SET007 Axioms: SET007+691.ax
%------------------------------------------------------------------------------
% File : SET007+691 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Fuzzy Relation
% 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 : fuzzy_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 33 ( 0 unt; 0 def)
% Number of atoms : 205 ( 27 equ)
% Maximal formula atoms : 13 ( 6 avg)
% Number of connectives : 249 ( 77 ~; 0 |; 26 &)
% ( 4 <=>; 142 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-2 aty)
% Number of functors : 23 ( 23 usr; 3 con; 0-7 aty)
% Number of variables : 159 ( 159 !; 0 ?)
% 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_fuzzy_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A) )
=> ~ v1_xboole_0(k2_relat_1(B)) ) ).
fof(t1_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ( v3_seq_4(k5_relset_1(A,k1_numbers,B))
& ! [C] :
( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
=> r1_xreal_0(k2_seq_1(A,k1_numbers,B,C),k3_pscomp_1(k5_relset_1(A,k1_numbers,B))) )
& ! [C] :
( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
=> r1_xreal_0(k4_pscomp_1(k5_relset_1(A,k1_numbers,B)),k2_seq_1(A,k1_numbers,B,C)) ) ) ) ) ).
fof(t2_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( ! [D] :
( r2_hidden(D,A)
=> r1_xreal_0(k2_seq_1(A,k1_numbers,B,D),k2_seq_1(A,k1_numbers,C,D)) )
=> r1_xreal_0(k3_pscomp_1(k5_relset_1(A,k1_numbers,B)),k3_pscomp_1(k5_relset_1(A,k1_numbers,C))) ) ) ) ) ).
fof(t3_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,k2_zfmisc_1(A,B))
=> ( r1_xreal_0(np__0,k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,C,D))
& r1_xreal_0(k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,C,D),np__1) ) ) ) ) ) ).
fof(t4_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D,E] :
( r2_hidden(k4_tarski(D,E),k2_zfmisc_1(A,B))
=> ( r1_xreal_0(np__0,k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,C,k4_tarski(D,E)))
& r1_xreal_0(k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,C,k4_tarski(D,E)),np__1) ) ) ) ) ) ).
fof(d1_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(B,A))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ( D = k1_fuzzy_4(A,B,C)
<=> ! [E,F] :
( r2_hidden(k4_tarski(E,F),k2_zfmisc_1(A,B))
=> k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,D,k4_tarski(E,F)) = k2_seq_1(k2_zfmisc_1(B,A),k1_numbers,C,k4_tarski(F,E)) ) ) ) ) ) ) ).
fof(t5_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> k1_fuzzy_4(A,B,k1_fuzzy_4(B,A,C)) = C ) ) ) ).
fof(t6_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> k3_fuzzy_1(k2_zfmisc_1(B,A),k1_fuzzy_4(B,A,C)) = k1_fuzzy_4(B,A,k3_fuzzy_1(k2_zfmisc_1(A,B),C)) ) ) ) ).
fof(t7_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> k1_fuzzy_4(B,A,k2_fuzzy_1(k2_zfmisc_1(A,B),C,D)) = k2_fuzzy_1(k2_zfmisc_1(B,A),k1_fuzzy_4(B,A,C),k1_fuzzy_4(B,A,D)) ) ) ) ) ).
fof(t8_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> k1_fuzzy_4(B,A,k1_fuzzy_1(k2_zfmisc_1(A,B),C,D)) = k1_fuzzy_1(k2_zfmisc_1(B,A),k1_fuzzy_4(B,A,C),k1_fuzzy_4(B,A,D)) ) ) ) ) ).
fof(t9_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ! [E,F] :
( ( r2_hidden(E,A)
& r2_hidden(F,B)
& r1_xreal_0(k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,C,k4_tarski(E,F)),k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,D,k4_tarski(E,F))) )
=> r1_xreal_0(k2_seq_1(k2_zfmisc_1(B,A),k1_numbers,k1_fuzzy_4(B,A,C),k4_tarski(F,E)),k2_seq_1(k2_zfmisc_1(B,A),k1_numbers,k1_fuzzy_4(B,A,D),k4_tarski(F,E))) ) ) ) ) ) ).
fof(t10_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ( r1_fuzzy_1(C,D)
=> r1_fuzzy_1(k1_fuzzy_4(B,A,C),k1_fuzzy_4(B,A,D)) ) ) ) ) ) ).
