TPTP Problem File: CAT021+2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : CAT021+2 : TPTP v8.2.0. Released v3.4.0.
% Domain : Category Theory
% Problem : Some Isomorphisms Between Functor Categories T11
% 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 : t11_isocat_2 [Urb08]
% Status : Theorem
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 4027 (1076 unt; 0 def)
% Number of atoms : 18258 (3434 equ)
% Maximal formula atoms : 52 ( 4 avg)
% Number of connectives : 15839 (1608 ~; 197 |;6462 &)
% ( 599 <=>;6973 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 208 ( 206 usr; 1 prp; 0-6 aty)
% Number of functors : 799 ( 799 usr; 292 con; 0-9 aty)
% Number of variables : 10804 (10294 !; 510 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Bushy version: includes all articles that contribute axioms to the
% Normal 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+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+18.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+21.ax').
include('Axioms/SET007/SET007+22.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+34.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+54.ax').
include('Axioms/SET007/SET007+55.ax').
include('Axioms/SET007/SET007+61.ax').
include('Axioms/SET007/SET007+64.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+202.ax').
include('Axioms/SET007/SET007+237.ax').
include('Axioms/SET007/SET007+299.ax').
include('Axioms/SET007/SET007+322.ax').
%------------------------------------------------------------------------------
fof(dt_k1_isocat_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,k1_fraenkel(B,C))
& m1_relset_1(D,A,k1_fraenkel(B,C)) )
=> ( v1_funct_1(k1_isocat_2(A,B,C,D))
& v1_funct_2(k1_isocat_2(A,B,C,D),k2_zfmisc_1(A,B),C)
& m2_relset_1(k1_isocat_2(A,B,C,D),k2_zfmisc_1(A,B),C) ) ) ).
fof(redefinition_k1_isocat_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,k1_fraenkel(B,C))
& m1_relset_1(D,A,k1_fraenkel(B,C)) )
=> k1_isocat_2(A,B,C,D) = k4_funct_5(D) ) ).
fof(dt_k2_isocat_2,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u1_cat_1(A)) )
=> m2_cat_1(k2_isocat_2(A,B,C),k12_nattra_1(A,B),B) ) ).
fof(dt_k3_isocat_2,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,k11_cat_2(A,B),C)
& m1_subset_1(E,u2_cat_1(A)) )
=> ( v1_funct_1(k3_isocat_2(A,B,C,D,E))
& v1_funct_2(k3_isocat_2(A,B,C,D,E),u2_cat_1(B),u2_cat_1(C))
& m2_relset_1(k3_isocat_2(A,B,C,D,E),u2_cat_1(B),u2_cat_1(C)) ) ) ).
fof(dt_k4_isocat_2,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,k11_cat_2(A,B),C)
& m1_subset_1(E,u2_cat_1(A)) )
=> m2_nattra_1(k4_isocat_2(A,B,C,D,E),B,C,k14_cat_2(A,B,C,D,k2_cat_1(A,E)),k14_cat_2(A,B,C,D,k3_cat_1(A,E))) ) ).
fof(dt_k5_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,k11_cat_2(A,B),C) )
=> m2_cat_1(k5_isocat_2(A,B,C,D),A,k12_nattra_1(B,C)) ) ).
fof(dt_k6_isocat_2,axiom,
! [A,B,C,D,E,F] :
( ( 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,k11_cat_2(A,B),C)
& m2_cat_1(E,k11_cat_2(A,B),C)
& m2_nattra_1(F,k11_cat_2(A,B),C,D,E) )
=> m2_nattra_1(k6_isocat_2(A,B,C,D,E,F),A,k12_nattra_1(B,C),k5_isocat_2(A,B,C,D),k5_isocat_2(A,B,C,E)) ) ).
fof(dt_k7_isocat_2,axiom,
! [A,B,C] :
( ( 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(k7_isocat_2(A,B,C),k12_nattra_1(k11_cat_2(A,B),C),k12_nattra_1(A,k12_nattra_1(B,C))) ) ).
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(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(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(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(dt_k10_isocat_2,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,A,C) )
=> m2_cat_1(k10_isocat_2(A,B,C,D,E),A,k11_cat_2(B,C)) ) ).
fof(redefinition_k10_isocat_2,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,A,C) )
=> k10_isocat_2(A,B,C,D,E) = k13_funct_3(D,E) ) ).
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_k13_isocat_2,axiom,
! [A,B,C,D,E,F] :
( ( 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(E,A,k11_cat_2(B,C))
& m2_nattra_1(F,A,k11_cat_2(B,C),D,E) )
=> m2_nattra_1(k13_isocat_2(A,B,C,D,E,F),A,B,k11_isocat_2(A,B,C,D),k11_isocat_2(A,B,C,E)) ) ).
fof(dt_k14_isocat_2,axiom,
! [A,B,C,D,E,F] :
( ( 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(E,A,k11_cat_2(B,C))
& m2_nattra_1(F,A,k11_cat_2(B,C),D,E) )
=> m2_nattra_1(k14_isocat_2(A,B,C,D,E,F),A,C,k12_isocat_2(A,B,C,D),k12_isocat_2(A,B,C,E)) ) ).
