TPTP Problem File: CAT033+3.p
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%------------------------------------------------------------------------------
% File : CAT033+3 : TPTP v8.2.0. 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 : 0.75 v8.2.0, 0.78 v8.1.0, 0.81 v7.4.0, 0.87 v7.3.0, 0.90 v7.1.0, 0.87 v7.0.0, 0.90 v6.4.0, 0.85 v6.3.0, 0.88 v6.2.0, 0.92 v6.1.0, 0.93 v6.0.0, 0.91 v5.5.0, 0.96 v5.4.0, 1.00 v5.2.0, 0.95 v5.0.0, 0.96 v4.1.0, 1.00 v3.4.0
% Syntax : Number of formulae : 16909 (2844 unt; 0 def)
% Number of atoms : 113752 (11340 equ)
% Maximal formula atoms : 62 ( 6 avg)
% Number of connectives : 111211 (14368 ~; 513 |;56735 &)
% (3150 <=>;36445 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 8 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 1054 (1052 usr; 2 prp; 0-6 aty)
% Number of functors : 2465 (2465 usr; 607 con; 0-10 aty)
% Number of variables : 43661 (41535 !;2126 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Chainy small version: includes all preceding MML articles that
% are included in any Bushy version.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
include('Axioms/SET007/SET007+0.ax').
include('Axioms/SET007/SET007+1.ax').
include('Axioms/SET007/SET007+2.ax').
include('Axioms/SET007/SET007+3.ax').
include('Axioms/SET007/SET007+4.ax').
include('Axioms/SET007/SET007+5.ax').
include('Axioms/SET007/SET007+6.ax').
include('Axioms/SET007/SET007+7.ax').
include('Axioms/SET007/SET007+8.ax').
include('Axioms/SET007/SET007+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+13.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+15.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+17.ax').
include('Axioms/SET007/SET007+18.ax').
include('Axioms/SET007/SET007+19.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+21.ax').
include('Axioms/SET007/SET007+22.ax').
include('Axioms/SET007/SET007+23.ax').
include('Axioms/SET007/SET007+24.ax').
include('Axioms/SET007/SET007+25.ax').
include('Axioms/SET007/SET007+26.ax').
include('Axioms/SET007/SET007+31.ax').
include('Axioms/SET007/SET007+32.ax').
include('Axioms/SET007/SET007+33.ax').
include('Axioms/SET007/SET007+34.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+40.ax').
include('Axioms/SET007/SET007+48.ax').
include('Axioms/SET007/SET007+50.ax').
include('Axioms/SET007/SET007+51.ax').
include('Axioms/SET007/SET007+54.ax').
include('Axioms/SET007/SET007+55.ax').
include('Axioms/SET007/SET007+59.ax').
include('Axioms/SET007/SET007+60.ax').
include('Axioms/SET007/SET007+61.ax').
include('Axioms/SET007/SET007+64.ax').
include('Axioms/SET007/SET007+66.ax').
include('Axioms/SET007/SET007+67.ax').
include('Axioms/SET007/SET007+68.ax').
include('Axioms/SET007/SET007+71.ax').
include('Axioms/SET007/SET007+75.ax').
include('Axioms/SET007/SET007+76.ax').
include('Axioms/SET007/SET007+77.ax').
include('Axioms/SET007/SET007+79.ax').
include('Axioms/SET007/SET007+80.ax').
include('Axioms/SET007/SET007+86.ax').
include('Axioms/SET007/SET007+91.ax').
include('Axioms/SET007/SET007+117.ax').
include('Axioms/SET007/SET007+125.ax').
include('Axioms/SET007/SET007+126.ax').
include('Axioms/SET007/SET007+148.ax').
include('Axioms/SET007/SET007+159.ax').
include('Axioms/SET007/SET007+165.ax').
include('Axioms/SET007/SET007+170.ax').
include('Axioms/SET007/SET007+182.ax').
include('Axioms/SET007/SET007+186.ax').
include('Axioms/SET007/SET007+188.ax').
include('Axioms/SET007/SET007+190.ax').
include('Axioms/SET007/SET007+200.ax').
include('Axioms/SET007/SET007+202.ax').
include('Axioms/SET007/SET007+205.ax').
include('Axioms/SET007/SET007+206.ax').
include('Axioms/SET007/SET007+207.ax').
include('Axioms/SET007/SET007+209.ax').
include('Axioms/SET007/SET007+210.ax').
include('Axioms/SET007/SET007+211.ax').
include('Axioms/SET007/SET007+212.ax').
include('Axioms/SET007/SET007+213.ax').
include('Axioms/SET007/SET007+217.ax').
include('Axioms/SET007/SET007+218.ax').
include('Axioms/SET007/SET007+223.ax').
include('Axioms/SET007/SET007+224.ax').
include('Axioms/SET007/SET007+225.ax').
include('Axioms/SET007/SET007+227.ax').
include('Axioms/SET007/SET007+237.ax').
include('Axioms/SET007/SET007+241.ax').
include('Axioms/SET007/SET007+242.ax').
include('Axioms/SET007/SET007+246.ax').
include('Axioms/SET007/SET007+247.ax').
include('Axioms/SET007/SET007+248.ax').
include('Axioms/SET007/SET007+252.ax').
include('Axioms/SET007/SET007+253.ax').
include('Axioms/SET007/SET007+255.ax').
include('Axioms/SET007/SET007+256.ax').
include('Axioms/SET007/SET007+276.ax').
include('Axioms/SET007/SET007+278.ax').
include('Axioms/SET007/SET007+279.ax').
include('Axioms/SET007/SET007+280.ax').
include('Axioms/SET007/SET007+281.ax').
