) ).
fof(t16_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,A)
=> ( ( v2_relat_2(C)
& r1_tarski(k3_relat_1(C),B) )
=> r2_relat_2(C,B) ) ) ) ) ).
fof(t17_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( r2_relat_2(B,A)
=> v2_relat_2(B) ) ) ) ).
fof(d3_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ( v2_oposet_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> k8_funct_2(A,A,B,k8_funct_2(A,A,B,C)) = C ) ) ) ) ).
fof(t18_oposet_1,axiom,
$true ).
fof(t19_oposet_1,axiom,
v2_oposet_1(k7_vectsp_2,k1_tarski(k1_xboole_0)) ).
fof(t20_oposet_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> v2_oposet_1(k6_partfun1(A),A) ) ).
fof(d4_oposet_1,axiom,
k3_oposet_1 = g1_oposet_1(k1_tarski(k1_xboole_0),k6_partfun1(k1_tarski(k1_xboole_0)),k7_vectsp_2) ).
fof(d5_oposet_1,axiom,
k4_oposet_1 = g1_oposet_1(k1_tarski(k1_xboole_0),k1_oposet_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0)),k7_vectsp_2) ).
fof(d6_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v3_oposet_1(A)
<=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& B = u1_robbins1(A)
& v2_oposet_1(B,u1_struct_0(A)) ) ) ) ).
fof(t21_oposet_1,axiom,
v3_oposet_1(k3_oposet_1) ).
fof(d7_oposet_1,axiom,
$true ).
fof(d8_oposet_1,axiom,
$true ).
fof(d9_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v4_oposet_1(A)
<=> v1_relat_2(u1_orders_2(A)) ) ) ).
fof(t22_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v2_orders_2(A)
=> v4_oposet_1(A) ) ) ).
fof(t23_oposet_1,axiom,
v2_orders_2(k3_oposet_1) ).
fof(d10_oposet_1,axiom,
$true ).
fof(d11_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v5_oposet_1(A)
<=> v2_relat_2(u1_orders_2(A)) ) ) ).
fof(d12_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v3_necklace(A)
<=> r2_relat_2(u1_orders_2(A),u1_struct_0(A)) ) ) ).
fof(t24_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v3_necklace(A)
=> v5_oposet_1(A) ) ) ).
fof(t25_oposet_1,axiom,
v3_necklace(k4_oposet_1) ).
fof(d13_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v6_oposet_1(A)
<=> ( v3_relat_2(u1_orders_2(A))
& m2_relset_1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(t26_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v1_necklace(A)
=> v6_oposet_1(A) ) ) ).
fof(t27_oposet_1,axiom,
v1_necklace(k3_oposet_1) ).
fof(d14_oposet_1,axiom,
$true ).
fof(d15_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v7_oposet_1(A)
<=> ( v4_relat_2(u1_orders_2(A))
& m2_relset_1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(t28_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v4_orders_2(A)
=> v7_oposet_1(A) ) ) ).
fof(d16_oposet_1,axiom,
$true ).
fof(d17_oposet_1,axiom,
$true ).
fof(d18_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v8_oposet_1(A)
<=> r5_relat_2(u1_orders_2(A),u1_struct_0(A)) ) ) ).
fof(t29_oposet_1,axiom,
$true ).
fof(t30_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v8_oposet_1(A)
=> v2_necklace(A) ) ) ).
fof(t31_oposet_1,axiom,
v8_oposet_1(k4_oposet_1) ).
fof(d19_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v9_oposet_1(A)
<=> ( v8_relat_2(u1_orders_2(A))
& m2_relset_1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(t32_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v3_orders_2(A)
=> v9_oposet_1(A) ) ) ).
fof(t33_oposet_1,axiom,
$true ).
fof(t34_oposet_1,axiom,
v3_orders_2(k4_oposet_1) ).
fof(t35_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v6_oposet_1(A)
& v9_oposet_1(A) )
=> v4_oposet_1(A) ) ) ).
fof(t36_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v5_oposet_1(A)
& v9_oposet_1(A) )
=> v2_necklace(A) ) ) ).
fof(t37_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v2_necklace(A)
=> v5_oposet_1(A) ) ) ).
fof(t38_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v2_orders_2(A)
& v6_oposet_1(A) )
=> v1_necklace(A) ) ) ).
fof(t39_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v2_orders_2(A)
& v7_oposet_1(A) )
=> v4_orders_2(A) ) ) ).
fof(t40_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v2_orders_2(A)
& v9_oposet_1(A) )
=> v3_orders_2(A) ) ) ).
fof(t41_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v3_necklace(A)
& v9_oposet_1(A) )
=> v3_orders_2(A) ) ) ).
fof(t42_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ( v3_necklace(A)
& v2_necklace(A) )
=> v8_oposet_1(A) ) ) ).
fof(d20_oposet_1,axiom,
$true ).
fof(d21_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v10_oposet_1(A)
<=> ( v4_oposet_1(A)
& v9_oposet_1(A) ) ) ) ).
fof(d22_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v11_oposet_1(A)
<=> ( v2_orders_2(A)
& v3_orders_2(A) ) ) ) ).
fof(t43_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v11_oposet_1(A)
=> v10_oposet_1(A) ) ) ).
fof(d23_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v12_oposet_1(A)
<=> ( v3_oposet_1(A)
& v11_oposet_1(A) ) ) ) ).
