TSTP Solution File: SEU352+1 by Leo-III---1.7.7
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- Process Solution
%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SEU352+1 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n032.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri May 19 11:58:26 EDT 2023
% Result : Theorem 219.29s 47.48s
% Output : Refutation 219.79s
% Verified :
% SZS Type : Refutation
% Derivation depth : 33
% Number of leaves : 124
% Syntax : Number of formulae : 1451 ( 425 unt; 63 typ; 0 def)
% Number of atoms : 3812 (1192 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 14274 (1978 ~;1560 |; 252 &;10326 @)
% ( 7 <=>; 151 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 97 ( 97 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 63 usr; 11 con; 0-6 aty)
% Number of variables : 2084 ( 0 ^;2043 !; 41 ?;2084 :)
% Comments :
%------------------------------------------------------------------------------
thf(empty_carrier_type,type,
empty_carrier: $i > $o ).
thf(lattice_type,type,
lattice: $i > $o ).
thf(complete_latt_str_type,type,
complete_latt_str: $i > $o ).
thf(latt_str_type,type,
latt_str: $i > $o ).
thf(lower_bounded_semilattstr_type,type,
lower_bounded_semilattstr: $i > $o ).
thf(bottom_of_semilattstr_type,type,
bottom_of_semilattstr: $i > $i ).
thf(join_of_latt_set_type,type,
join_of_latt_set: $i > $i > $i ).
thf(empty_set_type,type,
empty_set: $i ).
thf(relation_of2_type,type,
relation_of2: $i > $i > $i > $o ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(relation_of2_as_subset_type,type,
relation_of2_as_subset: $i > $i > $i > $o ).
thf(ordered_pair_type,type,
ordered_pair: $i > $i > $i ).
thf(unordered_pair_type,type,
unordered_pair: $i > $i > $i ).
thf(singleton_type,type,
singleton: $i > $i ).
thf(empty_type,type,
empty: $i > $o ).
thf(element_type,type,
element: $i > $i > $o ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(cartesian_product2_type,type,
cartesian_product2: $i > $i > $i ).
thf(relation_type,type,
relation: $i > $o ).
thf(function_type,type,
function: $i > $o ).
thf(apply_binary_type,type,
apply_binary: $i > $i > $i > $i ).
thf(apply_type,type,
apply: $i > $i > $i ).
thf(quasi_total_type,type,
quasi_total: $i > $i > $i > $o ).
thf(apply_binary_as_element_type,type,
apply_binary_as_element: $i > $i > $i > $i > $i > $i > $i ).
thf(in_type,type,
in: $i > $i > $o ).
thf(meet_semilatt_str_type,type,
meet_semilatt_str: $i > $o ).
thf(the_L_meet_type,type,
the_L_meet: $i > $i ).
thf(the_carrier_type,type,
the_carrier: $i > $i ).
thf(join_semilatt_str_type,type,
join_semilatt_str: $i > $o ).
thf(one_sorted_str_type,type,
one_sorted_str: $i > $o ).
thf(meet_commutative_type,type,
meet_commutative: $i > $o ).
thf(v1_binop_1_type,type,
v1_binop_1: $i > $i > $o ).
thf(v1_partfun1_type,type,
v1_partfun1: $i > $i > $i > $o ).
thf(meet_associative_type,type,
meet_associative: $i > $o ).
thf(v2_binop_1_type,type,
v2_binop_1: $i > $i > $o ).
thf(meet_type,type,
meet: $i > $i > $i > $i ).
thf(join_commutative_type,type,
join_commutative: $i > $o ).
thf(below_type,type,
below: $i > $i > $i > $o ).
thf(meet_absorbing_type,type,
meet_absorbing: $i > $o ).
thf(join_absorbing_type,type,
join_absorbing: $i > $o ).
thf(below_refl_type,type,
below_refl: $i > $i > $i > $o ).
thf(latt_element_smaller_type,type,
latt_element_smaller: $i > $i > $i > $o ).
thf(join_associative_type,type,
join_associative: $i > $o ).
thf(meet_commut_type,type,
meet_commut: $i > $i > $i > $i ).
thf(sk1_type,type,
sk1: $i ).
thf(sk2_type,type,
sk2: $i > $i > $i ).
thf(sk3_type,type,
sk3: $i > $i > $i ).
thf(sk4_type,type,
sk4: $i ).
thf(sk5_type,type,
sk5: $i ).
thf(sk6_type,type,
sk6: $i > $i ).
thf(sk7_type,type,
sk7: $i ).
thf(sk8_type,type,
sk8: $i ).
thf(sk9_type,type,
sk9: $i ).
thf(sk10_type,type,
sk10: $i > $i ).
thf(sk11_type,type,
sk11: $i > $i ).
thf(sk12_type,type,
sk12: $i > $i ).
thf(sk13_type,type,
sk13: $i > $i ).
thf(sk14_type,type,
sk14: $i > $i > $i ).
thf(sk15_type,type,
sk15: $i > $i > $i > $i ).
thf(sk16_type,type,
sk16: $i > $i > $i > $i ).
thf(sk17_type,type,
sk17: $i ).
thf(sk18_type,type,
sk18: $i ).
thf(sk19_type,type,
sk19: $i > $i > $i ).
thf(39,axiom,
empty @ empty_set,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_xboole_0) ).
thf(172,plain,
empty @ empty_set,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(56,axiom,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t6_boole) ).
thf(244,plain,
! [A: $i] :
( ( empty @ A )
=> ( A = empty_set ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(245,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( A = empty_set ) ),
inference(cnf,[status(esa)],[244]) ).
thf(246,plain,
! [A: $i] :
( ( A = empty_set )
| ~ ( empty @ A ) ),
inference(lifteq,[status(thm)],[245]) ).
thf(1,conjecture,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( lower_bounded_semilattstr @ A )
& ( latt_str @ A )
& ( ( bottom_of_semilattstr @ A )
= ( join_of_latt_set @ A @ empty_set ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t50_lattice3) ).
thf(2,negated_conjecture,
~ ! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( lower_bounded_semilattstr @ A )
& ( latt_str @ A )
& ( ( bottom_of_semilattstr @ A )
= ( join_of_latt_set @ A @ empty_set ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(63,plain,
~ ! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( lower_bounded_semilattstr @ A )
& ( latt_str @ A )
& ( ( bottom_of_semilattstr @ A )
= ( join_of_latt_set @ A @ empty_set ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(64,plain,
latt_str @ sk1,
inference(cnf,[status(esa)],[63]) ).
thf(46,axiom,
! [A: $i] :
( ( latt_str @ A )
=> ( ( meet_semilatt_str @ A )
& ( join_semilatt_str @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l3_lattices) ).
thf(211,plain,
! [A: $i] :
( ( latt_str @ A )
=> ( ( meet_semilatt_str @ A )
& ( join_semilatt_str @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(212,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( meet_semilatt_str @ A ) ),
inference(cnf,[status(esa)],[211]) ).
thf(444,plain,
! [A: $i] :
( ( meet_semilatt_str @ A )
| ( ( latt_str @ sk1 )
!= ( latt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[64,212]) ).
thf(445,plain,
meet_semilatt_str @ sk1,
inference(pattern_uni,[status(thm)],[444:[bind(A,$thf( sk1 ))]]) ).
thf(55,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k5_lattices) ).
thf(242,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(243,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[242]) ).
thf(26305,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk1 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[445,243]) ).
thf(26306,plain,
( ( empty_carrier @ sk1 )
| ( element @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[26305:[bind(A,$thf( sk1 ))]]) ).
thf(68,plain,
~ ( empty_carrier @ sk1 ),
inference(cnf,[status(esa)],[63]) ).
thf(26481,plain,
( $false
| ( element @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ) ) ),
inference(rewrite,[status(thm)],[26306,68]) ).
thf(26482,plain,
element @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ),
inference(simp,[status(thm)],[26481]) ).
thf(20,axiom,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_subset) ).
thf(118,plain,
! [A: $i,B: $i] :
( ( element @ A @ B )
=> ( ( empty @ B )
| ( in @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(119,plain,
! [B: $i,A: $i] :
( ~ ( element @ A @ B )
| ( empty @ B )
| ( in @ A @ B ) ),
inference(cnf,[status(esa)],[118]) ).
thf(26572,plain,
! [B: $i,A: $i] :
( ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[26482,119]) ).
thf(26573,plain,
( ( empty @ ( the_carrier @ sk1 ) )
| ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[26572:[bind(A,$thf( bottom_of_semilattstr @ sk1 )),bind(B,$thf( the_carrier @ sk1 ))]]) ).
thf(26,axiom,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l1_lattices) ).
thf(133,plain,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(134,plain,
! [A: $i] :
( ~ ( meet_semilatt_str @ A )
| ( one_sorted_str @ A ) ),
inference(cnf,[status(esa)],[133]) ).
thf(446,plain,
! [A: $i] :
( ( one_sorted_str @ A )
| ( ( meet_semilatt_str @ sk1 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[445,134]) ).
thf(447,plain,
one_sorted_str @ sk1,
inference(pattern_uni,[status(thm)],[446:[bind(A,$thf( sk1 ))]]) ).
thf(50,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ~ ( empty @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_struct_0) ).
thf(229,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ~ ( empty @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(230,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( one_sorted_str @ A )
| ~ ( empty @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[229]) ).
thf(22976,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( empty @ ( the_carrier @ A ) )
| ( ( one_sorted_str @ sk1 )
!= ( one_sorted_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[447,230]) ).
thf(22977,plain,
( ( empty_carrier @ sk1 )
| ~ ( empty @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[22976:[bind(A,$thf( sk1 ))]]) ).
thf(23389,plain,
( $false
| ~ ( empty @ ( the_carrier @ sk1 ) ) ),
inference(rewrite,[status(thm)],[22977,68]) ).
thf(23390,plain,
~ ( empty @ ( the_carrier @ sk1 ) ),
inference(simp,[status(thm)],[23389]) ).
thf(28147,plain,
( $false
| ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ) ) ),
inference(rewrite,[status(thm)],[26573,23390]) ).
thf(28148,plain,
in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ),
inference(simp,[status(thm)],[28147]) ).
thf(4,axiom,
! [A: $i,B: $i] : ( subset @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).
thf(72,plain,
! [A: $i] : ( subset @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(73,plain,
! [A: $i] : ( subset @ A @ A ),
inference(cnf,[status(esa)],[72]) ).
thf(27,axiom,
! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
<=> ( subset @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t3_subset) ).
thf(135,plain,
! [A: $i,B: $i] :
( ( ( element @ A @ ( powerset @ B ) )
=> ( subset @ A @ B ) )
& ( ( subset @ A @ B )
=> ( element @ A @ ( powerset @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(136,plain,
( ! [A: $i,B: $i] :
( ( element @ A @ ( powerset @ B ) )
=> ( subset @ A @ B ) )
& ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( element @ A @ ( powerset @ B ) ) ) ),
inference(miniscope,[status(thm)],[135]) ).
thf(137,plain,
! [B: $i,A: $i] :
( ~ ( subset @ A @ B )
| ( element @ A @ ( powerset @ B ) ) ),
inference(cnf,[status(esa)],[136]) ).
thf(139,plain,
! [B: $i,A: $i] :
( ~ ( subset @ A @ B )
| ( element @ A @ ( powerset @ B ) ) ),
inference(simp,[status(thm)],[137]) ).
thf(4004,plain,
! [C: $i,B: $i,A: $i] :
( ( element @ B @ ( powerset @ C ) )
| ( ( subset @ A @ A )
!= ( subset @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[73,139]) ).
thf(4005,plain,
! [A: $i] : ( element @ A @ ( powerset @ A ) ),
inference(pattern_uni,[status(thm)],[4004:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A ))]]) ).
thf(4051,plain,
! [C: $i,B: $i,A: $i] :
( ( empty @ C )
| ( in @ B @ C )
| ( ( element @ A @ ( powerset @ A ) )
!= ( element @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[4005,119]) ).
thf(4052,plain,
! [A: $i] :
( ( empty @ ( powerset @ A ) )
| ( in @ A @ ( powerset @ A ) ) ),
inference(pattern_uni,[status(thm)],[4051:[bind(A,$thf( D )),bind(B,$thf( D )),bind(C,$thf( powerset @ D ))]]) ).
thf(4063,plain,
! [A: $i] :
( ( empty @ ( powerset @ A ) )
| ( in @ A @ ( powerset @ A ) ) ),
inference(simp,[status(thm)],[4052]) ).
thf(22,axiom,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_subset_1) ).
thf(123,plain,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(124,plain,
~ ? [A: $i] : ( empty @ ( powerset @ A ) ),
inference(miniscope,[status(thm)],[123]) ).
thf(125,plain,
! [A: $i] :
~ ( empty @ ( powerset @ A ) ),
inference(cnf,[status(esa)],[124]) ).
thf(4088,plain,
! [A: $i] :
( $false
| ( in @ A @ ( powerset @ A ) ) ),
inference(rewrite,[status(thm)],[4063,125]) ).
thf(4089,plain,
! [A: $i] : ( in @ A @ ( powerset @ A ) ),
inference(simp,[status(thm)],[4088]) ).
thf(28,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).
thf(140,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ~ ( in @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(141,plain,
! [B: $i,A: $i] :
( ~ ( in @ A @ B )
| ~ ( in @ B @ A ) ),
inference(cnf,[status(esa)],[140]) ).
thf(4092,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( in @ C @ B )
| ( ( in @ A @ ( powerset @ A ) )
!= ( in @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[4089,141]) ).
thf(4093,plain,
! [A: $i] :
~ ( in @ ( powerset @ A ) @ A ),
inference(pattern_uni,[status(thm)],[4092:[bind(A,$thf( D )),bind(B,$thf( D )),bind(C,$thf( powerset @ D ))]]) ).
thf(4117,plain,
! [A: $i] :
~ ( in @ ( powerset @ A ) @ A ),
inference(simp,[status(thm)],[4093]) ).
thf(28188,plain,
! [A: $i] :
( ( in @ ( powerset @ A ) @ A )
!= ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[28148,4117]) ).
thf(28225,plain,
! [A: $i] :
( ( ( powerset @ A )
!= ( bottom_of_semilattstr @ sk1 ) )
| ( A
!= ( the_carrier @ sk1 ) ) ),
inference(simp,[status(thm)],[28188]) ).
thf(28233,plain,
( ( powerset @ ( the_carrier @ sk1 ) )
!= ( bottom_of_semilattstr @ sk1 ) ),
inference(simp,[status(thm)],[28225]) ).
thf(28443,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= ( bottom_of_semilattstr @ sk1 ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28233]) ).
thf(28444,plain,
( ~ ( empty @ sk1 )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= ( bottom_of_semilattstr @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[28443:[bind(A,$thf( sk1 ))]]) ).
thf(39050,plain,
( ( ( powerset @ ( the_carrier @ empty_set ) )
!= ( bottom_of_semilattstr @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,28444]) ).
thf(39081,plain,
( ( ( powerset @ ( the_carrier @ empty_set ) )
!= ( bottom_of_semilattstr @ sk1 ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[39050]) ).
thf(40,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ~ ( empty @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc5_struct_0) ).
thf(173,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( one_sorted_str @ A ) )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ ( the_carrier @ A ) ) )
& ~ ( empty @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(174,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( one_sorted_str @ A )
| ( element @ ( sk12 @ A ) @ ( powerset @ ( the_carrier @ A ) ) ) ),
inference(cnf,[status(esa)],[173]) ).
thf(7005,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( sk12 @ A ) @ ( powerset @ ( the_carrier @ A ) ) )
| ( ( one_sorted_str @ sk1 )
!= ( one_sorted_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[447,174]) ).
thf(7006,plain,
( ( empty_carrier @ sk1 )
| ( element @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) ) ),
inference(pattern_uni,[status(thm)],[7005:[bind(A,$thf( sk1 ))]]) ).
thf(13466,plain,
( $false
| ( element @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) ) ),
inference(rewrite,[status(thm)],[7006,68]) ).
thf(13467,plain,
element @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ),
inference(simp,[status(thm)],[13466]) ).
thf(13512,plain,
! [B: $i,A: $i] :
( ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[13467,119]) ).
thf(13513,plain,
( ( empty @ ( powerset @ ( the_carrier @ sk1 ) ) )
| ( in @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) ) ),
inference(pattern_uni,[status(thm)],[13512:[bind(A,$thf( sk12 @ sk1 )),bind(B,$thf( powerset @ ( the_carrier @ sk1 ) ))]]) ).
thf(14525,plain,
( $false
| ( in @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) ) ),
inference(rewrite,[status(thm)],[13513,125]) ).
thf(14526,plain,
in @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ),
inference(simp,[status(thm)],[14525]) ).
thf(492,plain,
! [B: $i,A: $i] :
( ~ ( in @ A @ B )
| ( ( in @ B @ A )
!= ( in @ A @ B ) )
| ~ $true ),
inference(eqfactor_ordered,[status(thm)],[141]) ).
thf(493,plain,
! [A: $i] :
~ ( in @ A @ A ),
inference(pattern_uni,[status(thm)],[492:[bind(A,$thf( B ))]]) ).
thf(494,plain,
! [A: $i] :
~ ( in @ A @ A ),
inference(simp,[status(thm)],[493]) ).
thf(14560,plain,
! [A: $i] :
( ( in @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) )
!= ( in @ A @ A ) ),
inference(paramod_ordered,[status(thm)],[14526,494]) ).
thf(14601,plain,
! [A: $i] :
( ( ( sk12 @ sk1 )
!= A )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= A ) ),
inference(simp,[status(thm)],[14560]) ).
thf(14606,plain,
( ( sk12 @ sk1 )
!= ( powerset @ ( the_carrier @ sk1 ) ) ),
inference(simp,[status(thm)],[14601]) ).
thf(591,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( meet_semilatt_str @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,445]) ).
thf(592,plain,
( ~ ( empty @ sk1 )
| ( meet_semilatt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[591:[bind(A,$thf( sk1 ))]]) ).
thf(1346,plain,
( ( meet_semilatt_str @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,592]) ).
thf(1361,plain,
( ( meet_semilatt_str @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[1346]) ).
thf(16,axiom,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_u1_lattices) ).
thf(106,plain,
! [A: $i] :
( ( meet_semilatt_str @ A )
=> ( ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( relation_of2_as_subset @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(107,plain,
! [A: $i] :
( ~ ( meet_semilatt_str @ A )
| ( function @ ( the_L_meet @ A ) ) ),
inference(cnf,[status(esa)],[106]) ).
thf(1826,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( function @ ( the_L_meet @ A ) )
| ( ( meet_semilatt_str @ empty_set )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[1361,107]) ).
thf(1827,plain,
( ( sk1 != empty_set )
| ( function @ ( the_L_meet @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[1826:[bind(A,$thf( empty_set ))]]) ).
thf(3398,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( sk1 != empty_set )
| ( function @ empty_set )
| ( A
!= ( the_L_meet @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[246,1827]) ).
thf(3399,plain,
( ~ ( empty @ ( the_L_meet @ empty_set ) )
| ( sk1 != empty_set )
| ( function @ empty_set ) ),
inference(pattern_uni,[status(thm)],[3398:[bind(A,$thf( the_L_meet @ empty_set ))]]) ).
thf(5,axiom,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2_as_subset @ C @ A @ B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m2_relset_1) ).
thf(74,plain,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2_as_subset @ C @ A @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(75,plain,
! [B: $i,A: $i] : ( relation_of2_as_subset @ ( sk3 @ B @ A ) @ A @ B ),
inference(cnf,[status(esa)],[74]) ).
thf(8,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
<=> ( relation_of2 @ C @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_m2_relset_1) ).
thf(82,plain,
! [A: $i,B: $i,C: $i] :
( ( ( relation_of2_as_subset @ C @ A @ B )
=> ( relation_of2 @ C @ A @ B ) )
& ( ( relation_of2 @ C @ A @ B )
=> ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(83,plain,
( ! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( relation_of2 @ C @ A @ B ) )
& ! [A: $i,B: $i,C: $i] :
( ( relation_of2 @ C @ A @ B )
=> ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(miniscope,[status(thm)],[82]) ).
thf(85,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( relation_of2_as_subset @ C @ A @ B )
| ( relation_of2 @ C @ A @ B ) ),
inference(cnf,[status(esa)],[83]) ).
thf(448,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( relation_of2 @ E @ C @ D )
| ( ( relation_of2_as_subset @ ( sk3 @ B @ A ) @ A @ B )
!= ( relation_of2_as_subset @ E @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[75,85]) ).
thf(449,plain,
! [B: $i,A: $i] : ( relation_of2 @ ( sk3 @ A @ B ) @ B @ A ),
inference(pattern_uni,[status(thm)],[448:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( G )),bind(D,$thf( F )),bind(E,$thf( sk3 @ F @ G ))]]) ).
thf(450,plain,
! [B: $i,A: $i] : ( relation_of2 @ ( sk3 @ A @ B ) @ B @ A ),
inference(simp,[status(thm)],[449]) ).
thf(84,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( relation_of2 @ C @ A @ B )
| ( relation_of2_as_subset @ C @ A @ B ) ),
inference(cnf,[status(esa)],[83]) ).
thf(86,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( relation_of2 @ C @ A @ B )
| ( relation_of2_as_subset @ C @ A @ B ) ),
inference(simp,[status(thm)],[84]) ).
thf(473,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( relation_of2_as_subset @ E @ C @ D )
| ( ( relation_of2 @ ( sk3 @ A @ B ) @ B @ A )
!= ( relation_of2 @ E @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[450,86]) ).
thf(474,plain,
! [B: $i,A: $i] : ( relation_of2_as_subset @ ( sk3 @ A @ B ) @ B @ A ),
inference(pattern_uni,[status(thm)],[473:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G )),bind(D,$thf( F )),bind(E,$thf( sk3 @ F @ G ))]]) ).
thf(479,plain,
! [B: $i,A: $i] : ( relation_of2_as_subset @ ( sk3 @ A @ B ) @ B @ A ),
inference(simp,[status(thm)],[474]) ).
thf(57,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( meet_commut @ A @ B @ C )
= ( meet @ A @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k4_lattices) ).
thf(247,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( meet_commut @ A @ B @ C )
= ( meet @ A @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(248,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( ( meet_commut @ A @ B @ C )
= ( meet @ A @ B @ C ) ) ),
inference(cnf,[status(esa)],[247]) ).
thf(249,plain,
! [C: $i,B: $i,A: $i] :
( ( ( meet_commut @ A @ B @ C )
= ( meet @ A @ B @ C ) )
| ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[248]) ).
thf(52,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( meet @ A @ B @ C )
= ( apply_binary_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_L_meet @ A ) @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_lattices) ).
thf(234,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( meet @ A @ B @ C )
= ( apply_binary_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_L_meet @ A ) @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(23,axiom,
? [A: $i] : ( meet_semilatt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_lattices) ).
thf(126,plain,
? [A: $i] : ( meet_semilatt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(127,plain,
meet_semilatt_str @ sk7,
inference(cnf,[status(esa)],[126]) ).
thf(329,plain,
! [A: $i] :
( ( one_sorted_str @ A )
| ( ( meet_semilatt_str @ sk7 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[127,134]) ).
thf(330,plain,
one_sorted_str @ sk7,
inference(pattern_uni,[status(thm)],[329:[bind(A,$thf( sk7 ))]]) ).
thf(6985,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( sk12 @ A ) @ ( powerset @ ( the_carrier @ A ) ) )
| ( ( one_sorted_str @ sk7 )
!= ( one_sorted_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[330,174]) ).
thf(6986,plain,
( ( empty_carrier @ sk7 )
| ( element @ ( sk12 @ sk7 ) @ ( powerset @ ( the_carrier @ sk7 ) ) ) ),
inference(pattern_uni,[status(thm)],[6985:[bind(A,$thf( sk7 ))]]) ).
thf(7,axiom,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_tarski) ).
thf(79,plain,
! [A: $i,B: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(80,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
inference(cnf,[status(esa)],[79]) ).
thf(81,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ A @ B )
= ( unordered_pair @ B @ A ) ),
inference(lifteq,[status(thm)],[80]) ).
thf(6,axiom,
! [A: $i,B: $i] :
( ( ordered_pair @ A @ B )
= ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_tarski) ).
thf(76,plain,
! [A: $i,B: $i] :
( ( ordered_pair @ A @ B )
= ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(77,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ A @ B )
= ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
inference(cnf,[status(esa)],[76]) ).
thf(78,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
= ( ordered_pair @ A @ B ) ),
inference(lifteq,[status(thm)],[77]) ).
thf(313,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[78,78]) ).
thf(314,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
= ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) ),
inference(pattern_uni,[status(thm)],[313:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ G @ F )),bind(D,$thf( singleton @ G ))]]) ).
thf(320,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
= ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) ),
inference(simp,[status(thm)],[314]) ).
thf(30,axiom,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_subset_1) ).
thf(144,plain,
! [A: $i,B: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(145,plain,
~ ? [A: $i,B: $i] : ( empty @ ( unordered_pair @ A @ B ) ),
inference(miniscope,[status(thm)],[144]) ).
thf(146,plain,
! [B: $i,A: $i] :
~ ( empty @ ( unordered_pair @ A @ B ) ),
inference(cnf,[status(esa)],[145]) ).
thf(299,plain,
! [B: $i,A: $i] :
( ( empty @ ( unordered_pair @ A @ B ) )
!= ( empty @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[172,146]) ).
thf(302,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ A @ B )
!= empty_set ),
inference(simp,[status(thm)],[299]) ).
thf(347,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ B @ A )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,302]) ).
thf(348,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ B @ A )
!= empty_set ),
inference(pattern_uni,[status(thm)],[347:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(33284,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[320,348]) ).
thf(33285,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33284:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( unordered_pair @ H @ I ) )),bind(D,$thf( ordered_pair @ H @ I ))]]) ).
thf(33460,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= empty_set ),
inference(simp,[status(thm)],[33285]) ).
thf(33581,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,33460]) ).
thf(33582,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33581:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(33658,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[320,33582]) ).
thf(33659,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33658:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( unordered_pair @ H @ I ) )),bind(D,$thf( ordered_pair @ H @ I ))]]) ).
thf(33767,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33659]) ).
thf(49317,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ C @ D ) @ ( singleton @ C ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,33767]) ).
thf(49318,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[49317:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(61,axiom,
? [A: $i] :
( ( one_sorted_str @ A )
& ~ ( empty_carrier @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc3_struct_0) ).
thf(262,plain,
? [A: $i] :
( ( one_sorted_str @ A )
& ~ ( empty_carrier @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(264,plain,
one_sorted_str @ sk18,
inference(cnf,[status(esa)],[262]) ).
thf(6981,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( sk12 @ A ) @ ( powerset @ ( the_carrier @ A ) ) )
| ( ( one_sorted_str @ sk18 )
!= ( one_sorted_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[264,174]) ).
thf(6982,plain,
( ( empty_carrier @ sk18 )
| ( element @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) ) ),
inference(pattern_uni,[status(thm)],[6981:[bind(A,$thf( sk18 ))]]) ).
thf(263,plain,
~ ( empty_carrier @ sk18 ),
inference(cnf,[status(esa)],[262]) ).
thf(11486,plain,
( $false
| ( element @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) ) ),
inference(rewrite,[status(thm)],[6982,263]) ).
thf(11487,plain,
element @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ),
inference(simp,[status(thm)],[11486]) ).
thf(11526,plain,
! [B: $i,A: $i] :
( ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[11487,119]) ).
thf(11527,plain,
( ( empty @ ( powerset @ ( the_carrier @ sk18 ) ) )
| ( in @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) ) ),
inference(pattern_uni,[status(thm)],[11526:[bind(A,$thf( sk12 @ sk18 )),bind(B,$thf( powerset @ ( the_carrier @ sk18 ) ))]]) ).
thf(14101,plain,
( $false
| ( in @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) ) ),
inference(rewrite,[status(thm)],[11527,125]) ).
thf(14102,plain,
in @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ),
inference(simp,[status(thm)],[14101]) ).
thf(14106,plain,
! [B: $i,A: $i] :
( ~ ( in @ B @ A )
| ( ( in @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) )
!= ( in @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[14102,141]) ).
