TSTP Solution File: SWW473+7 by Leo-III---1.7.7
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.7
% Problem : SWW473+7 : TPTP v8.1.2. Released v5.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n016.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 12:24:11 EDT 2023
% Result : Theorem 201.42s 52.60s
% Output : Refutation 203.44s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 1333
% Syntax : Number of formulae : 2591 ( 725 unt; 132 typ; 0 def)
% Number of atoms : 6700 (1903 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 112506 ( 650 ~; 271 |; 567 &;107615 @)
% ( 235 <=>;3168 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 11 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 124 ( 124 >; 0 *; 0 +; 0 <<)
% Number of symbols : 134 ( 132 usr; 28 con; 0-4 aty)
% Number of variables : 6254 ( 0 ^;6125 !; 129 ?;6254 :)
% Comments :
%------------------------------------------------------------------------------
thf(hBOOL_type,type,
hBOOL: $i > $o ).
thf(hAPP_type,type,
hAPP: $i > $i > $i > $i > $i ).
thf(fun_type,type,
fun: $i > $i > $i ).
thf(x_a_type,type,
x_a: $i ).
thf(bool_type,type,
bool: $i ).
thf(ord_less_eq_type,type,
ord_less_eq: $i > $i ).
thf(insert_type,type,
insert: $i > $i ).
thf(pname_type,type,
pname: $i ).
thf(mgt_call_type,type,
mgt_call: $i ).
thf(pn_type,type,
pn: $i ).
thf(g_type,type,
g: $i ).
thf(image_type,type,
image: $i > $i > $i ).
thf(u_type,type,
u: $i ).
thf(ring_type,type,
ring: $i > $o ).
thf(int_type,type,
int: $i ).
thf(semiri456707255roduct_type,type,
semiri456707255roduct: $i > $o ).
thf(nat_type,type,
nat: $i ).
thf(no_zero_divisors_type,type,
no_zero_divisors: $i > $o ).
thf(monoid_add_type,type,
monoid_add: $i > $o ).
thf(semiring_char_0_type,type,
semiring_char_0: $i > $o ).
thf(zero_neq_one_type,type,
zero_neq_one: $i > $o ).
thf(comm_monoid_add_type,type,
comm_monoid_add: $i > $o ).
thf(number_semiring_type,type,
number_semiring: $i > $o ).
thf(one_type,type,
one: $i > $o ).
thf(zero_type,type,
zero: $i > $o ).
thf(monoid_mult_type,type,
monoid_mult: $i > $o ).
thf(comm_semiring_1_type,type,
comm_semiring_1: $i > $o ).
thf(plus_plus_type,type,
plus_plus: $i > $i ).
thf(zero_zero_type,type,
zero_zero: $i > $i ).
thf(ti_type,type,
ti: $i > $i > $i ).
thf(number_ring_type,type,
number_ring: $i > $o ).
thf(number_number_of_type,type,
number_number_of: $i > $i ).
thf(pls_type,type,
pls: $i ).
thf(times_times_type,type,
times_times: $i > $i ).
thf(minus_minus_type,type,
minus_minus: $i > $i ).
thf(one_one_type,type,
one_one: $i > $i ).
thf(linord893533164strict_type,type,
linord893533164strict: $i > $o ).
thf(suc_type,type,
suc: $i ).
thf(minus_type,type,
minus: $i > $o ).
thf(cancel_semigroup_add_type,type,
cancel_semigroup_add: $i > $o ).
thf(bit1_type,type,
bit1: $i ).
thf(linord1278240602ring_1_type,type,
linord1278240602ring_1: $i > $o ).
thf(ordered_ab_group_add_type,type,
ordered_ab_group_add: $i > $o ).
thf(nat_case_type,type,
nat_case: $i > $i ).
thf(ordere779506340up_add_type,type,
ordere779506340up_add: $i > $o ).
thf(ordere453448008miring_type,type,
ordere453448008miring: $i > $o ).
thf(ord_less_type,type,
ord_less: $i > $i ).
thf(semiring_1_of_nat_type,type,
semiring_1_of_nat: $i > $i ).
thf(ring_char_0_type,type,
ring_char_0: $i > $o ).
thf(linorder_type,type,
linorder: $i > $o ).
thf(comm_monoid_mult_type,type,
comm_monoid_mult: $i > $o ).
thf(order_type,type,
order: $i > $o ).
thf(linord581940658strict_type,type,
linord581940658strict: $i > $o ).
thf(ord_type,type,
ord: $i > $o ).
thf(fFalse_type,type,
fFalse: $i ).
thf(cancel146912293up_add_type,type,
cancel146912293up_add: $i > $o ).
thf(semiring_1_type,type,
semiring_1: $i > $o ).
thf(semiri532925092at_aux_type,type,
semiri532925092at_aux: $i > $i ).
thf(linord20386208strict_type,type,
linord20386208strict: $i > $o ).
thf(number_type,type,
number: $i > $o ).
thf(semiring_type,type,
semiring: $i > $o ).
thf(finite_finite_type,type,
finite_finite: $i > $i ).
thf(finite_card_type,type,
finite_card: $i > $i ).
thf(finite_fold_image_type,type,
finite_fold_image: $i > $i > $i ).
thf(combk_type,type,
combk: $i > $i > $i ).
thf(abs_abs_type,type,
abs_abs: $i > $i ).
thf(ab_semigroup_mult_type,type,
ab_semigroup_mult: $i > $o ).
thf(member_type,type,
member: $i > $i ).
thf(collect_type,type,
collect: $i > $i ).
thf(combs_type,type,
combs: $i > $i > $i > $i ).
thf(combb_type,type,
combb: $i > $i > $i > $i ).
thf(fconj_type,type,
fconj: $i ).
thf(combc_type,type,
combc: $i > $i > $i > $i ).
thf(comm_semiring_type,type,
comm_semiring: $i > $o ).
thf(linordered_idom_type,type,
linordered_idom: $i > $o ).
thf(ring_1_Ints_type,type,
ring_1_Ints: $i > $i ).
thf(ordere236663937imp_le_type,type,
ordere236663937imp_le: $i > $o ).
thf(fTrue_type,type,
fTrue: $i ).
thf(bot_bot_type,type,
bot_bot: $i > $i ).
thf(nat_tsub_type,type,
nat_tsub: $i ).
thf(nat_neg_type,type,
nat_neg: $i ).
thf(succ_type,type,
succ: $i ).
thf(finite_finite_1_type,type,
finite_finite_1: $i > $o ).
thf(sum_sum_type,type,
sum_sum: $i > $i > $i ).
thf(linordered_semiring_type,type,
linordered_semiring: $i > $o ).
thf(ordere1490568538miring_type,type,
ordere1490568538miring: $i > $o ).
thf(ordered_ring_abs_type,type,
ordered_ring_abs: $i > $o ).
thf(ordere142940540dd_abs_type,type,
ordere142940540dd_abs: $i > $o ).
thf(sum_Plus_type,type,
sum_Plus: $i > $i > $i ).
thf(ab_semigroup_add_type,type,
ab_semigroup_add: $i > $o ).
thf(linord219039673up_add_type,type,
linord219039673up_add: $i > $o ).
thf(linordered_semidom_type,type,
linordered_semidom: $i > $o ).
thf(ab_group_add_type,type,
ab_group_add: $i > $o ).
thf(group_add_type,type,
group_add: $i > $o ).
thf(mult_zero_type,type,
mult_zero: $i > $o ).
thf(linordered_ring_type,type,
linordered_ring: $i > $o ).
thf(ordere216010020id_add_type,type,
ordere216010020id_add: $i > $o ).
thf(bot_type,type,
bot: $i > $o ).
thf(option_type,type,
option: $i > $i ).
thf(com_type,type,
com: $i ).
thf(the_1_type,type,
the_1: $i > $i ).
thf(ring_1_type,type,
ring_1: $i > $o ).
thf(undefined_type,type,
undefined: $i > $i ).
thf(ordered_semiring_type,type,
ordered_semiring: $i > $o ).
thf(ordere223160158up_add_type,type,
ordere223160158up_add: $i > $o ).
thf(preorder_type,type,
preorder: $i > $o ).
thf(ring_n68954251visors_type,type,
ring_n68954251visors: $i > $o ).
thf(ordered_ring_type,type,
ordered_ring: $i > $o ).
thf(fNot_type,type,
fNot: $i ).
thf(combi_type,type,
combi: $i > $i ).
thf(linord626643107strict_type,type,
linord626643107strict: $i > $o ).
thf(the_type,type,
the: $i > $i ).
thf(fimplies_type,type,
fimplies: $i ).
thf(fequal_type,type,
fequal: $i > $i ).
thf(na_type,type,
na: $i ).
thf(fdisj_type,type,
fdisj: $i ).
thf(p_type,type,
p: $i ).
thf(finite_comp_fun_idem_type,type,
finite_comp_fun_idem: $i > $i > $i ).
thf(finite2073411215e_idem_type,type,
finite2073411215e_idem: $i > $i ).
thf(finite1357897459simple_type,type,
finite1357897459simple: $i > $i > $i ).
thf(mgt_type,type,
mgt: $i ).
thf(if_type,type,
if: $i > $i ).
thf(finite908156982e_idem_type,type,
finite908156982e_idem: $i > $i > $i ).
thf(dense_linorder_type,type,
dense_linorder: $i > $o ).
thf(finite_folding_one_type,type,
finite_folding_one: $i > $i ).
thf(linord1117847801e_zero_type,type,
linord1117847801e_zero: $i > $o ).
thf(wt_type,type,
wt: $i ).
thf(the_elem_type,type,
the_elem: $i > $i ).
thf(linordered_field_type,type,
linordered_field: $i > $o ).
thf(body_type,type,
body: $i ).
