TPTP Problem File: SEU382+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU382+1 : TPTP v8.2.0. Released v3.3.0.
% Domain : Set theory
% Problem : MPTP bushy problem t4_waybel_7
% Version : [Urb07] axioms : Especial.
% English :
% Refs : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
% : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source : [Urb07]
% Names : bushy-t4_waybel_7 [Urb07]
% Status : Theorem
% Rating : 0.31 v8.2.0, 0.28 v7.5.0, 0.34 v7.4.0, 0.17 v7.3.0, 0.28 v7.1.0, 0.22 v7.0.0, 0.23 v6.4.0, 0.27 v6.3.0, 0.33 v6.2.0, 0.44 v6.1.0, 0.57 v6.0.0, 0.52 v5.5.0, 0.56 v5.4.0, 0.57 v5.3.0, 0.63 v5.2.0, 0.50 v5.1.0, 0.52 v5.0.0, 0.50 v4.1.0, 0.52 v4.0.0, 0.54 v3.7.0, 0.55 v3.5.0, 0.58 v3.3.0
% Syntax : Number of formulae : 89 ( 22 unt; 0 def)
% Number of atoms : 423 ( 10 equ)
% Maximal formula atoms : 22 ( 4 avg)
% Number of connectives : 408 ( 74 ~; 1 |; 262 &)
% ( 2 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 34 ( 32 usr; 1 prp; 0-3 aty)
% Number of functors : 10 ( 10 usr; 1 con; 0-2 aty)
% Number of variables : 113 ( 92 !; 21 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(abstractness_v1_orders_2,axiom,
! [A] :
( rel_str(A)
=> ( strict_rel_str(A)
=> A = rel_str_of(the_carrier(A),the_InternalRel(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc10_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& complete_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& up_complete_relstr(A)
& join_complete_relstr(A) ) ) ) ).
fof(cc10_waybel_1,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& boolean_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& upper_bounded_relstr(A)
& distributive_relstr(A)
& heyting_relstr(A) ) ) ) ).
fof(cc11_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& join_complete_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& lower_bounded_relstr(A) ) ) ) ).
fof(cc12_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& lower_bounded_relstr(A)
& up_complete_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& complete_relstr(A)
& lower_bounded_relstr(A)
& upper_bounded_relstr(A)
& bounded_relstr(A) ) ) ) ).
fof(cc13_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& antisymmetric_relstr(A)
& join_complete_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& antisymmetric_relstr(A)
& with_infima_relstr(A) ) ) ) ).
fof(cc14_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& antisymmetric_relstr(A)
& upper_bounded_relstr(A)
& join_complete_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& upper_bounded_relstr(A) ) ) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(cc1_lattice3,axiom,
! [A] :
( rel_str(A)
=> ( with_suprema_relstr(A)
=> ~ empty_carrier(A) ) ) ).
fof(cc1_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& complete_relstr(A) )
=> ( ~ empty_carrier(A)
& with_suprema_relstr(A)
& with_infima_relstr(A) ) ) ) ).
fof(cc1_yellow_2,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& upper_bounded_relstr(A)
& up_complete_relstr(A)
& join_complete_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& lower_bounded_relstr(A)
& upper_bounded_relstr(A)
& bounded_relstr(A)
& up_complete_relstr(A)
& join_complete_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& complete_relstr(A) ) ) ) ).
fof(cc1_yellow_3,axiom,
! [A] :
( rel_str(A)
=> ( empty_carrier(A)
=> v1_yellow_3(A) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(cc2_lattice3,axiom,
! [A] :
( rel_str(A)
=> ( with_infima_relstr(A)
=> ~ empty_carrier(A) ) ) ).
fof(cc2_yellow_3,axiom,
! [A] :
( rel_str(A)
=> ( ~ v1_yellow_3(A)
=> ~ empty_carrier(A) ) ) ).
fof(cc3_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& complete_relstr(A) )
=> ( ~ empty_carrier(A)
& bounded_relstr(A) ) ) ) ).
fof(cc3_yellow_3,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A) )
=> ( ~ empty_carrier(A)
& ~ v1_yellow_3(A) ) ) ) ).
fof(cc4_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( bounded_relstr(A)
=> ( lower_bounded_relstr(A)
& upper_bounded_relstr(A) ) ) ) ).
fof(cc5_waybel_1,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& heyting_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A) ) ) ) ).
fof(cc5_yellow_0,axiom,
! [A] :
( rel_str(A)
=> ( ( lower_bounded_relstr(A)
& upper_bounded_relstr(A) )
=> bounded_relstr(A) ) ) ).
