TPTP Problem File: CAT033+1.p
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%------------------------------------------------------------------------------
% File : CAT033+1 : TPTP v9.0.0. Released v3.4.0.
% Domain : Category Theory
% Problem : Yoneda Embedding T02
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Woj97] Wojciechowski (1997), Yoneda Embedding
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t2_yoneda_1 [Urb08]
% Status : Theorem
% Rating : 0.21 v9.0.0, 0.25 v7.5.0, 0.31 v7.4.0, 0.17 v7.3.0, 0.21 v7.2.0, 0.17 v7.1.0, 0.22 v7.0.0, 0.23 v6.4.0, 0.31 v6.3.0, 0.29 v6.2.0, 0.36 v6.1.0, 0.40 v6.0.0, 0.35 v5.5.0, 0.33 v5.4.0, 0.36 v5.3.0, 0.41 v5.2.0, 0.25 v5.1.0, 0.24 v5.0.0, 0.38 v4.1.0, 0.43 v4.0.1, 0.39 v4.0.0, 0.42 v3.7.0, 0.30 v3.5.0, 0.32 v3.4.0
% Syntax : Number of formulae : 53 ( 16 unt; 0 def)
% Number of atoms : 163 ( 11 equ)
% Maximal formula atoms : 20 ( 3 avg)
% Number of connectives : 128 ( 18 ~; 1 |; 71 &)
% ( 2 <=>; 36 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 15 ( 15 usr; 1 con; 0-6 aty)
% Number of variables : 90 ( 80 !; 10 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t2_yoneda_1,conjecture,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> m2_cat_1(k20_ens_1(A,B),A,k1_yoneda_1(A)) ) ) ).
fof(rc3_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ).
fof(cc2_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A) ) ).
fof(rc2_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) ) ).
fof(free_g1_cat_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,B,A)
& m1_relset_1(C,B,A)
& v1_funct_1(D)
& v1_funct_2(D,B,A)
& m1_relset_1(D,B,A)
& v1_funct_1(E)
& m1_relset_1(E,k2_zfmisc_1(B,B),B)
& v1_funct_1(F)
& v1_funct_2(F,A,B)
& m1_relset_1(F,A,B) )
=> ! [G,H,I,J,K,L] :
( g1_cat_1(A,B,C,D,E,F) = g1_cat_1(G,H,I,J,K,L)
=> ( A = G
& B = H
& C = I
& D = J
& E = K
& F = L ) ) ) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(dt_g1_cat_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,B,A)
& m1_relset_1(C,B,A)
& v1_funct_1(D)
& v1_funct_2(D,B,A)
& m1_relset_1(D,B,A)
& v1_funct_1(E)
& m1_relset_1(E,k2_zfmisc_1(B,B),B)
& v1_funct_1(F)
& v1_funct_2(F,A,B)
& m1_relset_1(F,A,B) )
=> ( v1_cat_1(g1_cat_1(A,B,C,D,E,F))
& l1_cat_1(g1_cat_1(A,B,C,D,E,F)) ) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_u3_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_funct_1(u3_cat_1(A))
& v1_funct_2(u3_cat_1(A),u2_cat_1(A),u1_cat_1(A))
& m2_relset_1(u3_cat_1(A),u2_cat_1(A),u1_cat_1(A)) ) ) ).
fof(dt_u4_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_funct_1(u4_cat_1(A))
& v1_funct_2(u4_cat_1(A),u2_cat_1(A),u1_cat_1(A))
& m2_relset_1(u4_cat_1(A),u2_cat_1(A),u1_cat_1(A)) ) ) ).
fof(dt_u5_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_funct_1(u5_cat_1(A))
& m2_relset_1(u5_cat_1(A),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(A)),u2_cat_1(A)) ) ) ).
fof(dt_u6_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_funct_1(u6_cat_1(A))
& v1_funct_2(u6_cat_1(A),u1_cat_1(A),u2_cat_1(A))
& m2_relset_1(u6_cat_1(A),u1_cat_1(A),u2_cat_1(A)) ) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(abstractness_v1_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ( v1_cat_1(A)
=> A = g1_cat_1(u1_cat_1(A),u2_cat_1(A),u3_cat_1(A),u4_cat_1(A),u5_cat_1(A),u6_cat_1(A)) ) ) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_ens_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(dt_u2_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ~ v1_xboole_0(u2_cat_1(A)) ) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(fc2_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k2_ens_1(A)) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(existence_l1_cat_1,axiom,
? [A] : l1_cat_1(A) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_cat_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ? [C] : m2_cat_1(C,A,B) ) ).
fof(dt_k12_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> l1_cat_1(k12_ens_1(A)) ) ).
fof(dt_k17_ens_1,axiom,
$true ).
fof(dt_k1_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v2_cat_1(k1_yoneda_1(A))
& l1_cat_1(k1_yoneda_1(A)) ) ) ).
fof(dt_k20_ens_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_funct_1(k20_ens_1(A,B))
& v1_funct_2(k20_ens_1(A,B),u2_cat_1(A),k2_ens_1(k17_ens_1(A)))
& m2_relset_1(k20_ens_1(A,B),u2_cat_1(A),k2_ens_1(k17_ens_1(A))) ) ) ).
fof(dt_l1_cat_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_cat_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m2_cat_1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,u2_cat_1(A),u2_cat_1(B))
& m2_relset_1(C,u2_cat_1(A),u2_cat_1(B)) ) ) ) ).
fof(dt_u1_cat_1,axiom,
! [A] :
( l1_cat_1(A)
=> ~ v1_xboole_0(u1_cat_1(A)) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_funct_1(A) ) ).
fof(fc4_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_cat_1(k12_ens_1(A))
& v2_cat_1(k12_ens_1(A)) ) ) ).
fof(fc5_ens_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ~ v1_xboole_0(k17_ens_1(A)) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(d1_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> k1_yoneda_1(A) = k12_ens_1(k17_ens_1(A)) ) ).
fof(t49_ens_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ( r1_tarski(k17_ens_1(B),A)
=> m2_cat_1(k20_ens_1(B,C),B,k12_ens_1(A)) ) ) ) ) ).
%------------------------------------------------------------------------------