thf(14107,plain,
~ ( in @ ( powerset @ ( the_carrier @ sk18 ) ) @ ( sk12 @ sk18 ) ),
inference(pattern_uni,[status(thm)],[14106:[bind(A,$thf( sk12 @ sk18 )),bind(B,$thf( powerset @ ( the_carrier @ sk18 ) ))]]) ).
thf(14476,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk18 ) )
| ( A != sk18 ) ),
inference(paramod_ordered,[status(thm)],[246,14107]) ).
thf(14477,plain,
( ~ ( empty @ sk18 )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk18 ) ) ),
inference(pattern_uni,[status(thm)],[14476:[bind(A,$thf( sk18 ))]]) ).
thf(42768,plain,
! [A: $i] :
( ~ ( empty @ sk18 )
| ( ( in @ A @ ( powerset @ A ) )
!= ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk18 ) ) ) ),
inference(paramod_ordered,[status(thm)],[4089,14477]) ).
thf(42822,plain,
! [A: $i] :
( ~ ( empty @ sk18 )
| ( A
!= ( powerset @ ( the_carrier @ empty_set ) ) )
| ( ( powerset @ A )
!= ( sk12 @ sk18 ) ) ),
inference(simp,[status(thm)],[42768]) ).
thf(42851,plain,
( ~ ( empty @ sk18 )
| ( ( sk12 @ sk18 )
!= ( powerset @ ( powerset @ ( the_carrier @ empty_set ) ) ) ) ),
inference(simp,[status(thm)],[42822]) ).
thf(35,axiom,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_subset_1) ).
thf(155,plain,
! [A: $i] :
? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(157,plain,
! [A: $i] : ( element @ ( sk10 @ A ) @ ( powerset @ A ) ),
inference(cnf,[status(esa)],[155]) ).
thf(156,plain,
! [A: $i] : ( empty @ ( sk10 @ A ) ),
inference(cnf,[status(esa)],[155]) ).
thf(571,plain,
! [B: $i,A: $i] :
( ( B = empty_set )
| ( ( empty @ ( sk10 @ A ) )
!= ( empty @ B ) ) ),
inference(paramod_ordered,[status(thm)],[156,246]) ).
thf(572,plain,
! [A: $i] :
( ( sk10 @ A )
= empty_set ),
inference(pattern_uni,[status(thm)],[571:[bind(A,$thf( C )),bind(B,$thf( sk10 @ C ))]]) ).
thf(694,plain,
! [A: $i] :
( ( sk10 @ A )
= empty_set ),
inference(simp,[status(thm)],[572]) ).
thf(1203,plain,
! [A: $i] : ( element @ empty_set @ ( powerset @ A ) ),
inference(rewrite,[status(thm)],[157,694]) ).
thf(11,axiom,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
=> ( relation @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_relset_1) ).
thf(93,plain,
! [A: $i,B: $i,C: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
=> ( relation @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(94,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
| ( relation @ C ) ),
inference(cnf,[status(esa)],[93]) ).
thf(1368,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( relation @ D )
| ( ( element @ empty_set @ ( powerset @ A ) )
!= ( element @ D @ ( powerset @ ( cartesian_product2 @ B @ C ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[1203,94]) ).
thf(1369,plain,
relation @ empty_set,
inference(pattern_uni,[status(thm)],[1368:[bind(A,$thf( cartesian_product2 @ E @ F )),bind(B,$thf( E )),bind(C,$thf( F )),bind(D,$thf( empty_set ))]]) ).
thf(138,plain,
! [B: $i,A: $i] :
( ~ ( element @ A @ ( powerset @ B ) )
| ( subset @ A @ B ) ),
inference(cnf,[status(esa)],[136]) ).
thf(28156,plain,
! [B: $i,A: $i] :
( ~ ( in @ B @ A )
| ( ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ) )
!= ( in @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[28148,141]) ).
thf(28157,plain,
~ ( in @ ( the_carrier @ sk1 ) @ ( bottom_of_semilattstr @ sk1 ) ),
inference(pattern_uni,[status(thm)],[28156:[bind(A,$thf( bottom_of_semilattstr @ sk1 )),bind(B,$thf( the_carrier @ sk1 ))]]) ).
thf(28323,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( in @ ( the_carrier @ sk1 ) @ ( bottom_of_semilattstr @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28157]) ).
thf(28324,plain,
( ~ ( empty @ sk1 )
| ~ ( in @ ( the_carrier @ sk1 ) @ ( bottom_of_semilattstr @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[28323:[bind(A,$thf( sk1 ))]]) ).
thf(175,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( one_sorted_str @ A )
| ~ ( empty @ ( sk12 @ A ) ) ),
inference(cnf,[status(esa)],[173]) ).
thf(7163,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( empty @ ( sk12 @ A ) )
| ( ( one_sorted_str @ sk1 )
!= ( one_sorted_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[447,175]) ).
thf(7164,plain,
( ( empty_carrier @ sk1 )
| ~ ( empty @ ( sk12 @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[7163:[bind(A,$thf( sk1 ))]]) ).
thf(7205,plain,
( $false
| ~ ( empty @ ( sk12 @ sk1 ) ) ),
inference(rewrite,[status(thm)],[7164,68]) ).
thf(7206,plain,
~ ( empty @ ( sk12 @ sk1 ) ),
inference(simp,[status(thm)],[7205]) ).
thf(7210,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ ( sk12 @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,7206]) ).
thf(7211,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ ( sk12 @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[7210:[bind(A,$thf( sk1 ))]]) ).
thf(7496,plain,
( ~ ( empty @ sk1 )
| ( ( empty @ ( sk12 @ empty_set ) )
!= ( empty @ sk1 ) )
| ~ $true ),
inference(eqfactor_ordered,[status(thm)],[7211]) ).
thf(7497,plain,
( ~ ( empty @ sk1 )
| ( ( empty @ ( sk12 @ empty_set ) )
!= ( empty @ sk1 ) ) ),
inference(simp,[status(thm)],[7496]) ).
thf(12311,plain,
( ( ( empty @ ( sk12 @ empty_set ) )
!= ( empty @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,7497]) ).
thf(12334,plain,
( ( ( empty @ ( sk12 @ empty_set ) )
!= ( empty @ sk1 ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[12311]) ).
thf(26429,plain,
( ( sk1 != empty_set )
| ( empty @ ( sk12 @ empty_set ) )
| ( empty @ sk1 ) ),
inference(bool_ext,[status(thm)],[12334]) ).
thf(7474,plain,
( ~ ( empty @ ( sk12 @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,7211]) ).
thf(7500,plain,
( ~ ( empty @ ( sk12 @ empty_set ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[7474]) ).
thf(43407,plain,
( ( sk1 != empty_set )
| ( empty @ sk1 )
| ( ( empty @ ( sk12 @ empty_set ) )
!= ( empty @ ( sk12 @ empty_set ) ) ) ),
inference(paramod_ordered,[status(thm)],[26429,7500]) ).
thf(43408,plain,
( ( sk1 != empty_set )
| ( empty @ sk1 ) ),
inference(pattern_uni,[status(thm)],[43407:[]]) ).
thf(18,axiom,
? [A: $i] :
~ ( empty @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_xboole_0) ).
thf(112,plain,
? [A: $i] :
~ ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(113,plain,
~ ! [A: $i] : ( empty @ A ),
inference(miniscope,[status(thm)],[112]) ).
thf(114,plain,
~ ( empty @ sk5 ),
inference(cnf,[status(esa)],[113]) ).
thf(44674,plain,
( ( sk1 != empty_set )
| ( ( empty @ sk5 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,114]) ).
thf(44719,plain,
( ( sk1 != empty_set )
| ( sk5 != sk1 ) ),
inference(simp,[status(thm)],[44674]) ).
thf(33356,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ D @ C ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
= ( ordered_pair @ ( unordered_pair @ D @ C ) @ ( singleton @ D ) ) )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[81,320]) ).
thf(33357,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
= ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
inference(pattern_uni,[status(thm)],[33356:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( A ))]]) ).
thf(337,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,78]) ).
thf(338,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
= ( ordered_pair @ A @ B ) ),
inference(pattern_uni,[status(thm)],[337:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(398,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[338,78]) ).
thf(399,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) ),
inference(pattern_uni,[status(thm)],[398:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( unordered_pair @ E @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(433,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) ),
inference(simp,[status(thm)],[399]) ).
thf(36256,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
= ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
inference(rewrite,[status(thm)],[33357,433]) ).
thf(49270,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ C @ D ) ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( ordered_pair @ ( unordered_pair @ C @ D ) @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[36256,33767]) ).
thf(49271,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[49270:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(9,axiom,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t8_boole) ).
thf(87,plain,
! [A: $i,B: $i] :
~ ( ( empty @ A )
& ( A != B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(88,plain,
~ ? [A: $i] :
( ( empty @ A )
& ? [B: $i] :
( ( A != B )
& ( empty @ B ) ) ),
inference(miniscope,[status(thm)],[87]) ).
thf(89,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ( A = B )
| ~ ( empty @ B ) ),
inference(cnf,[status(esa)],[88]) ).
thf(90,plain,
! [B: $i,A: $i] :
( ( A = B )
| ~ ( empty @ A )
| ~ ( empty @ B ) ),
inference(lifteq,[status(thm)],[89]) ).
thf(976,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( meet_semilatt_str @ A )
| ( B != sk7 ) ),
inference(paramod_ordered,[status(thm)],[90,127]) ).
thf(977,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk7 )
| ( meet_semilatt_str @ A ) ),
inference(pattern_uni,[status(thm)],[976:[bind(A,$thf( A )),bind(B,$thf( sk7 ))]]) ).
thf(424,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[338,338]) ).
thf(425,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
= ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) ),
inference(pattern_uni,[status(thm)],[424:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ E @ G ))]]) ).
thf(431,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
= ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) ),
inference(simp,[status(thm)],[425]) ).
thf(1915,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[431,348]) ).
thf(1916,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[1915:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ H ) )),bind(D,$thf( ordered_pair @ H @ F ))]]) ).
thf(1994,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= empty_set ),
inference(simp,[status(thm)],[1916]) ).
thf(33407,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[320,1994]) ).
thf(33408,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33407:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ I )),bind(D,$thf( singleton @ ( unordered_pair @ H @ I ) ))]]) ).
thf(33514,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33408]) ).
thf(49025,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ C @ D ) @ ( singleton @ C ) ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,33514]) ).
thf(49026,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[49025:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(109,plain,
! [A: $i] :
( ~ ( meet_semilatt_str @ A )
| ( relation_of2_as_subset @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[106]) ).
thf(2536,plain,
! [A: $i] :
( ( relation_of2_as_subset @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk1 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[445,109]) ).
thf(2537,plain,
relation_of2_as_subset @ ( the_L_meet @ sk1 ) @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ),
inference(pattern_uni,[status(thm)],[2536:[bind(A,$thf( sk1 ))]]) ).
thf(34,axiom,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_m2_relset_1) ).
thf(153,plain,
! [A: $i,B: $i,C: $i] :
( ( relation_of2_as_subset @ C @ A @ B )
=> ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(154,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( relation_of2_as_subset @ C @ A @ B )
| ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(cnf,[status(esa)],[153]) ).
thf(4338,plain,
! [C: $i,B: $i,A: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
| ( ( relation_of2_as_subset @ ( the_L_meet @ sk1 ) @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) )
!= ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[2537,154]) ).
thf(4339,plain,
element @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[4338:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) )),bind(B,$thf( the_carrier @ sk1 )),bind(C,$thf( the_L_meet @ sk1 ))]]) ).
thf(48,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( latt_element_smaller @ A @ B @ C )
<=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ D @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d17_lattice3) ).
thf(217,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( ( latt_element_smaller @ A @ B @ C )
=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ D @ B ) ) ) )
& ( ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ D @ B ) ) )
=> ( latt_element_smaller @ A @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(36,axiom,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_subset_1) ).
thf(158,plain,
! [A: $i] :
? [B: $i] : ( element @ B @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(159,plain,
! [A: $i] : ( element @ ( sk11 @ A ) @ A ),
inference(cnf,[status(esa)],[158]) ).
thf(3890,plain,
! [C: $i,B: $i,A: $i] :
( ( subset @ B @ C )
| ( ( element @ ( sk11 @ A ) @ A )
!= ( element @ B @ ( powerset @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[159,138]) ).
thf(3891,plain,
! [A: $i] : ( subset @ ( sk11 @ ( powerset @ A ) ) @ A ),
inference(pattern_uni,[status(thm)],[3890:[bind(A,$thf( powerset @ E )),bind(B,$thf( sk11 @ ( powerset @ E ) )),bind(C,$thf( E ))]]) ).
thf(3905,plain,
! [A: $i] : ( subset @ ( sk11 @ ( powerset @ A ) ) @ A ),
inference(simp,[status(thm)],[3891]) ).
thf(4554,plain,
! [B: $i,A: $i] :
( ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[4339,119]) ).
thf(4555,plain,
( ( empty @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) )
| ( in @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) ) ),
inference(pattern_uni,[status(thm)],[4554:[bind(A,$thf( the_L_meet @ sk1 )),bind(B,$thf( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ))]]) ).
thf(5287,plain,
( $false
| ( in @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) ) ),
inference(rewrite,[status(thm)],[4555,125]) ).
thf(5288,plain,
in @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ),
inference(simp,[status(thm)],[5287]) ).
thf(5291,plain,
! [B: $i,A: $i] :
( ~ ( in @ B @ A )
| ( ( in @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) )
!= ( in @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[5288,141]) ).
thf(5292,plain,
~ ( in @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) @ ( the_L_meet @ sk1 ) ),
inference(pattern_uni,[status(thm)],[5291:[bind(A,$thf( the_L_meet @ sk1 )),bind(B,$thf( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ))]]) ).
thf(2540,plain,
! [A: $i] :
( ( relation_of2_as_subset @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk7 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[127,109]) ).
thf(2541,plain,
relation_of2_as_subset @ ( the_L_meet @ sk7 ) @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ),
inference(pattern_uni,[status(thm)],[2540:[bind(A,$thf( sk7 ))]]) ).
thf(4336,plain,
! [C: $i,B: $i,A: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
| ( ( relation_of2_as_subset @ ( the_L_meet @ sk7 ) @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) )
!= ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[2541,154]) ).
thf(4337,plain,
element @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ),
inference(pattern_uni,[status(thm)],[4336:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) )),bind(B,$thf( the_carrier @ sk7 )),bind(C,$thf( the_L_meet @ sk7 ))]]) ).
thf(4390,plain,
! [C: $i,B: $i,A: $i] :
( ( relation @ C )
| ( ( element @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) )
!= ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[4337,94]) ).
thf(4391,plain,
relation @ ( the_L_meet @ sk7 ),
inference(pattern_uni,[status(thm)],[4390:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) )),bind(B,$thf( the_carrier @ sk7 )),bind(C,$thf( the_L_meet @ sk7 ))]]) ).
thf(28191,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28148]) ).
thf(28192,plain,
( ~ ( empty @ sk1 )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[28191:[bind(A,$thf( sk1 ))]]) ).
thf(587,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( latt_str @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,64]) ).
thf(588,plain,
( ~ ( empty @ sk1 )
| ( latt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[587:[bind(A,$thf( sk1 ))]]) ).
thf(1335,plain,
( ( latt_str @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,588]) ).
thf(1342,plain,
( ( latt_str @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[1335]) ).
thf(1727,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( latt_str @ A )
| ( sk1 != empty_set )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[246,1342]) ).
thf(1728,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( latt_str @ A )
| ( sk1 != empty_set ) ),
inference(pattern_uni,[status(thm)],[1727:[]]) ).
thf(3272,plain,
! [C: $i,B: $i,A: $i] :
( ( empty @ C )
| ( in @ B @ C )
| ( ( element @ empty_set @ ( powerset @ A ) )
!= ( element @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1203,119]) ).
thf(3273,plain,
! [A: $i] :
( ( empty @ ( powerset @ A ) )
| ( in @ empty_set @ ( powerset @ A ) ) ),
inference(pattern_uni,[status(thm)],[3272:[bind(A,$thf( D )),bind(B,$thf( empty_set )),bind(C,$thf( powerset @ D ))]]) ).
thf(3282,plain,
! [A: $i] :
( ( empty @ ( powerset @ A ) )
| ( in @ empty_set @ ( powerset @ A ) ) ),
inference(simp,[status(thm)],[3273]) ).
thf(3349,plain,
! [A: $i] :
( $false
| ( in @ empty_set @ ( powerset @ A ) ) ),
inference(rewrite,[status(thm)],[3282,125]) ).
thf(3350,plain,
! [A: $i] : ( in @ empty_set @ ( powerset @ A ) ),
inference(simp,[status(thm)],[3349]) ).
thf(4199,plain,
! [B: $i,A: $i] :
( ( in @ empty_set @ ( powerset @ A ) )
!= ( in @ ( powerset @ B ) @ B ) ),
inference(paramod_ordered,[status(thm)],[3350,4117]) ).
thf(4211,plain,
! [B: $i,A: $i] :
( ( ( powerset @ B )
!= empty_set )
| ( ( powerset @ A )
!= B ) ),
inference(simp,[status(thm)],[4199]) ).
thf(4215,plain,
! [A: $i] :
( ( powerset @ ( powerset @ A ) )
!= empty_set ),
inference(simp,[status(thm)],[4211]) ).
thf(15,axiom,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_boole) ).
thf(103,plain,
! [A: $i,B: $i] :
~ ( ( in @ A @ B )
& ( empty @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(104,plain,
~ ? [A: $i,B: $i] :
( ( in @ A @ B )
& ( empty @ B ) ),
inference(miniscope,[status(thm)],[103]) ).
thf(105,plain,
! [B: $i,A: $i] :
( ~ ( in @ A @ B )
| ~ ( empty @ B ) ),
inference(cnf,[status(esa)],[104]) ).
thf(44613,plain,
! [B: $i,A: $i] :
( ( sk1 != empty_set )
| ~ ( in @ A @ B )
| ( ( empty @ sk1 )
!= ( empty @ B ) ) ),
inference(paramod_ordered,[status(thm)],[43408,105]) ).
thf(44614,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ~ ( in @ A @ sk1 ) ),
inference(pattern_uni,[status(thm)],[44613:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(45623,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( in @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) )
!= ( in @ A @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[14102,44614]) ).
thf(45641,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( sk12 @ sk18 )
!= A )
| ( ( powerset @ ( the_carrier @ sk18 ) )
!= sk1 ) ),
inference(simp,[status(thm)],[45623]) ).
thf(45652,plain,
( ( sk1 != empty_set )
| ( ( powerset @ ( the_carrier @ sk18 ) )
!= sk1 ) ),
inference(simp,[status(thm)],[45641]) ).
thf(46707,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( powerset @ ( the_carrier @ sk18 ) )
!= sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,45652]) ).
thf(46708,plain,
( ~ ( empty @ sk1 )
| ( ( powerset @ ( the_carrier @ sk18 ) )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[46707:[bind(A,$thf( sk1 ))]]) ).
thf(65,plain,
lattice @ sk1,
inference(cnf,[status(esa)],[63]) ).
thf(60,axiom,
! [A: $i] :
( ( latt_str @ A )
=> ( ( ~ ( empty_carrier @ A )
& ( lattice @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc1_lattices) ).
thf(254,plain,
! [A: $i] :
( ( latt_str @ A )
=> ( ( ~ ( empty_carrier @ A )
& ( lattice @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(255,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ( meet_associative @ A ) ),
inference(cnf,[status(esa)],[254]) ).
thf(28078,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ( meet_associative @ A )
| ( ( lattice @ sk1 )
!= ( lattice @ A ) ) ),
inference(paramod_ordered,[status(thm)],[65,255]) ).
thf(28079,plain,
( ~ ( latt_str @ sk1 )
| ( empty_carrier @ sk1 )
| ( meet_associative @ sk1 ) ),
inference(pattern_uni,[status(thm)],[28078:[bind(A,$thf( sk1 ))]]) ).
thf(28618,plain,
( ~ $true
| $false
| ( meet_associative @ sk1 ) ),
inference(rewrite,[status(thm)],[28079,64,68]) ).
thf(28619,plain,
meet_associative @ sk1,
inference(simp,[status(thm)],[28618]) ).
thf(28630,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( meet_associative @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28619]) ).
thf(28631,plain,
( ~ ( empty @ sk1 )
| ( meet_associative @ empty_set ) ),
inference(pattern_uni,[status(thm)],[28630:[bind(A,$thf( sk1 ))]]) ).
thf(45,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ! [B: $i,C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( C
= ( join_of_latt_set @ A @ B ) )
<=> ( ( latt_element_smaller @ A @ C @ B )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_element_smaller @ A @ D @ B )
=> ( below @ A @ C @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d21_lattice3) ).
thf(195,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( ( ~ ( empty_carrier @ A )
& ( lattice @ A )
& ( complete_latt_str @ A )
& ( latt_str @ A ) )
=> ! [B: $i,C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( C
= ( join_of_latt_set @ A @ B ) )
=> ( ( latt_element_smaller @ A @ C @ B )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_element_smaller @ A @ D @ B )
=> ( below @ A @ C @ D ) ) ) ) )
& ( ( ( latt_element_smaller @ A @ C @ B )
& ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( latt_element_smaller @ A @ D @ B )
=> ( below @ A @ C @ D ) ) ) )
=> ( C
= ( join_of_latt_set @ A @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(196,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( C
!= ( join_of_latt_set @ A @ B ) )
| ~ ( element @ D @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ D @ B )
| ( below @ A @ C @ D ) ),
inference(cnf,[status(esa)],[195]) ).
thf(201,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( C
!= ( join_of_latt_set @ A @ B ) )
| ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( element @ D @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ D @ B )
| ( below @ A @ C @ D ) ),
inference(lifteq,[status(thm)],[196]) ).
thf(202,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ( below @ A @ ( join_of_latt_set @ A @ B ) @ C ) ),
inference(simp,[status(thm)],[201]) ).
thf(59,axiom,
? [A: $i] : ( latt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l3_lattices) ).
thf(252,plain,
? [A: $i] : ( latt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(253,plain,
latt_str @ sk17,
inference(cnf,[status(esa)],[252]) ).
thf(440,plain,
! [A: $i] :
( ( meet_semilatt_str @ A )
| ( ( latt_str @ sk17 )
!= ( latt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[253,212]) ).
thf(441,plain,
meet_semilatt_str @ sk17,
inference(pattern_uni,[status(thm)],[440:[bind(A,$thf( sk17 ))]]) ).
thf(451,plain,
! [A: $i] :
( ( one_sorted_str @ A )
| ( ( meet_semilatt_str @ sk17 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[441,134]) ).
thf(452,plain,
one_sorted_str @ sk17,
inference(pattern_uni,[status(thm)],[451:[bind(A,$thf( sk17 ))]]) ).
thf(6969,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( sk12 @ A ) @ ( powerset @ ( the_carrier @ A ) ) )
| ( ( one_sorted_str @ sk17 )
!= ( one_sorted_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[452,174]) ).
thf(6970,plain,
( ( empty_carrier @ sk17 )
| ( element @ ( sk12 @ sk17 ) @ ( powerset @ ( the_carrier @ sk17 ) ) ) ),
inference(pattern_uni,[status(thm)],[6969:[bind(A,$thf( sk17 ))]]) ).
thf(7082,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk17 )
| ( subset @ A @ B )
| ( ( element @ ( sk12 @ sk17 ) @ ( powerset @ ( the_carrier @ sk17 ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[6970,138]) ).
thf(7083,plain,
( ( empty_carrier @ sk17 )
| ( subset @ ( sk12 @ sk17 ) @ ( the_carrier @ sk17 ) ) ),
inference(pattern_uni,[status(thm)],[7082:[bind(A,$thf( sk12 @ sk17 )),bind(B,$thf( the_carrier @ sk17 ))]]) ).
thf(19,axiom,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_subset_1) ).
thf(115,plain,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(400,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[78,338]) ).
thf(401,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
= ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) ),
inference(pattern_uni,[status(thm)],[400:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ G @ F ))]]) ).
thf(434,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
= ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) ),
inference(simp,[status(thm)],[401]) ).
thf(2000,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
= ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) ),
inference(rewrite,[status(thm)],[434,431]) ).
thf(315,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ A @ B )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[78,302]) ).
thf(316,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ B @ A )
!= empty_set ),
inference(pattern_uni,[status(thm)],[315:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ G @ F )),bind(D,$thf( singleton @ G ))]]) ).
thf(321,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ B @ A )
!= empty_set ),
inference(simp,[status(thm)],[316]) ).
thf(2015,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[2000,321]) ).
thf(2016,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[2015:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ F ))]]) ).
thf(2104,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) )
!= empty_set ),
inference(simp,[status(thm)],[2016]) ).
thf(21,axiom,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_subset_1) ).
thf(120,plain,
! [A: $i] :
( ~ ( empty @ A )
=> ? [B: $i] :
( ( element @ B @ ( powerset @ A ) )
& ~ ( empty @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(66,plain,
complete_latt_str @ sk1,
inference(cnf,[status(esa)],[63]) ).
thf(613,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( complete_latt_str @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,66]) ).
thf(614,plain,
( ~ ( empty @ sk1 )
| ( complete_latt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[613:[bind(A,$thf( sk1 ))]]) ).
thf(23410,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ ( the_carrier @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,23390]) ).
thf(23411,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[23410:[bind(A,$thf( sk1 ))]]) ).
thf(24088,plain,
( ~ ( empty @ sk1 )
| ( ( empty @ ( the_carrier @ empty_set ) )
!= ( empty @ sk1 ) )
| ~ $true ),
inference(eqfactor_ordered,[status(thm)],[23411]) ).
thf(24091,plain,
( ~ ( empty @ sk1 )
| ( ( the_carrier @ empty_set )
!= sk1 ) ),
inference(simp,[status(thm)],[24088]) ).
thf(24642,plain,
( ( ( the_carrier @ empty_set )
!= sk1 )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,24091]) ).
thf(24666,plain,
( ( ( the_carrier @ empty_set )
!= sk1 )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[24642]) ).
thf(26332,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk17 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[441,243]) ).
thf(26333,plain,
( ( empty_carrier @ sk17 )
| ( element @ ( bottom_of_semilattstr @ sk17 ) @ ( the_carrier @ sk17 ) ) ),
inference(pattern_uni,[status(thm)],[26332:[bind(A,$thf( sk17 ))]]) ).
thf(29,axiom,
? [A: $i] : ( one_sorted_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l1_struct_0) ).
thf(142,plain,
? [A: $i] : ( one_sorted_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(49267,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ C @ D ) ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( ordered_pair @ ( unordered_pair @ C @ D ) @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[36256,33767]) ).
thf(49268,plain,
! [A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ A @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[49267:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(108,plain,
! [A: $i] :
( ~ ( meet_semilatt_str @ A )
| ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[106]) ).
thf(2140,plain,
! [A: $i] :
( ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk17 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[441,108]) ).
thf(2141,plain,
quasi_total @ ( the_L_meet @ sk17 ) @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ),
inference(pattern_uni,[status(thm)],[2140:[bind(A,$thf( sk17 ))]]) ).
thf(257,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ( join_commutative @ A ) ),
inference(cnf,[status(esa)],[254]) ).
thf(28691,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ( join_commutative @ A )
| ( ( lattice @ sk1 )
!= ( lattice @ A ) ) ),
inference(paramod_ordered,[status(thm)],[65,257]) ).
thf(28692,plain,
( ~ ( latt_str @ sk1 )
| ( empty_carrier @ sk1 )
| ( join_commutative @ sk1 ) ),
inference(pattern_uni,[status(thm)],[28691:[bind(A,$thf( sk1 ))]]) ).
thf(29729,plain,
( ~ $true
| $false
| ( join_commutative @ sk1 ) ),
inference(rewrite,[status(thm)],[28692,64,68]) ).
thf(29730,plain,
join_commutative @ sk1,
inference(simp,[status(thm)],[29729]) ).
thf(29739,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( join_commutative @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,29730]) ).
