SET007 Axioms: SET007+789.ax
%------------------------------------------------------------------------------
% File : SET007+789 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Lattice of Fuzzy Sets
% 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 : lfuzzy_0 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 82 ( 2 unt; 0 def)
% Number of atoms : 575 ( 53 equ)
% Maximal formula atoms : 24 ( 7 avg)
% Number of connectives : 566 ( 73 ~; 0 |; 312 &)
% ( 26 <=>; 155 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 49 ( 48 usr; 0 prp; 1-3 aty)
% Number of functors : 72 ( 72 usr; 26 con; 0-7 aty)
% Number of variables : 211 ( 183 !; 28 ?)
% 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(cc1_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( v2_lfuzzy_0(A)
=> ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A) ) ) ) ).
fof(cc2_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_struct_0(A)
=> v1_lfuzzy_0(A) ) ) ).
fof(rc1_lfuzzy_0,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_orders_2(A)
& v1_lfuzzy_0(A)
& v2_lfuzzy_0(A) ) ).
fof(cc3_lfuzzy_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_xcmplx_0(B)
& v1_xreal_0(B) ) ) ) ).
fof(fc1_lfuzzy_0,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( ~ v3_struct_0(k1_lfuzzy_0(A))
& v1_orders_2(k1_lfuzzy_0(A))
& v1_lfuzzy_0(k1_lfuzzy_0(A)) ) ) ).
fof(cc4_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_lfuzzy_0(A)
=> ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A) ) ) ) ).
fof(cc5_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v16_waybel_0(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v1_lfuzzy_0(A) ) ) ) ).
fof(cc6_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v16_waybel_0(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v1_lfuzzy_0(A) ) ) ) ).
fof(cc7_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v2_lfuzzy_0(A) )
=> ( ~ v3_struct_0(A)
& v3_yellow_0(A) ) ) ) ).
fof(cc8_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v2_lfuzzy_0(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v3_yellow_0(A)
& v16_waybel_0(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v2_waybel_3(A)
& v1_lfuzzy_0(A)
& v2_lfuzzy_0(A) ) ) ) ).
fof(cc9_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v16_waybel_0(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v16_waybel_0(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v2_waybel_1(A) ) ) ) ).
fof(cc10_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( ~ v3_struct_0(A)
& v2_lfuzzy_0(A) )
=> ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v2_yellow_0(A)
& v3_yellow_0(A)
& v16_waybel_0(A)
& v24_waybel_0(A)
& ~ v1_yellow_3(A)
& v1_waybel_2(A)
& v2_waybel_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v2_waybel_1(A)
& v9_waybel_1(A)
& v2_waybel_3(A)
& v1_lfuzzy_0(A)
& v2_lfuzzy_0(A) ) ) ) ).
fof(fc2_lfuzzy_0,axiom,
( ~ v1_xboole_0(k1_rcomp_1(np__0,np__1))
& v1_membered(k1_rcomp_1(np__0,np__1))
& v2_membered(k1_rcomp_1(np__0,np__1)) ) ).
fof(fc3_lfuzzy_0,axiom,
( ~ v3_struct_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v1_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v2_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v3_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v4_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v2_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v3_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v16_waybel_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v24_waybel_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& ~ v1_yellow_3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v1_waybel_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v2_waybel_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v1_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v2_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v3_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v2_waybel_1(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v9_waybel_1(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v2_waybel_3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v1_lfuzzy_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
& v2_lfuzzy_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1))) ) ).
fof(fc4_lfuzzy_0,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& v2_lattice3(B)
& v3_lattice3(B)
& v9_waybel_1(B)
& l1_orders_2(B) )
=> ( ~ v3_struct_0(k6_yellow_1(A,B))
& v1_orders_2(k6_yellow_1(A,B))
& v2_orders_2(k6_yellow_1(A,B))
& v3_orders_2(k6_yellow_1(A,B))
& v4_orders_2(k6_yellow_1(A,B))
& v2_yellow_0(k6_yellow_1(A,B))
& v24_waybel_0(k6_yellow_1(A,B))
& ~ v1_yellow_3(k6_yellow_1(A,B))
& v1_waybel_2(k6_yellow_1(A,B))
& v2_waybel_2(k6_yellow_1(A,B))
& v1_lattice3(k6_yellow_1(A,B))
& v2_lattice3(k6_yellow_1(A,B))
& v3_lattice3(k6_yellow_1(A,B))
& v2_waybel_1(k6_yellow_1(A,B))
& v9_waybel_1(k6_yellow_1(A,B)) ) ) ).