fof(t11_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> k1_fuzzy_4(B,A,k1_fuzzy_2(k2_zfmisc_1(A,B),C,D)) = k1_fuzzy_2(k2_zfmisc_1(B,A),k1_fuzzy_4(B,A,C),k1_fuzzy_4(B,A,D)) ) ) ) ) ).
fof(t12_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> k1_fuzzy_4(B,A,k6_fuzzy_1(k2_zfmisc_1(A,B),C,D)) = k6_fuzzy_1(k2_zfmisc_1(B,A),k1_fuzzy_4(B,A,C),k1_fuzzy_4(B,A,D)) ) ) ) ) ).
fof(d2_fuzzy_4,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,G] :
( ( r2_hidden(F,A)
& r2_hidden(G,C) )
=> ! [H] :
( m1_fuzzy_1(H,B)
=> ( H = k2_fuzzy_4(A,B,C,D,E,F,G)
<=> ! [I] :
( m1_subset_1(I,B)
=> k2_seq_1(B,k1_numbers,H,I) = k3_square_1(k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,D,k4_tarski(F,I)),k2_seq_1(k2_zfmisc_1(B,C),k1_numbers,E,k4_tarski(I,G))) ) ) ) ) ) ) ) ) ) ).
fof(d3_fuzzy_4,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_fuzzy_1(F,k2_zfmisc_1(A,C))
=> ( F = k3_fuzzy_4(A,B,C,D,E)
<=> ! [G,H] :
( r2_hidden(k4_tarski(G,H),k2_zfmisc_1(A,C))
=> k2_seq_1(k2_zfmisc_1(A,C),k1_numbers,F,k4_tarski(G,H)) = k3_pscomp_1(k5_relset_1(B,k1_numbers,k2_fuzzy_4(A,B,C,D,E,G,H))) ) ) ) ) ) ) ) ) ).
fof(t13_fuzzy_4,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_fuzzy_1(F,k2_zfmisc_1(B,C))
=> k3_fuzzy_4(A,B,C,D,k2_fuzzy_1(k2_zfmisc_1(B,C),E,F)) = k2_fuzzy_1(k2_zfmisc_1(A,C),k3_fuzzy_4(A,B,C,D,E),k3_fuzzy_4(A,B,C,D,F)) ) ) ) ) ) ) ).
fof(t14_fuzzy_4,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(A,B))
=> ! [F] :
( m1_fuzzy_1(F,k2_zfmisc_1(B,C))
=> k3_fuzzy_4(A,B,C,k2_fuzzy_1(k2_zfmisc_1(A,B),D,E),F) = k2_fuzzy_1(k2_zfmisc_1(A,C),k3_fuzzy_4(A,B,C,D,F),k3_fuzzy_4(A,B,C,E,F)) ) ) ) ) ) ) ).
fof(t15_fuzzy_4,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_fuzzy_1(F,k2_zfmisc_1(B,C))
=> r1_fuzzy_1(k3_fuzzy_4(A,B,C,D,k1_fuzzy_1(k2_zfmisc_1(B,C),E,F)),k1_fuzzy_1(k2_zfmisc_1(A,C),k3_fuzzy_4(A,B,C,D,E),k3_fuzzy_4(A,B,C,D,F))) ) ) ) ) ) ) ).
fof(t16_fuzzy_4,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(A,B))
=> ! [F] :
( m1_fuzzy_1(F,k2_zfmisc_1(B,C))
=> r1_fuzzy_1(k3_fuzzy_4(A,B,C,k1_fuzzy_1(k2_zfmisc_1(A,B),D,E),F),k1_fuzzy_1(k2_zfmisc_1(A,C),k3_fuzzy_4(A,B,C,D,F),k3_fuzzy_4(A,B,C,E,F))) ) ) ) ) ) ) ).
fof(t17_fuzzy_4,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))
=> k1_fuzzy_4(C,A,k3_fuzzy_4(A,B,C,D,E)) = k3_fuzzy_4(C,B,A,k1_fuzzy_4(C,B,E),k1_fuzzy_4(B,A,D)) ) ) ) ) ) ).