fof(cc6_waybel_1,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& heyting_relstr(A) )
=> ( ~ empty_carrier(A)
& distributive_relstr(A) ) ) ) ).
fof(cc7_waybel_1,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& heyting_relstr(A) )
=> ( ~ empty_carrier(A)
& upper_bounded_relstr(A) ) ) ) ).
fof(cc8_waybel_1,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& boolean_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& lower_bounded_relstr(A)
& upper_bounded_relstr(A)
& bounded_relstr(A)
& distributive_relstr(A)
& complemented_relstr(A) ) ) ) ).
fof(cc9_waybel_0,axiom,
! [A] :
( rel_str(A)
=> ( ( reflexive_relstr(A)
& with_suprema_relstr(A)
& up_complete_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& with_suprema_relstr(A)
& upper_bounded_relstr(A) ) ) ) ).
fof(cc9_waybel_1,axiom,
! [A] :
( rel_str(A)
=> ( ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& bounded_relstr(A)
& distributive_relstr(A)
& complemented_relstr(A) )
=> ( ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& lower_bounded_relstr(A)
& upper_bounded_relstr(A)
& bounded_relstr(A)
& distributive_relstr(A)
& complemented_relstr(A)
& boolean_relstr(A) ) ) ) ).
fof(d1_yellow_1,axiom,
! [A] : incl_POSet(A) = rel_str_of(A,inclusion_order(A)) ).
fof(dt_g1_orders_2,axiom,
! [A,B] :
( relation_of2(B,A,A)
=> ( strict_rel_str(rel_str_of(A,B))
& rel_str(rel_str_of(A,B)) ) ) ).
fof(dt_k1_wellord2,axiom,
! [A] : relation(inclusion_relation(A)) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_yellow_1,axiom,
! [A] :
( reflexive(inclusion_order(A))
& antisymmetric(inclusion_order(A))
& transitive(inclusion_order(A))
& v1_partfun1(inclusion_order(A),A,A)
& relation_of2_as_subset(inclusion_order(A),A,A) ) ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_yellow_1,axiom,
! [A] :
( strict_rel_str(incl_POSet(A))
& rel_str(incl_POSet(A)) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_yellow_1,axiom,
! [A] :
( strict_rel_str(boole_POSet(A))
& rel_str(boole_POSet(A)) ) ).
fof(dt_l1_orders_2,axiom,
! [A] :
( rel_str(A)
=> one_sorted_str(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
=> element(C,powerset(cartesian_product2(A,B))) ) ).
fof(dt_u1_orders_2,axiom,
! [A] :
( rel_str(A)
=> relation_of2_as_subset(the_InternalRel(A),the_carrier(A),the_carrier(A)) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_orders_2,axiom,
? [A] : rel_str(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : one_sorted_str(A) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : relation_of2(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : relation_of2_as_subset(C,A,B) ).
fof(fc13_yellow_3,axiom,
! [A] :
( ( ~ v1_yellow_3(A)
& rel_str(A) )
=> ( ~ empty(the_InternalRel(A))
& relation(the_InternalRel(A)) ) ) ).
fof(fc14_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(cartesian_product2(A,B)) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ~ empty(the_carrier(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_waybel_1,axiom,
! [A] :
( ~ empty_carrier(boole_POSet(A))
& strict_rel_str(boole_POSet(A))
& reflexive_relstr(boole_POSet(A))
& transitive_relstr(boole_POSet(A))
& antisymmetric_relstr(boole_POSet(A))
& with_suprema_relstr(boole_POSet(A))
& with_infima_relstr(boole_POSet(A))
& complete_relstr(boole_POSet(A))
& lower_bounded_relstr(boole_POSet(A))
& upper_bounded_relstr(boole_POSet(A))
& bounded_relstr(boole_POSet(A))
& up_complete_relstr(boole_POSet(A))
& join_complete_relstr(boole_POSet(A))
& distributive_relstr(boole_POSet(A)) ) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ empty(A)
& ~ empty(B) )
=> ~ empty(cartesian_product2(A,B)) ) ).
fof(fc5_yellow_1,axiom,
! [A] :
( strict_rel_str(incl_POSet(A))
& reflexive_relstr(incl_POSet(A))
& transitive_relstr(incl_POSet(A))
& antisymmetric_relstr(incl_POSet(A)) ) ).