fof(fc5_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k4_lfuzzy_0(A))
& v2_orders_2(k4_lfuzzy_0(A))
& v3_orders_2(k4_lfuzzy_0(A))
& v4_orders_2(k4_lfuzzy_0(A))
& v2_yellow_0(k4_lfuzzy_0(A))
& v24_waybel_0(k4_lfuzzy_0(A))
& ~ v1_yellow_3(k4_lfuzzy_0(A))
& v1_waybel_2(k4_lfuzzy_0(A))
& v2_waybel_2(k4_lfuzzy_0(A))
& v1_lattice3(k4_lfuzzy_0(A))
& v2_lattice3(k4_lfuzzy_0(A))
& v3_lattice3(k4_lfuzzy_0(A))
& v2_waybel_1(k4_lfuzzy_0(A))
& v9_waybel_1(k4_lfuzzy_0(A))
& v1_monoid_0(k4_lfuzzy_0(A)) ) ) ).
fof(d1_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_lfuzzy_0(A)
<=> ( r1_tarski(u1_struct_0(A),k1_numbers)
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r2_hidden(B,u1_struct_0(A))
& r2_hidden(C,u1_struct_0(A)) )
=> ( r2_hidden(k4_tarski(B,C),u1_orders_2(A))
<=> r1_xreal_0(B,C) ) ) ) ) ) ) ) ).
fof(d2_lfuzzy_0,axiom,
! [A] :
( l1_orders_2(A)
=> ( v2_lfuzzy_0(A)
<=> ( v1_lfuzzy_0(A)
& ? [B] :
( v1_xreal_0(B)
& ? [C] :
( v1_xreal_0(C)
& r1_xreal_0(B,C)
& u1_struct_0(A) = k1_rcomp_1(B,C) ) ) ) ) ) ).
fof(t1_lfuzzy_0,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ? [B] :
( v1_orders_2(B)
& l1_orders_2(B)
& u1_struct_0(B) = A
& v1_lfuzzy_0(B) ) ) ).
fof(t2_lfuzzy_0,axiom,
! [A] :
( ( v1_lfuzzy_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_lfuzzy_0(B)
& l1_orders_2(B) )
=> ( u1_struct_0(A) = u1_struct_0(B)
=> g1_orders_2(u1_struct_0(A),u1_orders_2(A)) = g1_orders_2(u1_struct_0(B),u1_orders_2(B)) ) ) ) ).
fof(d3_lfuzzy_0,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( ( v1_orders_2(B)
& v1_lfuzzy_0(B)
& l1_orders_2(B) )
=> ( B = k1_lfuzzy_0(A)
<=> u1_struct_0(B) = A ) ) ) ).
fof(t3_lfuzzy_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_xreal_0(B,C)
<=> r1_orders_2(A,B,C) ) ) ) ) ).
fof(t4_lfuzzy_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k13_lattice3(A,B,C) = k2_lfuzzy_0(A,B,C) ) ) ) ).
fof(t5_lfuzzy_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k12_lattice3(A,B,C) = k3_lfuzzy_0(A,B,C) ) ) ) ).
fof(t6_lfuzzy_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A) )
=> ( ? [B] :
( v1_xreal_0(B)
& r2_hidden(B,u1_struct_0(A))
& ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,u1_struct_0(A))
=> r1_xreal_0(B,C) ) ) )
<=> v1_yellow_0(A) ) ) ).
fof(t7_lfuzzy_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A) )
=> ( ? [B] :
( v1_xreal_0(B)
& r2_hidden(B,u1_struct_0(A))
& ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,u1_struct_0(A))
=> r1_xreal_0(C,B) ) ) )
<=> v2_yellow_0(A) ) ) ).
fof(t8_lfuzzy_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_lfuzzy_0(A)
& l1_orders_2(A) )
=> ! [B] : r1_yellow_0(A,B) ) ).
fof(t9_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v2_orders_2(k6_waybel_3(A,B,C))
& v3_orders_2(k6_waybel_3(A,B,C))
& v4_orders_2(k6_waybel_3(A,B,C))
& v1_lattice3(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) )
=> v1_lattice3(k5_yellow_1(A,B)) ) ) ) ).