fof(t18_fuzzy_4,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(A,B))
=> ! [F] :
( m1_fuzzy_1(F,k2_zfmisc_1(B,C))
=> ! [G] :
( m1_fuzzy_1(G,k2_zfmisc_1(B,C))
=> ! [H,I] :
( ( r2_hidden(H,A)
& r2_hidden(I,C)
& ! [J] :
( r2_hidden(J,B)
=> ( r1_xreal_0(k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,D,k4_tarski(H,J)),k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,E,k4_tarski(H,J)))
& r1_xreal_0(k2_seq_1(k2_zfmisc_1(B,C),k1_numbers,F,k4_tarski(J,I)),k2_seq_1(k2_zfmisc_1(B,C),k1_numbers,G,k4_tarski(J,I))) ) ) )
=> r1_xreal_0(k2_seq_1(k2_zfmisc_1(A,C),k1_numbers,k3_fuzzy_4(A,B,C,D,F),k4_tarski(H,I)),k2_seq_1(k2_zfmisc_1(A,C),k1_numbers,k3_fuzzy_4(A,B,C,E,G),k4_tarski(H,I))) ) ) ) ) ) ) ) ) ).
fof(t19_fuzzy_4,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(A,B))
=> ! [F] :
( m1_fuzzy_1(F,k2_zfmisc_1(B,C))
=> ! [G] :
( m1_fuzzy_1(G,k2_zfmisc_1(B,C))
=> ( ( r1_fuzzy_1(D,E)
& r1_fuzzy_1(F,G) )
=> r1_fuzzy_1(k3_fuzzy_4(A,B,C,D,F),k3_fuzzy_4(A,B,C,E,G)) ) ) ) ) ) ) ) ) ).
fof(d4_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ( C = k4_fuzzy_4(A,B)
<=> ! [D,E] :
( r2_hidden(k4_tarski(D,E),k2_zfmisc_1(A,B))
=> ( ( D = E
=> k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,C,k4_tarski(D,E)) = np__1 )
& ( D != E
=> k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,C,k4_tarski(D,E)) = np__0 ) ) ) ) ) ) ) ).
fof(t20_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ( k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,k1_fuzzy_3(A,B),C) = np__0
& k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,k2_fuzzy_3(A,B),C) = np__1 ) ) ) ) ).
fof(t21_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),k2_zfmisc_1(A,B))
=> ( k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,k1_fuzzy_3(A,B),k4_tarski(C,D)) = np__0
& k2_seq_1(k2_zfmisc_1(A,B),k1_numbers,k2_fuzzy_3(A,B),k4_tarski(C,D)) = np__1 ) ) ) ) ).
fof(t22_fuzzy_4,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))
=> k3_fuzzy_4(C,A,B,k1_fuzzy_3(C,A),D) = k1_fuzzy_3(C,B) ) ) ) ) ).
fof(t23_fuzzy_4,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))
=> k3_fuzzy_4(A,B,C,D,k1_fuzzy_3(B,C)) = k1_fuzzy_3(A,C) ) ) ) ) ).
fof(t24_fuzzy_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> k3_fuzzy_4(A,A,A,B,k1_fuzzy_3(A,A)) = k3_fuzzy_4(A,A,A,k1_fuzzy_3(A,A),B) ) ) ).
fof(dt_k1_fuzzy_4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_fuzzy_1(C,k2_zfmisc_1(B,A)) )
=> m1_fuzzy_1(k1_fuzzy_4(A,B,C),k2_zfmisc_1(A,B)) ) ).
fof(dt_k2_fuzzy_4,axiom,
! [A,B,C,D,E,F,G] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_fuzzy_1(D,k2_zfmisc_1(A,B))
& m1_fuzzy_1(E,k2_zfmisc_1(B,C)) )
=> m1_fuzzy_1(k2_fuzzy_4(A,B,C,D,E,F,G),B) ) ).
fof(dt_k3_fuzzy_4,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_fuzzy_1(D,k2_zfmisc_1(A,B))
& m1_fuzzy_1(E,k2_zfmisc_1(B,C)) )
=> m1_fuzzy_1(k3_fuzzy_4(A,B,C,D,E),k2_zfmisc_1(A,C)) ) ).
fof(dt_k4_fuzzy_4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> m1_fuzzy_1(k4_fuzzy_4(A,B),k2_zfmisc_1(A,B)) ) ).
%------------------------------------------------------------------------------