fof(fc6_yellow_1,axiom,
! [A] :
( ~ empty(A)
=> ( ~ empty_carrier(incl_POSet(A))
& strict_rel_str(incl_POSet(A))
& reflexive_relstr(incl_POSet(A))
& transitive_relstr(incl_POSet(A))
& antisymmetric_relstr(incl_POSet(A)) ) ) ).
fof(fc7_yellow_1,axiom,
! [A] :
( ~ empty_carrier(boole_POSet(A))
& strict_rel_str(boole_POSet(A))
& reflexive_relstr(boole_POSet(A))
& transitive_relstr(boole_POSet(A))
& antisymmetric_relstr(boole_POSet(A)) ) ).
fof(fc8_yellow_1,axiom,
! [A] :
( ~ empty_carrier(boole_POSet(A))
& strict_rel_str(boole_POSet(A))
& reflexive_relstr(boole_POSet(A))
& transitive_relstr(boole_POSet(A))
& antisymmetric_relstr(boole_POSet(A))
& lower_bounded_relstr(boole_POSet(A))
& upper_bounded_relstr(boole_POSet(A))
& bounded_relstr(boole_POSet(A))
& with_suprema_relstr(boole_POSet(A))
& with_infima_relstr(boole_POSet(A))
& complete_relstr(boole_POSet(A)) ) ).
fof(fc9_waybel_1,axiom,
! [A] :
( ~ empty_carrier(boole_POSet(A))
& strict_rel_str(boole_POSet(A))
& reflexive_relstr(boole_POSet(A))
& transitive_relstr(boole_POSet(A))
& antisymmetric_relstr(boole_POSet(A))
& with_suprema_relstr(boole_POSet(A))
& with_infima_relstr(boole_POSet(A))
& complete_relstr(boole_POSet(A))
& lower_bounded_relstr(boole_POSet(A))
& upper_bounded_relstr(boole_POSet(A))
& bounded_relstr(boole_POSet(A))
& up_complete_relstr(boole_POSet(A))
& join_complete_relstr(boole_POSet(A))
& distributive_relstr(boole_POSet(A))
& complemented_relstr(boole_POSet(A)) ) ).
fof(free_g1_orders_2,axiom,
! [A,B] :
( relation_of2(B,A,A)
=> ! [C,D] :
( rel_str_of(A,B) = rel_str_of(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(rc13_waybel_0,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& complete_relstr(A)
& lower_bounded_relstr(A)
& upper_bounded_relstr(A)
& bounded_relstr(A)
& up_complete_relstr(A)
& join_complete_relstr(A) ) ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc1_lattice3,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& complete_relstr(A) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc1_yellow_3,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& ~ v1_yellow_3(A) ) ).
fof(rc2_lattice3,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& complete_relstr(A) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc2_yellow_0,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& complete_relstr(A)
& lower_bounded_relstr(A)
& upper_bounded_relstr(A)
& bounded_relstr(A) ) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc3_struct_0,axiom,
? [A] :
( one_sorted_str(A)
& ~ empty_carrier(A) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc4_waybel_1,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& lower_bounded_relstr(A)
& upper_bounded_relstr(A)
& bounded_relstr(A)
& distributive_relstr(A)
& heyting_relstr(A)
& complemented_relstr(A)
& boolean_relstr(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ empty_carrier(A)
& one_sorted_str(A) )
=> ? [B] :
( element(B,powerset(the_carrier(A)))
& ~ empty(B) ) ) ).
fof(rc5_waybel_1,axiom,
? [A] :
( rel_str(A)
& ~ empty_carrier(A)
& strict_rel_str(A)
& reflexive_relstr(A)
& transitive_relstr(A)
& antisymmetric_relstr(A)
& with_suprema_relstr(A)
& with_infima_relstr(A)
& upper_bounded_relstr(A)
& distributive_relstr(A)
& heyting_relstr(A) ) ).
fof(redefinition_k1_yellow_1,axiom,
! [A] : inclusion_order(A) = inclusion_relation(A) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( relation_of2_as_subset(C,A,B)
<=> relation_of2(C,A,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t4_waybel_7,conjecture,
! [A] : the_carrier(boole_POSet(A)) = powerset(A) ).
fof(t4_yellow_1,axiom,
! [A] : boole_POSet(A) = incl_POSet(powerset(A)) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( in(A,B)
& element(B,powerset(C))
& empty(C) ) ).
fof(t6_boole,axiom,
! [A] :
( empty(A)
=> A = empty_set ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( in(A,B)
& empty(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( empty(A)
& A != B
& empty(B) ) ).
%------------------------------------------------------------------------------