thf(ab_sem1668676832m_mult_type,type,
ab_sem1668676832m_mult: $i > $o ).
thf(partial_flat_lub_type,type,
partial_flat_lub: $i > $i ).
thf(132,axiom,
comm_monoid_mult @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocomm__monoid__mult) ).
thf(1586,plain,
comm_monoid_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(82,axiom,
comm_semiring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ocomm__semiring) ).
thf(1411,plain,
comm_semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(1128,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_685_mult__strict__right__mono__neg) ).
thf(5116,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1128]) ).
thf(864,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ B @ bool ) ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_110_image__mono) ).
thf(4094,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ B @ bool ) ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).
thf(493,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_277_card_Oinsert) ).
thf(2778,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).
thf(845,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( one_one @ A ) ) @ ( ring_1_Ints @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_633_Ints__1) ).
thf(4022,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( one_one @ A ) ) @ ( ring_1_Ints @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).
thf(595,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_602_le__cube) ).
thf(3167,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).
thf(379,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_696_mult__less__mono2) ).
thf(2412,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).
thf(394,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i] :
? [C: $i] :
! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_821_pinf_I6_J) ).
thf(2463,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i] :
? [C: $i] :
! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).
thf(421,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_653_split__mult__neg__le) ).
thf(2544,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).
thf(357,axiom,
ordere216010020id_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__comm__monoid__add) ).
thf(2331,plain,
ordere216010020id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).
thf(1026,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) )
<=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B )
= C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_791_nat__eq__add__iff1) ).
thf(4728,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B )
= C ) )
& ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B )
= C )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).
thf(597,axiom,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) ) @ ( finite_folding_one @ A ) )
= ( finite_folding_one @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofolding__one_res) ).
thf(3172,plain,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) ) @ ( finite_folding_one @ A ) )
= ( finite_folding_one @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).
thf(334,axiom,
linord626643107strict @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semiring__1__strict) ).
thf(2250,plain,
linord626643107strict @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).
thf(551,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( zero_zero @ A ) ) @ ( ring_1_Ints @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_632_Ints__0) ).
thf(3021,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( zero_zero @ A ) ) @ ( ring_1_Ints @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).
thf(57,axiom,
linord20386208strict @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semiring__strict) ).
thf(1322,plain,
linord20386208strict @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(1118,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_folding__image__simple_Oinsert__remove) ).
thf(5088,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).
thf(1053,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) @ A )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_267_le__add__diff__inverse2) ).
thf(4835,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) @ A )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).
thf(348,axiom,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) )
<=> ( ( ti @ A @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_690_eq__add__iff2) ).
thf(2295,plain,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) )
=> ( ( ti @ A @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) ) )
& ( ( ( ti @ A @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).
thf(957,axiom,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ C @ F ) ) @ D ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ G ) @ H ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ G ) ) @ ( hAPP @ A @ B @ C @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_442_order__le__less__subst2) ).
thf(4450,plain,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ C @ F ) ) @ D ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ G ) @ H ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ G ) ) @ ( hAPP @ A @ B @ C @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).
thf(839,axiom,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_660_mult__left__mono) ).
thf(4005,plain,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).
thf(987,axiom,
! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_subset__insertI) ).
thf(4571,plain,
! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).
thf(557,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) ) @ ( finite1357897459simple @ A @ B ) )
= ( finite1357897459simple @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofolding__image__simple_res) ).
thf(3053,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) ) @ ( finite1357897459simple @ A @ B ) )
= ( finite1357897459simple @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).
thf(288,axiom,
( ( ti @ ( fun @ int @ int ) @ bit1 )
= bit1 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Int_OBit1_res) ).
thf(2109,plain,
( ( ti @ ( fun @ int @ int ) @ bit1 )
= bit1 ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(296,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_485_Suc__neq__Zero) ).
thf(2131,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(268,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) )
<=> ( ( B = C )
| ( A
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_587_mult__cancel1) ).
thf(2031,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) )
=> ( ( B = C )
| ( A
= ( zero_zero @ nat ) ) ) )
& ( ( ( B = C )
| ( A
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(911,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_521_diff__is__0__eq) ).
thf(4262,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).
thf(1196,axiom,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ ( hAPP @ B @ A @ E @ F ) ) )
=> ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ F ) @ C ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ G ) @ H ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ B @ A @ E @ G ) ) @ ( hAPP @ B @ A @ E @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ B @ A @ E @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_436_order__le__less__subst1) ).
thf(5329,plain,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ ( hAPP @ B @ A @ E @ F ) ) )
=> ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ F ) @ C ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ G ) @ H ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ B @ A @ E @ G ) ) @ ( hAPP @ B @ A @ E @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ B @ A @ E @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1196]) ).
thf(858,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_168_card__Suc__Diff1) ).
thf(4062,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).
thf(661,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( minus_minus @ ( fun @ B @ bool ) ) @ D ) @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_172_image__diff__subset) ).
thf(3399,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( minus_minus @ ( fun @ B @ bool ) ) @ D ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).
thf(256,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_244_add__Suc__shift) ).
thf(1995,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(1068,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) )
<=> ? [B: $i] :
! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_finite__nat__set__iff__bounded__le) ).
thf(4899,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) )
=> ? [B: $i] :
! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) ) ) )
& ( ? [B: $i] :
! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).
thf(777,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( zero_zero @ nat ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_455_le0) ).
thf(3798,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( zero_zero @ nat ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).
thf(750,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_683_mult__strict__left__mono) ).
thf(3698,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).
thf(831,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_316_order__less__asym_H) ).
thf(3979,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).
thf(786,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ti @ A @ D )
= ( hAPP @ B @ A @ E @ F ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ E ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_image__eqI) ).
thf(3830,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ti @ A @ D )
= ( hAPP @ B @ A @ E @ F ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ E ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).
thf(433,axiom,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_add__left__mono) ).
thf(2595,plain,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).
thf(210,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) ) @ A ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ A ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) ) @ A ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ succ @ B ) ) ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_943_Suc__nat__number__of__add) ).
thf(1850,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) ) @ A ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ A ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) ) @ A ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ succ @ B ) ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(398,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_Diff__cancel) ).
thf(2480,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).
thf(827,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_add__right__imp__eq) ).
thf(3969,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).
thf(251,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ suc @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_Suc__diff__diff) ).
thf(1982,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ suc @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(426,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( zero_zero @ nat ) ) )
<=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_457_le__0__eq) ).
thf(2559,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( zero_zero @ nat ) ) )
=> ( A
= ( zero_zero @ nat ) ) )
& ( ( A
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).
thf(843,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ B ) )
= ( ti @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_287_the__sym__eq__trivial) ).
thf(4015,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ B ) )
= ( ti @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).
thf(242,axiom,
! [A: $i] :
( ( zero_zero @ nat )
!= ( hAPP @ nat @ nat @ suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_486_Zero__neq__Suc) ).
thf(1956,plain,
! [A: $i] :
( ( zero_zero @ nat )
!= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(240,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ succ @ A )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_944_succ__def) ).
thf(1950,plain,
! [A: $i] :
( ( hAPP @ int @ int @ succ @ A )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(174,axiom,
! [A: $i] :
( pls
!= ( hAPP @ int @ int @ bit1 @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_979_rel__simps_I39_J) ).
thf(1747,plain,
! [A: $i] :
( pls
!= ( hAPP @ int @ int @ bit1 @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(1748,plain,
~ ? [A: $i] :
( pls
= ( hAPP @ int @ int @ bit1 @ A ) ),
inference(miniscope,[status(thm)],[1747]) ).
thf(1749,plain,
! [A: $i] :
( pls
!= ( hAPP @ int @ int @ bit1 @ A ) ),
inference(cnf,[status(esa)],[1748]) ).
thf(1750,plain,
! [A: $i] :
( ( hAPP @ int @ int @ bit1 @ A )
!= pls ),
inference(lifteq,[status(thm)],[1749]) ).
thf(10,axiom,
zero_neq_one @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ozero__neq__one) ).
thf(1211,plain,
zero_neq_one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(931,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( one_one @ int ) ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_780_zle__add1__eq__le) ).
thf(4339,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).
thf(861,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) )
<=> ? [B: $i] :
! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_387_finite__nat__set__iff__bounded) ).
thf(4075,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) )
=> ? [B: $i] :
! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) ) ) )
& ( ? [B: $i] :
! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).
thf(797,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_913_abs__triangle__ineq2) ).
thf(3858,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).
thf(1168,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_diff__single__insert) ).
thf(5242,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1168]) ).
thf(236,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ B )
= ( zero_zero @ A ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_929_abs__eq__0) ).
thf(1939,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ B )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ B )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(135,axiom,
linorder @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Olinorder) ).
thf(1591,plain,
linorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(508,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_771_zle__refl) ).
thf(2837,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).
thf(896,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_865_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) ).
thf(4215,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).
thf(49,axiom,
linord581940658strict @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__ring__strict) ).
thf(1305,plain,
linord581940658strict @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(91,axiom,
! [A: $i] :
( ( semiring_char_0 @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B )
= ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) )
<=> ( B = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_841_of__nat__eq__iff) ).
thf(1437,plain,
! [A: $i] :
( ( semiring_char_0 @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B )
= ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) )
=> ( B = C ) )
& ( ( B = C )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B )
= ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(655,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_338_order__less__imp__le) ).
thf(3382,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).
thf(279,axiom,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ti @ A @ F )
!= ( zero_zero @ A ) )
=> ( ( ( ( ti @ A @ D )
= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ F ) @ B ) )
!= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ F ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_687_add__scale__eq__noteq) ).