fof(t10_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v2_orders_2(k6_waybel_3(A,B,C))
& v3_orders_2(k6_waybel_3(A,B,C))
& v4_orders_2(k6_waybel_3(A,B,C))
& v2_lattice3(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) )
=> v2_lattice3(k5_yellow_1(A,B)) ) ) ) ).
fof(t11_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v2_orders_2(k6_waybel_3(A,B,C))
& v3_orders_2(k6_waybel_3(A,B,C))
& v4_orders_2(k6_waybel_3(A,B,C))
& v2_lattice3(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_yellow_1(A,B)))
=> ! [E] :
( m1_subset_1(E,A)
=> k7_waybel_3(A,B,k11_lattice3(k5_yellow_1(A,B),C,D),E) = k11_lattice3(k6_waybel_3(A,B,E),k7_waybel_3(A,B,C,E),k7_waybel_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t12_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v2_orders_2(k6_waybel_3(A,B,C))
& v3_orders_2(k6_waybel_3(A,B,C))
& v4_orders_2(k6_waybel_3(A,B,C))
& v1_lattice3(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k5_yellow_1(A,B)))
=> ! [E] :
( m1_subset_1(E,A)
=> k7_waybel_3(A,B,k10_lattice3(k5_yellow_1(A,B),C,D),E) = k10_lattice3(k6_waybel_3(A,B,E),k7_waybel_3(A,B,C,E),k7_waybel_3(A,B,D,E)) ) ) ) ) ) ) ).
fof(t13_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_yellow_1(B)
& v4_waybel_3(B)
& v5_waybel_3(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v2_orders_2(k6_waybel_3(A,B,C))
& v3_orders_2(k6_waybel_3(A,B,C))
& v4_orders_2(k6_waybel_3(A,B,C))
& v1_lattice3(k6_waybel_3(A,B,C))
& v2_lattice3(k6_waybel_3(A,B,C))
& v3_lattice3(k6_waybel_3(A,B,C))
& v9_waybel_1(k6_waybel_3(A,B,C))
& l1_orders_2(k6_waybel_3(A,B,C)) ) )
=> ( v3_lattice3(k5_yellow_1(A,B))
& v9_waybel_1(k5_yellow_1(A,B)) ) ) ) ) ).
fof(d4_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_lfuzzy_0(A) = k6_yellow_1(A,k1_lfuzzy_0(k1_rcomp_1(np__0,np__1))) ) ).
fof(t14_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> u1_struct_0(k4_lfuzzy_0(A)) = k1_funct_2(A,k1_rcomp_1(np__0,np__1)) ) ).
fof(d5_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
=> k5_lfuzzy_0(A,B) = B ) ) ).
fof(d6_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> k6_lfuzzy_0(A,B) = B ) ) ).
fof(t16_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( ! [D] :
( m1_subset_1(D,A)
=> r1_xreal_0(k7_lfuzzy_0(A,B,D),k7_lfuzzy_0(A,C,D)) )
<=> r1_orders_2(k4_lfuzzy_0(A),k6_lfuzzy_0(A,B),k6_lfuzzy_0(A,C)) ) ) ) ) ).
fof(t17_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_lfuzzy_0(A)))
=> ( r1_orders_2(k4_lfuzzy_0(A),B,C)
<=> ! [D] :
( m1_subset_1(D,A)
=> r1_xreal_0(k7_lfuzzy_0(A,k5_lfuzzy_0(A,B),D),k7_lfuzzy_0(A,k5_lfuzzy_0(A,C),D)) ) ) ) ) ) ).
fof(t18_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> k2_fuzzy_1(A,B,C) = k13_lattice3(k4_lfuzzy_0(A),k6_lfuzzy_0(A,B),k6_lfuzzy_0(A,C)) ) ) ) ).
fof(t19_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_lfuzzy_0(A)))
=> k13_lattice3(k4_lfuzzy_0(A),B,C) = k2_fuzzy_1(A,k5_lfuzzy_0(A,B),k5_lfuzzy_0(A,C)) ) ) ) ).
fof(t20_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> k1_fuzzy_1(A,B,C) = k12_lattice3(k4_lfuzzy_0(A),k6_lfuzzy_0(A,B),k6_lfuzzy_0(A,C)) ) ) ) ).