thf(29740,plain,
( ~ ( empty @ sk1 )
| ( join_commutative @ empty_set ) ),
inference(pattern_uni,[status(thm)],[29739:[bind(A,$thf( sk1 ))]]) ).
thf(33696,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[81,33582]) ).
thf(33697,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33696:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( A ))]]) ).
thf(7207,plain,
( ( empty @ ( sk12 @ sk1 ) )
!= ( empty @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[172,7206]) ).
thf(7220,plain,
( ( sk12 @ sk1 )
!= empty_set ),
inference(simp,[status(thm)],[7207]) ).
thf(26578,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( element @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,26482]) ).
thf(26579,plain,
( ~ ( empty @ sk1 )
| ( element @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[26578:[bind(A,$thf( sk1 ))]]) ).
thf(47,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k15_lattice3) ).
thf(214,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(215,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] : ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) ) ),
inference(miniscope,[status(thm)],[214]) ).
thf(216,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[215]) ).
thf(20452,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) )
| ( ( latt_str @ sk17 )
!= ( latt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[253,216]) ).
thf(20453,plain,
! [A: $i] :
( ( empty_carrier @ sk17 )
| ( element @ ( join_of_latt_set @ sk17 @ A ) @ ( the_carrier @ sk17 ) ) ),
inference(pattern_uni,[status(thm)],[20452:[bind(A,$thf( sk17 ))]]) ).
thf(20562,plain,
! [A: $i] :
( ( empty_carrier @ sk17 )
| ( element @ ( join_of_latt_set @ sk17 @ A ) @ ( the_carrier @ sk17 ) ) ),
inference(simp,[status(thm)],[20453]) ).
thf(21103,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ sk17 )
| ( empty @ C )
| ( in @ B @ C )
| ( ( element @ ( join_of_latt_set @ sk17 @ A ) @ ( the_carrier @ sk17 ) )
!= ( element @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[20562,119]) ).
thf(21104,plain,
! [A: $i] :
( ( empty_carrier @ sk17 )
| ( empty @ ( the_carrier @ sk17 ) )
| ( in @ ( join_of_latt_set @ sk17 @ A ) @ ( the_carrier @ sk17 ) ) ),
inference(pattern_uni,[status(thm)],[21103:[bind(A,$thf( E )),bind(B,$thf( join_of_latt_set @ sk17 @ E )),bind(C,$thf( the_carrier @ sk17 ))]]) ).
thf(21192,plain,
! [A: $i] :
( ( empty_carrier @ sk17 )
| ( empty @ ( the_carrier @ sk17 ) )
| ( in @ ( join_of_latt_set @ sk17 @ A ) @ ( the_carrier @ sk17 ) ) ),
inference(simp,[status(thm)],[21104]) ).
thf(4578,plain,
! [C: $i,B: $i,A: $i] :
( ( relation @ C )
| ( ( element @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) )
!= ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[4339,94]) ).
thf(4579,plain,
relation @ ( the_L_meet @ sk1 ),
inference(pattern_uni,[status(thm)],[4578:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) )),bind(B,$thf( the_carrier @ sk1 )),bind(C,$thf( the_L_meet @ sk1 ))]]) ).
thf(4646,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( relation @ ( the_L_meet @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,4579]) ).
thf(4647,plain,
( ~ ( empty @ sk1 )
| ( relation @ ( the_L_meet @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[4646:[bind(A,$thf( sk1 ))]]) ).
thf(3003,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[431,1994]) ).
thf(3004,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3003:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ F )),bind(D,$thf( singleton @ ( singleton @ H ) ))]]) ).
thf(3044,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[3004]) ).
thf(335,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ B @ A )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ ( unordered_pair @ C @ D ) @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[81,78]) ).
thf(336,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
= ( ordered_pair @ B @ A ) ),
inference(pattern_uni,[status(thm)],[335:[bind(A,$thf( unordered_pair @ G @ F )),bind(B,$thf( singleton @ G )),bind(C,$thf( G )),bind(D,$thf( F ))]]) ).
thf(355,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
= ( ordered_pair @ B @ A ) ),
inference(simp,[status(thm)],[336]) ).
thf(1265,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[355,78]) ).
thf(1266,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) )
= ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ),
inference(pattern_uni,[status(thm)],[1265:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ F )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(1329,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) )
= ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ),
inference(simp,[status(thm)],[1266]) ).
thf(3103,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) )
= ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ),
inference(rewrite,[status(thm)],[1329,431]) ).
thf(3009,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ A @ B ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[338,1994]) ).
thf(3010,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3009:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( unordered_pair @ E @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(3032,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) )
!= empty_set ),
inference(simp,[status(thm)],[3010]) ).
thf(7416,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ D ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[3103,3032]) ).
thf(7417,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[7416:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(7467,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[7417]) ).
thf(2538,plain,
! [A: $i] :
( ( relation_of2_as_subset @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk17 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[441,109]) ).
thf(2539,plain,
relation_of2_as_subset @ ( the_L_meet @ sk17 ) @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ),
inference(pattern_uni,[status(thm)],[2538:[bind(A,$thf( sk17 ))]]) ).
thf(4346,plain,
! [C: $i,B: $i,A: $i] :
( ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
| ( ( relation_of2_as_subset @ ( the_L_meet @ sk17 ) @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) )
!= ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[2539,154]) ).
thf(4347,plain,
element @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ),
inference(pattern_uni,[status(thm)],[4346:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) )),bind(B,$thf( the_carrier @ sk17 )),bind(C,$thf( the_L_meet @ sk17 ))]]) ).
thf(4855,plain,
! [C: $i,B: $i,A: $i] :
( ( relation @ C )
| ( ( element @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) )
!= ( element @ C @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[4347,94]) ).
thf(4856,plain,
relation @ ( the_L_meet @ sk17 ),
inference(pattern_uni,[status(thm)],[4855:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) )),bind(B,$thf( the_carrier @ sk17 )),bind(C,$thf( the_L_meet @ sk17 ))]]) ).
thf(912,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( meet_semilatt_str @ A )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,445]) ).
thf(913,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( meet_semilatt_str @ A ) ),
inference(pattern_uni,[status(thm)],[912:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(537,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( latt_str @ empty_set )
| ( A != sk17 ) ),
inference(paramod_ordered,[status(thm)],[246,253]) ).
thf(538,plain,
( ~ ( empty @ sk17 )
| ( latt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[537:[bind(A,$thf( sk17 ))]]) ).
thf(28187,plain,
! [A: $i] :
( ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ sk1 ) )
!= ( in @ A @ A ) ),
inference(paramod_ordered,[status(thm)],[28148,494]) ).
thf(28226,plain,
! [A: $i] :
( ( ( bottom_of_semilattstr @ sk1 )
!= A )
| ( ( the_carrier @ sk1 )
!= A ) ),
inference(simp,[status(thm)],[28187]) ).
thf(28234,plain,
( ( the_carrier @ sk1 )
!= ( bottom_of_semilattstr @ sk1 ) ),
inference(simp,[status(thm)],[28226]) ).
thf(1284,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ D ) @ ( ordered_pair @ B @ A ) )
= ( ordered_pair @ D @ C ) )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[355,355]) ).
thf(1285,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
= ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ),
inference(pattern_uni,[status(thm)],[1284:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ F ))]]) ).
thf(1316,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
= ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ),
inference(simp,[status(thm)],[1285]) ).
thf(418,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ A @ B )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ B @ A )
!= ( unordered_pair @ ( unordered_pair @ D @ C ) @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[81,338]) ).
thf(419,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
= ( ordered_pair @ B @ A ) ),
inference(pattern_uni,[status(thm)],[418:[bind(A,$thf( singleton @ G )),bind(B,$thf( unordered_pair @ E @ G )),bind(C,$thf( G )),bind(D,$thf( E ))]]) ).
thf(426,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
= ( ordered_pair @ B @ A ) ),
inference(simp,[status(thm)],[419]) ).
thf(2422,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ D @ C ) )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[1316,426]) ).
thf(2423,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) ),
inference(pattern_uni,[status(thm)],[2422:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ G ) )),bind(D,$thf( ordered_pair @ G @ H ))]]) ).
thf(2491,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) ),
inference(simp,[status(thm)],[2423]) ).
thf(286,plain,
! [A: $i] :
( ( empty @ ( powerset @ A ) )
!= ( empty @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[172,125]) ).
thf(288,plain,
! [A: $i] :
( ( powerset @ A )
!= empty_set ),
inference(simp,[status(thm)],[286]) ).
thf(1366,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( relation @ D )
| ( ( element @ ( sk11 @ A ) @ A )
!= ( element @ D @ ( powerset @ ( cartesian_product2 @ B @ C ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[159,94]) ).
thf(1367,plain,
! [B: $i,A: $i] : ( relation @ ( sk11 @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(pattern_uni,[status(thm)],[1366:[bind(A,$thf( powerset @ ( cartesian_product2 @ G @ H ) )),bind(B,$thf( G )),bind(C,$thf( H )),bind(D,$thf( sk11 @ ( powerset @ ( cartesian_product2 @ G @ H ) ) ))]]) ).
thf(1386,plain,
! [B: $i,A: $i] : ( relation @ ( sk11 @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(simp,[status(thm)],[1367]) ).
thf(49801,plain,
( ~ ( empty @ sk1 )
| ( ( in @ ( the_carrier @ sk1 ) @ ( bottom_of_semilattstr @ empty_set ) )
!= ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) ) ) ),
inference(paramod_ordered,[status(thm)],[28192,28324]) ).
thf(49936,plain,
( ~ ( empty @ sk1 )
| ( ( the_carrier @ sk1 )
!= ( bottom_of_semilattstr @ empty_set ) )
| ( ( the_carrier @ sk1 )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[49801]) ).
thf(49946,plain,
( ~ ( empty @ sk1 )
| ( ( the_carrier @ sk1 )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[49936]) ).
thf(50879,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,49946]) ).
thf(50880,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ sk1 )
| ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[50879:[bind(A,$thf( sk1 ))]]) ).
thf(50897,plain,
( ~ ( empty @ sk1 )
| ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[50880]) ).
thf(51552,plain,
( ( sk1 != empty_set )
| ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,50897]) ).
thf(51553,plain,
( ( sk1 != empty_set )
| ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[51552:[]]) ).
thf(213,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( join_semilatt_str @ A ) ),
inference(cnf,[status(esa)],[211]) ).
thf(469,plain,
! [A: $i] :
( ( join_semilatt_str @ A )
| ( ( latt_str @ sk1 )
!= ( latt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[64,213]) ).
thf(470,plain,
join_semilatt_str @ sk1,
inference(pattern_uni,[status(thm)],[469:[bind(A,$thf( sk1 ))]]) ).
thf(633,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( join_semilatt_str @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,470]) ).
thf(634,plain,
( ~ ( empty @ sk1 )
| ( join_semilatt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[633:[bind(A,$thf( sk1 ))]]) ).
thf(1467,plain,
( ( join_semilatt_str @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,634]) ).
thf(1482,plain,
( ( join_semilatt_str @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[1467]) ).
thf(43,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( lower_bounded_semilattstr @ A )
<=> ? [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
& ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d13_lattices) ).
thf(181,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( ( lower_bounded_semilattstr @ A )
=> ? [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
& ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) ) )
& ( ? [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
& ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) )
=> ( lower_bounded_semilattstr @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(185,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ( element @ ( sk13 @ A ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[181]) ).
thf(13468,plain,
! [B: $i,A: $i] :
( ( subset @ A @ B )
| ( ( element @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[13467,138]) ).
thf(13469,plain,
subset @ ( sk12 @ sk1 ) @ ( the_carrier @ sk1 ),
inference(pattern_uni,[status(thm)],[13468:[bind(A,$thf( sk12 @ sk1 )),bind(B,$thf( the_carrier @ sk1 ))]]) ).
thf(1834,plain,
! [A: $i] :
( ( function @ ( the_L_meet @ A ) )
| ( ( meet_semilatt_str @ sk7 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[127,107]) ).
thf(1835,plain,
function @ ( the_L_meet @ sk7 ),
inference(pattern_uni,[status(thm)],[1834:[bind(A,$thf( sk7 ))]]) ).
thf(972,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( meet_semilatt_str @ A )
| ( B != sk17 ) ),
inference(paramod_ordered,[status(thm)],[90,441]) ).
thf(973,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk17 )
| ( meet_semilatt_str @ A ) ),
inference(pattern_uni,[status(thm)],[972:[bind(A,$thf( A )),bind(B,$thf( sk17 ))]]) ).
thf(884,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( one_sorted_str @ A )
| ( B != sk18 ) ),
inference(paramod_ordered,[status(thm)],[90,264]) ).
thf(885,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk18 )
| ( one_sorted_str @ A ) ),
inference(pattern_uni,[status(thm)],[884:[bind(A,$thf( A )),bind(B,$thf( sk18 ))]]) ).
thf(4831,plain,
! [B: $i,A: $i] :
( ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[4347,119]) ).
thf(4832,plain,
( ( empty @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) )
| ( in @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) ) ),
inference(pattern_uni,[status(thm)],[4831:[bind(A,$thf( the_L_meet @ sk17 )),bind(B,$thf( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ))]]) ).
thf(5552,plain,
( $false
| ( in @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) ) ),
inference(rewrite,[status(thm)],[4832,125]) ).
thf(5553,plain,
in @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ),
inference(simp,[status(thm)],[5552]) ).
thf(14478,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( in @ ( powerset @ ( the_carrier @ sk18 ) ) @ ( sk12 @ empty_set ) )
| ( A != sk18 ) ),
inference(paramod_ordered,[status(thm)],[246,14107]) ).
thf(14479,plain,
( ~ ( empty @ sk18 )
| ~ ( in @ ( powerset @ ( the_carrier @ sk18 ) ) @ ( sk12 @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[14478:[bind(A,$thf( sk18 ))]]) ).
thf(42916,plain,
( ~ ( in @ ( powerset @ ( the_carrier @ sk18 ) ) @ ( sk12 @ empty_set ) )
| ( ( empty @ sk18 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,14479]) ).
thf(42991,plain,
( ~ ( in @ ( powerset @ ( the_carrier @ sk18 ) ) @ ( sk12 @ empty_set ) )
| ( sk18 != empty_set ) ),
inference(simp,[status(thm)],[42916]) ).
thf(143,plain,
one_sorted_str @ sk8,
inference(cnf,[status(esa)],[142]) ).
thf(503,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( one_sorted_str @ empty_set )
| ( A != sk8 ) ),
inference(paramod_ordered,[status(thm)],[246,143]) ).
thf(504,plain,
( ~ ( empty @ sk8 )
| ( one_sorted_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[503:[bind(A,$thf( sk8 ))]]) ).
thf(33,axiom,
? [A: $i] : ( join_semilatt_str @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_l2_lattices) ).
thf(151,plain,
? [A: $i] : ( join_semilatt_str @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(152,plain,
join_semilatt_str @ sk9,
inference(cnf,[status(esa)],[151]) ).
thf(25,axiom,
! [A: $i] :
( ( join_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l2_lattices) ).
thf(131,plain,
! [A: $i] :
( ( join_semilatt_str @ A )
=> ( one_sorted_str @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(132,plain,
! [A: $i] :
( ~ ( join_semilatt_str @ A )
| ( one_sorted_str @ A ) ),
inference(cnf,[status(esa)],[131]) ).
thf(296,plain,
! [A: $i] :
( ( one_sorted_str @ A )
| ( ( join_semilatt_str @ sk9 )
!= ( join_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[152,132]) ).
thf(297,plain,
one_sorted_str @ sk9,
inference(pattern_uni,[status(thm)],[296:[bind(A,$thf( sk9 ))]]) ).
thf(6973,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( sk12 @ A ) @ ( powerset @ ( the_carrier @ A ) ) )
| ( ( one_sorted_str @ sk9 )
!= ( one_sorted_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[297,174]) ).
thf(6974,plain,
( ( empty_carrier @ sk9 )
| ( element @ ( sk12 @ sk9 ) @ ( powerset @ ( the_carrier @ sk9 ) ) ) ),
inference(pattern_uni,[status(thm)],[6973:[bind(A,$thf( sk9 ))]]) ).
thf(7520,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk9 )
| ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( sk12 @ sk9 ) @ ( powerset @ ( the_carrier @ sk9 ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[6974,119]) ).
thf(7521,plain,
( ( empty_carrier @ sk9 )
| ( empty @ ( powerset @ ( the_carrier @ sk9 ) ) )
| ( in @ ( sk12 @ sk9 ) @ ( powerset @ ( the_carrier @ sk9 ) ) ) ),
inference(pattern_uni,[status(thm)],[7520:[bind(A,$thf( sk12 @ sk9 )),bind(B,$thf( powerset @ ( the_carrier @ sk9 ) ))]]) ).
thf(17982,plain,
( ( empty_carrier @ sk9 )
| $false
| ( in @ ( sk12 @ sk9 ) @ ( powerset @ ( the_carrier @ sk9 ) ) ) ),
inference(rewrite,[status(thm)],[7521,125]) ).
thf(17983,plain,
( ( empty_carrier @ sk9 )
| ( in @ ( sk12 @ sk9 ) @ ( powerset @ ( the_carrier @ sk9 ) ) ) ),
inference(simp,[status(thm)],[17982]) ).
thf(17,axiom,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_subset) ).
thf(110,plain,
! [A: $i,B: $i] :
( ( in @ A @ B )
=> ( element @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(1268,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ D ) @ ( ordered_pair @ A @ B ) )
= ( ordered_pair @ D @ C ) )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[78,355]) ).
thf(1269,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( unordered_pair @ B @ A ) ) @ ( ordered_pair @ B @ A ) )
= ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) ),
inference(pattern_uni,[status(thm)],[1268:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ G @ F ))]]) ).
thf(1330,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( unordered_pair @ B @ A ) ) @ ( ordered_pair @ B @ A ) )
= ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) ),
inference(simp,[status(thm)],[1269]) ).
thf(465,plain,
! [A: $i] :
( ( join_semilatt_str @ A )
| ( ( latt_str @ sk17 )
!= ( latt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[253,213]) ).
thf(466,plain,
join_semilatt_str @ sk17,
inference(pattern_uni,[status(thm)],[465:[bind(A,$thf( sk17 ))]]) ).
thf(916,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( join_semilatt_str @ A )
| ( B != sk17 ) ),
inference(paramod_ordered,[status(thm)],[90,466]) ).
thf(917,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk17 )
| ( join_semilatt_str @ A ) ),
inference(pattern_uni,[status(thm)],[916:[bind(A,$thf( A )),bind(B,$thf( sk17 ))]]) ).
thf(4344,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( element @ E @ ( powerset @ ( cartesian_product2 @ C @ D ) ) )
| ( ( relation_of2_as_subset @ ( sk3 @ B @ A ) @ A @ B )
!= ( relation_of2_as_subset @ E @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[75,154]) ).
thf(4345,plain,
! [B: $i,A: $i] : ( element @ ( sk3 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) ),
inference(pattern_uni,[status(thm)],[4344:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( G )),bind(D,$thf( F )),bind(E,$thf( sk3 @ F @ G ))]]) ).
thf(4349,plain,
! [B: $i,A: $i] : ( element @ ( sk3 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) ),
inference(simp,[status(thm)],[4345]) ).
thf(3268,plain,
! [C: $i,B: $i,A: $i] :
( ( empty @ C )
| ( in @ B @ C )
| ( ( element @ ( sk11 @ A ) @ A )
!= ( element @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[159,119]) ).
thf(3269,plain,
! [A: $i] :
( ( empty @ A )
| ( in @ ( sk11 @ A ) @ A ) ),
inference(pattern_uni,[status(thm)],[3268:[bind(A,$thf( D )),bind(B,$thf( sk11 @ D )),bind(C,$thf( D ))]]) ).
thf(3280,plain,
! [A: $i] :
( ( empty @ A )
| ( in @ ( sk11 @ A ) @ A ) ),
inference(simp,[status(thm)],[3269]) ).
thf(5564,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) )
!= ( in @ A @ A ) ),
inference(paramod_ordered,[status(thm)],[5553,494]) ).
thf(5637,plain,
! [A: $i] :
( ( ( the_L_meet @ sk17 )
!= A )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) )
!= A ) ),
inference(simp,[status(thm)],[5564]) ).
thf(5640,plain,
( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) )
!= ( the_L_meet @ sk17 ) ),
inference(simp,[status(thm)],[5637]) ).
thf(1844,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( function @ ( the_L_meet @ A ) )
| ( ( meet_semilatt_str @ empty_set )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[592,107]) ).
thf(1845,plain,
( ~ ( empty @ sk1 )
| ( function @ ( the_L_meet @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[1844:[bind(A,$thf( empty_set ))]]) ).
thf(3429,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( function @ empty_set )
| ( A
!= ( the_L_meet @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[246,1845]) ).
thf(3430,plain,
( ~ ( empty @ ( the_L_meet @ empty_set ) )
| ~ ( empty @ sk1 )
| ( function @ empty_set ) ),
inference(pattern_uni,[status(thm)],[3429:[bind(A,$thf( the_L_meet @ empty_set ))]]) ).
thf(21265,plain,
( ~ ( empty @ sk1 )
| ( function @ empty_set )
| ( ( empty @ ( the_L_meet @ empty_set ) )
!= ( empty @ sk1 ) )
| ~ $true ),
inference(eqfactor_ordered,[status(thm)],[3430]) ).
thf(21268,plain,
( ( function @ empty_set )
| ~ ( empty @ sk1 )
| ( ( the_L_meet @ empty_set )
!= sk1 ) ),
inference(simp,[status(thm)],[21265]) ).
thf(3007,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ B @ A ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[355,1994]) ).
thf(3008,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ A @ B ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3007:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ F )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(3031,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ A @ B ) )
!= empty_set ),
inference(simp,[status(thm)],[3008]) ).
thf(11762,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ C @ D ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[78,3031]) ).
thf(11763,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11762:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ G @ F )),bind(D,$thf( singleton @ G ))]]) ).
thf(11852,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[11763]) ).
thf(25676,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ D @ C ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ D ) ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[81,11852]) ).
thf(25677,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[25676:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( A ))]]) ).
thf(2500,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( join_semilatt_str @ A )
| ( sk1 != empty_set )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[246,1482]) ).
thf(2501,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( join_semilatt_str @ A )
| ( sk1 != empty_set ) ),
inference(pattern_uni,[status(thm)],[2500:[]]) ).
thf(16175,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( sk12 @ empty_set )
!= ( powerset @ ( the_carrier @ sk1 ) ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,14606]) ).
thf(16176,plain,
( ~ ( empty @ sk1 )
| ( ( sk12 @ empty_set )
!= ( powerset @ ( the_carrier @ sk1 ) ) ) ),
inference(pattern_uni,[status(thm)],[16175:[bind(A,$thf( sk1 ))]]) ).
thf(10,axiom,
? [A: $i] : ( empty @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_xboole_0) ).
thf(91,plain,
? [A: $i] : ( empty @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(92,plain,
empty @ sk4,
inference(cnf,[status(esa)],[91]) ).
thf(565,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( lattice @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,65]) ).
thf(566,plain,
( ~ ( empty @ sk1 )
| ( lattice @ empty_set ) ),
inference(pattern_uni,[status(thm)],[565:[bind(A,$thf( sk1 ))]]) ).
thf(714,plain,
( ( lattice @ empty_set )
| ( ( empty @ sk4 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[92,566]) ).
thf(719,plain,
( ( lattice @ empty_set )
| ( sk4 != sk1 ) ),
inference(simp,[status(thm)],[714]) ).
thf(563,plain,
! [A: $i] :
( ( A = empty_set )
| ( ( empty @ sk4 )
!= ( empty @ A ) ) ),
inference(paramod_ordered,[status(thm)],[92,246]) ).
thf(564,plain,
sk4 = empty_set,
inference(pattern_uni,[status(thm)],[563:[bind(A,$thf( sk4 ))]]) ).
thf(1545,plain,
( ( lattice @ empty_set )
| ( sk1 != empty_set ) ),
inference(rewrite,[status(thm)],[719,564]) ).
thf(4352,plain,
! [B: $i,A: $i] :
( ( subset @ A @ B )
| ( ( element @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[4337,138]) ).
thf(4353,plain,
subset @ ( the_L_meet @ sk7 ) @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ),
inference(pattern_uni,[status(thm)],[4352:[bind(A,$thf( the_L_meet @ sk7 )),bind(B,$thf( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ))]]) ).
thf(186,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( element @ ( sk14 @ B @ A ) @ ( the_carrier @ A ) )
| ( lower_bounded_semilattstr @ A ) ),
inference(cnf,[status(esa)],[181]) ).
thf(189,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( element @ ( sk14 @ B @ A ) @ ( the_carrier @ A ) )
| ( lower_bounded_semilattstr @ A ) ),
inference(simp,[status(thm)],[186]) ).
thf(48980,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ C @ D ) ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( ordered_pair @ ( unordered_pair @ C @ D ) @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[36256,33514]) ).
thf(48981,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[48980:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(5898,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( empty @ D )
| ( in @ C @ D )
| ( ( element @ ( sk3 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) )
!= ( element @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[4349,119]) ).
thf(5899,plain,
! [B: $i,A: $i] :
( ( empty @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
| ( in @ ( sk3 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(pattern_uni,[status(thm)],[5898:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( sk3 @ I @ H )),bind(D,$thf( powerset @ ( cartesian_product2 @ H @ I ) ))]]) ).
thf(5925,plain,
! [B: $i,A: $i] :
( ( empty @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
| ( in @ ( sk3 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(simp,[status(thm)],[5899]) ).
thf(30444,plain,
! [B: $i,A: $i] :
( $false
| ( in @ ( sk3 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(rewrite,[status(thm)],[5925,125]) ).
thf(30445,plain,
! [B: $i,A: $i] : ( in @ ( sk3 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ),
inference(simp,[status(thm)],[30444]) ).
thf(30485,plain,
! [C: $i,B: $i,A: $i] :
( ( in @ ( sk3 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
!= ( in @ ( powerset @ C ) @ C ) ),
inference(paramod_ordered,[status(thm)],[30445,4117]) ).
thf(30521,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk3 @ B @ A )
!= ( powerset @ C ) )
| ( ( powerset @ ( cartesian_product2 @ A @ B ) )
!= C ) ),
inference(simp,[status(thm)],[30485]) ).
thf(30558,plain,
! [B: $i,A: $i] :
( ( sk3 @ B @ A )
!= ( powerset @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(simp,[status(thm)],[30521]) ).
thf(33543,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[320,33460]) ).
thf(33544,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33543:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ I )),bind(D,$thf( singleton @ ( unordered_pair @ H @ I ) ))]]) ).
thf(33627,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33544]) ).
thf(67,plain,
( ( empty_carrier @ sk1 )
| ~ ( lattice @ sk1 )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ~ ( latt_str @ sk1 )
| ( ( bottom_of_semilattstr @ sk1 )
!= ( join_of_latt_set @ sk1 @ empty_set ) ) ),
inference(cnf,[status(esa)],[63]) ).
thf(69,plain,
( ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ( empty_carrier @ sk1 )
| ~ ( lattice @ sk1 )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ~ ( latt_str @ sk1 ) ),
inference(lifteq,[status(thm)],[67]) ).
thf(278,plain,
( ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| $false
| ~ $true
| ~ ( lower_bounded_semilattstr @ sk1 )
| ~ $true ),
inference(rewrite,[status(thm)],[69,64,65,68]) ).
thf(279,plain,
( ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 ) ),
inference(simp,[status(thm)],[278]) ).
thf(579,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,279]) ).
thf(580,plain,
( ~ ( empty @ sk1 )
| ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 ) ),
inference(pattern_uni,[status(thm)],[579:[bind(A,$thf( sk1 ))]]) ).
thf(3808,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,580]) ).
thf(3809,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ sk1 )
| ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set ) ),
inference(pattern_uni,[status(thm)],[3808:[bind(A,$thf( sk1 ))]]) ).
thf(3835,plain,
( ~ ( empty @ sk1 )
| ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set ) ),
inference(simp,[status(thm)],[3809]) ).
