TPTP Problem File: CAT026+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : CAT026+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Category Theory
% Problem : Some Isomorphisms Between Functor Categories T38
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Try92] Trybulec (1992), Some Isomorphisms Between Functor Cat
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t38_isocat_2 [Urb08]
% Status : Theorem
% Rating : 0.53 v8.1.0, 0.58 v7.5.0, 0.59 v7.4.0, 0.47 v7.3.0, 0.55 v7.2.0, 0.52 v7.1.0, 0.48 v7.0.0, 0.57 v6.4.0, 0.62 v6.3.0, 0.50 v6.2.0, 0.56 v6.1.0, 0.60 v6.0.0, 0.57 v5.5.0, 0.70 v5.4.0, 0.71 v5.3.0, 0.78 v5.2.0, 0.70 v5.1.0, 0.71 v5.0.0, 0.75 v4.1.0, 0.78 v4.0.1, 0.87 v4.0.0, 0.88 v3.7.0, 0.85 v3.5.0, 0.89 v3.4.0
% Syntax : Number of formulae : 50 ( 15 unt; 0 def)
% Number of atoms : 186 ( 7 equ)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 149 ( 13 ~; 1 |; 85 &)
% ( 2 <=>; 48 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 15 ( 13 usr; 1 prp; 0-4 aty)
% Number of functors : 13 ( 13 usr; 1 con; 0-5 aty)
% Number of variables : 113 ( 104 !; 9 ?)
% 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(t38_isocat_2,conjecture,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,A,k11_cat_2(B,C))
=> ! [E] :
( m2_cat_1(E,A,k11_cat_2(B,C))
=> ( r2_nattra_1(A,k11_cat_2(B,C),D,E)
=> ( r2_nattra_1(A,B,k11_isocat_2(A,B,C,D),k11_isocat_2(A,B,C,E))
& r2_nattra_1(A,C,k12_isocat_2(A,B,C,D),k12_isocat_2(A,B,C,E)) ) ) ) ) ) ) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
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(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(dt_k16_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k16_cat_2(A,B),k11_cat_2(A,B),A) ) ).
fof(dt_k17_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k17_cat_2(A,B),k11_cat_2(A,B),B) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
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(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
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(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(redefinition_k8_isocat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> k8_isocat_2(A,B) = k16_cat_2(A,B) ) ).
fof(redefinition_k9_isocat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> k9_isocat_2(A,B) = k17_cat_2(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_k5_relat_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> v1_relat_1(k5_relat_1(A,B)) ) ).
fof(dt_k8_isocat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k8_isocat_2(A,B),k11_cat_2(A,B),A) ) ).
fof(dt_k9_isocat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k9_isocat_2(A,B),k11_cat_2(A,B),B) ) ).
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(reflexivity_r2_nattra_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m2_cat_1(C,A,B)
& m2_cat_1(D,A,B) )
=> r2_nattra_1(A,B,C,C) ) ).
fof(existence_l1_cat_1,axiom,
? [A] : l1_cat_1(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(redefinition_k2_isocat_1,axiom,
! [A,B,C,D,E] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m2_cat_1(D,A,B)
& m2_cat_1(E,B,C) )
=> k2_isocat_1(A,B,C,D,E) = k5_relat_1(D,E) ) ).
fof(dt_k11_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ( v2_cat_1(k11_cat_2(A,B))
& l1_cat_1(k11_cat_2(A,B)) ) ) ).
fof(dt_k11_isocat_2,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m2_cat_1(D,A,k11_cat_2(B,C)) )
=> m2_cat_1(k11_isocat_2(A,B,C,D),A,B) ) ).
fof(dt_k12_isocat_2,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m2_cat_1(D,A,k11_cat_2(B,C)) )
=> m2_cat_1(k12_isocat_2(A,B,C,D),A,C) ) ).
fof(dt_k2_isocat_1,axiom,
! [A,B,C,D,E] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m2_cat_1(D,A,B)
& m2_cat_1(E,B,C) )
=> m2_cat_1(k2_isocat_1(A,B,C,D,E),A,C) ) ).
fof(dt_l1_cat_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(d7_isocat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,A,k11_cat_2(B,C))
=> k11_isocat_2(A,B,C,D) = k2_isocat_1(A,k11_cat_2(B,C),B,D,k8_isocat_2(B,C)) ) ) ) ) ).
fof(d8_isocat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,A,k11_cat_2(B,C))
=> k12_isocat_2(A,B,C,D) = k2_isocat_1(A,k11_cat_2(B,C),C,D,k9_isocat_2(B,C)) ) ) ) ) ).
fof(t27_isocat_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,B,C)
=> ! [G] :
( m2_cat_1(G,B,C)
=> ( ( r2_nattra_1(A,B,D,E)
& r2_nattra_1(B,C,F,G) )
=> r2_nattra_1(A,C,k2_isocat_1(A,B,C,D,F),k2_isocat_1(A,B,C,E,G)) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------