fof(t21_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_lfuzzy_0(A)))
=> k12_lattice3(k4_lfuzzy_0(A),B,C) = k1_fuzzy_1(A,k5_lfuzzy_0(A,B),k5_lfuzzy_0(A,C)) ) ) ) ).
fof(t23_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( m1_fuzzy_1(E,k2_zfmisc_1(A,B))
=> ! [F] :
( m1_fuzzy_1(F,k2_zfmisc_1(B,C))
=> ! [G] :
( m1_fuzzy_1(G,k2_zfmisc_1(C,D))
=> k3_fuzzy_4(A,C,D,k3_fuzzy_4(A,B,C,E,F),G) = k3_fuzzy_4(A,B,D,E,k3_fuzzy_4(B,C,D,F,G)) ) ) ) ) ) ) ) ).
fof(dt_k1_lfuzzy_0,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_orders_2(k1_lfuzzy_0(A))
& v1_lfuzzy_0(k1_lfuzzy_0(A))
& l1_orders_2(k1_lfuzzy_0(A)) ) ) ).
fof(dt_k2_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k2_lfuzzy_0(A,B,C),u1_struct_0(A)) ) ).
fof(commutativity_k2_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k2_lfuzzy_0(A,B,C) = k2_lfuzzy_0(A,C,B) ) ).
fof(idempotence_k2_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k2_lfuzzy_0(A,B,B) = B ) ).
fof(redefinition_k2_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k2_lfuzzy_0(A,B,C) = k2_square_1(B,C) ) ).
fof(dt_k3_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k3_lfuzzy_0(A,B,C),u1_struct_0(A)) ) ).
fof(commutativity_k3_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k3_lfuzzy_0(A,B,C) = k3_lfuzzy_0(A,C,B) ) ).
fof(idempotence_k3_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k3_lfuzzy_0(A,B,B) = B ) ).
fof(redefinition_k3_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_lfuzzy_0(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> k3_lfuzzy_0(A,B,C) = k1_square_1(B,C) ) ).
fof(dt_k4_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v2_orders_2(k4_lfuzzy_0(A))
& v3_orders_2(k4_lfuzzy_0(A))
& v4_orders_2(k4_lfuzzy_0(A))
& v1_lattice3(k4_lfuzzy_0(A))
& v2_lattice3(k4_lfuzzy_0(A))
& v3_lattice3(k4_lfuzzy_0(A))
& v9_waybel_1(k4_lfuzzy_0(A))
& l1_orders_2(k4_lfuzzy_0(A)) ) ) ).
fof(dt_k5_lfuzzy_0,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A))) )
=> m1_fuzzy_1(k5_lfuzzy_0(A,B),A) ) ).
fof(dt_k6_lfuzzy_0,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A) )
=> m1_subset_1(k6_lfuzzy_0(A,B),u1_struct_0(k4_lfuzzy_0(A))) ) ).
fof(dt_k7_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k7_lfuzzy_0(A,B,C),u1_struct_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))) ) ).
fof(redefinition_k7_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_subset_1(C,A) )
=> k7_lfuzzy_0(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k8_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
& m1_subset_1(C,A) )
=> m1_subset_1(k8_lfuzzy_0(A,B,C),u1_struct_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))) ) ).
fof(redefinition_k8_lfuzzy_0,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
& m1_subset_1(C,A) )
=> k8_lfuzzy_0(A,B,C) = k1_funct_1(B,C) ) ).
fof(t15_lfuzzy_0,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v9_waybel_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k12_lattice3(A,k1_yellow_0(A,B),C) = k1_yellow_0(A,a_3_0_lfuzzy_0(A,B,C)) ) ) ) ).
fof(t22_lfuzzy_0,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_fuzzy_1(E,k2_zfmisc_1(B,C))
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,C)
=> k7_lfuzzy_0(k2_zfmisc_1(A,C),k3_fuzzy_4(A,B,C,D,E),k1_domain_1(A,C,F,G)) = k1_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),a_7_0_lfuzzy_0(A,B,C,D,E,F,G)) ) ) ) ) ) ) ) ).
fof(s1_lfuzzy_0,axiom,
k1_yellow_0(f1_s1_lfuzzy_0,a_0_0_lfuzzy_0) = k1_yellow_0(f1_s1_lfuzzy_0,a_0_1_lfuzzy_0) ).