thf(261,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ( meet_absorbing @ A ) ),
inference(cnf,[status(esa)],[254]) ).
thf(7225,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( sk12 @ empty_set )
!= empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,7220]) ).
thf(7226,plain,
( ~ ( empty @ sk1 )
| ( ( sk12 @ empty_set )
!= empty_set ) ),
inference(pattern_uni,[status(thm)],[7225:[bind(A,$thf( sk1 ))]]) ).
thf(1012,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( one_sorted_str @ A )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,447]) ).
thf(1013,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( one_sorted_str @ A ) ),
inference(pattern_uni,[status(thm)],[1012:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(116,plain,
~ ? [A: $i] : ( empty @ ( singleton @ A ) ),
inference(miniscope,[status(thm)],[115]) ).
thf(117,plain,
! [A: $i] :
~ ( empty @ ( singleton @ A ) ),
inference(cnf,[status(esa)],[116]) ).
thf(282,plain,
! [A: $i] :
( ( empty @ ( singleton @ A ) )
!= ( empty @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[172,117]) ).
thf(284,plain,
! [A: $i] :
( ( singleton @ A )
!= empty_set ),
inference(simp,[status(thm)],[282]) ).
thf(37,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_meet @ A ) )
& ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v1_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_lattice2) ).
thf(160,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_meet @ A ) )
& ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v1_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(164,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_semilatt_str @ A )
| ( v1_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[160]) ).
thf(13601,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ sk1 ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,13469]) ).
thf(13602,plain,
( ~ ( empty @ sk1 )
| ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[13601:[bind(A,$thf( sk1 ))]]) ).
thf(41677,plain,
( ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,13602]) ).
thf(41717,plain,
( ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ sk1 ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[41677]) ).
thf(38469,plain,
( ~ ( in @ ( the_carrier @ sk1 ) @ ( bottom_of_semilattstr @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,28324]) ).
thf(38531,plain,
( ~ ( in @ ( the_carrier @ sk1 ) @ ( bottom_of_semilattstr @ empty_set ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[38469]) ).
thf(770,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( element @ A @ ( powerset @ C ) )
| ( B
!= ( sk10 @ C ) ) ),
inference(paramod_ordered,[status(thm)],[90,157]) ).
thf(771,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ ( sk10 @ B ) )
| ( element @ A @ ( powerset @ B ) ) ),
inference(pattern_uni,[status(thm)],[770:[bind(A,$thf( A )),bind(B,$thf( sk10 @ D )),bind(C,$thf( D ))]]) ).
thf(1054,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ ( sk10 @ B ) )
| ( element @ A @ ( powerset @ B ) ) ),
inference(simp,[status(thm)],[771]) ).
thf(64069,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ empty_set )
| ( element @ A @ ( powerset @ B ) ) ),
inference(rewrite,[status(thm)],[1054,694]) ).
thf(13228,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ D ) ) ) @ ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[338,3044]) ).
thf(13229,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) ) @ ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[13228:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( unordered_pair @ E @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(13295,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) ) @ ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[13229]) ).
thf(259,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ( join_associative @ A ) ),
inference(cnf,[status(esa)],[254]) ).
thf(28851,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ( join_associative @ A )
| ( ( lattice @ sk1 )
!= ( lattice @ A ) ) ),
inference(paramod_ordered,[status(thm)],[65,259]) ).
thf(28852,plain,
( ~ ( latt_str @ sk1 )
| ( empty_carrier @ sk1 )
| ( join_associative @ sk1 ) ),
inference(pattern_uni,[status(thm)],[28851:[bind(A,$thf( sk1 ))]]) ).
thf(33240,plain,
( ~ $true
| $false
| ( join_associative @ sk1 ) ),
inference(rewrite,[status(thm)],[28852,64,68]) ).
thf(33241,plain,
join_associative @ sk1,
inference(simp,[status(thm)],[33240]) ).
thf(53,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( meet_commut @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k4_lattices) ).
thf(237,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( meet_commut @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(238,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( element @ ( meet_commut @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[237]) ).
thf(1932,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ D @ C ) )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[431,355]) ).
thf(1933,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) ),
inference(pattern_uni,[status(thm)],[1932:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ H ) )),bind(D,$thf( ordered_pair @ H @ F ))]]) ).
thf(1978,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) ),
inference(simp,[status(thm)],[1933]) ).
thf(121,plain,
! [A: $i] :
( ( empty @ A )
| ( element @ ( sk6 @ A ) @ ( powerset @ A ) ) ),
inference(cnf,[status(esa)],[120]) ).
thf(2013,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[2000,321]) ).
thf(2014,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[2013:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(2102,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= empty_set ),
inference(simp,[status(thm)],[2014]) ).
thf(3060,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ B @ A ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[355,2102]) ).
thf(3061,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3060:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ F ))]]) ).
thf(3093,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= empty_set ),
inference(simp,[status(thm)],[3061]) ).
thf(13603,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( subset @ ( sk12 @ sk1 ) @ ( the_carrier @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,13469]) ).
thf(13604,plain,
( ~ ( empty @ sk1 )
| ( subset @ ( sk12 @ sk1 ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[13603:[bind(A,$thf( sk1 ))]]) ).
thf(29765,plain,
( ( join_commutative @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,29740]) ).
thf(29784,plain,
( ( join_commutative @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[29765]) ).
thf(442,plain,
! [B: $i,A: $i] :
( ~ ( latt_str @ A )
| ( one_sorted_str @ B )
| ( ( meet_semilatt_str @ A )
!= ( meet_semilatt_str @ B ) ) ),
inference(paramod_ordered,[status(thm)],[212,134]) ).
thf(443,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( one_sorted_str @ A ) ),
inference(pattern_uni,[status(thm)],[442:[bind(A,$thf( A )),bind(B,$thf( A ))]]) ).
thf(1456,plain,
( ( complete_latt_str @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,614]) ).
thf(1463,plain,
( ( complete_latt_str @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[1456]) ).
thf(2121,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( complete_latt_str @ A )
| ( sk1 != empty_set )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[246,1463]) ).
thf(2122,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( complete_latt_str @ A )
| ( sk1 != empty_set ) ),
inference(pattern_uni,[status(thm)],[2121:[]]) ).
thf(14136,plain,
! [A: $i] :
( ( in @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) )
!= ( in @ ( powerset @ A ) @ A ) ),
inference(paramod_ordered,[status(thm)],[14102,4117]) ).
thf(14171,plain,
! [A: $i] :
( ( ( sk12 @ sk18 )
!= ( powerset @ A ) )
| ( ( powerset @ ( the_carrier @ sk18 ) )
!= A ) ),
inference(simp,[status(thm)],[14136]) ).
thf(14183,plain,
( ( sk12 @ sk18 )
!= ( powerset @ ( powerset @ ( the_carrier @ sk18 ) ) ) ),
inference(simp,[status(thm)],[14171]) ).
thf(16203,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( sk12 @ empty_set )
!= ( powerset @ ( powerset @ ( the_carrier @ sk18 ) ) ) )
| ( A != sk18 ) ),
inference(paramod_ordered,[status(thm)],[246,14183]) ).
thf(16204,plain,
( ~ ( empty @ sk18 )
| ( ( sk12 @ empty_set )
!= ( powerset @ ( powerset @ ( the_carrier @ sk18 ) ) ) ) ),
inference(pattern_uni,[status(thm)],[16203:[bind(A,$thf( sk18 ))]]) ).
thf(904,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( latt_str @ A )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,64]) ).
thf(905,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( latt_str @ A ) ),
inference(pattern_uni,[status(thm)],[904:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(3,axiom,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2 @ C @ A @ B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',existence_m1_relset_1) ).
thf(70,plain,
! [A: $i,B: $i] :
? [C: $i] : ( relation_of2 @ C @ A @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(71,plain,
! [B: $i,A: $i] : ( relation_of2 @ ( sk2 @ B @ A ) @ A @ B ),
inference(cnf,[status(esa)],[70]) ).
thf(477,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( relation_of2_as_subset @ E @ C @ D )
| ( ( relation_of2 @ ( sk2 @ B @ A ) @ A @ B )
!= ( relation_of2 @ E @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[71,86]) ).
thf(478,plain,
! [B: $i,A: $i] : ( relation_of2_as_subset @ ( sk2 @ A @ B ) @ B @ A ),
inference(pattern_uni,[status(thm)],[477:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( G )),bind(D,$thf( F )),bind(E,$thf( sk2 @ F @ G ))]]) ).
thf(480,plain,
! [B: $i,A: $i] : ( relation_of2_as_subset @ ( sk2 @ A @ B ) @ B @ A ),
inference(simp,[status(thm)],[478]) ).
thf(2401,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1316,348]) ).
thf(2402,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[2401:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ G @ H )),bind(D,$thf( singleton @ ( singleton @ G ) ))]]) ).
thf(2483,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) )
!= empty_set ),
inference(simp,[status(thm)],[2402]) ).
thf(57966,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ C ) @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[1978,2483]) ).
thf(57967,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( ordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[57966:[bind(A,$thf( K )),bind(B,$thf( L )),bind(C,$thf( singleton @ ( ordered_pair @ L @ K ) )),bind(D,$thf( ordered_pair @ ( singleton @ L ) @ ( unordered_pair @ K @ L ) ))]]) ).
thf(58365,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( ordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[57967]) ).
thf(7498,plain,
( ~ ( empty @ sk1 )
| ( ( sk12 @ empty_set )
!= sk1 ) ),
inference(simp,[status(thm)],[7496]) ).
thf(11197,plain,
( ( ( sk12 @ empty_set )
!= sk1 )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,7498]) ).
thf(11218,plain,
( ( ( sk12 @ empty_set )
!= sk1 )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[11197]) ).
thf(40329,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( in @ A @ ( powerset @ A ) )
!= ( in @ ( the_carrier @ sk1 ) @ ( bottom_of_semilattstr @ empty_set ) ) ) ),
inference(paramod_ordered,[status(thm)],[4089,38531]) ).
thf(40375,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( A
!= ( the_carrier @ sk1 ) )
| ( ( powerset @ A )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[40329]) ).
thf(40403,plain,
( ( sk1 != empty_set )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[40375]) ).
thf(3005,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ B @ A ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[426,1994]) ).
thf(3006,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ B @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3005:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(3045,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ B @ A ) )
!= empty_set ),
inference(simp,[status(thm)],[3006]) ).
thf(11892,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ D @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[431,3045]) ).
thf(11893,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11892:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ F )),bind(D,$thf( singleton @ ( singleton @ H ) ))]]) ).
thf(11954,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[11893]) ).
thf(198,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ~ ( below @ A @ C @ ( sk15 @ C @ B @ A ) )
| ( C
= ( join_of_latt_set @ A @ B ) ) ),
inference(cnf,[status(esa)],[195]) ).
thf(205,plain,
! [C: $i,B: $i,A: $i] :
( ( C
= ( join_of_latt_set @ A @ B ) )
| ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ~ ( below @ A @ C @ ( sk15 @ C @ B @ A ) ) ),
inference(lifteq,[status(thm)],[198]) ).
thf(206,plain,
! [C: $i,B: $i,A: $i] :
( ( C
= ( join_of_latt_set @ A @ B ) )
| ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ~ ( below @ A @ C @ ( sk15 @ C @ B @ A ) ) ),
inference(simp,[status(thm)],[205]) ).
thf(4055,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( relation @ D )
| ( ( element @ A @ ( powerset @ A ) )
!= ( element @ D @ ( powerset @ ( cartesian_product2 @ B @ C ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[4005,94]) ).
thf(4056,plain,
! [B: $i,A: $i] : ( relation @ ( cartesian_product2 @ A @ B ) ),
inference(pattern_uni,[status(thm)],[4055:[bind(A,$thf( cartesian_product2 @ E @ F )),bind(B,$thf( E )),bind(C,$thf( F )),bind(D,$thf( cartesian_product2 @ E @ F ))]]) ).
thf(4065,plain,
! [B: $i,A: $i] : ( relation @ ( cartesian_product2 @ A @ B ) ),
inference(simp,[status(thm)],[4056]) ).
thf(31,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t4_subset) ).
thf(147,plain,
! [A: $i,B: $i,C: $i] :
( ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) ) )
=> ( element @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(5906,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( relation @ E )
| ( ( element @ ( sk3 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) )
!= ( element @ E @ ( powerset @ ( cartesian_product2 @ C @ D ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[4349,94]) ).
thf(5907,plain,
! [B: $i,A: $i] : ( relation @ ( sk3 @ A @ B ) ),
inference(pattern_uni,[status(thm)],[5906:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G )),bind(D,$thf( F )),bind(E,$thf( sk3 @ F @ G ))]]) ).
thf(5929,plain,
! [B: $i,A: $i] : ( relation @ ( sk3 @ A @ B ) ),
inference(simp,[status(thm)],[5907]) ).
thf(6991,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( sk12 @ A ) @ ( powerset @ ( the_carrier @ A ) ) )
| ( ( one_sorted_str @ sk8 )
!= ( one_sorted_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[143,174]) ).
thf(6992,plain,
( ( empty_carrier @ sk8 )
| ( element @ ( sk12 @ sk8 ) @ ( powerset @ ( the_carrier @ sk8 ) ) ) ),
inference(pattern_uni,[status(thm)],[6991:[bind(A,$thf( sk8 ))]]) ).
thf(12669,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk8 )
| ( subset @ A @ B )
| ( ( element @ ( sk12 @ sk8 ) @ ( powerset @ ( the_carrier @ sk8 ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[6992,138]) ).
thf(12670,plain,
( ( empty_carrier @ sk8 )
| ( subset @ ( sk12 @ sk8 ) @ ( the_carrier @ sk8 ) ) ),
inference(pattern_uni,[status(thm)],[12669:[bind(A,$thf( sk12 @ sk8 )),bind(B,$thf( the_carrier @ sk8 ))]]) ).
thf(29309,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ( meet_absorbing @ A )
| ( ( lattice @ sk1 )
!= ( lattice @ A ) ) ),
inference(paramod_ordered,[status(thm)],[65,261]) ).
thf(29310,plain,
( ~ ( latt_str @ sk1 )
| ( empty_carrier @ sk1 )
| ( meet_absorbing @ sk1 ) ),
inference(pattern_uni,[status(thm)],[29309:[bind(A,$thf( sk1 ))]]) ).
thf(35720,plain,
( ~ $true
| $false
| ( meet_absorbing @ sk1 ) ),
inference(rewrite,[status(thm)],[29310,64,68]) ).
thf(35721,plain,
meet_absorbing @ sk1,
inference(simp,[status(thm)],[35720]) ).
thf(35737,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( meet_absorbing @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,35721]) ).
thf(35738,plain,
( ~ ( empty @ sk1 )
| ( meet_absorbing @ empty_set ) ),
inference(pattern_uni,[status(thm)],[35737:[bind(A,$thf( sk1 ))]]) ).
thf(7504,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk9 )
| ( subset @ A @ B )
| ( ( element @ ( sk12 @ sk9 ) @ ( powerset @ ( the_carrier @ sk9 ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[6974,138]) ).
thf(7505,plain,
( ( empty_carrier @ sk9 )
| ( subset @ ( sk12 @ sk9 ) @ ( the_carrier @ sk9 ) ) ),
inference(pattern_uni,[status(thm)],[7504:[bind(A,$thf( sk12 @ sk9 )),bind(B,$thf( the_carrier @ sk9 ))]]) ).
thf(62,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( lower_bounded_semilattstr @ A )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( B
= ( bottom_of_semilattstr @ A ) )
<=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d16_lattices) ).
thf(265,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( lower_bounded_semilattstr @ A )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ( ( B
= ( bottom_of_semilattstr @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) ) )
& ( ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( meet @ A @ B @ C )
= B )
& ( ( meet @ A @ C @ B )
= B ) ) )
=> ( B
= ( bottom_of_semilattstr @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(848,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( one_sorted_str @ A )
| ( B != sk9 ) ),
inference(paramod_ordered,[status(thm)],[90,297]) ).
thf(849,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk9 )
| ( one_sorted_str @ A ) ),
inference(pattern_uni,[status(thm)],[848:[bind(A,$thf( A )),bind(B,$thf( sk9 ))]]) ).
thf(44591,plain,
( ( sk1 != empty_set )
| ( ( empty @ ( the_carrier @ sk1 ) )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,23390]) ).
thf(44709,plain,
( ( sk1 != empty_set )
| ( ( the_carrier @ sk1 )
!= sk1 ) ),
inference(simp,[status(thm)],[44591]) ).
thf(45108,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( the_carrier @ sk1 )
!= sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,44709]) ).
thf(45109,plain,
( ~ ( empty @ sk1 )
| ( ( the_carrier @ sk1 )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[45108:[bind(A,$thf( sk1 ))]]) ).
thf(33587,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[426,33460]) ).
thf(33588,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33587:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(33651,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33588]) ).
thf(3064,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1316,2102]) ).
thf(3065,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3064:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ G @ H )),bind(D,$thf( singleton @ ( singleton @ G ) ))]]) ).
thf(3095,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[3065]) ).
thf(13370,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ C ) ) ) @ ( ordered_pair @ ( singleton @ C ) @ ( ordered_pair @ B @ A ) ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[355,3095]) ).
thf(13371,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) ) @ ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[13370:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ F )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(13440,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) ) @ ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[13371]) ).
thf(14135,plain,
! [A: $i] :
( ( in @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) )
!= ( in @ A @ A ) ),
inference(paramod_ordered,[status(thm)],[14102,494]) ).
thf(14173,plain,
! [A: $i] :
( ( ( sk12 @ sk18 )
!= A )
| ( ( powerset @ ( the_carrier @ sk18 ) )
!= A ) ),
inference(simp,[status(thm)],[14135]) ).
thf(14184,plain,
( ( sk12 @ sk18 )
!= ( powerset @ ( the_carrier @ sk18 ) ) ),
inference(simp,[status(thm)],[14173]) ).
thf(14427,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( sk12 @ empty_set )
!= ( powerset @ ( the_carrier @ sk18 ) ) )
| ( A != sk18 ) ),
inference(paramod_ordered,[status(thm)],[246,14184]) ).
thf(14428,plain,
( ~ ( empty @ sk18 )
| ( ( sk12 @ empty_set )
!= ( powerset @ ( the_carrier @ sk18 ) ) ) ),
inference(pattern_uni,[status(thm)],[14427:[bind(A,$thf( sk18 ))]]) ).
thf(26969,plain,
( ( ( sk12 @ empty_set )
!= ( powerset @ ( the_carrier @ sk18 ) ) )
| ( ( empty @ sk18 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,14428]) ).
thf(26993,plain,
( ( ( sk12 @ empty_set )
!= ( powerset @ ( the_carrier @ sk18 ) ) )
| ( sk18 != empty_set ) ),
inference(simp,[status(thm)],[26969]) ).
thf(43036,plain,
( ( function @ empty_set )
| ( ( the_L_meet @ empty_set )
!= sk1 )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,21268]) ).
thf(43060,plain,
( ( function @ empty_set )
| ( ( the_L_meet @ empty_set )
!= sk1 )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[43036]) ).
thf(50850,plain,
( ( sk1 != empty_set )
| ( ( the_carrier @ sk1 )
!= ( bottom_of_semilattstr @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,49946]) ).
thf(50851,plain,
( ( sk1 != empty_set )
| ( ( the_carrier @ sk1 )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[50850:[]]) ).
thf(24090,plain,
( ~ ( empty @ sk1 )
| ( ( empty @ ( the_carrier @ empty_set ) )
!= ( empty @ sk1 ) ) ),
inference(simp,[status(thm)],[24088]) ).
thf(14530,plain,
! [B: $i,A: $i] :
( ~ ( in @ B @ A )
| ( ( in @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) )
!= ( in @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[14526,141]) ).
thf(14531,plain,
~ ( in @ ( powerset @ ( the_carrier @ sk1 ) ) @ ( sk12 @ sk1 ) ),
inference(pattern_uni,[status(thm)],[14530:[bind(A,$thf( sk12 @ sk1 )),bind(B,$thf( powerset @ ( the_carrier @ sk1 ) ))]]) ).
thf(16355,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( in @ ( powerset @ ( the_carrier @ sk1 ) ) @ ( sk12 @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,14531]) ).
thf(16356,plain,
( ~ ( empty @ sk1 )
| ~ ( in @ ( powerset @ ( the_carrier @ sk1 ) ) @ ( sk12 @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[16355:[bind(A,$thf( sk1 ))]]) ).
thf(28321,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ sk1 ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28157]) ).
thf(28322,plain,
( ~ ( empty @ sk1 )
| ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[28321:[bind(A,$thf( sk1 ))]]) ).
thf(37895,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28322]) ).
thf(37896,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ sk1 )
| ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[37895:[bind(A,$thf( sk1 ))]]) ).
thf(37956,plain,
( ~ ( empty @ sk1 )
| ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[37896]) ).
thf(40129,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( ( in @ A @ ( powerset @ A ) )
!= ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ empty_set ) ) ) ),
inference(paramod_ordered,[status(thm)],[4089,37956]) ).
thf(40180,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( A
!= ( the_carrier @ empty_set ) )
| ( ( powerset @ A )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[40129]) ).
thf(40211,plain,
( ~ ( empty @ sk1 )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[40180]) ).
thf(14,axiom,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t5_subset) ).
thf(100,plain,
! [A: $i,B: $i,C: $i] :
~ ( ( in @ A @ B )
& ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(101,plain,
~ ? [A: $i,B: $i] :
( ( in @ A @ B )
& ? [C: $i] :
( ( element @ B @ ( powerset @ C ) )
& ( empty @ C ) ) ),
inference(miniscope,[status(thm)],[100]) ).
thf(102,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( in @ A @ B )
| ~ ( element @ B @ ( powerset @ C ) )
| ~ ( empty @ C ) ),
inference(cnf,[status(esa)],[101]) ).
thf(51,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( meet_commut @ A @ B @ C )
= ( meet_commut @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k4_lattices) ).
thf(231,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( meet_commut @ A @ B @ C )
= ( meet_commut @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(232,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( ( meet_commut @ A @ B @ C )
= ( meet_commut @ A @ C @ B ) ) ),
inference(cnf,[status(esa)],[231]) ).
thf(233,plain,
! [C: $i,B: $i,A: $i] :
( ( ( meet_commut @ A @ B @ C )
= ( meet_commut @ A @ C @ B ) )
| ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[232]) ).
thf(886,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( B
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( A
!= ( join_of_latt_set @ sk1 @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[90,279]) ).
thf(887,plain,
! [A: $i] :
( ~ ( empty @ ( join_of_latt_set @ sk1 @ empty_set ) )
| ~ ( empty @ A )
| ( A
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 ) ),
inference(pattern_uni,[status(thm)],[886:[bind(A,$thf( join_of_latt_set @ sk1 @ empty_set )),bind(B,$thf( B ))]]) ).
thf(1112,plain,
( ~ ( empty @ ( join_of_latt_set @ sk1 @ empty_set ) )
| ~ ( empty @ ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 ) ),
inference(simp,[status(thm)],[887]) ).
thf(5833,plain,
( ~ ( empty @ ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( ( empty @ ( join_of_latt_set @ sk1 @ empty_set ) )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,1112]) ).
thf(5875,plain,
( ~ ( empty @ ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( ( join_of_latt_set @ sk1 @ empty_set )
!= empty_set ) ),
inference(simp,[status(thm)],[5833]) ).
thf(26576,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,26482]) ).
thf(26577,plain,
( ~ ( empty @ sk1 )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[26576:[bind(A,$thf( sk1 ))]]) ).
thf(48415,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,26577]) ).
thf(48416,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ sk1 )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[48415:[bind(A,$thf( sk1 ))]]) ).
thf(48479,plain,
( ~ ( empty @ sk1 )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(simp,[status(thm)],[48416]) ).
thf(63617,plain,
( ( sk1 != empty_set )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,48479]) ).
thf(63618,plain,
( ( sk1 != empty_set )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[63617:[]]) ).
thf(956,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( one_sorted_str @ A )
| ( B != sk7 ) ),
inference(paramod_ordered,[status(thm)],[90,330]) ).
thf(957,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk7 )
| ( one_sorted_str @ A ) ),
inference(pattern_uni,[status(thm)],[956:[bind(A,$thf( A )),bind(B,$thf( sk7 ))]]) ).
thf(260,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ( join_absorbing @ A ) ),
inference(cnf,[status(esa)],[254]) ).
thf(308,plain,
! [B: $i,A: $i] :
( ~ ( in @ A @ B )
| ( ( empty @ empty_set )
!= ( empty @ B ) ) ),
inference(paramod_ordered,[status(thm)],[172,105]) ).
thf(309,plain,
! [A: $i] :
~ ( in @ A @ empty_set ),
inference(pattern_uni,[status(thm)],[308:[bind(A,$thf( A )),bind(B,$thf( empty_set ))]]) ).
thf(14595,plain,
! [A: $i] :
( ( in @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) )
!= ( in @ A @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[14526,309]) ).
thf(14600,plain,
! [A: $i] :
( ( ( sk12 @ sk1 )
!= A )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= empty_set ) ),
inference(simp,[status(thm)],[14595]) ).
thf(14605,plain,
( ( powerset @ ( the_carrier @ sk1 ) )
!= empty_set ),
inference(simp,[status(thm)],[14600]) ).
thf(44,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( below_refl @ A @ B @ C )
<=> ( below @ A @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_r3_lattices) ).
thf(192,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( ( ( below_refl @ A @ B @ C )
=> ( below @ A @ B @ C ) )
& ( ( below @ A @ B @ C )
=> ( below_refl @ A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(269,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( element @ ( sk19 @ B @ A ) @ ( the_carrier @ A ) )
| ( B
= ( bottom_of_semilattstr @ A ) ) ),
inference(cnf,[status(esa)],[265]) ).
thf(275,plain,
! [B: $i,A: $i] :
( ( B
= ( bottom_of_semilattstr @ A ) )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( element @ ( sk19 @ B @ A ) @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[269]) ).
thf(33589,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[355,33460]) ).
thf(33590,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33589:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ F )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(33652,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33590]) ).
thf(36607,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ C ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( ordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[36256,33652]) ).
thf(36608,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[36607:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( unordered_pair @ E @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(36838,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[36608]) ).
thf(43354,plain,
( ( ( sk12 @ empty_set )
!= ( powerset @ ( powerset @ ( the_carrier @ sk18 ) ) ) )
| ( ( empty @ sk18 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,16204]) ).
thf(43378,plain,
( ( ( sk12 @ empty_set )
!= ( powerset @ ( powerset @ ( the_carrier @ sk18 ) ) ) )
| ( sk18 != empty_set ) ),
inference(simp,[status(thm)],[43354]) ).
thf(14472,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( in @ ( powerset @ empty_set ) @ ( sk12 @ sk18 ) )
| ( A
!= ( the_carrier @ sk18 ) ) ),
inference(paramod_ordered,[status(thm)],[246,14107]) ).
thf(14473,plain,
( ~ ( empty @ ( the_carrier @ sk18 ) )
| ~ ( in @ ( powerset @ empty_set ) @ ( sk12 @ sk18 ) ) ),
inference(pattern_uni,[status(thm)],[14472:[bind(A,$thf( the_carrier @ sk18 ))]]) ).
thf(41956,plain,
( ~ ( in @ ( powerset @ empty_set ) @ ( sk12 @ sk18 ) )
| ( ( empty @ ( the_carrier @ sk18 ) )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,14473]) ).
thf(42023,plain,
( ~ ( in @ ( powerset @ empty_set ) @ ( sk12 @ sk18 ) )
| ( ( the_carrier @ sk18 )
!= empty_set ) ),
inference(simp,[status(thm)],[41956]) ).
thf(40125,plain,
( ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,37956]) ).
thf(40198,plain,
( ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ empty_set ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[40125]) ).
thf(41402,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( in @ A @ ( powerset @ A ) )
!= ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ empty_set ) ) ) ),
inference(paramod_ordered,[status(thm)],[4089,40198]) ).
thf(41450,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( A
!= ( the_carrier @ empty_set ) )
| ( ( powerset @ A )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[41402]) ).
thf(41484,plain,
( ( sk1 != empty_set )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[41450]) ).