thf(2077,plain,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ti @ A @ F )
!= ( zero_zero @ A ) )
=> ( ( ( ( ti @ A @ D )
= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ F ) @ B ) )
!= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ F ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(854,axiom,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ D ) @ E ) )
=> ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
<=> ( ( ti @ A @ D )
= ( ti @ A @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178_diff__eq__diff__eq) ).
thf(4043,plain,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ D ) @ E ) )
=> ( ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( ( ti @ A @ D )
= ( ti @ A @ E ) ) )
& ( ( ( ti @ A @ D )
= ( ti @ A @ E ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).
thf(703,axiom,
! [A: $i] :
( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ A )
= ( hAPP @ ( fun @ int @ bool ) @ nat @ ( finite_card @ int ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_860_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J) ).
thf(3534,plain,
! [A: $i] :
( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ A )
= ( hAPP @ ( fun @ int @ bool ) @ nat @ ( finite_card @ int ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).
thf(852,axiom,
! [A: $i,B: $i] :
( ( bot @ B )
=> ! [C: $i] :
( ( hAPP @ A @ B @ ( bot_bot @ ( fun @ A @ B ) ) @ C )
= ( bot_bot @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_bot__fun__def) ).
thf(4037,plain,
! [A: $i,B: $i] :
( ( bot @ B )
=> ! [C: $i] :
( ( hAPP @ A @ B @ ( bot_bot @ ( fun @ A @ B ) ) @ C )
= ( bot_bot @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).
thf(407,axiom,
( ( ti @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ fdisj )
= fdisj ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_fdisj_res) ).
thf(2512,plain,
( ( ti @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ fdisj )
= fdisj ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).
thf(231,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
<=> ? [C: $i] :
( B
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ suc @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_870_zless__iff__Suc__zadd) ).
thf(1924,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ suc @ C ) ) ) ) )
& ( ? [C: $i] :
( B
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ suc @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(320,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_diff__Suc__Suc) ).
thf(2210,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).
thf(286,axiom,
! [A: $i,B: $i,C: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ B @ ( fun @ A @ C ) ) ) @ ( combc @ A @ B @ C ) )
= ( combc @ A @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_COMBC_res) ).
thf(2104,plain,
! [A: $i,B: $i,C: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ B @ ( fun @ A @ C ) ) ) @ ( combc @ A @ B @ C ) )
= ( combc @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(1163,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ suc @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_Suc__le__D) ).
thf(5225,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ suc @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1163]) ).
thf(768,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_847_zless__int) ).
thf(3762,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).
thf(447,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_662_mult__nonpos__nonpos) ).
thf(2644,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).
thf(196,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_703_mult__Suc) ).
thf(1808,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(668,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_787_nat__mult__less__cancel1) ).
thf(3417,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).
thf(811,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_431_psubsetD) ).
thf(3912,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).
thf(24,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_561_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) ).
thf(1225,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(1226,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(cnf,[status(esa)],[1225]) ).
thf(1227,plain,
! [B: $i,A: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) )
| ~ ( comm_semiring_1 @ A ) ),
inference(lifteq,[status(thm)],[1226]) ).
thf(1,conjecture,
hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_6) ).
thf(2,negated_conjecture,
~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(1203,plain,
~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(5396,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( ti @ A @ B ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
!= ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ) ),
inference(paramod_ordered,[status(thm)],[1227,1203]) ).
thf(5402,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( ti @ A @ B ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( A
!= ( fun @ x_a @ bool ) )
| ( A
!= ( fun @ x_a @ bool ) )
| ( ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) )
!= ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) )
| ( B != g ) ),
inference(simp,[status(thm)],[5396]) ).
thf(5415,plain,
( ~ ( comm_semiring_1 @ ( fun @ x_a @ bool ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( ti @ ( fun @ x_a @ bool ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( plus_plus @ ( fun @ x_a @ bool ) ) @ ( zero_zero @ ( fun @ x_a @ bool ) ) )
!= ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) ) ),
inference(simp,[status(thm)],[5402]) ).
thf(5562,plain,
( ~ ( comm_semiring_1 @ ( fun @ x_a @ bool ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( ti @ ( fun @ x_a @ bool ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( fun @ x_a @ bool )
!= x_a )
| ( ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) )
!= ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) )
| ( ( plus_plus @ ( fun @ x_a @ bool ) )
!= ( insert @ x_a ) )
| ( ( hAPP @ pname @ x_a @ mgt_call @ pn )
!= ( zero_zero @ ( fun @ x_a @ bool ) ) ) ),
inference(simp,[status(thm)],[5415]) ).
thf(5625,plain,
( ~ ( comm_semiring_1 @ ( fun @ x_a @ bool ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( ti @ ( fun @ x_a @ bool ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( fun @ x_a @ bool )
!= x_a )
| ( ( plus_plus @ ( fun @ x_a @ bool ) )
!= ( insert @ x_a ) )
| ( ( hAPP @ pname @ x_a @ mgt_call @ pn )
!= ( zero_zero @ ( fun @ x_a @ bool ) ) ) ),
inference(simp,[status(thm)],[5562]) ).
thf(457,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_578_Ints__diff) ).
thf(2671,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).
thf(459,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_405_add__diff__inverse) ).
thf(2676,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).
thf(995,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_398_Suc__le__eq) ).
thf(4603,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).
thf(4,axiom,
semiri456707255roduct @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) ).
thf(1205,plain,
semiri456707255roduct @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(191,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_749_times__numeral__code_I5_J) ).
thf(1793,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(514,axiom,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_416_not__psubset__empty) ).
thf(2854,plain,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).
thf(1166,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( hAPP @ bool @ ( fun @ A @ ( fun @ A @ A ) ) @ ( if @ A ) @ fFalse ) @ B ) @ C )
= ( ti @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_2_1_T) ).
thf(5237,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( hAPP @ bool @ ( fun @ A @ ( fun @ A @ A ) ) @ ( if @ A ) @ fFalse ) @ B ) @ C )
= ( ti @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1166]) ).
thf(946,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
<=> ? [B: $i] :
( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_510_gr0__conv__Suc) ).
thf(4408,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ? [B: $i] :
( A
= ( hAPP @ nat @ nat @ suc @ B ) ) )
& ( ? [B: $i] :
( A
= ( hAPP @ nat @ nat @ suc @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).
thf(1087,axiom,
hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ g ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_1) ).
thf(4973,plain,
hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ g ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).
thf(106,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_598_add__mult__distrib2) ).
thf(1486,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(977,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_725_le__add__iff2) ).
thf(4532,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).
thf(550,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( one_one @ int ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_809_int__gr__induct) ).
thf(3017,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( one_one @ int ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).
thf(903,axiom,
ord @ bool,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Oord) ).
thf(4234,plain,
ord @ bool,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).
thf(578,axiom,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( finite_finite @ A ) )
= ( finite_finite @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofinite_res) ).
thf(3120,plain,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( finite_finite @ A ) )
= ( finite_finite @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).
thf(763,axiom,
! [A: $i] :
( ( linordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_712_mult__right__less__imp__less) ).
thf(3745,plain,
! [A: $i] :
( ( linordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).
thf(463,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fdisj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_insert__compr) ).
thf(2686,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fdisj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).
thf(651,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_singleton__inject) ).
thf(3367,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).
thf(847,axiom,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( member @ A ) )
= ( member @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_member_res) ).
thf(4026,plain,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( member @ A ) )
= ( member @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).
thf(642,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_insert__Diff1) ).
thf(3338,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).
thf(386,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( insert @ B ) @ ( hAPP @ A @ B @ C @ D ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_image__insert) ).
thf(2434,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( insert @ B ) @ ( hAPP @ A @ B @ C @ D ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).
thf(774,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_554_even__less__0__iff) ).
thf(3781,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).
thf(635,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_427_bounded__nat__set__is__finite) ).
thf(3313,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).
thf(511,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_add__le__cancel__right) ).
thf(2844,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).
thf(1038,axiom,
! [A: $i] :
( ( ord @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( C = D )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_ord__eq__le__trans) ).
thf(4762,plain,
! [A: $i] :
( ( ord @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( C = D )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).
thf(898,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_384_termination__basic__simps_I5_J) ).
thf(4220,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).
thf(598,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ C )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ! [D: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ E )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_219_finite__ne__induct) ).
thf(3175,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ C )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ! [D: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ E )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).
thf(38,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( zero_zero @ int ) ) @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_761_zadd__0) ).
thf(1275,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( zero_zero @ int ) ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(1191,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ E ) ) @ ( hAPP @ nat @ A @ D @ ( hAPP @ nat @ nat @ suc @ E ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ B ) ) @ ( hAPP @ nat @ A @ D @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_412_lift__Suc__mono__less__iff) ).
thf(5316,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ E ) ) @ ( hAPP @ nat @ A @ D @ ( hAPP @ nat @ nat @ suc @ E ) ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ B ) ) @ ( hAPP @ nat @ A @ D @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ B ) ) @ ( hAPP @ nat @ A @ D @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1191]) ).
thf(60,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) )
<=> ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_477_add__is__0) ).
thf(1327,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( zero_zero @ nat ) ) ) )
& ( ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(749,axiom,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_diff__eq__diff__less__eq) ).
thf(3693,plain,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).
thf(179,axiom,
linordered_ring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__ring) ).
thf(1761,plain,
linordered_ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(63,axiom,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) ) ) @ ( nat_case @ A ) )
= ( nat_case @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Nat_Onat_Onat__case_res) ).
thf(1342,plain,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) ) ) @ ( nat_case @ A ) )
= ( nat_case @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(683,axiom,
hBOOL @ ( hAPP @ ( fun @ pname @ bool ) @ bool @ ( hAPP @ pname @ ( fun @ ( fun @ pname @ bool ) @ bool ) @ ( member @ pname ) @ pn ) @ u ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_4) ).