include('Axioms/SET007/SET007+293.ax').
include('Axioms/SET007/SET007+295.ax').
include('Axioms/SET007/SET007+297.ax').
include('Axioms/SET007/SET007+298.ax').
include('Axioms/SET007/SET007+299.ax').
include('Axioms/SET007/SET007+301.ax').
include('Axioms/SET007/SET007+308.ax').
include('Axioms/SET007/SET007+309.ax').
include('Axioms/SET007/SET007+311.ax').
include('Axioms/SET007/SET007+312.ax').
include('Axioms/SET007/SET007+317.ax').
include('Axioms/SET007/SET007+321.ax').
include('Axioms/SET007/SET007+322.ax').
include('Axioms/SET007/SET007+327.ax').
include('Axioms/SET007/SET007+335.ax').
include('Axioms/SET007/SET007+338.ax').
include('Axioms/SET007/SET007+339.ax').
include('Axioms/SET007/SET007+354.ax').
include('Axioms/SET007/SET007+363.ax').
include('Axioms/SET007/SET007+365.ax').
include('Axioms/SET007/SET007+370.ax').
include('Axioms/SET007/SET007+375.ax').
include('Axioms/SET007/SET007+377.ax').
include('Axioms/SET007/SET007+384.ax').
include('Axioms/SET007/SET007+387.ax').
include('Axioms/SET007/SET007+388.ax').
include('Axioms/SET007/SET007+393.ax').
include('Axioms/SET007/SET007+394.ax').
include('Axioms/SET007/SET007+395.ax').
include('Axioms/SET007/SET007+396.ax').
include('Axioms/SET007/SET007+399.ax').
include('Axioms/SET007/SET007+401.ax').
include('Axioms/SET007/SET007+405.ax').
include('Axioms/SET007/SET007+406.ax').
include('Axioms/SET007/SET007+407.ax').
include('Axioms/SET007/SET007+411.ax').
include('Axioms/SET007/SET007+412.ax').
include('Axioms/SET007/SET007+426.ax').
include('Axioms/SET007/SET007+427.ax').
include('Axioms/SET007/SET007+432.ax').
include('Axioms/SET007/SET007+433.ax').
include('Axioms/SET007/SET007+438.ax').
include('Axioms/SET007/SET007+441.ax').
include('Axioms/SET007/SET007+445.ax').
include('Axioms/SET007/SET007+448.ax').
include('Axioms/SET007/SET007+449.ax').
include('Axioms/SET007/SET007+455.ax').
include('Axioms/SET007/SET007+463.ax').
include('Axioms/SET007/SET007+464.ax').
include('Axioms/SET007/SET007+466.ax').
include('Axioms/SET007/SET007+480.ax').
include('Axioms/SET007/SET007+481.ax').
include('Axioms/SET007/SET007+483.ax').
include('Axioms/SET007/SET007+484.ax').
include('Axioms/SET007/SET007+485.ax').
include('Axioms/SET007/SET007+486.ax').
include('Axioms/SET007/SET007+487.ax').
include('Axioms/SET007/SET007+488.ax').
include('Axioms/SET007/SET007+489.ax').
include('Axioms/SET007/SET007+490.ax').
include('Axioms/SET007/SET007+492.ax').
include('Axioms/SET007/SET007+493.ax').
include('Axioms/SET007/SET007+494.ax').
include('Axioms/SET007/SET007+495.ax').
include('Axioms/SET007/SET007+496.ax').
include('Axioms/SET007/SET007+497.ax').
include('Axioms/SET007/SET007+498.ax').
include('Axioms/SET007/SET007+500.ax').
include('Axioms/SET007/SET007+503.ax').
include('Axioms/SET007/SET007+505.ax').
include('Axioms/SET007/SET007+506.ax').
include('Axioms/SET007/SET007+509.ax').
include('Axioms/SET007/SET007+513.ax').
%------------------------------------------------------------------------------
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_k2_yoneda_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> m2_cat_1(k2_yoneda_1(A,B),A,k1_yoneda_1(A)) ) ).
fof(dt_k3_yoneda_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m2_nattra_1(k3_yoneda_1(A,B),A,k1_yoneda_1(A),k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B))) ) ).
fof(dt_k4_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> m1_oppcat_1(k4_yoneda_1(A),A,k12_nattra_1(A,k1_yoneda_1(A))) ) ).
fof(dt_k5_yoneda_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_oppcat_1(C,A,B)
& m1_subset_1(D,u1_cat_1(A)) )
=> m1_subset_1(k5_yoneda_1(A,B,C,D),u1_cat_1(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(t1_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( m1_subset_1(D,u2_cat_1(k1_yoneda_1(A)))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(k1_yoneda_1(A)))
=> ( ( k3_cat_1(k1_yoneda_1(A),D) = k2_cat_1(k1_yoneda_1(A),E)
& k4_tarski(k12_cat_2(k1_yoneda_1(A),k1_yoneda_1(A),k2_cat_1(k1_yoneda_1(A),D),k3_cat_1(k1_yoneda_1(A),D)),B) = D
& k4_tarski(k12_cat_2(k1_yoneda_1(A),k1_yoneda_1(A),k2_cat_1(k1_yoneda_1(A),E),k3_cat_1(k1_yoneda_1(A),E)),C) = E )
=> k4_tarski(k12_cat_2(k1_yoneda_1(A),k1_yoneda_1(A),k2_cat_1(k1_yoneda_1(A),D),k3_cat_1(k1_yoneda_1(A),E)),k5_relat_1(B,C)) = k4_cat_1(k1_yoneda_1(A),D,E) ) ) ) ) ) ) ).
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)) ) ) ).
%------------------------------------------------------------------------------