thf(41682,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,13602]) ).
thf(41683,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ sk1 )
| ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[41682:[bind(A,$thf( sk1 ))]]) ).
thf(41718,plain,
( ~ ( empty @ sk1 )
| ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(simp,[status(thm)],[41683]) ).
thf(56853,plain,
( ( sk1 != empty_set )
| ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,41718]) ).
thf(56854,plain,
( ( sk1 != empty_set )
| ( subset @ ( sk12 @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[56853:[]]) ).
thf(4334,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( element @ E @ ( powerset @ ( cartesian_product2 @ C @ D ) ) )
| ( ( relation_of2_as_subset @ ( sk2 @ A @ B ) @ B @ A )
!= ( relation_of2_as_subset @ E @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[480,154]) ).
thf(4335,plain,
! [B: $i,A: $i] : ( element @ ( sk2 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) ),
inference(pattern_uni,[status(thm)],[4334:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G )),bind(D,$thf( F )),bind(E,$thf( sk2 @ F @ G ))]]) ).
thf(4351,plain,
! [B: $i,A: $i] : ( element @ ( sk2 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) ),
inference(simp,[status(thm)],[4335]) ).
thf(6686,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( empty @ D )
| ( in @ C @ D )
| ( ( element @ ( sk2 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) )
!= ( element @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[4351,119]) ).
thf(6687,plain,
! [B: $i,A: $i] :
( ( empty @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
| ( in @ ( sk2 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(pattern_uni,[status(thm)],[6686:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( sk2 @ I @ H )),bind(D,$thf( powerset @ ( cartesian_product2 @ H @ I ) ))]]) ).
thf(6725,plain,
! [B: $i,A: $i] :
( ( empty @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
| ( in @ ( sk2 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(simp,[status(thm)],[6687]) ).
thf(33776,plain,
! [B: $i,A: $i] :
( $false
| ( in @ ( sk2 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(rewrite,[status(thm)],[6725,125]) ).
thf(33777,plain,
! [B: $i,A: $i] : ( in @ ( sk2 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ),
inference(simp,[status(thm)],[33776]) ).
thf(33823,plain,
! [C: $i,B: $i,A: $i] :
( ( in @ ( sk2 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
!= ( in @ ( powerset @ C ) @ C ) ),
inference(paramod_ordered,[status(thm)],[33777,4117]) ).
thf(33859,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ B @ A )
!= ( powerset @ C ) )
| ( ( powerset @ ( cartesian_product2 @ A @ B ) )
!= C ) ),
inference(simp,[status(thm)],[33823]) ).
thf(33869,plain,
! [B: $i,A: $i] :
( ( sk2 @ B @ A )
!= ( powerset @ ( powerset @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(simp,[status(thm)],[33859]) ).
thf(20390,plain,
( ( sk1 != empty_set )
| ( function @ empty_set )
| ( ( empty @ ( the_L_meet @ empty_set ) )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,3399]) ).
thf(20427,plain,
( ( function @ empty_set )
| ( sk1 != empty_set )
| ( ( the_L_meet @ empty_set )
!= empty_set ) ),
inference(simp,[status(thm)],[20390]) ).
thf(12,axiom,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( apply_binary @ A @ B @ C )
= ( apply @ A @ ( ordered_pair @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_binop_1) ).
thf(95,plain,
! [A: $i] :
( ( ( relation @ A )
& ( function @ A ) )
=> ! [B: $i,C: $i] :
( ( apply_binary @ A @ B @ C )
= ( apply @ A @ ( ordered_pair @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(96,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( relation @ A )
| ~ ( function @ A )
| ( ( apply_binary @ A @ B @ C )
= ( apply @ A @ ( ordered_pair @ B @ C ) ) ) ),
inference(cnf,[status(esa)],[95]) ).
thf(97,plain,
! [C: $i,B: $i,A: $i] :
( ( ( apply_binary @ A @ B @ C )
= ( apply @ A @ ( ordered_pair @ B @ C ) ) )
| ~ ( relation @ A )
| ~ ( function @ A ) ),
inference(lifteq,[status(thm)],[96]) ).
thf(1912,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[431,78]) ).
thf(1913,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
= ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) ),
inference(pattern_uni,[status(thm)],[1912:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ F )),bind(D,$thf( singleton @ ( singleton @ H ) ))]]) ).
thf(1993,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
= ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) ),
inference(simp,[status(thm)],[1913]) ).
thf(59012,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1993,33582]) ).
thf(59013,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) @ ( singleton @ ( singleton @ ( ordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[59012:[bind(A,$thf( L )),bind(B,$thf( K )),bind(C,$thf( singleton @ ( ordered_pair @ K @ L ) )),bind(D,$thf( ordered_pair @ ( singleton @ K ) @ ( unordered_pair @ L @ K ) ))]]) ).
thf(59122,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) @ ( singleton @ ( singleton @ ( ordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[59013]) ).
thf(505,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( empty @ A )
| ( relation_of2 @ empty_set @ C @ B )
| ( A
!= ( sk3 @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[246,450]) ).
thf(506,plain,
! [B: $i,A: $i] :
( ~ ( empty @ ( sk3 @ A @ B ) )
| ( relation_of2 @ empty_set @ B @ A ) ),
inference(pattern_uni,[status(thm)],[505:[bind(A,$thf( sk3 @ D @ E )),bind(B,$thf( D )),bind(C,$thf( E ))]]) ).
thf(710,plain,
! [B: $i,A: $i] :
( ~ ( empty @ ( sk3 @ A @ B ) )
| ( relation_of2 @ empty_set @ B @ A ) ),
inference(simp,[status(thm)],[506]) ).
thf(3011,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[1316,1994]) ).
thf(3012,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3011:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ G ) )),bind(D,$thf( ordered_pair @ G @ H ))]]) ).
thf(3033,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[3012]) ).
thf(33249,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( join_associative @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,33241]) ).
thf(33250,plain,
( ~ ( empty @ sk1 )
| ( join_associative @ empty_set ) ),
inference(pattern_uni,[status(thm)],[33249:[bind(A,$thf( sk1 ))]]) ).
thf(218,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ( ! [C: $i] :
( ( latt_element_smaller @ A @ B @ C )
=> ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ D @ B ) ) ) )
& ! [C: $i] :
( ! [D: $i] :
( ( element @ D @ ( the_carrier @ A ) )
=> ( ( in @ D @ C )
=> ( below @ A @ D @ B ) ) )
=> ( latt_element_smaller @ A @ B @ C ) ) ) ) ),
inference(miniscope,[status(thm)],[217]) ).
thf(220,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( in @ ( sk16 @ C @ B @ A ) @ C )
| ( latt_element_smaller @ A @ B @ C ) ),
inference(cnf,[status(esa)],[218]) ).
thf(224,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( in @ ( sk16 @ C @ B @ A ) @ C )
| ( latt_element_smaller @ A @ B @ C ) ),
inference(simp,[status(thm)],[220]) ).
thf(28189,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( in @ empty_set @ ( the_carrier @ sk1 ) )
| ( A
!= ( bottom_of_semilattstr @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[246,28148]) ).
thf(28190,plain,
( ~ ( empty @ ( bottom_of_semilattstr @ sk1 ) )
| ( in @ empty_set @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[28189:[bind(A,$thf( bottom_of_semilattstr @ sk1 ))]]) ).
thf(48816,plain,
( ( in @ empty_set @ ( the_carrier @ sk1 ) )
| ( ( empty @ ( bottom_of_semilattstr @ sk1 ) )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,28190]) ).
thf(48889,plain,
( ( in @ empty_set @ ( the_carrier @ sk1 ) )
| ( ( bottom_of_semilattstr @ sk1 )
!= empty_set ) ),
inference(simp,[status(thm)],[48816]) ).
thf(27096,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk17 )
| ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( bottom_of_semilattstr @ sk17 ) @ ( the_carrier @ sk17 ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[26333,119]) ).
thf(27097,plain,
( ( empty_carrier @ sk17 )
| ( empty @ ( the_carrier @ sk17 ) )
| ( in @ ( bottom_of_semilattstr @ sk17 ) @ ( the_carrier @ sk17 ) ) ),
inference(pattern_uni,[status(thm)],[27096:[bind(A,$thf( bottom_of_semilattstr @ sk17 )),bind(B,$thf( the_carrier @ sk17 ))]]) ).
thf(577,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( bottom_of_semilattstr @ sk1 )
!= empty_set )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( A
!= ( join_of_latt_set @ sk1 @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[246,279]) ).
thf(578,plain,
( ~ ( empty @ ( join_of_latt_set @ sk1 @ empty_set ) )
| ( ( bottom_of_semilattstr @ sk1 )
!= empty_set )
| ~ ( lower_bounded_semilattstr @ sk1 ) ),
inference(pattern_uni,[status(thm)],[577:[bind(A,$thf( join_of_latt_set @ sk1 @ empty_set ))]]) ).
thf(595,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ( element @ empty_set @ B )
| ( A
!= ( sk11 @ B ) ) ),
inference(paramod_ordered,[status(thm)],[246,159]) ).
thf(596,plain,
! [A: $i] :
( ~ ( empty @ ( sk11 @ A ) )
| ( element @ empty_set @ A ) ),
inference(pattern_uni,[status(thm)],[595:[bind(A,$thf( sk11 @ C )),bind(B,$thf( C ))]]) ).
thf(697,plain,
! [A: $i] :
( ~ ( empty @ ( sk11 @ A ) )
| ( element @ empty_set @ A ) ),
inference(simp,[status(thm)],[596]) ).
thf(619,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty_carrier @ empty_set )
| ( A != sk18 ) ),
inference(paramod_ordered,[status(thm)],[246,263]) ).
thf(620,plain,
( ~ ( empty @ sk18 )
| ~ ( empty_carrier @ empty_set ) ),
inference(pattern_uni,[status(thm)],[619:[bind(A,$thf( sk18 ))]]) ).
thf(1244,plain,
( ~ ( empty_carrier @ empty_set )
| ( ( empty @ sk18 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,620]) ).
thf(1259,plain,
( ~ ( empty_carrier @ empty_set )
| ( sk18 != empty_set ) ),
inference(simp,[status(thm)],[1244]) ).
thf(1830,plain,
! [A: $i] :
( ( function @ ( the_L_meet @ A ) )
| ( ( meet_semilatt_str @ sk1 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[445,107]) ).
thf(1831,plain,
function @ ( the_L_meet @ sk1 ),
inference(pattern_uni,[status(thm)],[1830:[bind(A,$thf( sk1 ))]]) ).
thf(1855,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( function @ empty_set )
| ( A
!= ( the_L_meet @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[246,1831]) ).
thf(1856,plain,
( ~ ( empty @ ( the_L_meet @ sk1 ) )
| ( function @ empty_set ) ),
inference(pattern_uni,[status(thm)],[1855:[bind(A,$thf( the_L_meet @ sk1 ))]]) ).
thf(34572,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ D ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[3103,33651]) ).
thf(34573,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[34572:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(34666,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[34573]) ).
thf(182,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( ( meet @ A @ ( sk13 @ A ) @ B )
= ( sk13 @ A ) ) ),
inference(cnf,[status(esa)],[181]) ).
thf(188,plain,
! [B: $i,A: $i] :
( ( ( meet @ A @ ( sk13 @ A ) @ B )
= ( sk13 @ A ) )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[182]) ).
thf(38,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_associative @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_meet @ A ) )
& ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v2_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc5_lattice2) ).
thf(166,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_associative @ A )
& ( meet_semilatt_str @ A ) )
=> ( ( relation @ ( the_L_meet @ A ) )
& ( function @ ( the_L_meet @ A ) )
& ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
& ( v2_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) )
& ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(169,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_associative @ A )
| ~ ( meet_semilatt_str @ A )
| ( relation @ ( the_L_meet @ A ) ) ),
inference(cnf,[status(esa)],[166]) ).
thf(57906,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) @ ( singleton @ D ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1978,33697]) ).
thf(57907,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) @ ( singleton @ ( singleton @ ( ordered_pair @ B @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[57906:[bind(A,$thf( K )),bind(B,$thf( L )),bind(C,$thf( ordered_pair @ ( singleton @ L ) @ ( unordered_pair @ K @ L ) )),bind(D,$thf( singleton @ ( ordered_pair @ L @ K ) ))]]) ).
thf(58339,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) @ ( singleton @ ( singleton @ ( ordered_pair @ B @ A ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[57907]) ).
thf(3046,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ A @ B ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[78,2102]) ).
thf(3047,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( unordered_pair @ B @ A ) ) @ ( ordered_pair @ B @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3046:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ G @ F ))]]) ).
thf(3087,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( unordered_pair @ B @ A ) ) @ ( ordered_pair @ B @ A ) )
!= empty_set ),
inference(simp,[status(thm)],[3047]) ).
thf(221,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( below @ A @ ( sk16 @ C @ B @ A ) @ B )
| ( latt_element_smaller @ A @ B @ C ) ),
inference(cnf,[status(esa)],[218]) ).
thf(225,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( below @ A @ ( sk16 @ C @ B @ A ) @ B )
| ( latt_element_smaller @ A @ B @ C ) ),
inference(simp,[status(thm)],[221]) ).
thf(641,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( one_sorted_str @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,447]) ).
thf(642,plain,
( ~ ( empty @ sk1 )
| ( one_sorted_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[641:[bind(A,$thf( sk1 ))]]) ).
thf(525,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( join_semilatt_str @ empty_set )
| ( A != sk9 ) ),
inference(paramod_ordered,[status(thm)],[246,152]) ).
thf(526,plain,
( ~ ( empty @ sk9 )
| ( join_semilatt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[525:[bind(A,$thf( sk9 ))]]) ).
thf(2001,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ D ) ) )
= ( ordered_pair @ ( singleton @ D ) @ ( unordered_pair @ C @ D ) ) )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[2000,431]) ).
thf(2002,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) ) ),
inference(pattern_uni,[status(thm)],[2001:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(2090,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) ) ),
inference(simp,[status(thm)],[2002]) ).
thf(1935,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[431,338]) ).
thf(1936,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) ),
inference(pattern_uni,[status(thm)],[1935:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ H ) )),bind(D,$thf( ordered_pair @ H @ F ))]]) ).
thf(1979,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) ),
inference(simp,[status(thm)],[1936]) ).
thf(53093,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) ),
inference(rewrite,[status(thm)],[1979,431]) ).
thf(67749,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) ),
inference(rewrite,[status(thm)],[2090,53093,431]) ).
thf(653,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty_carrier @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,68]) ).
thf(654,plain,
( ~ ( empty @ sk1 )
| ~ ( empty_carrier @ empty_set ) ),
inference(pattern_uni,[status(thm)],[653:[bind(A,$thf( sk1 ))]]) ).
thf(1409,plain,
( ~ ( empty_carrier @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,654]) ).
thf(1424,plain,
( ~ ( empty_carrier @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[1409]) ).
thf(1790,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty_carrier @ A )
| ( sk1 != empty_set )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[246,1424]) ).
thf(1791,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty_carrier @ A )
| ( sk1 != empty_set ) ),
inference(pattern_uni,[status(thm)],[1790:[]]) ).
thf(45586,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( in @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) )
!= ( in @ A @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[14526,44614]) ).
thf(45640,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( sk12 @ sk1 )
!= A )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= sk1 ) ),
inference(simp,[status(thm)],[45586]) ).
thf(45651,plain,
( ( sk1 != empty_set )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= sk1 ) ),
inference(simp,[status(thm)],[45640]) ).
thf(46333,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,45651]) ).
thf(46334,plain,
( ~ ( empty @ sk1 )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[46333:[bind(A,$thf( sk1 ))]]) ).
thf(5299,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) )
!= ( in @ A @ A ) ),
inference(paramod_ordered,[status(thm)],[5288,494]) ).
thf(5371,plain,
! [A: $i] :
( ( ( the_L_meet @ sk1 )
!= A )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) )
!= A ) ),
inference(simp,[status(thm)],[5299]) ).
thf(5387,plain,
( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) )
!= ( the_L_meet @ sk1 ) ),
inference(simp,[status(thm)],[5371]) ).
thf(256,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ( meet_commutative @ A ) ),
inference(cnf,[status(esa)],[254]) ).
thf(486,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( relation_of2 @ E @ C @ D )
| ( ( relation_of2_as_subset @ ( sk2 @ A @ B ) @ B @ A )
!= ( relation_of2_as_subset @ E @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[480,85]) ).
thf(487,plain,
! [B: $i,A: $i] : ( relation_of2 @ ( sk2 @ A @ B ) @ B @ A ),
inference(pattern_uni,[status(thm)],[486:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G )),bind(D,$thf( F )),bind(E,$thf( sk2 @ F @ G ))]]) ).
thf(488,plain,
! [B: $i,A: $i] : ( relation_of2 @ ( sk2 @ A @ B ) @ B @ A ),
inference(simp,[status(thm)],[487]) ).
thf(29220,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ( join_absorbing @ A )
| ( ( lattice @ sk1 )
!= ( lattice @ A ) ) ),
inference(paramod_ordered,[status(thm)],[65,260]) ).
thf(29221,plain,
( ~ ( latt_str @ sk1 )
| ( empty_carrier @ sk1 )
| ( join_absorbing @ sk1 ) ),
inference(pattern_uni,[status(thm)],[29220:[bind(A,$thf( sk1 ))]]) ).
thf(34541,plain,
( ~ $true
| $false
| ( join_absorbing @ sk1 ) ),
inference(rewrite,[status(thm)],[29221,64,68]) ).
thf(34542,plain,
join_absorbing @ sk1,
inference(simp,[status(thm)],[34541]) ).
thf(34556,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( join_absorbing @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,34542]) ).
thf(34557,plain,
( ~ ( empty @ sk1 )
| ( join_absorbing @ empty_set ) ),
inference(pattern_uni,[status(thm)],[34556:[bind(A,$thf( sk1 ))]]) ).
thf(553,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( one_sorted_str @ empty_set )
| ( A != sk17 ) ),
inference(paramod_ordered,[status(thm)],[246,452]) ).
thf(554,plain,
( ~ ( empty @ sk17 )
| ( one_sorted_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[553:[bind(A,$thf( sk17 ))]]) ).
thf(193,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_absorbing @ A )
| ~ ( join_absorbing @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( below_refl @ A @ B @ C )
| ( below @ A @ B @ C ) ),
inference(cnf,[status(esa)],[192]) ).
thf(900,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ A )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,279]) ).
thf(901,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ A ) ),
inference(pattern_uni,[status(thm)],[900:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(545,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( empty @ A )
| ( relation_of2_as_subset @ empty_set @ C @ B )
| ( A
!= ( sk2 @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[246,480]) ).
thf(546,plain,
! [B: $i,A: $i] :
( ~ ( empty @ ( sk2 @ A @ B ) )
| ( relation_of2_as_subset @ empty_set @ B @ A ) ),
inference(pattern_uni,[status(thm)],[545:[bind(A,$thf( sk2 @ D @ E )),bind(B,$thf( D )),bind(C,$thf( E ))]]) ).
thf(686,plain,
! [B: $i,A: $i] :
( ~ ( empty @ ( sk2 @ A @ B ) )
| ( relation_of2_as_subset @ empty_set @ B @ A ) ),
inference(simp,[status(thm)],[546]) ).
thf(33585,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[431,33460]) ).
thf(33586,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33585:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ F )),bind(D,$thf( singleton @ ( singleton @ H ) ))]]) ).
thf(33650,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33586]) ).
thf(836,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( one_sorted_str @ A )
| ( B != sk17 ) ),
inference(paramod_ordered,[status(thm)],[90,452]) ).
thf(837,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk17 )
| ( one_sorted_str @ A ) ),
inference(pattern_uni,[status(thm)],[836:[bind(A,$thf( A )),bind(B,$thf( sk17 ))]]) ).
thf(41960,plain,
! [A: $i] :
( ~ ( empty @ ( the_carrier @ sk18 ) )
| ( ( in @ A @ ( powerset @ A ) )
!= ( in @ ( powerset @ empty_set ) @ ( sk12 @ sk18 ) ) ) ),
inference(paramod_ordered,[status(thm)],[4089,14473]) ).
thf(42019,plain,
! [A: $i] :
( ~ ( empty @ ( the_carrier @ sk18 ) )
| ( A
!= ( powerset @ empty_set ) )
| ( ( powerset @ A )
!= ( sk12 @ sk18 ) ) ),
inference(simp,[status(thm)],[41960]) ).
thf(42041,plain,
( ~ ( empty @ ( the_carrier @ sk18 ) )
| ( ( sk12 @ sk18 )
!= ( powerset @ ( powerset @ empty_set ) ) ) ),
inference(simp,[status(thm)],[42019]) ).
thf(36610,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ C ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( ordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[36256,33652]) ).
thf(36611,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[36610:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ G @ F )),bind(D,$thf( singleton @ G ))]]) ).
thf(36839,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[36611]) ).
thf(37884,plain,
( ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,28322]) ).
thf(37946,plain,
( ~ ( in @ ( the_carrier @ empty_set ) @ ( bottom_of_semilattstr @ sk1 ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[37884]) ).
thf(11790,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ C @ D ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[431,3031]) ).
thf(11791,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11790:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ F )),bind(D,$thf( singleton @ ( singleton @ H ) ))]]) ).
thf(11833,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[11791]) ).
thf(33593,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[1316,33460]) ).
thf(33594,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33593:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ G ) )),bind(D,$thf( ordered_pair @ G @ H ))]]) ).
thf(33654,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33594]) ).
thf(39334,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ C ) ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= ( ordered_pair @ ( singleton @ C ) @ ( unordered_pair @ C @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[2000,33654]) ).
thf(39335,plain,
! [A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[39334:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(46786,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,46334]) ).
thf(46787,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ sk1 )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[46786:[bind(A,$thf( sk1 ))]]) ).
thf(46809,plain,
( ~ ( empty @ sk1 )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= sk1 ) ),
inference(simp,[status(thm)],[46787]) ).
thf(5634,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) )
!= ( in @ A @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[5553,309]) ).
thf(5636,plain,
! [A: $i] :
( ( ( the_L_meet @ sk17 )
!= A )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) )
!= empty_set ) ),
inference(simp,[status(thm)],[5634]) ).
thf(5639,plain,
( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) )
!= empty_set ),
inference(simp,[status(thm)],[5636]) ).
thf(20482,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) )
| ( ( latt_str @ sk1 )
!= ( latt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[64,216]) ).
thf(20483,plain,
! [A: $i] :
( ( empty_carrier @ sk1 )
| ( element @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[20482:[bind(A,$thf( sk1 ))]]) ).
thf(20542,plain,
! [A: $i] :
( ( empty_carrier @ sk1 )
| ( element @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) ) ),
inference(simp,[status(thm)],[20483]) ).
thf(20592,plain,
! [A: $i] :
( $false
| ( element @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) ) ),
inference(rewrite,[status(thm)],[20542,68]) ).
thf(20593,plain,
! [A: $i] : ( element @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) ),
inference(simp,[status(thm)],[20592]) ).
thf(35768,plain,
( ( meet_absorbing @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,35738]) ).
thf(35793,plain,
( ( meet_absorbing @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[35768]) ).
thf(14561,plain,
! [A: $i] :
( ( in @ ( sk12 @ sk1 ) @ ( powerset @ ( the_carrier @ sk1 ) ) )
!= ( in @ ( powerset @ A ) @ A ) ),
inference(paramod_ordered,[status(thm)],[14526,4117]) ).
thf(14599,plain,
! [A: $i] :
( ( ( sk12 @ sk1 )
!= ( powerset @ A ) )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= A ) ),
inference(simp,[status(thm)],[14561]) ).
thf(14604,plain,
( ( sk12 @ sk1 )
!= ( powerset @ ( powerset @ ( the_carrier @ sk1 ) ) ) ),
inference(simp,[status(thm)],[14599]) ).
thf(17962,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( sk12 @ empty_set )
!= ( powerset @ ( powerset @ ( the_carrier @ sk1 ) ) ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,14604]) ).
thf(17963,plain,
( ~ ( empty @ sk1 )
| ( ( sk12 @ empty_set )
!= ( powerset @ ( powerset @ ( the_carrier @ sk1 ) ) ) ) ),
inference(pattern_uni,[status(thm)],[17962:[bind(A,$thf( sk1 ))]]) ).
thf(33583,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ B @ A )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,33460]) ).
thf(33584,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33583:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( A ))]]) ).
thf(804,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( latt_str @ A )
| ( B != sk17 ) ),
inference(paramod_ordered,[status(thm)],[90,253]) ).
thf(805,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk17 )
| ( latt_str @ A ) ),
inference(pattern_uni,[status(thm)],[804:[bind(A,$thf( A )),bind(B,$thf( sk17 ))]]) ).
thf(6672,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( subset @ C @ D )
| ( ( element @ ( sk2 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) )
!= ( element @ C @ ( powerset @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[4351,138]) ).
thf(6673,plain,
! [B: $i,A: $i] : ( subset @ ( sk2 @ B @ A ) @ ( cartesian_product2 @ A @ B ) ),
inference(pattern_uni,[status(thm)],[6672:[bind(A,$thf( H )),bind(B,$thf( G )),bind(C,$thf( sk2 @ H @ G )),bind(D,$thf( cartesian_product2 @ G @ H ))]]) ).
thf(6716,plain,
! [B: $i,A: $i] : ( subset @ ( sk2 @ B @ A ) @ ( cartesian_product2 @ A @ B ) ),
inference(simp,[status(thm)],[6673]) ).
thf(36295,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ C @ D ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( ordered_pair @ ( unordered_pair @ D @ C ) @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[36256,25677]) ).
thf(36296,plain,
! [A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ A ) ) @ ( ordered_pair @ ( unordered_pair @ A @ A ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[36295:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(14170,plain,
! [A: $i] :
( ( in @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) )
!= ( in @ A @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[14102,309]) ).
thf(14175,plain,
! [A: $i] :
( ( ( sk12 @ sk18 )
!= A )
| ( ( powerset @ ( the_carrier @ sk18 ) )
!= empty_set ) ),
inference(simp,[status(thm)],[14170]) ).
thf(14185,plain,
( ( powerset @ ( the_carrier @ sk18 ) )
!= empty_set ),
inference(simp,[status(thm)],[14175]) ).
thf(30455,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( in @ D @ C )
| ( ( in @ ( sk3 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
!= ( in @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[30445,141]) ).
thf(30456,plain,
! [B: $i,A: $i] :
~ ( in @ ( powerset @ ( cartesian_product2 @ A @ B ) ) @ ( sk3 @ B @ A ) ),
inference(pattern_uni,[status(thm)],[30455:[bind(A,$thf( H )),bind(B,$thf( I )),bind(C,$thf( sk3 @ I @ H )),bind(D,$thf( powerset @ ( cartesian_product2 @ H @ I ) ))]]) ).
thf(30557,plain,
! [B: $i,A: $i] :
~ ( in @ ( powerset @ ( cartesian_product2 @ A @ B ) ) @ ( sk3 @ B @ A ) ),
inference(simp,[status(thm)],[30456]) ).
thf(952,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( complete_latt_str @ A )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,66]) ).
thf(953,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( complete_latt_str @ A ) ),
inference(pattern_uni,[status(thm)],[952:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(585,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( join_of_latt_set @ sk1 @ A )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[246,279]) ).
thf(586,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( join_of_latt_set @ sk1 @ A )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 ) ),
inference(pattern_uni,[status(thm)],[585:[]]) ).
thf(661,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( in @ B @ A )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[246,309]) ).
thf(662,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( in @ B @ A ) ),
inference(pattern_uni,[status(thm)],[661:[]]) ).
thf(5556,plain,
! [B: $i,A: $i] :
( ~ ( in @ B @ A )
| ( ( in @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) )
!= ( in @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[5553,141]) ).
thf(5557,plain,
~ ( in @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) @ ( the_L_meet @ sk17 ) ),
inference(pattern_uni,[status(thm)],[5556:[bind(A,$thf( the_L_meet @ sk17 )),bind(B,$thf( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ))]]) ).