fof(dt_k15_isocat_2,axiom,
! [A,B,C,D,E,F,G,H,I] :
( ( 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,A,B)
& m2_cat_1(F,A,C)
& m2_cat_1(G,A,C)
& m1_nattra_1(H,A,B,D,E)
& m1_nattra_1(I,A,C,F,G) )
=> m1_nattra_1(k15_isocat_2(A,B,C,D,E,F,G,H,I),A,k11_cat_2(B,C),k10_isocat_2(A,B,C,D,F),k10_isocat_2(A,B,C,E,G)) ) ).
fof(dt_k16_isocat_2,axiom,
! [A,B,C,D,E,F,G,H,I] :
( ( 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,A,B)
& m2_cat_1(F,A,C)
& m2_cat_1(G,A,C)
& m2_nattra_1(H,A,B,D,E)
& m2_nattra_1(I,A,C,F,G) )
=> m2_nattra_1(k16_isocat_2(A,B,C,D,E,F,G,H,I),A,k11_cat_2(B,C),k10_isocat_2(A,B,C,D,F),k10_isocat_2(A,B,C,E,G)) ) ).
fof(dt_k17_isocat_2,axiom,
! [A,B,C] :
( ( 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(k17_isocat_2(A,B,C),k12_nattra_1(A,k11_cat_2(B,C)),k11_cat_2(k12_nattra_1(A,B),k12_nattra_1(A,C))) ) ).
fof(t1_isocat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_fraenkel(B,C))
& m2_relset_1(D,A,k1_fraenkel(B,C)) )
=> r4_nattra_1(A,k1_fraenkel(B,C),A,k1_fraenkel(B,C),k2_cat_2(A,B,C,k1_isocat_2(A,B,C,D)),D) ) ) ) ) ).
fof(t2_isocat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_fraenkel(B,C))
& m2_relset_1(D,A,k1_fraenkel(B,C)) )
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> k1_funct_1(k1_isocat_2(A,B,C,D),k4_tarski(E,F)) = k8_funct_2(B,C,k1_cat_2(A,B,C,k1_fraenkel(B,C),D,E),F) ) ) ) ) ) ) ).
fof(t3_isocat_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_tarski(A),B)
& m2_relset_1(C,k1_tarski(A),B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_tarski(A),B)
& m2_relset_1(D,k1_tarski(A),B) )
=> ( k1_funct_1(C,A) = k1_funct_1(D,A)
=> r4_nattra_1(k1_tarski(A),B,k1_tarski(A),B,C,D) ) ) ) ) ).
fof(t4_isocat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> r2_hidden(k8_funct_2(A,B,D,C),k2_relat_1(D)) ) ) ) ) ).
fof(t5_isocat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(B,C))
& m2_relset_1(D,A,k2_zfmisc_1(B,C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,k2_zfmisc_1(B,C))
& m2_relset_1(E,A,k2_zfmisc_1(B,C)) )
=> ( ( r4_nattra_1(A,B,A,B,k7_funct_2(A,k2_zfmisc_1(B,C),B,D,k9_funct_3(B,C)),k7_funct_2(A,k2_zfmisc_1(B,C),B,E,k9_funct_3(B,C)))
& r4_nattra_1(A,C,A,C,k7_funct_2(A,k2_zfmisc_1(B,C),C,D,k10_funct_3(B,C)),k7_funct_2(A,k2_zfmisc_1(B,C),C,E,k10_funct_3(B,C))) )
=> r4_nattra_1(A,k2_zfmisc_1(B,C),A,k2_zfmisc_1(B,C),D,E) ) ) ) ) ) ) ).
fof(t6_isocat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> k4_cat_1(A,B,k10_cat_1(A,k3_cat_1(A,B))) = B ) ) ).
fof(t7_isocat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> k4_cat_1(A,k10_cat_1(A,k2_cat_1(A,B)),B) = B ) ) ).
fof(t8_isocat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(k12_nattra_1(A,B)))
<=> m2_cat_1(C,A,B) ) ) ) ).
fof(t9_isocat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(k12_nattra_1(A,B)))
=> ? [D] :
( m2_cat_1(D,A,B)
& ? [E] :
( m2_cat_1(E,A,B)
& ? [F] :
( m2_nattra_1(F,A,B,D,E)
& r2_nattra_1(A,B,D,E)
& k2_cat_1(k12_nattra_1(A,B),C) = D
& k3_cat_1(k12_nattra_1(A,B),C) = E
& C = k4_tarski(k4_tarski(D,E),F) ) ) ) ) ) ) ).
fof(d1_isocat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m2_cat_1(D,k12_nattra_1(A,B),B)
=> ( D = k2_isocat_2(A,B,C)
<=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,A,B)
=> ! [G] :
( m2_nattra_1(G,A,B,E,F)
=> ( r2_nattra_1(A,B,E,F)
=> k1_funct_1(D,k4_tarski(k4_tarski(E,F),G)) = k5_nattra_1(A,B,E,F,G,C) ) ) ) ) ) ) ) ) ) ).
fof(l11_isocat_2,axiom,
! [A,B] :
( u1_cat_1(k8_cat_1(A,B)) = k1_tarski(A)
& u2_cat_1(k8_cat_1(A,B)) = k1_tarski(B) ) ).
fof(t10_isocat_2,axiom,
$true ).
fof(t11_isocat_2,conjecture,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B,C] : r1_isocat_1(k12_nattra_1(k8_cat_1(B,C),A),A) ) ).
%------------------------------------------------------------------------------