TPTP Problem File: SEU103+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SEU103+1 : TPTP v8.2.0. Released v3.2.0.
% Domain : Set theory
% Problem : Boolean domains, theorem 14
% Version : [Urb06] axioms : Especial.
% English :
% Refs : [TD90] Trybulec & Darmochwal (1990), Boolean Domains
% : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source : [Urb06]
% Names : finsub_1__t14_finsub_1 [Urb06]
% Status : Theorem
% Rating : 0.39 v8.2.0, 0.44 v8.1.0, 0.42 v7.5.0, 0.50 v7.4.0, 0.33 v7.3.0, 0.45 v7.2.0, 0.41 v7.1.0, 0.35 v7.0.0, 0.40 v6.4.0, 0.42 v6.2.0, 0.56 v6.1.0, 0.63 v6.0.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.71 v5.3.0, 0.74 v5.2.0, 0.55 v5.1.0, 0.52 v5.0.0, 0.62 v4.1.0, 0.61 v4.0.1, 0.65 v4.0.0, 0.67 v3.7.0, 0.60 v3.5.0, 0.63 v3.4.0, 0.58 v3.3.0, 0.43 v3.2.0
% Syntax : Number of formulae : 46 ( 14 unt; 0 def)
% Number of atoms : 124 ( 14 equ)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 104 ( 26 ~; 1 |; 50 &)
% ( 1 <=>; 26 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 17 ( 16 usr; 0 prp; 1-2 aty)
% Number of functors : 7 ( 7 usr; 1 con; 0-3 aty)
% Number of variables : 86 ( 76 !; 10 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(cc1_finset_1,axiom,
! [A] :
( empty(A)
=> finite(A) ) ).
fof(cc1_finsub_1,axiom,
! [A] :
( preboolean(A)
=> ( cup_closed(A)
& diff_closed(A) ) ) ).
fof(cc2_finset_1,axiom,
! [A] :
( finite(A)
=> ! [B] :
( element(B,powerset(A))
=> finite(B) ) ) ).
fof(cc2_finsub_1,axiom,
! [A] :
( ( cup_closed(A)
& diff_closed(A) )
=> preboolean(A) ) ).
fof(commutativity_k1_finsub_1,axiom,
! [A,B,C] :
( ( ~ empty(A)
& preboolean(A)
& element(B,A)
& element(C,A) )
=> prebool_union2(A,B,C) = prebool_union2(A,C,B) ) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(commutativity_k5_xboole_0,axiom,
! [A,B] : symmetric_difference(A,B) = symmetric_difference(B,A) ).
fof(d6_xboole_0,axiom,
! [A,B] : symmetric_difference(A,B) = set_union2(set_difference(A,B),set_difference(B,A)) ).
fof(dt_k1_finsub_1,axiom,
! [A,B,C] :
( ( ~ empty(A)
& preboolean(A)
& element(B,A)
& element(C,A) )
=> element(prebool_union2(A,B,C),A) ) ).
fof(dt_k2_finsub_1,axiom,
! [A,B,C] :
( ( ~ empty(A)
& preboolean(A)
& element(B,A)
& element(C,A) )
=> element(prebool_difference(A,B,C),A) ) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc12_finset_1,axiom,
! [A,B] :
( finite(A)
=> finite(set_difference(A,B)) ) ).
fof(fc17_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(symmetric_difference(A,B)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(fc9_finset_1,axiom,
! [A,B] :
( ( finite(A)
& finite(B) )
=> finite(set_union2(A,B)) ) ).
fof(idempotence_k1_finsub_1,axiom,
! [A,B,C] :
( ( ~ empty(A)
& preboolean(A)
& element(B,A)
& element(C,A) )
=> prebool_union2(A,B,B) = B ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(rc1_finset_1,axiom,
? [A] :
( ~ empty(A)
& finite(A) ) ).
fof(rc1_finsub_1,axiom,
? [A] :
( ~ empty(A)
& cup_closed(A)
& cap_closed(A)
& diff_closed(A)
& preboolean(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(rc2_finset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B)
& relation(B)
& function(B)
& one_to_one(B)
& epsilon_transitive(B)
& epsilon_connected(B)
& ordinal(B)
& natural(B)
& finite(B) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( element(B,powerset(A))
& empty(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(rc3_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(rc4_finset_1,axiom,
! [A] :
( ~ empty(A)
=> ? [B] :
( element(B,powerset(A))
& ~ empty(B)
& finite(B) ) ) ).
fof(redefinition_k1_finsub_1,axiom,
! [A,B,C] :
( ( ~ empty(A)
& preboolean(A)
& element(B,A)
& element(C,A) )
=> prebool_union2(A,B,C) = set_union2(B,C) ) ).
fof(redefinition_k2_finsub_1,axiom,
! [A,B,C] :
( ( ~ empty(A)
& preboolean(A)
& element(B,A)
& element(C,A) )
=> prebool_difference(A,B,C) = set_difference(B,C) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(t14_finsub_1,conjecture,
! [A,B,C] :
( ( ~ empty(C)
& preboolean(C) )
=> ( ( element(A,C)
& element(B,C) )
=> element(symmetric_difference(A,B),C) ) ) ).
fof(t1_boole,axiom,
! [A] : set_union2(A,empty_set) = 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_boole,axiom,
! [A] : set_difference(A,empty_set) = A ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_boole,axiom,
! [A] : set_difference(empty_set,A) = empty_set ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t5_boole,axiom,
! [A] : symmetric_difference(A,empty_set) = 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) ) ).
%------------------------------------------------------------------------------