fof(s2_lfuzzy_0,axiom,
k1_yellow_0(f1_s2_lfuzzy_0,a_0_2_lfuzzy_0) = k1_yellow_0(f1_s2_lfuzzy_0,a_0_3_lfuzzy_0) ).
fof(s3_lfuzzy_0,axiom,
( ! [A] :
( m1_subset_1(A,f1_s3_lfuzzy_0)
=> ! [B] :
( m1_subset_1(B,f2_s3_lfuzzy_0)
=> ( p1_s3_lfuzzy_0(A,B)
=> f3_s3_lfuzzy_0(A,B) = f4_s3_lfuzzy_0(A,B) ) ) )
=> a_0_4_lfuzzy_0 = a_0_5_lfuzzy_0 ) ).
fof(s4_lfuzzy_0,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s4_lfuzzy_0)
=> ! [B] :
( m1_subset_1(B,f2_s4_lfuzzy_0)
=> ( p1_s4_lfuzzy_0(A,B)
<=> p2_s4_lfuzzy_0(A,B) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s4_lfuzzy_0)
=> ! [B] :
( m1_subset_1(B,f2_s4_lfuzzy_0)
=> ( p1_s4_lfuzzy_0(A,B)
=> f3_s4_lfuzzy_0(A,B) = f4_s4_lfuzzy_0(A,B) ) ) ) )
=> a_0_6_lfuzzy_0 = a_0_7_lfuzzy_0 ) ).
fof(s5_lfuzzy_0,axiom,
( ! [A] :
( m1_subset_1(A,f2_s5_lfuzzy_0)
=> ! [B] :
( m1_subset_1(B,f3_s5_lfuzzy_0)
=> ( ( p1_s5_lfuzzy_0(A)
& p2_s5_lfuzzy_0(B) )
=> f4_s5_lfuzzy_0(A,B) = f5_s5_lfuzzy_0(A,B) ) ) )
=> k1_yellow_0(f1_s5_lfuzzy_0,a_0_8_lfuzzy_0) = k1_yellow_0(f1_s5_lfuzzy_0,a_0_9_lfuzzy_0) ) ).
fof(fraenkel_a_3_0_lfuzzy_0,axiom,
! [A,B,C,D] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& v2_lattice3(B)
& v3_lattice3(B)
& v9_waybel_1(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_lfuzzy_0(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k12_lattice3(B,E,D)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_7_0_lfuzzy_0,axiom,
! [A,B,C,D,E,F,G,H] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& m1_fuzzy_1(E,k2_zfmisc_1(B,C))
& m1_fuzzy_1(F,k2_zfmisc_1(C,D))
& m1_subset_1(G,B)
& m1_subset_1(H,D) )
=> ( r2_hidden(A,a_7_0_lfuzzy_0(B,C,D,E,F,G,H))
<=> ? [I] :
( m1_subset_1(I,C)
& A = k12_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k7_lfuzzy_0(k2_zfmisc_1(B,C),E,k1_domain_1(B,C,G,I)),k7_lfuzzy_0(k2_zfmisc_1(C,D),F,k1_domain_1(C,D,I,H))) ) ) ) ).
fof(fraenkel_a_0_0_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_0_lfuzzy_0)
<=> ? [B] :
( m1_subset_1(B,f2_s1_lfuzzy_0)
& A = k1_yellow_0(f1_s1_lfuzzy_0,a_1_0_lfuzzy_0(B))
& p1_s1_lfuzzy_0(B) ) ) ).
fof(fraenkel_a_1_0_lfuzzy_0,axiom,
! [A,B] :
( m1_subset_1(B,f2_s1_lfuzzy_0)
=> ( r2_hidden(A,a_1_0_lfuzzy_0(B))
<=> ? [C] :
( m1_subset_1(C,f3_s1_lfuzzy_0)
& A = f4_s1_lfuzzy_0(B,C)
& p2_s1_lfuzzy_0(C) ) ) ) ).
fof(fraenkel_a_0_1_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_1_lfuzzy_0)
<=> ? [B,C] :
( m1_subset_1(B,f2_s1_lfuzzy_0)
& m1_subset_1(C,f3_s1_lfuzzy_0)
& A = f4_s1_lfuzzy_0(B,C)
& p1_s1_lfuzzy_0(B)
& p2_s1_lfuzzy_0(C) ) ) ).
