SET007 Axioms: SET007+464.ax
%------------------------------------------------------------------------------
% File : SET007+464 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Properties of Functor Structures
% 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 : functor1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 27 ( 2 unt; 0 def)
% Number of atoms : 311 ( 16 equ)
% Maximal formula atoms : 30 ( 11 avg)
% Number of connectives : 349 ( 65 ~; 0 |; 169 &)
% ( 3 <=>; 112 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 11 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 29 ( 27 usr; 1 prp; 0-3 aty)
% Number of functors : 19 ( 19 usr; 0 con; 1-6 aty)
% Number of variables : 105 ( 103 !; 2 ?)
% 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_functor1,axiom,
? [A] :
( l2_altcat_1(A)
& ~ v3_struct_0(A)
& v2_altcat_1(A)
& v12_altcat_1(A)
& v1_altcat_2(A) ) ).
fof(rc2_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_altcat_2(A)
& l2_altcat_1(A) )
=> ? [B] :
( m1_altcat_2(B,A)
& ~ v3_struct_0(B)
& v1_altcat_2(B) ) ) ).
fof(fc1_functor1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v1_altcat_2(A)
& l2_altcat_1(A)
& ~ v3_struct_0(B)
& v1_altcat_2(B)
& l2_altcat_1(B)
& v8_functor0(C,A,B)
& l2_functor0(C,A,B)
& ~ v3_struct_0(D)
& v1_altcat_2(D)
& m1_altcat_2(D,A) )
=> v8_functor0(k14_functor0(A,B,D,C),D,B) ) ).
fof(t1_functor1,axiom,
! [A] : v2_funct_2(k6_partfun1(A),A,A) ).
fof(t2_functor1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> ( C = D
=> k6_funct_3(A,C) = k7_funct_2(D,B,A,k6_funct_3(B,D),k6_funct_3(A,B)) ) ) ) ) ) ).
fof(t3_functor1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( v3_funct_2(C,A,B)
=> ( v1_funct_1(k2_funct_1(C))
& v1_funct_2(k2_funct_1(C),B,A)
& m2_relset_1(k2_funct_1(C),B,A) ) ) ) ).
fof(t4_functor1,axiom,
! [A,B,C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,C)
& m2_relset_1(E,B,C) )
=> ~ ( v3_funct_2(D,A,B)
& v3_funct_2(E,B,C)
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,C)
& m2_relset_1(F,A,C) )
=> ~ ( F = k1_partfun1(A,B,B,C,D,E)
& v3_funct_2(F,A,C) ) ) ) ) ) ) ).
fof(t5_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_altcat_2(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& m1_altcat_2(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_altcat_2(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_altcat_2(D,B) )
=> ( C = D
=> k10_functor0(A,C) = k13_functor0(D,B,A,k10_functor0(B,D),k10_functor0(A,B)) ) ) ) ) ) ).
fof(t6_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_altcat_1(B) )
=> ! [C] :
( l2_functor0(C,A,B)
=> ( v21_functor0(C,A,B)
=> ( v3_funct_2(u1_functor0(A,B,C),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)))
& v1_msualg_3(u2_functor0(A,B,C)) ) ) ) ) ) ).
fof(t7_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l1_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& l1_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( l2_functor0(E,B,C)
=> ( ( v4_functor0(D,A,B)
& v4_functor0(E,B,C) )
=> v4_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) ).
fof(t8_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l1_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& l1_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( l2_functor0(E,B,C)
=> ( ( v17_functor0(D,A,B)
& v17_functor0(E,B,C) )
=> v17_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) ).
fof(t9_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l1_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& l1_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( l2_functor0(E,B,C)
=> ( ( v5_functor0(D,A,B)
& v5_functor0(E,B,C) )
=> v5_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) ).
fof(t10_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l1_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& l1_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( l2_functor0(E,B,C)
=> ( ( v18_functor0(D,A,B)
& v18_functor0(E,B,C) )
=> v18_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) ).
fof(t11_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l1_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& l1_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( l2_functor0(E,B,C)
=> ( ( v19_functor0(D,A,B)
& v19_functor0(E,B,C) )
=> v19_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) ).
fof(t12_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l1_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& l1_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( l2_functor0(E,B,C)
=> ( ( v20_functor0(D,A,B)
& v20_functor0(E,B,C) )
=> v20_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) ).
fof(t13_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l1_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& l1_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( l2_functor0(E,B,C)
=> ( ( v21_functor0(D,A,B)
& v21_functor0(E,B,C) )
=> v21_functor0(k13_functor0(A,B,C,D,E),A,C) ) ) ) ) ) ) ).