fof(d24_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v13_oposet_1(A)
<=> ( v2_orders_2(A)
& v7_oposet_1(A)
& v9_oposet_1(A) ) ) ) ).
fof(d25_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v14_oposet_1(A)
<=> ( v2_orders_2(A)
& v4_orders_2(A)
& v3_orders_2(A) ) ) ) ).
fof(t44_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v13_oposet_1(A)
<=> v14_oposet_1(A) ) ) ).
fof(d26_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v15_oposet_1(A)
<=> ( v3_oposet_1(A)
& v14_oposet_1(A) ) ) ) ).
fof(d27_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v16_oposet_1(A)
<=> ( v2_necklace(A)
& v9_oposet_1(A) ) ) ) ).
fof(d28_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v17_oposet_1(A)
<=> ( v8_oposet_1(A)
& v3_orders_2(A) ) ) ) ).
fof(t45_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v17_oposet_1(A)
=> v16_oposet_1(A) ) ) ).
fof(t46_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v16_oposet_1(A)
=> v5_oposet_1(A) ) ) ).
fof(t47_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( ( v3_necklace(A)
& v16_oposet_1(A) )
=> v17_oposet_1(A) ) ) ).
fof(t48_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v17_oposet_1(A)
=> v3_necklace(A) ) ) ).
fof(t49_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v11_oposet_1(A)
& l1_oposet_1(A) )
=> ( v7_oposet_1(A)
=> v14_oposet_1(A) ) ) ).
fof(d29_oposet_1,axiom,
$true ).
fof(d30_oposet_1,axiom,
$true ).
fof(d31_oposet_1,axiom,
$true ).
fof(d32_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( v18_oposet_1(B,A)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A))
& C = B
& v2_oposet_1(C,u1_struct_0(A))
& v5_waybel_0(C,A,A) ) ) ) ) ).
fof(d33_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v19_oposet_1(A)
<=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& B = u1_robbins1(A)
& v18_oposet_1(B,A) ) ) ) ).
fof(t50_oposet_1,axiom,
$true ).
fof(t51_oposet_1,axiom,
v18_oposet_1(u1_robbins1(k3_oposet_1),k3_oposet_1) ).
fof(d34_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( r1_oposet_1(A,B)
<=> ( v18_oposet_1(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_yellow_0(A,k2_struct_0(A,C,k1_waybel_0(A,A,B,C)))
& r2_yellow_0(A,k2_struct_0(A,C,k1_waybel_0(A,A,B,C))) ) ) ) ) ) ) ).
fof(d35_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v20_oposet_1(A)
<=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& B = u1_robbins1(A)
& r1_oposet_1(A,B) ) ) ) ).
fof(t52_oposet_1,axiom,
v20_oposet_1(k3_oposet_1) ).
fof(d36_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( r2_oposet_1(A,B)
<=> ( v18_oposet_1(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_yellow_0(A,k2_struct_0(A,C,k1_waybel_0(A,A,B,C)))
& r2_yellow_0(A,k2_struct_0(A,C,k1_waybel_0(A,A,B,C)))
& r4_waybel_1(A,k1_yellow_0(A,k2_struct_0(A,C,k1_waybel_0(A,A,B,C))),u1_struct_0(A))
& r3_waybel_1(A,k2_yellow_0(A,k2_struct_0(A,C,k1_waybel_0(A,A,B,C))),u1_struct_0(A)) ) ) ) ) ) ) ).
fof(d37_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ( v21_oposet_1(A)
<=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A))
& B = u1_robbins1(A)
& r2_oposet_1(A,B) ) ) ) ).
fof(t53_oposet_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_oposet_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( r2_oposet_1(A,B)
=> r1_oposet_1(A,B) ) ) ) ).
fof(t54_oposet_1,axiom,
v21_oposet_1(k3_oposet_1) ).
fof(dt_l1_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( l1_orders_2(A)
& l1_robbins1(A) ) ) ).
fof(existence_l1_oposet_1,axiom,
? [A] : l1_oposet_1(A) ).
fof(abstractness_v1_oposet_1,axiom,
! [A] :
( l1_oposet_1(A)
=> ( v1_oposet_1(A)
=> A = g1_oposet_1(u1_struct_0(A),u1_orders_2(A),u1_robbins1(A)) ) ) ).
fof(dt_k1_oposet_1,axiom,
! [A,B] : m2_relset_1(k1_oposet_1(A,B),A,B) ).
fof(dt_k2_oposet_1,axiom,
! [A,B] : m2_relset_1(k2_oposet_1(A,B),A,B) ).
fof(dt_k3_oposet_1,axiom,
( v1_oposet_1(k3_oposet_1)
& l1_oposet_1(k3_oposet_1) ) ).
fof(dt_k4_oposet_1,axiom,
( v1_oposet_1(k4_oposet_1)
& l1_oposet_1(k4_oposet_1) ) ).
fof(dt_g1_oposet_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> ( v1_oposet_1(g1_oposet_1(A,B,C))
& l1_oposet_1(g1_oposet_1(A,B,C)) ) ) ).
fof(free_g1_oposet_1,axiom,
! [A,B,C] :
( ( m1_relset_1(B,A,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> ! [D,E,F] :
( g1_oposet_1(A,B,C) = g1_oposet_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
%------------------------------------------------------------------------------