fof(fraenkel_a_0_2_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_2_lfuzzy_0)
<=> ? [B] :
( m1_subset_1(B,f3_s2_lfuzzy_0)
& A = k1_yellow_0(f1_s2_lfuzzy_0,a_1_1_lfuzzy_0(B))
& p2_s2_lfuzzy_0(B) ) ) ).
fof(fraenkel_a_1_1_lfuzzy_0,axiom,
! [A,B] :
( m1_subset_1(B,f3_s2_lfuzzy_0)
=> ( r2_hidden(A,a_1_1_lfuzzy_0(B))
<=> ? [C] :
( m1_subset_1(C,f2_s2_lfuzzy_0)
& A = f4_s2_lfuzzy_0(C,B)
& p1_s2_lfuzzy_0(C) ) ) ) ).
fof(fraenkel_a_0_3_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_3_lfuzzy_0)
<=> ? [B,C] :
( m1_subset_1(B,f2_s2_lfuzzy_0)
& m1_subset_1(C,f3_s2_lfuzzy_0)
& A = f4_s2_lfuzzy_0(B,C)
& p1_s2_lfuzzy_0(B)
& p2_s2_lfuzzy_0(C) ) ) ).
fof(fraenkel_a_0_4_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_4_lfuzzy_0)
<=> ? [B,C] :
( m1_subset_1(B,f1_s3_lfuzzy_0)
& m1_subset_1(C,f2_s3_lfuzzy_0)
& A = f3_s3_lfuzzy_0(B,C)
& p1_s3_lfuzzy_0(B,C) ) ) ).
fof(fraenkel_a_0_5_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_5_lfuzzy_0)
<=> ? [B,C] :
( m1_subset_1(B,f1_s3_lfuzzy_0)
& m1_subset_1(C,f2_s3_lfuzzy_0)
& A = f4_s3_lfuzzy_0(B,C)
& p1_s3_lfuzzy_0(B,C) ) ) ).
fof(fraenkel_a_0_6_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_6_lfuzzy_0)
<=> ? [B,C] :
( m1_subset_1(B,f1_s4_lfuzzy_0)
& m1_subset_1(C,f2_s4_lfuzzy_0)
& A = f3_s4_lfuzzy_0(B,C)
& p1_s4_lfuzzy_0(B,C) ) ) ).
fof(fraenkel_a_0_7_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_7_lfuzzy_0)
<=> ? [B,C] :
( m1_subset_1(B,f1_s4_lfuzzy_0)
& m1_subset_1(C,f2_s4_lfuzzy_0)
& A = f4_s4_lfuzzy_0(B,C)
& p2_s4_lfuzzy_0(B,C) ) ) ).
fof(fraenkel_a_0_8_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_8_lfuzzy_0)
<=> ? [B] :
( m1_subset_1(B,f2_s5_lfuzzy_0)
& A = k1_yellow_0(f1_s5_lfuzzy_0,a_1_2_lfuzzy_0(B))
& p1_s5_lfuzzy_0(B) ) ) ).
fof(fraenkel_a_1_2_lfuzzy_0,axiom,
! [A,B] :
( m1_subset_1(B,f2_s5_lfuzzy_0)
=> ( r2_hidden(A,a_1_2_lfuzzy_0(B))
<=> ? [C] :
( m1_subset_1(C,f3_s5_lfuzzy_0)
& A = f4_s5_lfuzzy_0(B,C)
& p2_s5_lfuzzy_0(C) ) ) ) ).
fof(fraenkel_a_0_9_lfuzzy_0,axiom,
! [A] :
( r2_hidden(A,a_0_9_lfuzzy_0)
<=> ? [B] :
( m1_subset_1(B,f3_s5_lfuzzy_0)
& A = k1_yellow_0(f1_s5_lfuzzy_0,a_1_3_lfuzzy_0(B))
& p2_s5_lfuzzy_0(B) ) ) ).
fof(fraenkel_a_1_3_lfuzzy_0,axiom,
! [A,B] :
( m1_subset_1(B,f3_s5_lfuzzy_0)
=> ( r2_hidden(A,a_1_3_lfuzzy_0(B))
<=> ? [C] :
( m1_subset_1(C,f2_s5_lfuzzy_0)
& A = f5_s5_lfuzzy_0(C,B)
& p1_s5_lfuzzy_0(C) ) ) ) ).
%------------------------------------------------------------------------------