thf(3481,plain,
hBOOL @ ( hAPP @ ( fun @ pname @ bool ) @ bool @ ( hAPP @ pname @ ( fun @ ( fun @ pname @ bool ) @ bool ) @ ( member @ pname ) @ pn ) @ u ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).
thf(112,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ succ @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_945_number__of__succ) ).
thf(1507,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ succ @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(21,axiom,
semiri456707255roduct @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) ).
thf(1222,plain,
semiri456707255roduct @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(754,axiom,
preorder @ bool,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Opreorder) ).
thf(3717,plain,
preorder @ bool,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).
thf(368,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) )
<=> ( ( ti @ A @ B )
!= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_910_zero__less__abs__iff) ).
thf(2360,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) )
=> ( ( ti @ A @ B )
!= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).
thf(652,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_614_less__number__of) ).
thf(3370,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).
thf(1131,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_845_less__imp__of__nat__less) ).
thf(5123,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1131]) ).
thf(618,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_679_mult__less__cancel__left__neg) ).
thf(3249,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( zero_zero @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).
thf(140,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_623_number__of__diff) ).
thf(1613,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(230,axiom,
! [A: $i] :
( ( zero @ A )
=> ( ( ti @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Ozero__class_Ozero_res) ).
thf(1921,plain,
! [A: $i] :
( ( zero @ A )
=> ( ( ti @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(781,axiom,
! [A: $i,B: $i] :
( ( comm_monoid_mult @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( combs @ A @ B @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( combb @ B @ ( fun @ B @ B ) @ A ) @ ( times_times @ B ) ) @ C ) ) @ D ) ) @ ( one_one @ B ) ) @ E )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( times_times @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ ( one_one @ B ) ) @ E ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ D ) @ ( one_one @ B ) ) @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_736_fold__image__distrib) ).
thf(3809,plain,
! [A: $i,B: $i] :
( ( comm_monoid_mult @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( combs @ A @ B @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ ( fun @ B @ B ) ) ) @ ( combb @ B @ ( fun @ B @ B ) @ A ) @ ( times_times @ B ) ) @ C ) ) @ D ) ) @ ( one_one @ B ) ) @ E )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( times_times @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ ( one_one @ B ) ) @ E ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ D ) @ ( one_one @ B ) ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).
thf(503,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_603_le__square) ).
thf(2825,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).
thf(802,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ C ) @ ( one_one @ int ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_806_int__less__induct) ).
thf(3880,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ C ) @ ( one_one @ int ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).
thf(377,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_456_zero__less__Suc) ).
thf(2405,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).
thf(537,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
<=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
& ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_421_psubset__eq) ).
thf(2957,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
& ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( ti @ ( fun @ A @ bool ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
& ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( ti @ ( fun @ A @ bool ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).
thf(731,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_551_ex__least__nat__le) ).
thf(3634,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).
thf(876,axiom,
! [A: $i] :
( ( ring_char_0 @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B )
= ( zero_zero @ A ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_706_Ints__double__eq__0__iff) ).
thf(4142,plain,
! [A: $i] :
( ( ring_char_0 @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B )
= ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).
thf(1172,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_263_le__add__diff) ).
thf(5252,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1172]) ).
thf(385,axiom,
! [A: $i] :
( ( ordere1490568538miring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_659_comm__mult__left__mono) ).
thf(2432,plain,
! [A: $i] :
( ( ordere1490568538miring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).
thf(835,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_insertI2) ).
thf(3994,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).
thf(401,axiom,
( ( ti @ pname @ pn )
= pn ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_v_pn_res) ).
thf(2491,plain,
( ( ti @ pname @ pn )
= pn ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).
thf(28,axiom,
! [A: $i] :
( ( zero_zero @ nat )
!= ( hAPP @ nat @ nat @ suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_490_Zero__not__Suc) ).
thf(1240,plain,
! [A: $i] :
( ( zero_zero @ nat )
!= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(1241,plain,
~ ? [A: $i] :
( ( zero_zero @ nat )
= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(miniscope,[status(thm)],[1240]) ).
thf(1242,plain,
! [A: $i] :
( ( zero_zero @ nat )
!= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(cnf,[status(esa)],[1241]) ).
thf(1243,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= ( zero_zero @ nat ) ),
inference(lifteq,[status(thm)],[1242]) ).
thf(263,axiom,
cancel_semigroup_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__semigroup__add) ).
thf(2020,plain,
cancel_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(694,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_subset__refl) ).
thf(3513,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).
thf(1012,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ int @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( member @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_888_transfer__nat__int__set__relations_I2_J) ).
thf(4679,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ int @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( member @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ int @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( member @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).
thf(1000,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) )
<=> ~ ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
| ? [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_540_nat__diff__split__asm) ).
thf(4618,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) )
=> ~ ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
| ? [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) ) )
& ( ~ ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
| ? [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).
thf(568,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_add__le__imp__le__right) ).
thf(3088,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).
thf(772,axiom,
( ( ti @ ( fun @ com @ bool ) @ wt )
= wt ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_v_wt_res) ).
thf(3777,plain,
( ( ti @ ( fun @ com @ bool ) @ wt )
= wt ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).
thf(633,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_324_order__less__asym) ).
thf(3308,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).
thf(262,axiom,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( ( ti @ A @ D )
!= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
<=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) )
!= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_645_crossproduct__noteq) ).
thf(2010,plain,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( ( ( ti @ A @ D )
!= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) )
!= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ B ) ) ) )
& ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) )
!= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ B ) ) )
=> ( ( ( ti @ A @ D )
!= ( ti @ A @ E ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(234,axiom,
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( one_one @ nat ) )
= ( one_one @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_837_int__1) ).
thf(1935,plain,
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( one_one @ nat ) )
= ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(1072,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_Diff__insert2) ).
thf(4916,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).
thf(69,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_260_diff__cancel2) ).
thf(1359,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(586,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_615_Ints__number__of) ).
thf(3140,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).
thf(101,axiom,
minus @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ominus) ).
thf(1475,plain,
minus @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(801,axiom,
( ( ti @ ( fun @ pname @ x_a ) @ mgt_call )
= mgt_call ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_v_mgt__call_res) ).
thf(3878,plain,
( ( ti @ ( fun @ pname @ x_a ) @ mgt_call )
= mgt_call ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).
thf(225,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ pls )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_982_add__Pls__right) ).
thf(1904,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ pls )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(25,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_973_add__numeral__0__right) ).
thf(1228,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(142,axiom,
ab_semigroup_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oab__semigroup__add) ).
thf(1623,plain,
ab_semigroup_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(1101,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_singleton__conv) ).
thf(5024,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).
thf(1198,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
<=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) @ ( zero_zero @ int ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ int @ int @ ( number_number_of @ int ) @ A )
= ( zero_zero @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( A = B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_936_eq__nat__number__of) ).
thf(5342,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) @ ( zero_zero @ int ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ int @ int @ ( number_number_of @ int ) @ A )
= ( zero_zero @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( A = B ) ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) @ ( zero_zero @ int ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ int @ int @ ( number_number_of @ int ) @ A )
= ( zero_zero @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( A = B ) ) ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1198]) ).
thf(221,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= A )
=> ( B
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_476_add__eq__self__zero) ).
thf(1887,plain,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= A )
=> ( B
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(1082,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ! [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ B @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ suc @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_560_nat__induct) ).
thf(4956,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ! [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ B @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ suc @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).
thf(947,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_840_Ints__of__nat) ).
thf(4415,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).
thf(1037,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( one_one @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_762_int__0__less__1) ).
thf(4761,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).
thf(58,axiom,
comm_monoid_mult @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ocomm__monoid__mult) ).
thf(1323,plain,
comm_monoid_mult @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(340,axiom,
semiring @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring) ).
thf(2274,plain,
semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).
thf(792,axiom,
minus @ bool,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Groups_Ominus) ).
thf(3845,plain,
minus @ bool,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).
thf(255,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ ( one_one @ nat ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_576_expand__Suc) ).
thf(1992,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ ( one_one @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(784,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_340_order__le__imp__less__or__eq) ).
thf(3820,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).
thf(805,axiom,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ D @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ E ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ H ) @ G ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ D @ H ) ) @ ( hAPP @ A @ B @ D @ G ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ D @ C ) ) @ F ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_435_xt4) ).
thf(3891,plain,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ D @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ E ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ H ) @ G ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ D @ H ) ) @ ( hAPP @ A @ B @ D @ G ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ D @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).
thf(451,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_924_abs__add__abs) ).
thf(2654,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).
thf(905,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_407_less__diff__iff) ).
thf(4238,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).
thf(604,axiom,
finite_finite_1 @ bool,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Finite__Set_Ofinite) ).
thf(3204,plain,
finite_finite_1 @ bool,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).
thf(622,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_526_add__nonneg__pos) ).
thf(3263,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).
thf(608,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ D @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_538_predicate1D) ).
thf(3221,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ D @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).
thf(830,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ C )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_equals0D) ).
thf(3976,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ C )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).
thf(356,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_520_diff__add__0) ).
thf(2328,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).
thf(819,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_set__rev__mp) ).
thf(3938,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).
thf(1029,axiom,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_720_sum__squares__ge__zero) ).
thf(4743,plain,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).
thf(985,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_265_add__diff__assoc) ).
thf(4563,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).
thf(252,axiom,
number_ring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber__ring) ).
thf(1985,plain,
number_ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(860,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) @ ( one_one @ nat ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_card__Diff__singleton__if) ).
thf(4070,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) @ ( one_one @ nat ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).
thf(455,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_708_mult__right__le__imp__le) ).
thf(2667,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).
thf(1016,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_set__mp) ).
thf(4692,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).
thf(1182,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_328_linorder__le__less__linear) ).
thf(5284,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1182]) ).
thf(747,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_501_add__pos__pos) ).
thf(3689,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).