fof(t14_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_altcat_2(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l2_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& m1_altcat_2(C,A) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& m1_altcat_2(D,A) )
=> ! [E] :
( ( ~ v3_struct_0(E)
& m1_altcat_2(E,C) )
=> ( D = E
=> ! [F] :
( l2_functor0(F,A,B)
=> k14_functor0(A,B,D,F) = k14_functor0(C,B,E,k14_functor0(A,B,C,F)) ) ) ) ) ) ) ) ).
fof(t15_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_altcat_2(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_altcat_2(B)
& l2_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_altcat_2(C)
& l2_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( l2_functor0(E,B,C)
=> ! [F] :
( ( ~ v3_struct_0(F)
& v1_altcat_2(F)
& m1_altcat_2(F,A) )
=> k14_functor0(A,C,F,k13_functor0(A,B,C,D,E)) = k13_functor0(F,B,C,k14_functor0(A,B,F,D),E) ) ) ) ) ) ) ).
fof(t16_functor1,axiom,
$true ).
fof(t17_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_altcat_2(B,A) )
=> ( v2_altcat_2(B,A)
<=> v18_functor0(k10_functor0(A,B),B,A) ) ) ) ).
fof(t18_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_altcat_1(B) )
=> ! [C] :
( ( v10_functor0(C,A,B)
& l2_functor0(C,A,B) )
=> ( v18_functor0(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> v2_funct_2(k5_functor0(A,B,C,D,E),k1_altcat_1(A,D,E),k1_altcat_1(B,k3_functor0(A,B,C,D),k3_functor0(A,B,C,E))) ) ) ) ) ) ) ).
fof(t19_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l2_altcat_1(B) )
=> ! [C] :
( ( v10_functor0(C,A,B)
& l2_functor0(C,A,B) )
=> ( v17_functor0(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> v2_funct_1(k5_functor0(A,B,C,D,E)) ) ) ) ) ) ) ).
fof(t20_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> k15_functor0(A,A,k12_functor0(A)) = k12_functor0(A) ) ).
fof(t21_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v12_altcat_1(A)
& v1_altcat_2(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& v12_altcat_1(B)
& v1_altcat_2(B)
& l2_altcat_1(B) )
=> ! [C] :
( ( v8_functor0(C,A,B)
& l2_functor0(C,A,B) )
=> ( v21_functor0(C,A,B)
=> ! [D] :
( ( v8_functor0(D,B,A)
& l2_functor0(D,B,A) )
=> ( g2_functor0(B,A,u1_functor0(B,A,D),u2_functor0(B,A,D)) = k15_functor0(A,B,C)
=> k13_functor0(B,A,B,D,C) = k12_functor0(B) ) ) ) ) ) ) ).
fof(t22_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v12_altcat_1(A)
& v1_altcat_2(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& v12_altcat_1(B)
& v1_altcat_2(B)
& l2_altcat_1(B) )
=> ! [C] :
( ( v8_functor0(C,A,B)
& l2_functor0(C,A,B) )
=> ( v21_functor0(C,A,B)
=> k13_functor0(A,B,A,C,k15_functor0(A,B,C)) = k12_functor0(A) ) ) ) ) ).
fof(t23_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v12_altcat_1(A)
& v1_altcat_2(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& v12_altcat_1(B)
& v1_altcat_2(B)
& l2_altcat_1(B) )
=> ! [C] :
( ( v8_functor0(C,A,B)
& l2_functor0(C,A,B) )
=> ( v21_functor0(C,A,B)
=> k15_functor0(B,A,k15_functor0(A,B,C)) = g2_functor0(A,B,u1_functor0(A,B,C),u2_functor0(A,B,C)) ) ) ) ) ).
fof(t24_functor1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v12_altcat_1(A)
& v1_altcat_2(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& v12_altcat_1(B)
& v1_altcat_2(B)
& l2_altcat_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_altcat_1(C)
& v12_altcat_1(C)
& v1_altcat_2(C)
& l2_altcat_1(C) )
=> ! [D] :
( ( v8_functor0(D,A,B)
& l2_functor0(D,A,B) )
=> ! [E] :
( ( v8_functor0(E,B,C)
& l2_functor0(E,B,C) )
=> ! [F] :
( ( v8_functor0(F,B,A)
& l2_functor0(F,B,A) )
=> ! [G] :
( ( v8_functor0(G,C,B)
& l2_functor0(G,C,B) )
=> ( ( v21_functor0(E,B,C)
& v21_functor0(D,A,B)
& v21_functor0(G,C,B)
& v21_functor0(F,B,A)
& g2_functor0(B,A,u1_functor0(B,A,F),u2_functor0(B,A,F)) = k15_functor0(A,B,D)
& g2_functor0(C,B,u1_functor0(C,B,G),u2_functor0(C,B,G)) = k15_functor0(B,C,E) )
=> k15_functor0(A,C,k13_functor0(A,B,C,D,E)) = k13_functor0(C,B,A,G,F) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------