thf(24,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( function @ D )
& ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( element @ E @ A )
& ( element @ F @ B ) )
=> ( ( apply_binary_as_element @ A @ B @ C @ D @ E @ F )
= ( apply_binary @ D @ E @ F ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',redefinition_k2_binop_1) ).
thf(128,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( function @ D )
& ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( element @ E @ A )
& ( element @ F @ B ) )
=> ( ( apply_binary_as_element @ A @ B @ C @ D @ E @ F )
= ( apply_binary @ D @ E @ F ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(3799,plain,
( ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,580]) ).
thf(3834,plain,
( ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[3799]) ).
thf(1290,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ D ) @ ( ordered_pair @ A @ B ) )
= ( ordered_pair @ D @ C ) )
| ( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[338,355]) ).
thf(1291,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ B @ A ) )
= ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) ),
inference(pattern_uni,[status(thm)],[1290:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ E @ G ))]]) ).
thf(1318,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ B @ A ) )
= ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) ),
inference(simp,[status(thm)],[1291]) ).
thf(11786,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( unordered_pair @ B @ A ) ) @ ( ordered_pair @ C @ D ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,3031]) ).
thf(11787,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( unordered_pair @ B @ A ) ) @ ( ordered_pair @ A @ B ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11786:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(7096,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk17 )
| ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( sk12 @ sk17 ) @ ( powerset @ ( the_carrier @ sk17 ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[6970,119]) ).
thf(7097,plain,
( ( empty_carrier @ sk17 )
| ( empty @ ( powerset @ ( the_carrier @ sk17 ) ) )
| ( in @ ( sk12 @ sk17 ) @ ( powerset @ ( the_carrier @ sk17 ) ) ) ),
inference(pattern_uni,[status(thm)],[7096:[bind(A,$thf( sk12 @ sk17 )),bind(B,$thf( powerset @ ( the_carrier @ sk17 ) ))]]) ).
thf(16219,plain,
( ( empty_carrier @ sk17 )
| $false
| ( in @ ( sk12 @ sk17 ) @ ( powerset @ ( the_carrier @ sk17 ) ) ) ),
inference(rewrite,[status(thm)],[7097,125]) ).
thf(16220,plain,
( ( empty_carrier @ sk17 )
| ( in @ ( sk12 @ sk17 ) @ ( powerset @ ( the_carrier @ sk17 ) ) ) ),
inference(simp,[status(thm)],[16219]) ).
thf(964,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ~ ( empty_carrier @ A )
| ( B != sk18 ) ),
inference(paramod_ordered,[status(thm)],[90,263]) ).
thf(965,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk18 )
| ~ ( empty_carrier @ A ) ),
inference(pattern_uni,[status(thm)],[964:[bind(A,$thf( A )),bind(B,$thf( sk18 ))]]) ).
thf(161,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_semilatt_str @ A )
| ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[160]) ).
thf(3500,plain,
( ( function @ empty_set )
| ( ( empty @ ( the_L_meet @ sk1 ) )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,1856]) ).
thf(3523,plain,
( ( function @ empty_set )
| ( ( the_L_meet @ sk1 )
!= empty_set ) ),
inference(simp,[status(thm)],[3500]) ).
thf(44769,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( sk5 != sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,44719]) ).
thf(44770,plain,
( ~ ( empty @ sk1 )
| ( sk5 != sk1 ) ),
inference(pattern_uni,[status(thm)],[44769:[bind(A,$thf( sk1 ))]]) ).
thf(32,axiom,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc4_subset_1) ).
thf(149,plain,
! [A: $i,B: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B ) )
=> ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(46960,plain,
( ( sk1 != empty_set )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= sk1 )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,46809]) ).
thf(46961,plain,
( ( sk1 != empty_set )
| ( ( powerset @ ( the_carrier @ empty_set ) )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[46960:[]]) ).
thf(11798,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ C @ D ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[1316,3031]) ).
thf(11799,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11798:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ G ) )),bind(D,$thf( ordered_pair @ G @ H ))]]) ).
thf(11837,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[11799]) ).
thf(33547,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[78,33460]) ).
thf(33548,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33547:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ G @ F )),bind(D,$thf( singleton @ G ))]]) ).
thf(33630,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33548]) ).
thf(38121,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ D @ C ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[81,33630]) ).
thf(38122,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[38121:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( B )),bind(D,$thf( A ))]]) ).
thf(148,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( in @ A @ B )
| ~ ( element @ B @ ( powerset @ C ) )
| ( element @ A @ C ) ),
inference(cnf,[status(esa)],[147]) ).
thf(35035,plain,
( ( join_absorbing @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,34557]) ).
thf(35058,plain,
( ( join_absorbing @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[35035]) ).
thf(42,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( below @ A @ B @ C )
& ( below @ A @ C @ B ) )
=> ( B = C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t26_lattices) ).
thf(178,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_semilatt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( ( ( below @ A @ B @ C )
& ( below @ A @ C @ B ) )
=> ( B = C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(179,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( join_commutative @ A )
| ~ ( join_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( below @ A @ B @ C )
| ~ ( below @ A @ C @ B )
| ( B = C ) ),
inference(cnf,[status(esa)],[178]) ).
thf(180,plain,
! [C: $i,B: $i,A: $i] :
( ( B = C )
| ( empty_carrier @ A )
| ~ ( join_commutative @ A )
| ~ ( join_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( below @ A @ B @ C )
| ~ ( below @ A @ C @ B ) ),
inference(lifteq,[status(thm)],[179]) ).
thf(583,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,279]) ).
thf(584,plain,
( ~ ( empty @ sk1 )
| ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set ) ),
inference(pattern_uni,[status(thm)],[583:[bind(A,$thf( sk1 ))]]) ).
thf(183,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( ( meet @ A @ B @ ( sk14 @ B @ A ) )
!= B )
| ( ( meet @ A @ ( sk14 @ B @ A ) @ B )
!= B )
| ( lower_bounded_semilattstr @ A ) ),
inference(cnf,[status(esa)],[181]) ).
thf(190,plain,
! [B: $i,A: $i] :
( ( ( meet @ A @ B @ ( sk14 @ B @ A ) )
!= B )
| ( ( meet @ A @ ( sk14 @ B @ A ) @ B )
!= B )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( lower_bounded_semilattstr @ A ) ),
inference(lifteq,[status(thm)],[183]) ).
thf(191,plain,
! [B: $i,A: $i] :
( ( ( meet @ A @ B @ ( sk14 @ B @ A ) )
!= B )
| ( ( meet @ A @ ( sk14 @ B @ A ) @ B )
!= B )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( lower_bounded_semilattstr @ A ) ),
inference(simp,[status(thm)],[190]) ).
thf(34170,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[433,33582]) ).
thf(34171,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[34170:[bind(A,$thf( H )),bind(B,$thf( I )),bind(C,$thf( singleton @ ( unordered_pair @ H @ I ) )),bind(D,$thf( ordered_pair @ I @ H ))]]) ).
thf(34243,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[34171]) ).
thf(44602,plain,
( ( sk1 != empty_set )
| ( function @ empty_set )
| ( ( empty @ ( the_L_meet @ sk1 ) )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,1856]) ).
thf(44722,plain,
( ( function @ empty_set )
| ( sk1 != empty_set )
| ( ( the_L_meet @ sk1 )
!= sk1 ) ),
inference(simp,[status(thm)],[44602]) ).
thf(11796,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ C @ D ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[338,3031]) ).
thf(11797,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11796:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( unordered_pair @ E @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(11836,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[11797]) ).
thf(30484,plain,
! [C: $i,B: $i,A: $i] :
( ( in @ ( sk3 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
!= ( in @ C @ C ) ),
inference(paramod_ordered,[status(thm)],[30445,494]) ).
thf(30518,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk3 @ B @ A )
!= C )
| ( ( powerset @ ( cartesian_product2 @ A @ B ) )
!= C ) ),
inference(simp,[status(thm)],[30484]) ).
thf(30556,plain,
! [B: $i,A: $i] :
( ( sk3 @ B @ A )
!= ( powerset @ ( cartesian_product2 @ A @ B ) ) ),
inference(simp,[status(thm)],[30518]) ).
thf(33822,plain,
! [C: $i,B: $i,A: $i] :
( ( in @ ( sk2 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
!= ( in @ C @ C ) ),
inference(paramod_ordered,[status(thm)],[33777,494]) ).
thf(33858,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk2 @ B @ A )
!= C )
| ( ( powerset @ ( cartesian_product2 @ A @ B ) )
!= C ) ),
inference(simp,[status(thm)],[33822]) ).
thf(33868,plain,
! [B: $i,A: $i] :
( ( sk2 @ B @ A )
!= ( powerset @ ( cartesian_product2 @ A @ B ) ) ),
inference(simp,[status(thm)],[33858]) ).
thf(49876,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28192]) ).
thf(49877,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ sk1 )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[49876:[bind(A,$thf( sk1 ))]]) ).
thf(49957,plain,
( ~ ( empty @ sk1 )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(simp,[status(thm)],[49877]) ).
thf(65531,plain,
( ( sk1 != empty_set )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,49957]) ).
thf(65532,plain,
( ( sk1 != empty_set )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[65531:[]]) ).
thf(41,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( meet @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_lattices) ).
thf(176,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_semilatt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( element @ ( meet @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(1832,plain,
! [A: $i] :
( ( function @ ( the_L_meet @ A ) )
| ( ( meet_semilatt_str @ sk17 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[441,107]) ).
thf(1833,plain,
function @ ( the_L_meet @ sk17 ),
inference(pattern_uni,[status(thm)],[1832:[bind(A,$thf( sk17 ))]]) ).
thf(24063,plain,
( ~ ( empty @ ( the_carrier @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,23411]) ).
thf(24089,plain,
( ~ ( empty @ ( the_carrier @ empty_set ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[24063]) ).
thf(33708,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1316,33582]) ).
thf(33709,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33708:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ G @ H )),bind(D,$thf( singleton @ ( singleton @ G ) ))]]) ).
thf(33760,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33709]) ).
thf(43889,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( ordered_pair @ C @ D ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) )
!= ( ordered_pair @ ( singleton @ C ) @ ( unordered_pair @ C @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[3103,33760]) ).
thf(43890,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[43889:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(33591,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[338,33460]) ).
thf(33592,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33591:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( unordered_pair @ E @ G )),bind(D,$thf( singleton @ G ))]]) ).
thf(33653,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33592]) ).
thf(280,plain,
( ( empty @ sk5 )
!= ( empty @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[172,114]) ).
thf(281,plain,
sk5 != empty_set,
inference(simp,[status(thm)],[280]) ).
thf(197,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ( latt_element_smaller @ A @ ( sk15 @ C @ B @ A ) @ B )
| ( C
= ( join_of_latt_set @ A @ B ) ) ),
inference(cnf,[status(esa)],[195]) ).
thf(203,plain,
! [C: $i,B: $i,A: $i] :
( ( C
= ( join_of_latt_set @ A @ B ) )
| ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ( latt_element_smaller @ A @ ( sk15 @ C @ B @ A ) @ B ) ),
inference(lifteq,[status(thm)],[197]) ).
thf(204,plain,
! [C: $i,B: $i,A: $i] :
( ( C
= ( join_of_latt_set @ A @ B ) )
| ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ( latt_element_smaller @ A @ ( sk15 @ C @ B @ A ) @ B ) ),
inference(simp,[status(thm)],[203]) ).
thf(26334,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk7 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[127,243]) ).
thf(26335,plain,
( ( empty_carrier @ sk7 )
| ( element @ ( bottom_of_semilattstr @ sk7 ) @ ( the_carrier @ sk7 ) ) ),
inference(pattern_uni,[status(thm)],[26334:[bind(A,$thf( sk7 ))]]) ).
thf(12066,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk7 )
| ( subset @ A @ B )
| ( ( element @ ( sk12 @ sk7 ) @ ( powerset @ ( the_carrier @ sk7 ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[6986,138]) ).
thf(12067,plain,
( ( empty_carrier @ sk7 )
| ( subset @ ( sk12 @ sk7 ) @ ( the_carrier @ sk7 ) ) ),
inference(pattern_uni,[status(thm)],[12066:[bind(A,$thf( sk12 @ sk7 )),bind(B,$thf( the_carrier @ sk7 ))]]) ).
thf(43130,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) @ ( singleton @ ( ordered_pair @ C @ D ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( ordered_pair @ ( unordered_pair @ C @ D ) @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[36256,33627]) ).
thf(43131,plain,
! [A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( ordered_pair @ A @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[43130:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(235,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( ( meet @ A @ B @ C )
= ( apply_binary_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_L_meet @ A ) @ B @ C ) ) ),
inference(cnf,[status(esa)],[234]) ).
thf(236,plain,
! [C: $i,B: $i,A: $i] :
( ( ( apply_binary_as_element @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_carrier @ A ) @ ( the_L_meet @ A ) @ B @ C )
= ( meet @ A @ B @ C ) )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[235]) ).
thf(38065,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ D @ C ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[78,33630]) ).
thf(38066,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[38065:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ G @ F ))]]) ).
thf(38202,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[38066]) ).
thf(28193,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28148]) ).
thf(28194,plain,
( ~ ( empty @ sk1 )
| ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[28193:[bind(A,$thf( sk1 ))]]) ).
thf(38247,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ D @ C ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[78,33653]) ).
thf(38248,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[38247:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ G @ F )),bind(D,$thf( singleton @ G ))]]) ).
thf(38408,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[38248]) ).
thf(44637,plain,
( ( sk1 != empty_set )
| ( ( empty @ ( sk12 @ sk1 ) )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,7206]) ).
thf(44744,plain,
( ( sk1 != empty_set )
| ( ( sk12 @ sk1 )
!= sk1 ) ),
inference(simp,[status(thm)],[44637]) ).
thf(3056,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[431,2102]) ).
thf(3057,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[3056:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ H ) )),bind(D,$thf( ordered_pair @ H @ F ))]]) ).
thf(3091,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[3057]) ).
thf(184,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( ( meet @ A @ B @ ( sk13 @ A ) )
= ( sk13 @ A ) ) ),
inference(cnf,[status(esa)],[181]) ).
thf(187,plain,
! [B: $i,A: $i] :
( ( ( meet @ A @ B @ ( sk13 @ A ) )
= ( sk13 @ A ) )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[184]) ).
thf(2720,plain,
! [C: $i,B: $i,A: $i] :
( ( relation_of2 @ C @ A @ B )
| ( ( relation_of2_as_subset @ ( the_L_meet @ sk7 ) @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) )
!= ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[2541,85]) ).
thf(2721,plain,
relation_of2 @ ( the_L_meet @ sk7 ) @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ),
inference(pattern_uni,[status(thm)],[2720:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) )),bind(B,$thf( the_carrier @ sk7 )),bind(C,$thf( the_L_meet @ sk7 ))]]) ).
thf(20669,plain,
! [C: $i,B: $i,A: $i] :
( ( empty @ C )
| ( in @ B @ C )
| ( ( element @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) )
!= ( element @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[20593,119]) ).
thf(20670,plain,
! [A: $i] :
( ( empty @ ( the_carrier @ sk1 ) )
| ( in @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[20669:[bind(A,$thf( E )),bind(B,$thf( join_of_latt_set @ sk1 @ E )),bind(C,$thf( the_carrier @ sk1 ))]]) ).
thf(20734,plain,
! [A: $i] :
( ( empty @ ( the_carrier @ sk1 ) )
| ( in @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) ) ),
inference(simp,[status(thm)],[20670]) ).
thf(23391,plain,
! [A: $i] :
( $false
| ( in @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) ) ),
inference(rewrite,[status(thm)],[20734,23390]) ).
thf(23392,plain,
! [A: $i] : ( in @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) ),
inference(simp,[status(thm)],[23391]) ).
thf(23785,plain,
! [B: $i,A: $i] :
( ( in @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) )
!= ( in @ ( powerset @ B ) @ B ) ),
inference(paramod_ordered,[status(thm)],[23392,4117]) ).
thf(23827,plain,
! [B: $i,A: $i] :
( ( ( join_of_latt_set @ sk1 @ A )
!= ( powerset @ B ) )
| ( ( the_carrier @ sk1 )
!= B ) ),
inference(simp,[status(thm)],[23785]) ).
thf(23848,plain,
! [A: $i] :
( ( join_of_latt_set @ sk1 @ A )
!= ( powerset @ ( the_carrier @ sk1 ) ) ),
inference(simp,[status(thm)],[23827]) ).
thf(268,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( ( meet @ A @ B @ ( sk19 @ B @ A ) )
!= B )
| ( ( meet @ A @ ( sk19 @ B @ A ) @ B )
!= B )
| ( B
= ( bottom_of_semilattstr @ A ) ) ),
inference(cnf,[status(esa)],[265]) ).
thf(274,plain,
! [B: $i,A: $i] :
( ( ( meet @ A @ B @ ( sk19 @ B @ A ) )
!= B )
| ( ( meet @ A @ ( sk19 @ B @ A ) @ B )
!= B )
| ( B
= ( bottom_of_semilattstr @ A ) )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[268]) ).
thf(6795,plain,
( ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,3835]) ).
thf(6832,plain,
( ( ( join_of_latt_set @ empty_set @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[6795]) ).
thf(36399,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ D ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[36256,3032]) ).
thf(36400,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[36399:[bind(A,$thf( G )),bind(B,$thf( E )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ E @ G ))]]) ).
thf(36766,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[36400]) ).
thf(23751,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( in @ C @ B )
| ( ( in @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) )
!= ( in @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[23392,141]) ).
thf(23752,plain,
! [A: $i] :
~ ( in @ ( the_carrier @ sk1 ) @ ( join_of_latt_set @ sk1 @ A ) ),
inference(pattern_uni,[status(thm)],[23751:[bind(A,$thf( E )),bind(B,$thf( join_of_latt_set @ sk1 @ E )),bind(C,$thf( the_carrier @ sk1 ))]]) ).
thf(23838,plain,
! [A: $i] :
~ ( in @ ( the_carrier @ sk1 ) @ ( join_of_latt_set @ sk1 @ A ) ),
inference(simp,[status(thm)],[23752]) ).
thf(4366,plain,
! [B: $i,A: $i] :
( ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[4337,119]) ).
thf(4367,plain,
( ( empty @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) )
| ( in @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) ) ),
inference(pattern_uni,[status(thm)],[4366:[bind(A,$thf( the_L_meet @ sk7 )),bind(B,$thf( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ))]]) ).
thf(5009,plain,
( $false
| ( in @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) ) ),
inference(rewrite,[status(thm)],[4367,125]) ).
thf(5010,plain,
in @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ),
inference(simp,[status(thm)],[5009]) ).
thf(5091,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) )
!= ( in @ A @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[5010,309]) ).
thf(5092,plain,
! [A: $i] :
( ( ( the_L_meet @ sk7 )
!= A )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) )
!= empty_set ) ),
inference(simp,[status(thm)],[5091]) ).
thf(5103,plain,
( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) )
!= empty_set ),
inference(simp,[status(thm)],[5092]) ).
thf(2429,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ C ) )
= ( ordered_pair @ C @ D ) )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1316,338]) ).
thf(2430,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) ),
inference(pattern_uni,[status(thm)],[2429:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ G @ H )),bind(D,$thf( singleton @ ( singleton @ G ) ))]]) ).
thf(2493,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) ),
inference(simp,[status(thm)],[2430]) ).
thf(1527,plain,
( ( one_sorted_str @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,642]) ).
thf(1542,plain,
( ( one_sorted_str @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[1527]) ).
thf(28260,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,28234]) ).
thf(28261,plain,
( ~ ( empty @ sk1 )
| ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[28260:[bind(A,$thf( sk1 ))]]) ).
thf(28487,plain,
( ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,28261]) ).
thf(28511,plain,
( ( ( the_carrier @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[28487]) ).
thf(58939,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[1993,1994]) ).
thf(58940,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( ordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[58939:[bind(A,$thf( L )),bind(B,$thf( K )),bind(C,$thf( ordered_pair @ ( singleton @ K ) @ ( unordered_pair @ L @ K ) )),bind(D,$thf( singleton @ ( ordered_pair @ K @ L ) ))]]) ).
thf(59089,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( ordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[58940]) ).
thf(736,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( one_sorted_str @ A )
| ( B != sk8 ) ),
inference(paramod_ordered,[status(thm)],[90,143]) ).
thf(737,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk8 )
| ( one_sorted_str @ A ) ),
inference(pattern_uni,[status(thm)],[736:[bind(A,$thf( A )),bind(B,$thf( sk8 ))]]) ).
thf(13,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( function @ D )
& ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( element @ E @ A )
& ( element @ F @ B ) )
=> ( element @ ( apply_binary_as_element @ A @ B @ C @ D @ E @ F ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k2_binop_1) ).
thf(98,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ~ ( empty @ A )
& ~ ( empty @ B )
& ( function @ D )
& ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
& ( element @ E @ A )
& ( element @ F @ B ) )
=> ( element @ ( apply_binary_as_element @ A @ B @ C @ D @ E @ F ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(615,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( one_sorted_str @ empty_set )
| ( A != sk7 ) ),
inference(paramod_ordered,[status(thm)],[246,330]) ).
thf(616,plain,
( ~ ( empty @ sk7 )
| ( one_sorted_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[615:[bind(A,$thf( sk7 ))]]) ).
thf(266,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( B
!= ( bottom_of_semilattstr @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( ( meet @ A @ B @ C )
= B ) ),
inference(cnf,[status(esa)],[265]) ).
thf(270,plain,
! [C: $i,B: $i,A: $i] :
( ( B
!= ( bottom_of_semilattstr @ A ) )
| ( ( meet @ A @ B @ C )
= B )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[266]) ).
thf(271,plain,
! [B: $i,A: $i] :
( ( ( meet @ A @ ( bottom_of_semilattstr @ A ) @ B )
= ( bottom_of_semilattstr @ A ) )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) )
| ~ ( element @ B @ ( the_carrier @ A ) ) ),
inference(simp,[status(thm)],[270]) ).
thf(199,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( C
!= ( join_of_latt_set @ A @ B ) )
| ( latt_element_smaller @ A @ C @ B ) ),
inference(cnf,[status(esa)],[195]) ).
thf(207,plain,
! [C: $i,B: $i,A: $i] :
( ( C
!= ( join_of_latt_set @ A @ B ) )
| ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( latt_element_smaller @ A @ C @ B ) ),
inference(lifteq,[status(thm)],[199]) ).
thf(208,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( element @ ( join_of_latt_set @ A @ B ) @ ( the_carrier @ A ) )
| ( latt_element_smaller @ A @ ( join_of_latt_set @ A @ B ) @ B ) ),
inference(simp,[status(thm)],[207]) ).
thf(4540,plain,
! [B: $i,A: $i] :
( ( subset @ A @ B )
| ( ( element @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[4339,138]) ).
thf(4541,plain,
subset @ ( the_L_meet @ sk1 ) @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ),
inference(pattern_uni,[status(thm)],[4540:[bind(A,$thf( the_L_meet @ sk1 )),bind(B,$thf( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ))]]) ).
thf(625,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( meet_semilatt_str @ empty_set )
| ( A != sk7 ) ),
inference(paramod_ordered,[status(thm)],[246,127]) ).
thf(626,plain,
( ~ ( empty @ sk7 )
| ( meet_semilatt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[625:[bind(A,$thf( sk7 ))]]) ).
thf(34134,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[433,1994]) ).
thf(34135,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[34134:[bind(A,$thf( H )),bind(B,$thf( I )),bind(C,$thf( ordered_pair @ I @ H )),bind(D,$thf( singleton @ ( unordered_pair @ H @ I ) ))]]) ).
thf(34225,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[34135]) ).
thf(3892,plain,
! [C: $i,B: $i,A: $i] :
( ( subset @ B @ C )
| ( ( element @ empty_set @ ( powerset @ A ) )
!= ( element @ B @ ( powerset @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[1203,138]) ).
thf(3893,plain,
! [A: $i] : ( subset @ empty_set @ A ),
inference(pattern_uni,[status(thm)],[3892:[bind(A,$thf( A )),bind(B,$thf( empty_set )),bind(C,$thf( A ))]]) ).
thf(4817,plain,
! [B: $i,A: $i] :
( ( subset @ A @ B )
| ( ( element @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[4347,138]) ).
thf(4818,plain,
subset @ ( the_L_meet @ sk17 ) @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ),
inference(pattern_uni,[status(thm)],[4817:[bind(A,$thf( the_L_meet @ sk17 )),bind(B,$thf( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ))]]) ).
thf(593,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( join_semilatt_str @ empty_set )
| ( A != sk17 ) ),
inference(paramod_ordered,[status(thm)],[246,466]) ).
thf(594,plain,
( ~ ( empty @ sk17 )
| ( join_semilatt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[593:[bind(A,$thf( sk17 ))]]) ).
thf(3942,plain,
( ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,584]) ).
thf(3977,plain,
( ( ( join_of_latt_set @ sk1 @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[3942]) ).
thf(58,axiom,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( below @ A @ ( meet_commut @ A @ B @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t23_lattices) ).
thf(250,plain,
! [A: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( latt_str @ A ) )
=> ! [B: $i] :
( ( element @ B @ ( the_carrier @ A ) )
=> ! [C: $i] :
( ( element @ C @ ( the_carrier @ A ) )
=> ( below @ A @ ( meet_commut @ A @ B @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(251,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_absorbing @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( below @ A @ ( meet_commut @ A @ B @ C ) @ B ) ),
inference(cnf,[status(esa)],[250]) ).
thf(33700,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[431,33582]) ).
thf(33701,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33700:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ H ) )),bind(D,$thf( ordered_pair @ H @ F ))]]) ).
thf(33756,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33701]) ).
thf(12105,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk7 )
| ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( sk12 @ sk7 ) @ ( powerset @ ( the_carrier @ sk7 ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[6986,119]) ).
thf(12106,plain,
( ( empty_carrier @ sk7 )
| ( empty @ ( powerset @ ( the_carrier @ sk7 ) ) )
| ( in @ ( sk12 @ sk7 ) @ ( powerset @ ( the_carrier @ sk7 ) ) ) ),
inference(pattern_uni,[status(thm)],[12105:[bind(A,$thf( sk12 @ sk7 )),bind(B,$thf( powerset @ ( the_carrier @ sk7 ) ))]]) ).
thf(18242,plain,
( ( empty_carrier @ sk7 )
| $false
| ( in @ ( sk12 @ sk7 ) @ ( powerset @ ( the_carrier @ sk7 ) ) ) ),
inference(rewrite,[status(thm)],[12106,125]) ).
thf(18243,plain,
( ( empty_carrier @ sk7 )
| ( in @ ( sk12 @ sk7 ) @ ( powerset @ ( the_carrier @ sk7 ) ) ) ),
inference(simp,[status(thm)],[18242]) ).
thf(34903,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ B @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1318,33582]) ).
thf(34904,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[34903:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ I )),bind(D,$thf( singleton @ ( unordered_pair @ I @ H ) ))]]) ).
thf(34963,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( ordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[34904]) ).
thf(2138,plain,
! [A: $i] :
( ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk1 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[445,108]) ).
thf(2139,plain,
quasi_total @ ( the_L_meet @ sk1 ) @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ),
inference(pattern_uni,[status(thm)],[2138:[bind(A,$thf( sk1 ))]]) ).
thf(219,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( element @ ( sk16 @ C @ B @ A ) @ ( the_carrier @ A ) )
| ( latt_element_smaller @ A @ B @ C ) ),
inference(cnf,[status(esa)],[218]) ).
thf(223,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( element @ ( sk16 @ C @ B @ A ) @ ( the_carrier @ A ) )
| ( latt_element_smaller @ A @ B @ C ) ),
inference(simp,[status(thm)],[219]) ).
thf(49,axiom,
! [A: $i] :
( ( latt_str @ A )
=> ( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',cc2_lattices) ).
thf(226,plain,
! [A: $i] :
( ( latt_str @ A )
=> ( ( ~ ( empty_carrier @ A )
& ( join_commutative @ A )
& ( join_associative @ A )
& ( meet_commutative @ A )
& ( meet_associative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A ) )
=> ( ~ ( empty_carrier @ A )
& ( lattice @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(21633,plain,
! [B: $i,A: $i] :
( ( empty @ ( the_carrier @ sk1 ) )
| ( ( in @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) )
!= ( in @ B @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[20734,309]) ).