thf(900,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ D ) @ E ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ ( hAPP @ B @ A @ C @ D ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_insert__image) ).
thf(4224,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ D ) @ E ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ ( hAPP @ B @ A @ C @ D ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).
thf(217,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ pls ) @ A )
= pls ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_983_mult__Pls) ).
thf(1871,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ pls ) @ A )
= pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(226,axiom,
! [A: $i] :
( ( ti @ A @ ( undefined @ A ) )
= ( undefined @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_HOL_Oundefined_res) ).
thf(1907,plain,
! [A: $i] :
( ( ti @ A @ ( undefined @ A ) )
= ( undefined @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(1180,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ D )
= ( bot_bot @ ( fun @ B @ bool ) ) )
<=> ( ( ti @ ( fun @ A @ bool ) @ D )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_image__is__empty) ).
thf(5274,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ D )
= ( bot_bot @ ( fun @ B @ bool ) ) )
=> ( ( ti @ ( fun @ A @ bool ) @ D )
= ( bot_bot @ ( fun @ A @ bool ) ) ) )
& ( ( ( ti @ ( fun @ A @ bool ) @ D )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ D )
= ( bot_bot @ ( fun @ B @ bool ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1180]) ).
thf(725,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_566_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) ).
thf(3617,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).
thf(929,axiom,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_656_mult__mono_H) ).
thf(4334,plain,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).
thf(323,axiom,
ordere1490568538miring @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Oordered__comm__semiring) ).
thf(2217,plain,
ordere1490568538miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).
thf(580,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( ( ti @ A @ C )
= ( ti @ A @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_321_xt1_I2_J) ).
thf(3125,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( ( ti @ A @ C )
= ( ti @ A @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).
thf(1140,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ B @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ D ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_448_finite__Plus) ).
thf(5150,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ B @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ D ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1140]) ).
thf(54,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( ti @ ( fun @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) ) @ ( semiri532925092at_aux @ A ) )
= ( semiri532925092at_aux @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Nat_Osemiring__1__class_Oof__nat__aux_res) ).
thf(1313,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( ti @ ( fun @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) ) @ ( semiri532925092at_aux @ A ) )
= ( semiri532925092at_aux @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(933,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_909_abs__not__less__zero) ).
thf(4349,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).
thf(40,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_876_zmult__zless__mono2__lemma) ).
thf(1279,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(915,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
<=> ( ( A
= ( zero_zero @ nat ) )
| ? [C: $i] :
( ( A
= ( hAPP @ nat @ nat @ suc @ C ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_512_less__Suc__eq__0__disj) ).
thf(4275,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( ( A
= ( zero_zero @ nat ) )
| ? [C: $i] :
( ( A
= ( hAPP @ nat @ nat @ suc @ C ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) ) ) ) )
& ( ( ( A
= ( zero_zero @ nat ) )
| ? [C: $i] :
( ( A
= ( hAPP @ nat @ nat @ suc @ C ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).
thf(1134,axiom,
! [A: $i] :
( ( cancel146912293up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_225_add__imp__eq) ).
thf(5130,plain,
! [A: $i] :
( ( cancel146912293up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1134]) ).
thf(1144,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_707_mult__left__le__imp__le) ).
thf(5160,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1144]) ).
thf(95,axiom,
linord20386208strict @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__semiring__strict) ).
thf(1455,plain,
linord20386208strict @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(1039,axiom,
! [A: $i] :
( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less_eq @ nat ) ) @ A ) ) )
= ( hAPP @ nat @ nat @ suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_card__Collect__le__nat) ).
thf(4766,plain,
! [A: $i] :
( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less_eq @ nat ) ) @ A ) ) )
= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).
thf(98,axiom,
! [A: $i,B: $i] :
( ( ti @ A @ ( ti @ A @ B ) )
= ( ti @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_ti_idem) ).
thf(1462,plain,
! [A: $i,B: $i] :
( ( ti @ A @ ( ti @ A @ B ) )
= ( ti @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(796,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_31_le__SucE) ).
thf(3855,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).
thf(382,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_987_rel__simps_I34_J) ).
thf(2420,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).
thf(573,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_234_add__le__cancel__left) ).
thf(3099,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).
thf(696,axiom,
! [A: $i,B: $i] :
( ( hAPP @ A @ A @ ( combi @ A ) @ B )
= ( ti @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBI_1_1_U) ).
thf(3517,plain,
! [A: $i,B: $i] :
( ( hAPP @ A @ A @ ( combi @ A ) @ B )
= ( ti @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).
thf(886,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_400_Suc__leI) ).
thf(4179,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).
thf(103,axiom,
semiring_1 @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring__1) ).
thf(1479,plain,
semiring_1 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(1075,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_358_less__Suc__eq) ).
thf(4928,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).
thf(531,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_856_transfer__int__nat__relations_I2_J) ).
thf(2929,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).
thf(549,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_269_diff__add__assoc) ).
thf(3014,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).
thf(822,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( hAPP @ ( fun @ A @ C ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ ( fun @ C @ B ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ A @ B ) ) @ ( combs @ A @ C @ B ) @ D ) @ E ) @ F )
= ( hAPP @ C @ B @ ( hAPP @ A @ ( fun @ C @ B ) @ D @ F ) @ ( hAPP @ A @ C @ E @ F ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBS_1_1_U) ).
thf(3951,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( hAPP @ ( fun @ A @ C ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ ( fun @ C @ B ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ A @ B ) ) @ ( combs @ A @ C @ B ) @ D ) @ E ) @ F )
= ( hAPP @ C @ B @ ( hAPP @ A @ ( fun @ C @ B ) @ D @ F ) @ ( hAPP @ A @ C @ E @ F ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).
thf(185,axiom,
( ( hAPP @ int @ int @ succ @ pls )
= ( hAPP @ int @ int @ bit1 @ pls ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_949_succ__Pls) ).
thf(1773,plain,
( ( hAPP @ int @ int @ succ @ pls )
= ( hAPP @ int @ int @ bit1 @ pls ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(1119,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ C )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_218_folding__one__idem_Oinsert__idem) ).
thf(5091,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ C )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).
thf(1041,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ C @ ( ti @ A @ D ) )
= ( hAPP @ A @ B @ C @ D ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_hAPP_arg2) ).
thf(4771,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ C @ ( ti @ A @ D ) )
= ( hAPP @ A @ B @ C @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).
thf(209,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_604_diff__mult__distrib) ).
thf(1847,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(1035,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_274_folding__one__idem_Oin__idem) ).
thf(4756,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).
thf(790,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ ( fun @ B @ B ) ) @ bool @ ( finite_comp_fun_idem @ A @ B ) @ E ) )
=> ( ( hAPP @ B @ B @ ( hAPP @ A @ ( fun @ B @ B ) @ E @ C ) @ ( hAPP @ B @ B @ ( hAPP @ A @ ( fun @ B @ B ) @ E @ C ) @ D ) )
= ( hAPP @ B @ B @ ( hAPP @ A @ ( fun @ B @ B ) @ E @ C ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_285_comp__fun__idem_Ofun__left__idem) ).
thf(3840,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ ( fun @ B @ B ) ) @ bool @ ( finite_comp_fun_idem @ A @ B ) @ E ) )
=> ( ( hAPP @ B @ B @ ( hAPP @ A @ ( fun @ B @ B ) @ E @ C ) @ ( hAPP @ B @ B @ ( hAPP @ A @ ( fun @ B @ B ) @ E @ C ) @ D ) )
= ( hAPP @ B @ B @ ( hAPP @ A @ ( fun @ B @ B ) @ E @ C ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).
thf(1174,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_988_less__eq__int__code_I16_J) ).
thf(5256,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1174]) ).
thf(515,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_finite__insert) ).
thf(2857,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).
thf(1058,axiom,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ D @ C ) ) @ ( hAPP @ A @ B @ E @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_le__funE) ).
thf(4862,plain,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ D @ C ) ) @ ( hAPP @ A @ B @ E @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).
thf(305,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_962_semiring__numeral__1__eq__1) ).
thf(2169,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).
thf(732,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_500_double__add__less__zero__iff__single__add__less__zero) ).
thf(3638,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).
thf(115,axiom,
( ( one_one @ int )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( one_one @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_862_transfer__int__nat__numerals_I2_J) ).
thf(1514,plain,
( ( one_one @ int )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( one_one @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(891,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_linorder__linear) ).
thf(4194,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).
thf(30,axiom,
cancel_semigroup_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ocancel__semigroup__add) ).
thf(1247,plain,
cancel_semigroup_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(1133,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_313_order__less__imp__not__eq) ).
thf(5127,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1133]) ).
thf(423,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_subset__trans) ).
thf(2552,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).
thf(56,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_752_zdiff__zmult__distrib) ).
thf(1319,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(222,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) )
= ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_925_abs__idempotent) ).
thf(1890,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) )
= ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(172,axiom,
! [A: $i,B: $i,C: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ C ) ) ) @ ( combs @ A @ B @ C ) )
= ( combs @ A @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_COMBS_res) ).
thf(1741,plain,
! [A: $i,B: $i,C: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ B @ C ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ C ) ) ) @ ( combs @ A @ B @ C ) )
= ( combs @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(77,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) )
= ( one_one @ int ) )
=> ( ( hAPP @ int @ int @ ( abs_abs @ int ) @ A )
= ( one_one @ int ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_931_abs__zmult__eq__1) ).
thf(1398,plain,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) )
= ( one_one @ int ) )
=> ( ( hAPP @ int @ int @ ( abs_abs @ int ) @ A )
= ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(923,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_411_less__imp__Suc__add) ).
thf(4317,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).
thf(983,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_824_int__induct) ).
thf(4551,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).
thf(232,axiom,
comm_semiring_1 @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ocomm__semiring__1) ).
thf(1931,plain,
comm_semiring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(265,axiom,
number @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Int_Onumber) ).
thf(2024,plain,
number @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(788,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) @ ( one_one @ nat ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_card__Diff__singleton) ).
thf(3835,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) @ ( one_one @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).
thf(543,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_637_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) ).
thf(2989,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).
thf(1114,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_352_add__less__imp__less__right) ).
thf(5076,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).
thf(70,axiom,
( ( zero_zero @ int )
!= ( one_one @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_799_int__0__neq__1) ).
thf(1362,plain,
( ( zero_zero @ int )
!= ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(972,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_le__Suc__eq) ).
thf(4507,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( A
= ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).
thf(707,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_423_linorder__neqE__linordered__idom) ).
thf(3557,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).
thf(308,axiom,
! [A: $i] :
( ( number @ A )
=> ( ( ti @ ( fun @ int @ A ) @ ( number_number_of @ A ) )
= ( number_number_of @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Int_Onumber__class_Onumber__of_res) ).
thf(2178,plain,
! [A: $i] :
( ( number @ A )
=> ( ( ti @ ( fun @ int @ A ) @ ( number_number_of @ A ) )
= ( number_number_of @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(152,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_756_zadd__zmult__distrib) ).
thf(1664,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(991,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ D ) @ E ) @ F ) @ G ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ! [H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ H ) @ C ) )
=> ( ( hAPP @ A @ B @ F @ H )
= ( ti @ B @ E ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ G @ C )
= ( ti @ B @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_214_folding__image__simple_Oneutral) ).
thf(4583,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ D ) @ E ) @ F ) @ G ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ! [H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ H ) @ C ) )
=> ( ( hAPP @ A @ B @ F @ H )
= ( ti @ B @ E ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ G @ C )
= ( ti @ B @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).
thf(884,axiom,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( hAPP @ B @ A @ D @ E ) ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ H ) @ G ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ B @ A @ D @ H ) ) @ ( hAPP @ B @ A @ D @ G ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( hAPP @ B @ A @ D @ F ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_441_xt7) ).
thf(4173,plain,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( hAPP @ B @ A @ D @ E ) ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ H ) @ G ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ B @ A @ D @ H ) ) @ ( hAPP @ B @ A @ D @ G ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( hAPP @ B @ A @ D @ F ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).
thf(757,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_empty__subsetI) ).
thf(3724,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).
thf(228,axiom,
! [A: $i] :
( ( zero @ A )
=> ! [B: $i] :
( ( ( zero_zero @ A )
= ( ti @ A @ B ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_474_zero__reorient) ).
thf(1911,plain,
! [A: $i] :
( ( zero @ A )
=> ! [B: $i] :
( ( ( ( zero_zero @ A )
= ( ti @ A @ B ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( zero_zero @ A )
= ( ti @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(449,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_406_diff__less__mono) ).
thf(2650,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).
thf(700,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_in__mono) ).
thf(3527,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).
thf(1104,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_984_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J) ).
thf(5032,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1104]) ).
thf(243,axiom,
order @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Oorder) ).
thf(1960,plain,
order @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(1044,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_255_add__leE) ).
thf(4783,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ~ ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).
thf(107,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_487_nat_Osimps_I3_J) ).
thf(1489,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(1018,axiom,
! [A: $i,B: $i] :
( ? [C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
<=> ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_ex__in__conv) ).
thf(4696,plain,
! [A: $i,B: $i] :
( ( ? [C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) ) )
& ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ? [C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).
thf(894,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ A ) @ ( partial_flat_lub @ A ) @ C ) @ B )
= ( ti @ A @ C ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ A ) @ ( partial_flat_lub @ A ) @ C ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_281_flat__lub__def) ).
thf(4205,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ A ) @ ( partial_flat_lub @ A ) @ C ) @ B )
= ( ti @ A @ C ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ A ) @ ( partial_flat_lub @ A ) @ C ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).
thf(84,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_873_of__nat__Suc) ).
thf(1415,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(1176,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_374_add__less__mono) ).
thf(5264,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1176]) ).
thf(639,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ( ti @ A @ C )
!= ( ti @ A @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_342_xt1_I11_J) ).
thf(3322,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ( ti @ A @ C )
!= ( ti @ A @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).
thf(440,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_297_less__not__refl) ).
thf(2616,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).
thf(237,axiom,
number @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber) ).
thf(1946,plain,
number @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(99,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ! [E: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ E ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ E ) @ ( one_one @ nat ) ) ) ) @ ( hAPP @ nat @ int @ B @ E ) ) ) ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ B @ D ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ nat @ int @ B @ C ) ) )
=> ? [E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ E ) @ C ) )
& ( ( hAPP @ nat @ int @ B @ E )
= A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_942_nat__intermed__int__val) ).
thf(1465,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ! [E: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ E ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ E ) @ ( one_one @ nat ) ) ) ) @ ( hAPP @ nat @ int @ B @ E ) ) ) ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ B @ D ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ nat @ int @ B @ C ) ) )
=> ? [E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ E ) @ C ) )
& ( ( hAPP @ nat @ int @ B @ E )
= A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(954,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ pls ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_989_rel__simps_I2_J) ).
thf(4443,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ pls ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).
thf(564,axiom,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i,C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_721_not__sum__squares__lt__zero) ).
thf(3075,plain,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i,C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).
thf(197,axiom,
( ( ti @ ( fun @ ( option @ com ) @ com ) @ ( the_1 @ com ) )
= ( the_1 @ com ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Option_Othe_res) ).
thf(1811,plain,
( ( ti @ ( fun @ ( option @ com ) @ com ) @ ( the_1 @ com ) )
= ( the_1 @ com ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(844,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_810_int__ge__induct) ).
thf(4018,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).
thf(1102,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ B )
= ( ti @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_906_abs__of__nonneg) ).
thf(5027,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ B )
= ( ti @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).
thf(524,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ A @ B ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) @ ( sum_Plus @ A @ B ) @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_446_finite__PlusD_I2_J) ).
thf(2893,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ A @ B ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) @ ( sum_Plus @ A @ B ) @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).
thf(396,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_698_mult__less__cancel2) ).
thf(2469,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).
thf(741,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_678_zero__less__mult__pos2) ).
thf(3670,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).
thf(837,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_finite__Diff) ).
thf(4001,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).
thf(85,axiom,
semiring_1 @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Osemiring__1) ).
thf(1418,plain,
semiring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(390,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_697_mult__less__mono1) ).
thf(2451,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).
thf(212,axiom,
ordere142940540dd_abs @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__ab__group__add__abs) ).
thf(1858,plain,
ordere142940540dd_abs @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(674,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_830_zero__zle__int) ).
thf(3438,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).
thf(877,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ bool ) @ ( finite_comp_fun_idem @ A @ B ) )
= ( finite_comp_fun_idem @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ocomp__fun__idem_res) ).
thf(4147,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ B @ B ) ) @ bool ) @ ( finite_comp_fun_idem @ A @ B ) )
= ( finite_comp_fun_idem @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).
thf(699,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2_finite__imageI) ).
thf(3525,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).
thf(1049,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_667_mult__le__0__iff) ).
thf(4815,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).
thf(1024,axiom,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ D @ C ) ) @ ( hAPP @ A @ B @ E @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_le__funD) ).
thf(4724,plain,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ D @ C ) ) @ ( hAPP @ A @ B @ E @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).
thf(648,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ nat @ bool ) ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ int @ bool ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_885_transfer__nat__int__set__relations_I5_J) ).
thf(3357,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ nat @ bool ) ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ int @ bool ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ int @ bool ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ nat @ bool ) ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).
thf(336,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ D )
=> ( ( ti @ A @ B )
= ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ D ) ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ ( hAPP @ bool @ bool @ fNot @ D ) ) ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ C ) ) ) ) ) )
& ( ~ ( hBOOL @ D )
=> ( ( ti @ A @ C )
= ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ D ) ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ ( hAPP @ bool @ bool @ fNot @ D ) ) ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_283_If__def) ).
thf(2254,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ D )
=> ( ( ti @ A @ B )
= ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ D ) ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ ( hAPP @ bool @ bool @ fNot @ D ) ) ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ C ) ) ) ) ) )
& ( ~ ( hBOOL @ D )
=> ( ( ti @ A @ C )
= ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ D ) ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ ( hAPP @ bool @ bool @ fNot @ D ) ) ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).
thf(711,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_522_diff__is__0__eq_H) ).
thf(3569,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).
thf(418,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( zero_zero @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_812_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J) ).
thf(2540,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).
thf(1061,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) @ A ) ) @ ( zero_zero @ int ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_766_odd__less__0) ).
thf(4871,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) @ A ) ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) @ A ) ) @ ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).
thf(1048,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_726_le__add__iff1) ).
thf(4810,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).
thf(881,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_774_zle__antisym) ).
thf(4162,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).
thf(658,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_742_nat__less__add__iff2) ).
thf(3389,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).
thf(274,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
= ( zero_zero @ A ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_899_of__nat__number__of__eq) ).
thf(2058,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
= ( zero_zero @ A ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(14,axiom,
number_semiring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber__semiring) ).
thf(1215,plain,
number_semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(399,axiom,
! [A: $i] :
( ! [B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) )
<=> ! [B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_867_transfer__int__nat__quantifiers_I1_J) ).
thf(2483,plain,
! [A: $i] :
( ( ! [B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) )
& ( ! [B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).
thf(971,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_922_abs__mult__self) ).
thf(4504,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).
thf(66,axiom,
( ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_952_transfer__int__nat__numerals_I4_J) ).
thf(1351,plain,
( ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(1050,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ A ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_778_zadd__strict__right__mono) ).
thf(4825,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ A ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).