thf(21638,plain,
! [B: $i,A: $i] :
( ( empty @ ( the_carrier @ sk1 ) )
| ( ( join_of_latt_set @ sk1 @ A )
!= B )
| ( ( the_carrier @ sk1 )
!= empty_set ) ),
inference(simp,[status(thm)],[21633]) ).
thf(21648,plain,
( ( empty @ ( the_carrier @ sk1 ) )
| ( ( the_carrier @ sk1 )
!= empty_set ) ),
inference(simp,[status(thm)],[21638]) ).
thf(23393,plain,
( $false
| ( ( the_carrier @ sk1 )
!= empty_set ) ),
inference(rewrite,[status(thm)],[21648,23390]) ).
thf(23394,plain,
( ( the_carrier @ sk1 )
!= empty_set ),
inference(simp,[status(thm)],[23393]) ).
thf(3258,plain,
! [C: $i,B: $i,A: $i] :
( ( empty @ A )
| ( empty @ C )
| ( in @ B @ C )
| ( ( element @ ( sk6 @ A ) @ ( powerset @ A ) )
!= ( element @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[121,119]) ).
thf(3259,plain,
! [A: $i] :
( ( empty @ A )
| ( empty @ ( powerset @ A ) )
| ( in @ ( sk6 @ A ) @ ( powerset @ A ) ) ),
inference(pattern_uni,[status(thm)],[3258:[bind(A,$thf( E )),bind(B,$thf( sk6 @ E )),bind(C,$thf( powerset @ E ))]]) ).
thf(3283,plain,
! [A: $i] :
( ( empty @ A )
| ( empty @ ( powerset @ A ) )
| ( in @ ( sk6 @ A ) @ ( powerset @ A ) ) ),
inference(simp,[status(thm)],[3259]) ).
thf(4281,plain,
! [A: $i] :
( ( empty @ A )
| $false
| ( in @ ( sk6 @ A ) @ ( powerset @ A ) ) ),
inference(rewrite,[status(thm)],[3283,125]) ).
thf(4282,plain,
! [A: $i] :
( ( empty @ A )
| ( in @ ( sk6 @ A ) @ ( powerset @ A ) ) ),
inference(simp,[status(thm)],[4281]) ).
thf(54,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( below_refl @ A @ B @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r3_lattices) ).
thf(239,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ( element @ C @ ( the_carrier @ A ) ) )
=> ( below_refl @ A @ B @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(240,plain,
! [A: $i,B: $i] :
( ( ~ ( empty_carrier @ A )
& ( meet_commutative @ A )
& ( meet_absorbing @ A )
& ( join_absorbing @ A )
& ( latt_str @ A )
& ( element @ B @ ( the_carrier @ A ) )
& ? [C: $i] : ( element @ C @ ( the_carrier @ A ) ) )
=> ( below_refl @ A @ B @ B ) ),
inference(miniscope,[status(thm)],[239]) ).
thf(241,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_absorbing @ A )
| ~ ( join_absorbing @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( below_refl @ A @ B @ B ) ),
inference(cnf,[status(esa)],[240]) ).
thf(129,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( empty @ A )
| ( empty @ B )
| ~ ( function @ D )
| ~ ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
| ~ ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
| ~ ( element @ E @ A )
| ~ ( element @ F @ B )
| ( ( apply_binary_as_element @ A @ B @ C @ D @ E @ F )
= ( apply_binary @ D @ E @ F ) ) ),
inference(cnf,[status(esa)],[128]) ).
thf(130,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( ( apply_binary_as_element @ A @ B @ C @ D @ E @ F )
= ( apply_binary @ D @ E @ F ) )
| ( empty @ A )
| ( empty @ B )
| ~ ( function @ D )
| ~ ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
| ~ ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
| ~ ( element @ E @ A )
| ~ ( element @ F @ B ) ),
inference(lifteq,[status(thm)],[129]) ).
thf(573,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( empty @ A )
| ( relation_of2 @ empty_set @ C @ B )
| ( A
!= ( sk2 @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[246,488]) ).
thf(574,plain,
! [B: $i,A: $i] :
( ~ ( empty @ ( sk2 @ A @ B ) )
| ( relation_of2 @ empty_set @ B @ A ) ),
inference(pattern_uni,[status(thm)],[573:[bind(A,$thf( sk2 @ D @ E )),bind(B,$thf( D )),bind(C,$thf( E ))]]) ).
thf(695,plain,
! [B: $i,A: $i] :
( ~ ( empty @ ( sk2 @ A @ B ) )
| ( relation_of2 @ empty_set @ B @ A ) ),
inference(simp,[status(thm)],[574]) ).
thf(48977,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ C @ D ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( ordered_pair @ ( unordered_pair @ C @ D ) @ ( singleton @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[36256,33514]) ).
thf(48978,plain,
! [A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ A @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ A ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[48977:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(26574,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( element @ empty_set @ ( the_carrier @ sk1 ) )
| ( A
!= ( bottom_of_semilattstr @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[246,26482]) ).
thf(26575,plain,
( ~ ( empty @ ( bottom_of_semilattstr @ sk1 ) )
| ( element @ empty_set @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[26574:[bind(A,$thf( bottom_of_semilattstr @ sk1 ))]]) ).
thf(48136,plain,
( ( element @ empty_set @ ( the_carrier @ sk1 ) )
| ( ( empty @ ( bottom_of_semilattstr @ sk1 ) )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,26575]) ).
thf(48226,plain,
( ( element @ empty_set @ ( the_carrier @ sk1 ) )
| ( ( bottom_of_semilattstr @ sk1 )
!= empty_set ) ),
inference(simp,[status(thm)],[48136]) ).
thf(3875,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( function @ empty_set )
| ( ( the_L_meet @ empty_set )
!= empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,3523]) ).
thf(3876,plain,
( ~ ( empty @ sk1 )
| ( function @ empty_set )
| ( ( the_L_meet @ empty_set )
!= empty_set ) ),
inference(pattern_uni,[status(thm)],[3875:[bind(A,$thf( sk1 ))]]) ).
thf(167,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_associative @ A )
| ~ ( meet_semilatt_str @ A )
| ( v1_partfun1 @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[166]) ).
thf(23441,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( the_carrier @ empty_set )
!= empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,23394]) ).
thf(23442,plain,
( ~ ( empty @ sk1 )
| ( ( the_carrier @ empty_set )
!= empty_set ) ),
inference(pattern_uni,[status(thm)],[23441:[bind(A,$thf( sk1 ))]]) ).
thf(24119,plain,
( ( ( the_carrier @ empty_set )
!= empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,23442]) ).
thf(24143,plain,
( ( ( the_carrier @ empty_set )
!= empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[24119]) ).
thf(48519,plain,
( ( sk1 != empty_set )
| ( element @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,26579]) ).
thf(48520,plain,
( ( sk1 != empty_set )
| ( element @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[48519:[]]) ).
thf(1550,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( lattice @ A )
| ( sk1 != empty_set )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[246,1545]) ).
thf(1551,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( lattice @ A )
| ( sk1 != empty_set ) ),
inference(pattern_uni,[status(thm)],[1550:[]]) ).
thf(194,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_absorbing @ A )
| ~ ( join_absorbing @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( below @ A @ B @ C )
| ( below_refl @ A @ B @ C ) ),
inference(cnf,[status(esa)],[192]) ).
thf(30517,plain,
! [C: $i,B: $i,A: $i] :
( ( in @ ( sk3 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
!= ( in @ C @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[30445,309]) ).
thf(30522,plain,
! [C: $i,B: $i,A: $i] :
( ( ( sk3 @ B @ A )
!= C )
| ( ( powerset @ ( cartesian_product2 @ A @ B ) )
!= empty_set ) ),
inference(simp,[status(thm)],[30517]) ).
thf(30560,plain,
! [B: $i,A: $i] :
( ( powerset @ ( cartesian_product2 @ A @ B ) )
!= empty_set ),
inference(simp,[status(thm)],[30522]) ).
thf(11896,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ D @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[355,3045]) ).
thf(11897,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11896:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ F )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(11956,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[11897]) ).
thf(28392,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ( meet_commutative @ A )
| ( ( lattice @ sk1 )
!= ( lattice @ A ) ) ),
inference(paramod_ordered,[status(thm)],[65,256]) ).
thf(28393,plain,
( ~ ( latt_str @ sk1 )
| ( empty_carrier @ sk1 )
| ( meet_commutative @ sk1 ) ),
inference(pattern_uni,[status(thm)],[28392:[bind(A,$thf( sk1 ))]]) ).
thf(29105,plain,
( ~ $true
| $false
| ( meet_commutative @ sk1 ) ),
inference(rewrite,[status(thm)],[28393,64,68]) ).
thf(29106,plain,
meet_commutative @ sk1,
inference(simp,[status(thm)],[29105]) ).
thf(29131,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( meet_commutative @ empty_set )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,29106]) ).
thf(29132,plain,
( ~ ( empty @ sk1 )
| ( meet_commutative @ empty_set ) ),
inference(pattern_uni,[status(thm)],[29131:[bind(A,$thf( sk1 ))]]) ).
thf(29175,plain,
( ( meet_commutative @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,29132]) ).
thf(29202,plain,
( ( meet_commutative @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[29175]) ).
thf(992,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( join_semilatt_str @ A )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,470]) ).
thf(993,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( join_semilatt_str @ A ) ),
inference(pattern_uni,[status(thm)],[992:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(34816,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ B @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[1318,33460]) ).
thf(34817,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[34816:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( unordered_pair @ I @ H ) )),bind(D,$thf( ordered_pair @ H @ I ))]]) ).
thf(34926,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[34817]) ).
thf(5300,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) )
!= ( in @ ( powerset @ A ) @ A ) ),
inference(paramod_ordered,[status(thm)],[5288,4117]) ).
thf(5372,plain,
! [A: $i] :
( ( ( the_L_meet @ sk1 )
!= ( powerset @ A ) )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) )
!= A ) ),
inference(simp,[status(thm)],[5300]) ).
thf(5388,plain,
( ( the_L_meet @ sk1 )
!= ( powerset @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) ) ),
inference(simp,[status(thm)],[5372]) ).
thf(150,plain,
! [B: $i,A: $i] :
( ( empty @ A )
| ( empty @ B )
| ~ ( empty @ ( cartesian_product2 @ A @ B ) ) ),
inference(cnf,[status(esa)],[149]) ).
thf(39407,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ C ) @ ( ordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ C ) ) ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[426,33654]) ).
thf(39408,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[39407:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(39555,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[39408]) ).
thf(38473,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( ( in @ A @ ( powerset @ A ) )
!= ( in @ ( the_carrier @ sk1 ) @ ( bottom_of_semilattstr @ empty_set ) ) ) ),
inference(paramod_ordered,[status(thm)],[4089,28324]) ).
thf(38527,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( A
!= ( the_carrier @ sk1 ) )
| ( ( powerset @ A )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[38473]) ).
thf(38556,plain,
( ~ ( empty @ sk1 )
| ( ( powerset @ ( the_carrier @ sk1 ) )
!= ( bottom_of_semilattstr @ empty_set ) ) ),
inference(simp,[status(thm)],[38527]) ).
thf(2142,plain,
! [A: $i] :
( ( quasi_total @ ( the_L_meet @ A ) @ ( cartesian_product2 @ ( the_carrier @ A ) @ ( the_carrier @ A ) ) @ ( the_carrier @ A ) )
| ( ( meet_semilatt_str @ sk7 )
!= ( meet_semilatt_str @ A ) ) ),
inference(paramod_ordered,[status(thm)],[127,108]) ).
thf(2143,plain,
quasi_total @ ( the_L_meet @ sk7 ) @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ),
inference(pattern_uni,[status(thm)],[2142:[bind(A,$thf( sk7 ))]]) ).
thf(3888,plain,
! [C: $i,B: $i,A: $i] :
( ( empty @ A )
| ( subset @ B @ C )
| ( ( element @ ( sk6 @ A ) @ ( powerset @ A ) )
!= ( element @ B @ ( powerset @ C ) ) ) ),
inference(paramod_ordered,[status(thm)],[121,138]) ).
thf(3889,plain,
! [A: $i] :
( ( empty @ A )
| ( subset @ ( sk6 @ A ) @ A ) ),
inference(pattern_uni,[status(thm)],[3888:[bind(A,$thf( D )),bind(B,$thf( sk6 @ D )),bind(C,$thf( D ))]]) ).
thf(3904,plain,
! [A: $i] :
( ( empty @ A )
| ( subset @ ( sk6 @ A ) @ A ) ),
inference(simp,[status(thm)],[3889]) ).
thf(2655,plain,
! [C: $i,B: $i,A: $i] :
( ( relation_of2 @ C @ A @ B )
| ( ( relation_of2_as_subset @ ( the_L_meet @ sk17 ) @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) )
!= ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[2539,85]) ).
thf(2656,plain,
relation_of2 @ ( the_L_meet @ sk17 ) @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ),
inference(pattern_uni,[status(thm)],[2655:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) )),bind(B,$thf( the_carrier @ sk17 )),bind(C,$thf( the_L_meet @ sk17 ))]]) ).
thf(45161,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( ( sk12 @ sk1 )
!= sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,44744]) ).
thf(45162,plain,
( ~ ( empty @ sk1 )
| ( ( sk12 @ sk1 )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[45161:[bind(A,$thf( sk1 ))]]) ).
thf(3752,plain,
( ( ( bottom_of_semilattstr @ sk1 )
!= empty_set )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( ( empty @ ( join_of_latt_set @ sk1 @ empty_set ) )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,578]) ).
thf(3791,plain,
( ( ( bottom_of_semilattstr @ sk1 )
!= empty_set )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( ( join_of_latt_set @ sk1 @ empty_set )
!= empty_set ) ),
inference(simp,[status(thm)],[3752]) ).
thf(177,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( element @ ( meet @ A @ B @ C ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[176]) ).
thf(48289,plain,
( ( sk1 != empty_set )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,26577]) ).
thf(48290,plain,
( ( sk1 != empty_set )
| ( element @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[48289:[]]) ).
thf(16353,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk1 ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,14531]) ).
thf(16354,plain,
( ~ ( empty @ sk1 )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[16353:[bind(A,$thf( sk1 ))]]) ).
thf(44454,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ empty_set ) )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,16354]) ).
thf(44455,plain,
( ~ ( empty @ sk1 )
| ~ ( empty @ sk1 )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[44454:[bind(A,$thf( sk1 ))]]) ).
thf(44528,plain,
( ~ ( empty @ sk1 )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ empty_set ) ) ),
inference(simp,[status(thm)],[44455]) ).
thf(33917,plain,
( ( join_associative @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,33250]) ).
thf(33936,plain,
( ( join_associative @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[33917]) ).
thf(2426,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ D @ C ) )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1316,355]) ).
thf(2427,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) ),
inference(pattern_uni,[status(thm)],[2426:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ G @ H )),bind(D,$thf( singleton @ ( singleton @ G ) ))]]) ).
thf(2492,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) ),
inference(simp,[status(thm)],[2427]) ).
thf(57037,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( unordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) ),
inference(rewrite,[status(thm)],[2492,1316]) ).
thf(575,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( one_sorted_str @ empty_set )
| ( A != sk18 ) ),
inference(paramod_ordered,[status(thm)],[246,264]) ).
thf(576,plain,
( ~ ( empty @ sk18 )
| ( one_sorted_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[575:[bind(A,$thf( sk18 ))]]) ).
thf(5565,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk17 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) )
!= ( in @ ( powerset @ A ) @ A ) ),
inference(paramod_ordered,[status(thm)],[5553,4117]) ).
thf(5635,plain,
! [A: $i] :
( ( ( the_L_meet @ sk17 )
!= ( powerset @ A ) )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) )
!= A ) ),
inference(simp,[status(thm)],[5565]) ).
thf(5638,plain,
( ( the_L_meet @ sk17 )
!= ( powerset @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk17 ) @ ( the_carrier @ sk17 ) ) @ ( the_carrier @ sk17 ) ) ) ) ),
inference(simp,[status(thm)],[5635]) ).
thf(41767,plain,
( ( subset @ ( sk12 @ sk1 ) @ ( the_carrier @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,13604]) ).
thf(41807,plain,
( ( subset @ ( sk12 @ sk1 ) @ ( the_carrier @ empty_set ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[41767]) ).
thf(7425,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ D ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[2000,3032]) ).
thf(7426,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[7425:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ F ))]]) ).
thf(7456,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[7426]) ).
thf(61579,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( function @ empty_set )
| ( ( the_L_meet @ sk1 )
!= sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,44722]) ).
thf(61580,plain,
( ~ ( empty @ sk1 )
| ( function @ empty_set )
| ( ( the_L_meet @ sk1 )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[61579:[bind(A,$thf( sk1 ))]]) ).
thf(36402,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ D ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[36256,3032]) ).
thf(36403,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[36402:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ G )),bind(D,$thf( unordered_pair @ G @ F ))]]) ).
thf(36767,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[36403]) ).
thf(99,plain,
! [F: $i,E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( empty @ A )
| ( empty @ B )
| ~ ( function @ D )
| ~ ( quasi_total @ D @ ( cartesian_product2 @ A @ B ) @ C )
| ~ ( relation_of2 @ D @ ( cartesian_product2 @ A @ B ) @ C )
| ~ ( element @ E @ A )
| ~ ( element @ F @ B )
| ( element @ ( apply_binary_as_element @ A @ B @ C @ D @ E @ F ) @ C ) ),
inference(cnf,[status(esa)],[98]) ).
thf(7568,plain,
( ( ( sk12 @ empty_set )
!= empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,7226]) ).
thf(7589,plain,
( ( ( sk12 @ empty_set )
!= empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[7568]) ).
thf(42764,plain,
( ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk18 ) )
| ( ( empty @ sk18 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,14477]) ).
thf(42825,plain,
( ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk18 ) )
| ( sk18 != empty_set ) ),
inference(simp,[status(thm)],[42764]) ).
thf(6694,plain,
! [E: $i,D: $i,C: $i,B: $i,A: $i] :
( ( relation @ E )
| ( ( element @ ( sk2 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) )
!= ( element @ E @ ( powerset @ ( cartesian_product2 @ C @ D ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[4351,94]) ).
thf(6695,plain,
! [B: $i,A: $i] : ( relation @ ( sk2 @ A @ B ) ),
inference(pattern_uni,[status(thm)],[6694:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( G )),bind(D,$thf( F )),bind(E,$thf( sk2 @ F @ G ))]]) ).
thf(6711,plain,
! [B: $i,A: $i] : ( relation @ ( sk2 @ A @ B ) ),
inference(simp,[status(thm)],[6695]) ).
thf(5369,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk1 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) ) )
!= ( in @ A @ empty_set ) ),
inference(paramod_ordered,[status(thm)],[5288,309]) ).
thf(5370,plain,
! [A: $i] :
( ( ( the_L_meet @ sk1 )
!= A )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) )
!= empty_set ) ),
inference(simp,[status(thm)],[5369]) ).
thf(5386,plain,
( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) ) )
!= empty_set ),
inference(simp,[status(thm)],[5370]) ).
thf(5884,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( subset @ C @ D )
| ( ( element @ ( sk3 @ A @ B ) @ ( powerset @ ( cartesian_product2 @ B @ A ) ) )
!= ( element @ C @ ( powerset @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[4349,138]) ).
thf(5885,plain,
! [B: $i,A: $i] : ( subset @ ( sk3 @ B @ A ) @ ( cartesian_product2 @ A @ B ) ),
inference(pattern_uni,[status(thm)],[5884:[bind(A,$thf( H )),bind(B,$thf( G )),bind(C,$thf( sk3 @ H @ G )),bind(D,$thf( cartesian_product2 @ G @ H ))]]) ).
thf(5934,plain,
! [B: $i,A: $i] : ( subset @ ( sk3 @ B @ A ) @ ( cartesian_product2 @ A @ B ) ),
inference(simp,[status(thm)],[5885]) ).
thf(49799,plain,
( ( sk1 != empty_set )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,28192]) ).
thf(49800,plain,
( ( sk1 != empty_set )
| ( in @ ( bottom_of_semilattstr @ empty_set ) @ ( the_carrier @ sk1 ) ) ),
inference(pattern_uni,[status(thm)],[49799:[]]) ).
thf(2590,plain,
! [C: $i,B: $i,A: $i] :
( ( relation_of2 @ C @ A @ B )
| ( ( relation_of2_as_subset @ ( the_L_meet @ sk1 ) @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ) )
!= ( relation_of2_as_subset @ C @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[2537,85]) ).
thf(2591,plain,
relation_of2 @ ( the_L_meet @ sk1 ) @ ( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) ) @ ( the_carrier @ sk1 ),
inference(pattern_uni,[status(thm)],[2590:[bind(A,$thf( cartesian_product2 @ ( the_carrier @ sk1 ) @ ( the_carrier @ sk1 ) )),bind(B,$thf( the_carrier @ sk1 )),bind(C,$thf( the_L_meet @ sk1 ))]]) ).
thf(34575,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ D ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) )
!= ( ordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[3103,33651]) ).
thf(34576,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[34575:[bind(A,$thf( F )),bind(B,$thf( G )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ F ))]]) ).
thf(34669,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[34576]) ).
thf(52585,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ D @ C ) @ ( singleton @ D ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,36838]) ).
thf(52586,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[52585:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(4662,plain,
( ( relation @ ( the_L_meet @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,4647]) ).
thf(4683,plain,
( ( relation @ ( the_L_meet @ empty_set ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[4662]) ).
thf(13187,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ D ) ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( ordered_pair @ ( singleton @ D ) @ ( unordered_pair @ C @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[2000,3044]) ).
thf(13188,plain,
! [A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[13187:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(5021,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) )
!= ( in @ A @ A ) ),
inference(paramod_ordered,[status(thm)],[5010,494]) ).
thf(5093,plain,
! [A: $i] :
( ( ( the_L_meet @ sk7 )
!= A )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) )
!= A ) ),
inference(simp,[status(thm)],[5021]) ).
thf(5104,plain,
( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) )
!= ( the_L_meet @ sk7 ) ),
inference(simp,[status(thm)],[5093]) ).
thf(780,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( join_semilatt_str @ A )
| ( B != sk9 ) ),
inference(paramod_ordered,[status(thm)],[90,152]) ).
thf(781,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk9 )
| ( join_semilatt_str @ A ) ),
inference(pattern_uni,[status(thm)],[780:[bind(A,$thf( A )),bind(B,$thf( sk9 ))]]) ).
thf(311,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( empty @ ( ordered_pair @ A @ B ) )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[78,146]) ).
thf(312,plain,
! [B: $i,A: $i] :
~ ( empty @ ( ordered_pair @ B @ A ) ),
inference(pattern_uni,[status(thm)],[311:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ G @ F )),bind(D,$thf( singleton @ G ))]]) ).
thf(319,plain,
! [B: $i,A: $i] :
~ ( empty @ ( ordered_pair @ B @ A ) ),
inference(simp,[status(thm)],[312]) ).
thf(44567,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( empty @ ( powerset @ A ) )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,125]) ).
thf(44723,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( powerset @ A )
!= sk1 ) ),
inference(simp,[status(thm)],[44567]) ).
thf(23002,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ C @ D ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) )
!= ( ordered_pair @ ( singleton @ C ) @ ( unordered_pair @ C @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[3103,3033]) ).
thf(23003,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[23002:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(12710,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk8 )
| ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( sk12 @ sk8 ) @ ( powerset @ ( the_carrier @ sk8 ) ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[6992,119]) ).
thf(12711,plain,
( ( empty_carrier @ sk8 )
| ( empty @ ( powerset @ ( the_carrier @ sk8 ) ) )
| ( in @ ( sk12 @ sk8 ) @ ( powerset @ ( the_carrier @ sk8 ) ) ) ),
inference(pattern_uni,[status(thm)],[12710:[bind(A,$thf( sk12 @ sk8 )),bind(B,$thf( powerset @ ( the_carrier @ sk8 ) ))]]) ).
thf(20254,plain,
( ( empty_carrier @ sk8 )
| $false
| ( in @ ( sk12 @ sk8 ) @ ( powerset @ ( the_carrier @ sk8 ) ) ) ),
inference(rewrite,[status(thm)],[12711,125]) ).
thf(20255,plain,
( ( empty_carrier @ sk8 )
| ( in @ ( sk12 @ sk8 ) @ ( powerset @ ( the_carrier @ sk8 ) ) ) ),
inference(simp,[status(thm)],[20254]) ).
thf(23784,plain,
! [B: $i,A: $i] :
( ( in @ ( join_of_latt_set @ sk1 @ A ) @ ( the_carrier @ sk1 ) )
!= ( in @ B @ B ) ),
inference(paramod_ordered,[status(thm)],[23392,494]) ).
thf(23826,plain,
! [B: $i,A: $i] :
( ( ( join_of_latt_set @ sk1 @ A )
!= B )
| ( ( the_carrier @ sk1 )
!= B ) ),
inference(simp,[status(thm)],[23784]) ).
thf(23847,plain,
! [A: $i] :
( ( join_of_latt_set @ sk1 @ A )
!= ( the_carrier @ sk1 ) ),
inference(simp,[status(thm)],[23826]) ).
thf(34087,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[433,33460]) ).
thf(34088,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[34087:[bind(A,$thf( H )),bind(B,$thf( I )),bind(C,$thf( ordered_pair @ I @ H )),bind(D,$thf( singleton @ ( unordered_pair @ H @ I ) ))]]) ).
thf(34207,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) @ ( singleton @ ( ordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[34088]) ).
thf(4201,plain,
! [B: $i,A: $i] :
( ( in @ A @ ( powerset @ A ) )
!= ( in @ ( powerset @ B ) @ B ) ),
inference(paramod_ordered,[status(thm)],[4089,4117]) ).
thf(4210,plain,
! [B: $i,A: $i] :
( ( A
!= ( powerset @ B ) )
| ( ( powerset @ A )
!= B ) ),
inference(simp,[status(thm)],[4201]) ).
thf(4214,plain,
! [A: $i] :
( ( powerset @ ( powerset @ A ) )
!= A ),
inference(simp,[status(thm)],[4210]) ).
thf(51669,plain,
( ( sk1 != empty_set )
| ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ empty_set ) )
| ( ( empty @ sk1 )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,28194]) ).
thf(51670,plain,
( ( sk1 != empty_set )
| ( in @ ( bottom_of_semilattstr @ sk1 ) @ ( the_carrier @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[51669:[]]) ).
thf(28112,plain,
( ( ( sk12 @ empty_set )
!= ( powerset @ ( the_carrier @ sk1 ) ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,16176]) ).
thf(28136,plain,
( ( ( sk12 @ empty_set )
!= ( powerset @ ( the_carrier @ sk1 ) ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[28112]) ).
thf(33360,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( ordered_pair @ D @ C ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( unordered_pair @ D @ C ) @ ( singleton @ D ) ) )
| ( ( unordered_pair @ B @ A )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[81,320]) ).
thf(33361,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) ),
inference(pattern_uni,[status(thm)],[33360:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(37127,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) )
= ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) ),
inference(rewrite,[status(thm)],[33361,433]) ).
thf(1928,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( unordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ D @ C ) )
| ( ( unordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ ( singleton @ B ) ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[431,426]) ).
thf(1929,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) ),
inference(pattern_uni,[status(thm)],[1928:[bind(A,$thf( F )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ F )),bind(D,$thf( singleton @ ( singleton @ H ) ))]]) ).
thf(1977,plain,
! [B: $i,A: $i] :
( ( unordered_pair @ ( singleton @ ( singleton @ ( singleton @ B ) ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) ),
inference(simp,[status(thm)],[1929]) ).
thf(50258,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) )
= ( ordered_pair @ ( singleton @ ( singleton @ B ) ) @ ( ordered_pair @ B @ A ) ) ),
inference(rewrite,[status(thm)],[1977,1316]) ).