thf(853,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ? [B: $i] :
( A
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_895_zero__le__imp__eq__int) ).
thf(4040,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ? [B: $i] :
( A
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).
thf(610,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_567_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) ).
thf(3226,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).
thf(538,axiom,
! [A: $i,B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_1_1_T) ).
thf(2965,plain,
! [A: $i,B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).
thf(1066,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ C ) @ A ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_779_zadd__zless__mono) ).
thf(4892,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ C ) @ A ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).
thf(992,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( hAPP @ int @ int @ bit1 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_986_rel__simps_I22_J) ).
thf(4588,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( hAPP @ int @ int @ bit1 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ ( hAPP @ int @ int @ bit1 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).
thf(654,axiom,
! [A: $i] :
( ( linord1117847801e_zero @ A )
=> ! [B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_740_field__le__mult__one__interval) ).
thf(3378,plain,
! [A: $i] :
( ( linord1117847801e_zero @ A )
=> ! [B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).
thf(369,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_finite__induct) ).
thf(2367,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).
thf(387,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_359_Suc__less__eq) ).
thf(2437,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).
thf(704,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
<=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_305_linorder__neq__iff) ).
thf(3537,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) )
=> ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).
thf(850,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ pname @ bool ) @ bool @ ( hAPP @ pname @ ( fun @ ( fun @ pname @ bool ) @ bool ) @ ( member @ pname ) @ A ) @ u ) )
=> ( hBOOL @ ( hAPP @ com @ bool @ wt @ ( hAPP @ ( option @ com ) @ com @ ( the_1 @ com ) @ ( hAPP @ pname @ ( option @ com ) @ body @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_189_assms_I4_J) ).
thf(4033,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ pname @ bool ) @ bool @ ( hAPP @ pname @ ( fun @ ( fun @ pname @ bool ) @ bool ) @ ( member @ pname ) @ A ) @ u ) )
=> ( hBOOL @ ( hAPP @ com @ bool @ wt @ ( hAPP @ ( option @ com ) @ com @ ( the_1 @ com ) @ ( hAPP @ pname @ ( option @ com ) @ body @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).
thf(506,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
| ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_subset__singletonD) ).
thf(2833,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
| ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).
thf(600,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_350_add__strict__left__mono) ).
thf(3189,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).
thf(760,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ E ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_Diff__mono) ).
thf(3739,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ E ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).
thf(343,axiom,
! [A: $i] :
( ( monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_469_add__0__left) ).
thf(2281,plain,
! [A: $i] :
( ( monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).
thf(414,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0_assms_I1_J) ).
thf(2530,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).
thf(411,axiom,
! [A: $i] :
( ( linordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_710_mult__left__less__imp__less) ).
thf(2523,plain,
! [A: $i] :
( ( linordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).
thf(192,axiom,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ C )
!= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B )
!= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_582_no__zero__divisors) ).
thf(1796,plain,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ C )
!= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B )
!= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(1096,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_498_add__nonpos__nonpos) ).
thf(4996,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).
thf(1185,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_finite_OinsertI) ).
thf(5296,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1185]) ).
thf(491,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) ).
thf(2772,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).
thf(592,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_card__insert__disjoint) ).
thf(3157,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).
thf(1170,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_Suc__leD) ).
thf(5246,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1170]) ).
thf(1193,axiom,
! [A: $i] :
( ( ring_char_0 @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ B ) ) @ B )
!= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_739_Ints__odd__nonzero) ).
thf(5321,plain,
! [A: $i] :
( ( ring_char_0 @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ B ) ) @ B )
!= ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1193]) ).
thf(744,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_730_n__less__n__mult__m) ).
thf(3676,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).
thf(64,axiom,
! [A: $i] :
( ( ( number @ A )
& ( semiring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_631_left__distrib__number__of) ).
thf(1345,plain,
! [A: $i] :
( ( ( number @ A )
& ( semiring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(445,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_832_of__nat__0__le__iff) ).
thf(2639,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).
thf(285,axiom,
linord893533164strict @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict) ).
thf(2103,plain,
linord893533164strict @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(483,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_713_mult__le__less__imp__less) ).
thf(2750,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).
thf(80,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( one_one @ nat ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_611_nat__mult__1__right) ).
thf(1405,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( one_one @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(664,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_nat__add__commute) ).
thf(3406,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).
thf(317,axiom,
ordere223160158up_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(2205,plain,
ordere223160158up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).
thf(862,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
<=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_Diff__iff) ).
thf(4082,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).
thf(859,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_452_pred__subset__eq) ).
thf(4065,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).
thf(572,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_802_q__pos__lemma) ).
thf(3097,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).
thf(5,axiom,
no_zero_divisors @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ono__zero__divisors) ).
thf(1206,plain,
no_zero_divisors @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(1143,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_Diff__insert) ).
thf(5157,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1143]) ).
thf(631,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_347_add__less__cancel__right) ).
thf(3301,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).
thf(702,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ B ) ) @ A )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_Suc__diff__le) ).
thf(3531,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ B ) ) @ A )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).
thf(599,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ ( fun @ B @ bool ) @ A ) ) ) ) @ ( finite_fold_image @ A @ B ) )
= ( finite_fold_image @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofold__image_res) ).
thf(3186,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ A @ ( fun @ ( fun @ B @ bool ) @ A ) ) ) ) @ ( finite_fold_image @ A @ B ) )
= ( finite_fold_image @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).
thf(1137,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) )
= B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_264_le__add__diff__inverse) ).
thf(5142,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1137]) ).
thf(1184,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_704_Suc__mult__le__cancel1) ).
thf(5291,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1184]) ).
thf(637,axiom,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_669_zero__le__square) ).
thf(3318,plain,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).
thf(936,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
<=> ? [C: $i] :
( B
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_851_zle__iff__zadd) ).
thf(4370,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ C ) ) ) )
& ( ? [C: $i] :
( B
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).
thf(446,axiom,
! [A: $i,B: $i] :
( ( minus @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hAPP @ A @ B @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( minus_minus @ ( fun @ A @ B ) ) @ C ) @ D ) @ E )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( minus_minus @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_fun__diff__def) ).
thf(2641,plain,
! [A: $i,B: $i] :
( ( minus @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hAPP @ A @ B @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( minus_minus @ ( fun @ A @ B ) ) @ C ) @ D ) @ E )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( minus_minus @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).
thf(167,axiom,
linord219039673up_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Olinordered__ab__group__add) ).
thf(1730,plain,
linord219039673up_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(1186,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) )
<=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_add__right__cancel) ).
thf(5298,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ D ) ) )
& ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1186]) ).
thf(673,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ A @ B ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) @ ( sum_Plus @ A @ B ) @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_447_finite__PlusD_I1_J) ).
thf(3435,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ A @ B ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) @ ( sum_Plus @ A @ B ) @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).
thf(714,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( one_one @ int ) ) ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_781_add1__zle__eq) ).
thf(3582,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( one_one @ int ) ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( one_one @ int ) ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).
thf(611,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_482_less__nat__zero__code) ).
thf(3229,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).
thf(676,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_657_mult__left__mono__neg) ).
thf(3450,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).
thf(691,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C )
= ( ti @ A @ B ) ) )
& ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_folding__one_Oremove) ).
thf(3503,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C )
= ( ti @ A @ B ) ) )
& ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).
thf(527,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ C @ D ) ) @ ( hAPP @ nat @ nat @ C @ E ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ C @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ C @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_409_mono__nat__linear__lb) ).
thf(2910,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ C @ D ) ) @ ( hAPP @ nat @ nat @ C @ E ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ C @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ C @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).
thf(925,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_346_xt1_I8_J) ).
thf(4323,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).
thf(716,axiom,
! [A: $i,B: $i] :
( ( order @ B )
=> ( order @ ( fun @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun___Orderings_Oorder) ).
thf(3591,plain,
! [A: $i,B: $i] :
( ( order @ B )
=> ( order @ ( fun @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).
thf(710,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_xt1_I6_J) ).
thf(3567,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).
thf(161,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ D ) @ ( one_one @ nat ) ) ) ) @ ( hAPP @ nat @ int @ B @ D ) ) ) ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ B @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ nat @ int @ B @ C ) ) )
=> ? [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ C ) )
& ( ( hAPP @ nat @ int @ B @ D )
= A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_941_nat0__intermed__int__val) ).
thf(1691,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ D ) @ ( one_one @ nat ) ) ) ) @ ( hAPP @ nat @ int @ B @ D ) ) ) ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ B @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ nat @ int @ B @ C ) ) )
=> ? [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ C ) )
& ( ( hAPP @ nat @ int @ B @ D )
= A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(833,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_card__insert__if) ).
thf(3984,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).
thf(596,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_834_of__nat__less__0__iff) ).
thf(3169,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).
thf(964,axiom,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less @ ( fun @ A @ B ) ) @ C ) @ D ) )
<=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ D ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_415_less__fun__def) ).
thf(4480,plain,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less @ ( fun @ A @ B ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ D ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less @ ( fun @ A @ B ) ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).
thf(623,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fdisj @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fdisj_2_1_U) ).
thf(3265,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fdisj @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).
thf(448,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ nat @ A ) @ C ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_388_nat__seg__image__imp__finite) ).
thf(2646,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ nat @ A ) @ C ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).
thf(795,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_folding__image__simple_Oremove) ).
thf(3852,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).
thf(50,axiom,
ord @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Oord) ).
thf(1306,plain,
ord @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(1147,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) )
= ( ti @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_284_the__eq__trivial) ).
thf(5167,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) )
= ( ti @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1147]) ).
thf(292,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_966_add__special_I2_J) ).
thf(2118,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(769,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_877_of__nat__0__less__iff) ).
thf(3767,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).
thf(329,axiom,
! [A: $i,B: $i] :
( ( ( one_one @ nat )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) )
<=> ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_612_nat__1__eq__mult__iff) ).