thf(39409,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ C ) @ ( ordered_pair @ B @ A ) ) @ ( singleton @ ( singleton @ ( singleton @ C ) ) ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[355,33654]) ).
thf(39410,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[39409:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( singleton @ F )),bind(D,$thf( unordered_pair @ F @ G ))]]) ).
thf(39556,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) ) @ ( singleton @ ( singleton @ ( singleton @ ( singleton @ A ) ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[39410]) ).
thf(168,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_associative @ A )
| ~ ( meet_semilatt_str @ A )
| ( v2_binop_1 @ ( the_L_meet @ A ) @ ( the_carrier @ A ) ) ),
inference(cnf,[status(esa)],[166]) ).
thf(42777,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk18 )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ empty_set ) )
| ( A != sk18 ) ),
inference(paramod_ordered,[status(thm)],[246,14477]) ).
thf(42778,plain,
( ~ ( empty @ sk18 )
| ~ ( empty @ sk18 )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ empty_set ) ) ),
inference(pattern_uni,[status(thm)],[42777:[bind(A,$thf( sk18 ))]]) ).
thf(42843,plain,
( ~ ( empty @ sk18 )
| ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ empty_set ) ) ),
inference(simp,[status(thm)],[42778]) ).
thf(44447,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( ( in @ A @ ( powerset @ A ) )
!= ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk1 ) ) ) ),
inference(paramod_ordered,[status(thm)],[4089,16354]) ).
thf(44506,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( A
!= ( powerset @ ( the_carrier @ empty_set ) ) )
| ( ( powerset @ A )
!= ( sk12 @ sk1 ) ) ),
inference(simp,[status(thm)],[44447]) ).
thf(44537,plain,
( ~ ( empty @ sk1 )
| ( ( sk12 @ sk1 )
!= ( powerset @ ( powerset @ ( the_carrier @ empty_set ) ) ) ) ),
inference(simp,[status(thm)],[44506]) ).
thf(267,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ( B
!= ( bottom_of_semilattstr @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ( ( meet @ A @ C @ B )
= B ) ),
inference(cnf,[status(esa)],[265]) ).
thf(272,plain,
! [C: $i,B: $i,A: $i] :
( ( B
!= ( bottom_of_semilattstr @ A ) )
| ( ( meet @ A @ C @ B )
= B )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( element @ C @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[267]) ).
thf(273,plain,
! [B: $i,A: $i] :
( ( ( meet @ A @ B @ ( bottom_of_semilattstr @ A ) )
= ( bottom_of_semilattstr @ A ) )
| ( empty_carrier @ A )
| ~ ( meet_semilatt_str @ A )
| ~ ( lower_bounded_semilattstr @ A )
| ~ ( element @ ( bottom_of_semilattstr @ A ) @ ( the_carrier @ A ) )
| ~ ( element @ B @ ( the_carrier @ A ) ) ),
inference(simp,[status(thm)],[272]) ).
thf(111,plain,
! [B: $i,A: $i] :
( ~ ( in @ A @ B )
| ( element @ A @ B ) ),
inference(cnf,[status(esa)],[110]) ).
thf(3485,plain,
! [C: $i,B: $i,A: $i] :
( ( empty @ A )
| ( element @ B @ C )
| ( ( in @ ( sk11 @ A ) @ A )
!= ( in @ B @ C ) ) ),
inference(paramod_ordered,[status(thm)],[3280,111]) ).
thf(3486,plain,
! [A: $i] :
( ( empty @ A )
| ( element @ ( sk11 @ A ) @ A ) ),
inference(pattern_uni,[status(thm)],[3485:[bind(A,$thf( D )),bind(B,$thf( sk11 @ D )),bind(C,$thf( D ))]]) ).
thf(3496,plain,
! [A: $i] :
( ( empty @ A )
| ( element @ ( sk11 @ A ) @ A ) ),
inference(simp,[status(thm)],[3486]) ).
thf(5013,plain,
! [B: $i,A: $i] :
( ~ ( in @ B @ A )
| ( ( in @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) )
!= ( in @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[5010,141]) ).
thf(5014,plain,
~ ( in @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) @ ( the_L_meet @ sk7 ) ),
inference(pattern_uni,[status(thm)],[5013:[bind(A,$thf( the_L_meet @ sk7 )),bind(B,$thf( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ))]]) ).
thf(864,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( lattice @ A )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,65]) ).
thf(865,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( lattice @ A ) ),
inference(pattern_uni,[status(thm)],[864:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(11864,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ B ) ) @ ( ordered_pair @ D @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[78,3045]) ).
thf(11865,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11864:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ G @ F )),bind(D,$thf( singleton @ G ))]]) ).
thf(11943,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ B @ A ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[11865]) ).
thf(200,plain,
! [C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ( element @ ( sk15 @ C @ B @ A ) @ ( the_carrier @ A ) )
| ( C
= ( join_of_latt_set @ A @ B ) ) ),
inference(cnf,[status(esa)],[195]) ).
thf(209,plain,
! [C: $i,B: $i,A: $i] :
( ( C
= ( join_of_latt_set @ A @ B ) )
| ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ( element @ ( sk15 @ C @ B @ A ) @ ( the_carrier @ A ) ) ),
inference(lifteq,[status(thm)],[200]) ).
thf(210,plain,
! [C: $i,B: $i,A: $i] :
( ( C
= ( join_of_latt_set @ A @ B ) )
| ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( lattice @ A )
| ~ ( complete_latt_str @ A )
| ~ ( element @ C @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ C @ B )
| ( element @ ( sk15 @ C @ B @ A ) @ ( the_carrier @ A ) ) ),
inference(simp,[status(thm)],[209]) ).
thf(5022,plain,
! [A: $i] :
( ( in @ ( the_L_meet @ sk7 ) @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) )
!= ( in @ ( powerset @ A ) @ A ) ),
inference(paramod_ordered,[status(thm)],[5010,4117]) ).
thf(5094,plain,
! [A: $i] :
( ( ( the_L_meet @ sk7 )
!= ( powerset @ A ) )
| ( ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) )
!= A ) ),
inference(simp,[status(thm)],[5022]) ).
thf(5105,plain,
( ( the_L_meet @ sk7 )
!= ( powerset @ ( powerset @ ( cartesian_product2 @ ( cartesian_product2 @ ( the_carrier @ sk7 ) @ ( the_carrier @ sk7 ) ) @ ( the_carrier @ sk7 ) ) ) ) ),
inference(simp,[status(thm)],[5094]) ).
thf(896,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ( ( join_of_latt_set @ A @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,279]) ).
thf(897,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ( ( join_of_latt_set @ A @ empty_set )
!= ( bottom_of_semilattstr @ sk1 ) )
| ~ ( lower_bounded_semilattstr @ sk1 ) ),
inference(pattern_uni,[status(thm)],[896:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(639,plain,
! [C: $i,B: $i,A: $i] :
( ~ ( empty @ A )
| ( relation_of2_as_subset @ empty_set @ B @ C )
| ( A
!= ( sk3 @ C @ B ) ) ),
inference(paramod_ordered,[status(thm)],[246,75]) ).
thf(640,plain,
! [B: $i,A: $i] :
( ~ ( empty @ ( sk3 @ A @ B ) )
| ( relation_of2_as_subset @ empty_set @ B @ A ) ),
inference(pattern_uni,[status(thm)],[639:[bind(A,$thf( sk3 @ D @ E )),bind(B,$thf( E )),bind(C,$thf( D ))]]) ).
thf(664,plain,
! [B: $i,A: $i] :
( ~ ( empty @ ( sk3 @ A @ B ) )
| ( relation_of2_as_subset @ empty_set @ B @ A ) ),
inference(simp,[status(thm)],[640]) ).
thf(46270,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ( ( powerset @ B )
!= sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,44723]) ).
thf(46271,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( ( powerset @ A )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[46270:[bind(A,$thf( sk1 ))]]) ).
thf(46298,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( ( powerset @ A )
!= sk1 ) ),
inference(simp,[status(thm)],[46271]) ).
thf(623,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( meet_semilatt_str @ empty_set )
| ( A != sk17 ) ),
inference(paramod_ordered,[status(thm)],[246,441]) ).
thf(624,plain,
( ~ ( empty @ sk17 )
| ( meet_semilatt_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[623:[bind(A,$thf( sk17 ))]]) ).
thf(11900,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ D @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( singleton @ A ) ) @ ( ordered_pair @ A @ B ) )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[1316,3045]) ).
thf(11901,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[11900:[bind(A,$thf( G )),bind(B,$thf( H )),bind(C,$thf( singleton @ ( singleton @ G ) )),bind(D,$thf( ordered_pair @ G @ H ))]]) ).
thf(11958,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ B ) ) ) @ ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( singleton @ A ) ) ) )
!= empty_set ),
inference(simp,[status(thm)],[11901]) ).
thf(4098,plain,
! [B: $i,A: $i] :
( ( in @ A @ ( powerset @ A ) )
!= ( in @ B @ B ) ),
inference(paramod_ordered,[status(thm)],[4089,494]) ).
thf(4109,plain,
! [B: $i,A: $i] :
( ( A != B )
| ( ( powerset @ A )
!= B ) ),
inference(simp,[status(thm)],[4098]) ).
thf(4118,plain,
! [A: $i] :
( ( powerset @ A )
!= A ),
inference(simp,[status(thm)],[4109]) ).
thf(27977,plain,
! [B: $i,A: $i] :
( ( empty_carrier @ sk7 )
| ( empty @ B )
| ( in @ A @ B )
| ( ( element @ ( bottom_of_semilattstr @ sk7 ) @ ( the_carrier @ sk7 ) )
!= ( element @ A @ B ) ) ),
inference(paramod_ordered,[status(thm)],[26335,119]) ).
thf(27978,plain,
( ( empty_carrier @ sk7 )
| ( empty @ ( the_carrier @ sk7 ) )
| ( in @ ( bottom_of_semilattstr @ sk7 ) @ ( the_carrier @ sk7 ) ) ),
inference(pattern_uni,[status(thm)],[27977:[bind(A,$thf( bottom_of_semilattstr @ sk7 )),bind(B,$thf( the_carrier @ sk7 ))]]) ).
thf(43661,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) ) @ ( singleton @ ( ordered_pair @ D @ C ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( ordered_pair @ ( singleton @ D ) @ ( unordered_pair @ C @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[2000,33650]) ).
thf(43662,plain,
! [A: $i] :
( ( ordered_pair @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ A ) ) @ ( singleton @ ( ordered_pair @ A @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[43661:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(44443,plain,
( ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,16354]) ).
thf(44510,plain,
( ~ ( in @ ( powerset @ ( the_carrier @ empty_set ) ) @ ( sk12 @ sk1 ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[44443]) ).
thf(44590,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( empty @ ( singleton @ A ) )
!= ( empty @ sk1 ) ) ),
inference(paramod_ordered,[status(thm)],[43408,117]) ).
thf(44729,plain,
! [A: $i] :
( ( sk1 != empty_set )
| ( ( singleton @ A )
!= sk1 ) ),
inference(simp,[status(thm)],[44590]) ).
thf(163,plain,
! [A: $i] :
( ( empty_carrier @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_semilatt_str @ A )
| ( relation @ ( the_L_meet @ A ) ) ),
inference(cnf,[status(esa)],[160]) ).
thf(1364,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( empty @ A )
| ( relation @ D )
| ( ( element @ ( sk6 @ A ) @ ( powerset @ A ) )
!= ( element @ D @ ( powerset @ ( cartesian_product2 @ B @ C ) ) ) ) ),
inference(paramod_ordered,[status(thm)],[121,94]) ).
thf(1365,plain,
! [B: $i,A: $i] :
( ( empty @ ( cartesian_product2 @ A @ B ) )
| ( relation @ ( sk6 @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(pattern_uni,[status(thm)],[1364:[bind(A,$thf( cartesian_product2 @ F @ G )),bind(B,$thf( F )),bind(C,$thf( G )),bind(D,$thf( sk6 @ ( cartesian_product2 @ F @ G ) ))]]) ).
thf(1384,plain,
! [B: $i,A: $i] :
( ( empty @ ( cartesian_product2 @ A @ B ) )
| ( relation @ ( sk6 @ ( cartesian_product2 @ A @ B ) ) ) ),
inference(simp,[status(thm)],[1365]) ).
thf(36259,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) )
!= empty_set )
| ( ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) )
!= ( ordered_pair @ ( unordered_pair @ D @ C ) @ ( singleton @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[36256,33697]) ).
thf(36260,plain,
! [A: $i] :
( ( ordered_pair @ ( unordered_pair @ A @ A ) @ ( singleton @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[36259:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(11488,plain,
! [B: $i,A: $i] :
( ( subset @ A @ B )
| ( ( element @ ( sk12 @ sk18 ) @ ( powerset @ ( the_carrier @ sk18 ) ) )
!= ( element @ A @ ( powerset @ B ) ) ) ),
inference(paramod_ordered,[status(thm)],[11487,138]) ).
thf(11489,plain,
subset @ ( sk12 @ sk18 ) @ ( the_carrier @ sk18 ),
inference(pattern_uni,[status(thm)],[11488:[bind(A,$thf( sk12 @ sk18 )),bind(B,$thf( the_carrier @ sk18 ))]]) ).
thf(33793,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ~ ( in @ D @ C )
| ( ( in @ ( sk2 @ B @ A ) @ ( powerset @ ( cartesian_product2 @ A @ B ) ) )
!= ( in @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[33777,141]) ).
thf(33794,plain,
! [B: $i,A: $i] :
~ ( in @ ( powerset @ ( cartesian_product2 @ A @ B ) ) @ ( sk2 @ B @ A ) ),
inference(pattern_uni,[status(thm)],[33793:[bind(A,$thf( H )),bind(B,$thf( I )),bind(C,$thf( sk2 @ I @ H )),bind(D,$thf( powerset @ ( cartesian_product2 @ H @ I ) ))]]) ).
thf(33867,plain,
! [B: $i,A: $i] :
~ ( in @ ( powerset @ ( cartesian_product2 @ A @ B ) ) @ ( sk2 @ B @ A ) ),
inference(simp,[status(thm)],[33794]) ).
thf(33704,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( ordered_pair @ B @ A ) @ ( singleton @ C ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[355,33582]) ).
thf(33705,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[33704:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ F ))]]) ).
thf(33758,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( ordered_pair @ A @ B ) @ ( singleton @ ( unordered_pair @ A @ B ) ) )
!= empty_set ),
inference(simp,[status(thm)],[33705]) ).
thf(122,plain,
! [A: $i] :
( ( empty @ A )
| ~ ( empty @ ( sk6 @ A ) ) ),
inference(cnf,[status(esa)],[120]) ).
thf(51997,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ D @ C ) @ ( singleton @ D ) ) )
!= empty_set )
| ( ( unordered_pair @ A @ B )
!= ( unordered_pair @ C @ D ) ) ),
inference(paramod_ordered,[status(thm)],[81,36766]) ).
thf(51998,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ B ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[51997:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).
thf(29690,plain,
( ( ( empty @ ( the_carrier @ empty_set ) )
!= ( empty @ sk1 ) )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,24090]) ).
thf(29716,plain,
( ( ( empty @ ( the_carrier @ empty_set ) )
!= ( empty @ sk1 ) )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[29690]) ).
thf(46392,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ( ( singleton @ B )
!= sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,44729]) ).
thf(46393,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( ( singleton @ A )
!= sk1 ) ),
inference(pattern_uni,[status(thm)],[46392:[bind(A,$thf( sk1 ))]]) ).
thf(46418,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ( ( singleton @ A )
!= sk1 ) ),
inference(simp,[status(thm)],[46393]) ).
thf(1811,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( meet_semilatt_str @ A )
| ( sk1 != empty_set )
| ( empty_set != empty_set ) ),
inference(paramod_ordered,[status(thm)],[246,1361]) ).
thf(1812,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( meet_semilatt_str @ A )
| ( sk1 != empty_set ) ),
inference(pattern_uni,[status(thm)],[1811:[]]) ).
thf(222,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( empty_carrier @ A )
| ~ ( latt_str @ A )
| ~ ( element @ B @ ( the_carrier @ A ) )
| ~ ( latt_element_smaller @ A @ B @ C )
| ~ ( element @ D @ ( the_carrier @ A ) )
| ~ ( in @ D @ C )
| ( below @ A @ D @ B ) ),
inference(cnf,[status(esa)],[218]) ).
thf(45606,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( in @ B @ sk1 )
| ( A != sk1 ) ),
inference(paramod_ordered,[status(thm)],[246,44614]) ).
thf(45607,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ~ ( in @ A @ sk1 ) ),
inference(pattern_uni,[status(thm)],[45606:[bind(A,$thf( sk1 ))]]) ).
thf(45659,plain,
! [A: $i] :
( ~ ( empty @ sk1 )
| ~ ( in @ A @ sk1 ) ),
inference(simp,[status(thm)],[45607]) ).
thf(28657,plain,
( ( meet_associative @ empty_set )
| ( ( empty @ sk1 )
!= ( empty @ empty_set ) ) ),
inference(paramod_ordered,[status(thm)],[172,28631]) ).
thf(28680,plain,
( ( meet_associative @ empty_set )
| ( sk1 != empty_set ) ),
inference(simp,[status(thm)],[28657]) ).
thf(1270,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ B @ A )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[355,348]) ).
thf(1271,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ A @ B )
!= empty_set ),
inference(pattern_uni,[status(thm)],[1270:[bind(A,$thf( G )),bind(B,$thf( F )),bind(C,$thf( unordered_pair @ F @ G )),bind(D,$thf( singleton @ F ))]]) ).
thf(1331,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ A @ B )
!= empty_set ),
inference(simp,[status(thm)],[1271]) ).
thf(227,plain,
! [A: $i] :
( ~ ( latt_str @ A )
| ( empty_carrier @ A )
| ~ ( join_commutative @ A )
| ~ ( join_associative @ A )
| ~ ( meet_commutative @ A )
| ~ ( meet_associative @ A )
| ~ ( meet_absorbing @ A )
| ~ ( join_absorbing @ A )
| ( lattice @ A ) ),
inference(cnf,[status(esa)],[226]) ).
thf(2997,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) )
!= ( ordered_pair @ ( singleton @ D ) @ ( unordered_pair @ C @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[2000,1994]) ).
thf(2998,plain,
! [A: $i] :
( ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ A ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[2997:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(34871,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ D ) @ ( ordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ B ) ) )
!= empty_set )
| ( ( unordered_pair @ ( singleton @ ( unordered_pair @ A @ B ) ) @ ( ordered_pair @ B @ A ) )
!= ( unordered_pair @ D @ C ) ) ),
inference(paramod_ordered,[status(thm)],[1318,2102]) ).
thf(34872,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[34871:[bind(A,$thf( I )),bind(B,$thf( H )),bind(C,$thf( ordered_pair @ H @ I )),bind(D,$thf( singleton @ ( unordered_pair @ I @ H ) ))]]) ).
thf(34947,plain,
! [B: $i,A: $i] :
( ( ordered_pair @ ( singleton @ ( singleton @ ( unordered_pair @ B @ A ) ) ) @ ( ordered_pair @ ( unordered_pair @ B @ A ) @ ( singleton @ A ) ) )
!= empty_set ),
inference(simp,[status(thm)],[34872]) ).
thf(1036,plain,
! [B: $i,A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ B )
| ~ ( empty_carrier @ A )
| ( B != sk1 ) ),
inference(paramod_ordered,[status(thm)],[90,68]) ).
thf(1037,plain,
! [A: $i] :
( ~ ( empty @ A )
| ~ ( empty @ sk1 )
| ~ ( empty_carrier @ A ) ),
inference(pattern_uni,[status(thm)],[1036:[bind(A,$thf( A )),bind(B,$thf( sk1 ))]]) ).
thf(559,plain,
! [A: $i] :
( ~ ( empty @ A )
| ( one_sorted_str @ empty_set )
| ( A != sk9 ) ),
inference(paramod_ordered,[status(thm)],[246,297]) ).
thf(560,plain,
( ~ ( empty @ sk9 )
| ( one_sorted_str @ empty_set ) ),
inference(pattern_uni,[status(thm)],[559:[bind(A,$thf( sk9 ))]]) ).
thf(23021,plain,
! [D: $i,C: $i,B: $i,A: $i] :
( ( ( ordered_pair @ ( singleton @ ( ordered_pair @ C @ D ) ) @ ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ B @ A ) ) )
!= empty_set )
| ( ( ordered_pair @ ( singleton @ B ) @ ( unordered_pair @ A @ B ) )
!= ( ordered_pair @ ( singleton @ C ) @ ( unordered_pair @ C @ D ) ) ) ),
inference(paramod_ordered,[status(thm)],[2000,3033]) ).
thf(23022,plain,
! [A: $i] :
( ( ordered_pair @ ( singleton @ ( ordered_pair @ A @ A ) ) @ ( ordered_pair @ ( singleton @ A ) @ ( unordered_pair @ A @ A ) ) )
!= empty_set ),
inference(pattern_uni,[status(thm)],[23021:[bind(A,$thf( A )),bind(B,$thf( A )),bind(C,$thf( A )),bind(D,$thf( A ))]]) ).
thf(95846,plain,
$false,
inference(e,[status(thm)],[39081,4005,14606,3399,479,249,234,6986,49318,42851,1369,138,28324,44719,43408,49271,977,49026,12334,4339,217,3905,5292,4391,28192,1728,4215,46708,28631,202,7083,115,2104,120,247,614,24666,26333,142,49268,2141,29740,33697,28444,7220,26579,21192,4647,3044,5288,7467,4856,1342,913,538,153,28234,174,3350,24091,2491,288,1386,320,51553,1482,257,3103,185,13469,1835,973,885,5553,42991,23411,504,17983,110,1330,917,4349,3280,5640,21268,25677,2501,16176,125,1545,4353,189,48981,30558,93,33627,152,3835,78,261,7500,7226,1013,284,216,20562,164,41717,38531,321,64069,13295,33241,238,1978,121,348,3093,13604,29784,443,211,253,2122,106,16204,905,480,58365,11218,28619,634,40403,132,133,11954,74,206,4065,147,5929,12670,2483,35738,243,7505,265,849,45109,33651,13440,14184,26993,43060,85,50851,24090,16356,40211,102,11852,233,452,5875,117,2000,63618,957,260,160,23390,14605,192,275,36838,43378,42023,41484,2541,56854,302,297,70,33869,37956,252,65,14477,20427,97,59122,710,3033,33250,224,229,566,48889,27097,578,697,1259,1856,34666,188,2537,169,58339,141,3087,1361,225,41718,642,526,67749,3032,1791,46334,5387,212,109,256,488,34557,554,14473,193,14428,901,686,33650,1424,40198,33777,837,42041,36839,1831,173,37946,11833,39335,46809,5639,654,20593,4347,35793,14526,17963,134,33584,805,33654,6716,36296,14185,30557,953,586,662,73,5557,128,237,3834,1318,105,244,11787,16220,14107,48479,965,166,45652,36256,161,3523,44770,149,46961,11837,38122,34542,148,33767,264,279,64,35058,180,584,191,34243,44722,11836,7497,30556,33868,28190,118,65532,259,176,1833,24089,694,43890,33653,281,204,71,26335,12067,43131,236,181,445,38202,28194,38408,144,14183,466,44744,3091,86,26577,159,33760,187,2721,172,23848,274,13467,81,6832,36766,4337,28148,23838,5103,2493,230,1542,76,28511,59089,737,33460,7206,11487,98,616,271,208,494,4541,103,213,91,66,626,28261,155,28322,34225,3893,1112,1993,4818,594,140,6974,3977,251,108,330,33756,18243,34963,2139,223,135,226,23394,4282,241,35721,130,695,48978,48226,431,3876,167,45651,24143,3031,426,1203,255,48520,123,355,1551,194,29730,112,1845,580,63,30560,11956,29202,263,993,34926,7211,29106,5388,150,39555,95,38556,2143,3904,2656,14604,45162,3791,177,48290,44528,127,13602,3095,33936,441,50897,4351,338,57037,470,576,5638,4117,41807,7456,16354,29132,154,72,175,7498,143,61580,6992,23442,33630,36767,14479,99,6970,7589,42825,6711,250,231,1316,450,26482,87,44709,5386,5934,49800,2591,588,34669,52586,4089,4683,158,49946,23392,13188,5104,781,319,44723,33514,114,139,5010,23003,119,246,3430,20255,23847,82,214,75,34207,433,4214,51670,1994,28136,151,2102,447,37127,28157,50258,39556,26575,620,168,262,28233,146,42843,44537,30445,273,3496,4579,44614,5014,865,564,11943,210,107,126,79,195,5105,897,664,3045,94,46298,239,242,624,1827,11958,4118,27978,43662,44510,49957,44729,131,163,178,1384,36260,2539,68,11489,33867,33758,53093,122,51998,29716,46418,1812,222,592,45659,28680,100,90,111,1331,254,227,14531,33582,309,1463,2998,33652,34947,1037,560,23022,14102]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU352+1 : TPTP v8.1.2. Released v3.3.0.
% 0.10/0.11 % Command : run_Leo-III %s %d
% 0.11/0.30 % Computer : n032.cluster.edu
% 0.11/0.30 % Model : x86_64 x86_64
% 0.11/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.30 % Memory : 8042.1875MB
% 0.11/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.30 % CPULimit : 300
% 0.11/0.30 % WCLimit : 300
% 0.11/0.30 % DateTime : Thu May 18 13:10:50 EDT 2023
% 0.11/0.30 % CPUTime :
% 0.70/0.77 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.16/0.94 % [INFO] Parsing done (174ms).
% 1.16/0.95 % [INFO] Running in sequential loop mode.
% 1.68/1.20 % [INFO] eprover registered as external prover.
% 1.68/1.20 % [INFO] cvc4 registered as external prover.
% 1.68/1.21 % [INFO] Scanning for conjecture ...
% 1.76/1.29 % [INFO] Found a conjecture and 72 axioms. Running axiom selection ...
% 1.91/1.37 % [INFO] Axiom selection finished. Selected 60 axioms (removed 12 axioms).
% 2.12/1.45 % [INFO] Problem is first-order (TPTP FOF).
% 2.12/1.46 % [INFO] Type checking passed.
% 2.12/1.46 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 219.29/47.47 % External prover 'e' found a proof!
% 219.29/47.47 % [INFO] Killing All external provers ...
% 219.29/47.48 % Time passed: 47057ms (effective reasoning time: 46518ms)
% 219.29/47.48 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 219.29/47.48 % Axioms used in derivation (60): redefinition_m2_relset_1, d13_lattices, dt_m2_relset_1, dt_k2_binop_1, dt_l2_lattices, redefinition_r3_lattices, existence_m2_relset_1, redefinition_k2_binop_1, d2_lattices, fc1_xboole_0, d16_lattices, existence_l1_lattices, d5_tarski, t26_lattices, t1_subset, fc4_subset_1, existence_l3_lattices, dt_k5_lattices, fc4_lattice2, reflexivity_r3_lattices, dt_l1_lattices, t6_boole, rc2_subset_1, existence_l2_lattices, fc3_subset_1, t5_subset, existence_m1_relset_1, dt_l3_lattices, fc2_subset_1, d21_lattice3, t4_subset, t7_boole, d17_lattice3, rc3_struct_0, commutativity_k4_lattices, fc1_subset_1, fc5_lattice2, rc1_subset_1, cc1_lattices, dt_k15_lattice3, t3_subset, cc1_relset_1, dt_u1_lattices, existence_m1_subset_1, commutativity_k2_tarski, fc1_struct_0, dt_k4_lattices, reflexivity_r1_tarski, t8_boole, rc1_xboole_0, rc5_struct_0, t2_subset, existence_l1_struct_0, dt_k2_lattices, d1_binop_1, rc2_xboole_0, t23_lattices, antisymmetry_r2_hidden, cc2_lattices, redefinition_k4_lattices
% 219.29/47.48 % No. of inferences in proof: 1388
% 219.29/47.48 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 47057 ms resp. 46518 ms w/o parsing
% 219.79/47.68 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 219.79/47.69 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------