thf(2232,plain,
! [A: $i,B: $i] :
( ( ( ( one_one @ nat )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) )
=> ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) )
& ( ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) )
=> ( ( one_one @ nat )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).
thf(989,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( ti @ A @ B ) ) )
& ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_folding__one_Oinsert__remove) ).
thf(4576,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( ti @ A @ B ) ) )
& ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).
thf(281,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_620_mult__number__of__left) ).
thf(2083,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(849,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_681_mult__neg__neg) ).
thf(4031,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).
thf(867,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_744_nat__mult__assoc) ).
thf(4100,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).
thf(921,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ B ) ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_364_less__trans__Suc) ).
thf(4312,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ B ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).
thf(453,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_nat__add__left__cancel__le) ).
thf(2660,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).
thf(857,axiom,
! [A: $i] :
( ( ab_sem1668676832m_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_606_times_Oidem) ).
thf(4059,plain,
! [A: $i] :
( ( ab_sem1668676832m_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).
thf(216,axiom,
! [A: $i] :
( ( comm_monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_468_add__0) ).
thf(1868,plain,
! [A: $i] :
( ( comm_monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(372,axiom,
( ( ti @ ( fun @ bool @ bool ) @ fNot )
= fNot ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_fNot_res) ).
thf(2382,plain,
( ( ti @ ( fun @ bool @ bool ) @ fNot )
= fNot ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).
thf(706,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
<=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_subset__empty) ).
thf(3550,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ) )
& ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).
thf(1181,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ D )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_375_less__add__eq__less) ).
thf(5281,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ D ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ D )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1181]) ).
thf(146,axiom,
! [A: $i] :
( ( one @ A )
=> ! [B: $i] :
( ( ( one_one @ A )
= ( ti @ A @ B ) )
<=> ( ( ti @ A @ B )
= ( one_one @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_one__reorient) ).
thf(1636,plain,
! [A: $i] :
( ( one @ A )
=> ! [B: $i] :
( ( ( ( one_one @ A )
= ( ti @ A @ B ) )
=> ( ( ti @ A @ B )
= ( one_one @ A ) ) )
& ( ( ( ti @ A @ B )
= ( one_one @ A ) )
=> ( ( one_one @ A )
= ( ti @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(431,axiom,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( ord @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ D @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ E ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ H ) @ G ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ D @ H ) ) @ ( hAPP @ A @ B @ D @ G ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ D @ C ) ) @ F ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_434_xt6) ).
thf(2589,plain,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( ord @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ D @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ E ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ H ) @ G ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ D @ H ) ) @ ( hAPP @ A @ B @ D @ G ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ D @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).
thf(419,axiom,
hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ na ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ nat @ ( finite_card @ x_a ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_2) ).
thf(2541,plain,
hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ na ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ nat @ ( finite_card @ x_a ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).
thf(1093,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_600_mult__le__mono2) ).
thf(4989,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1093]) ).
thf(175,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_1_res) ).
thf(1751,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(932,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_843_of__nat__le__iff) ).
thf(4344,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).
thf(969,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_729_n__less__m__mult__n) ).
thf(4499,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).
thf(202,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( hAPP @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) ) @ ( combb @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ nat ) @ ( member @ nat ) ) @ suc ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ A ) ) ) )
= ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_543_card__less__Suc2) ).
thf(1824,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( hAPP @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) ) @ ( combb @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ nat ) @ ( member @ nat ) ) @ suc ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ A ) ) ) )
= ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(746,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( ti @ ( fun @ B @ bool ) @ D )
= ( bot_bot @ ( fun @ B @ bool ) ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ ( hAPP @ A @ ( fun @ B @ A ) @ ( combk @ A @ B ) @ C ) ) @ D )
= ( bot_bot @ ( fun @ A @ bool ) ) ) )
& ( ( ( ti @ ( fun @ B @ bool ) @ D )
!= ( bot_bot @ ( fun @ B @ bool ) ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ ( hAPP @ A @ ( fun @ B @ A ) @ ( combk @ A @ B ) @ C ) ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_image__constant__conv) ).
thf(3682,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( ti @ ( fun @ B @ bool ) @ D )
= ( bot_bot @ ( fun @ B @ bool ) ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ ( hAPP @ A @ ( fun @ B @ A ) @ ( combk @ A @ B ) @ C ) ) @ D )
= ( bot_bot @ ( fun @ A @ bool ) ) ) )
& ( ( ( ti @ ( fun @ B @ bool ) @ D )
!= ( bot_bot @ ( fun @ B @ bool ) ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ ( hAPP @ A @ ( fun @ B @ A ) @ ( combk @ A @ B ) @ C ) ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).
thf(26,axiom,
! [A: $i,B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( zero_zero @ nat ) ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_738_mult__eq__if) ).
thf(1231,plain,
! [A: $i,B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( zero_zero @ nat ) ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(1232,plain,
( ! [A: $i,B: $i] :
( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( zero_zero @ nat ) ) )
& ! [A: $i,B: $i] :
( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ) ),
inference(miniscope,[status(thm)],[1231]) ).
thf(1234,plain,
! [B: $i,A: $i] :
( ( B
!= ( zero_zero @ nat ) )
| ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( zero_zero @ nat ) ) ),
inference(cnf,[status(esa)],[1232]) ).
thf(1237,plain,
! [B: $i,A: $i] :
( ( B
!= ( zero_zero @ nat ) )
| ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( zero_zero @ nat ) ) ),
inference(lifteq,[status(thm)],[1234]) ).
thf(1238,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
inference(simp,[status(thm)],[1237]) ).
thf(37,axiom,
ordere779506340up_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__ab__semigroup__add) ).
thf(1274,plain,
ordere779506340up_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(458,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_nat__add__left__commute) ).
thf(2673,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).
thf(469,axiom,
( ( ti @ ( fun @ com @ x_a ) @ mgt )
= mgt ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_v_mgt_res) ).
thf(2705,plain,
( ( ti @ ( fun @ com @ x_a ) @ mgt )
= mgt ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).
thf(338,axiom,
( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ pls ) )
= ( one_one @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_959_nat__numeral__1__eq__1) ).
thf(2264,plain,
( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ pls ) )
= ( one_one @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).
thf(22,axiom,
comm_monoid_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ocomm__monoid__add) ).
thf(1223,plain,
comm_monoid_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(1022,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_636_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) ).
thf(4712,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).
thf(752,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_793_nat__diff__add__eq2) ).
thf(3711,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).
thf(828,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_295_Suc__mono) ).
thf(3972,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).
thf(35,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( zero_zero @ nat ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_461_minus__nat_Odiff__0) ).
thf(1255,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( zero_zero @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(953,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
<=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
| ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_420_subset__iff__psubset__eq) ).
thf(4434,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
| ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
| ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).
thf(16,axiom,
semiring_char_0 @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Nat_Osemiring__char__0) ).
thf(1217,plain,
semiring_char_0 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(1177,axiom,
( ( ti @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ fconj )
= fconj ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_fconj_res) ).
thf(5266,plain,
( ( ti @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ fconj )
= fconj ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1177]) ).
thf(765,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_872_zdiff__int) ).
thf(3752,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).
thf(834,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
<=> ( ( ti @ A @ C )
= ( ti @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_order__antisym__conv) ).
thf(3989,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ B ) ) )
& ( ( ( ti @ A @ C )
= ( ti @ A @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).
thf(874,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
<=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
| ? [C: $i,D: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_finite_Osimps) ).
thf(4128,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
| ? [C: $i,D: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) ) ) )
& ( ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
| ? [C: $i,D: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).
thf(803,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ suc @ D ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_410_inc__induct) ).
thf(3884,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ suc @ D ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).
thf(533,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_422_psubset__card__mono) ).
thf(2937,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).
thf(523,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_224_add__left__imp__eq) ).
thf(2890,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).
thf(380,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ ( image @ A @ B ) )
= ( image @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Set_Oimage_res) ).
thf(2414,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) ) @ ( image @ A @ B ) )
= ( image @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).
thf(1007,axiom,
( ( bot_bot @ ( fun @ int @ bool ) )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ ( bot_bot @ ( fun @ nat @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_886_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J) ).
thf(4661,plain,
( ( bot_bot @ ( fun @ int @ bool ) )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ ( bot_bot @ ( fun @ nat @ bool ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).
thf(413,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_773_zle__trans) ).
thf(2528,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).
thf(601,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_131_empty__Diff) ).
thf(3191,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).
thf(207,axiom,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ A ) )
= ( zero_zero @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ A ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_947_nat__number__of__Bit1) ).
thf(1837,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ A ) )
= ( zero_zero @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ A ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(875,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_527_add__strict__increasing) ).
thf(4140,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).
thf(996,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_565_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) ).
thf(4608,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).
thf(1095,axiom,
( ( hAPP @ ( fun @ x_a @ bool ) @ nat @ ( finite_card @ x_a ) @ g )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ x_a @ bool ) @ nat @ ( finite_card @ x_a ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) ) @ ( hAPP @ nat @ nat @ suc @ na ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_3) ).
thf(4994,plain,
( ( hAPP @ ( fun @ x_a @ bool ) @ nat @ ( finite_card @ x_a ) @ g )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ x_a @ bool ) @ nat @ ( finite_card @ x_a ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) ) @ ( hAPP @ nat @ nat @ suc @ na ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).
thf(518,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ B @ A @ C @ D ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_imageI) ).
thf(2872,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ B @ A @ C @ D ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).
thf(755,axiom,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_369_not__add__less2) ).
thf(3718,plain,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).
thf(315,axiom,
! [A: $i] :
( ( ( ti @ bool @ A )
= fTrue )
| ( ( ti @ bool @ A )
= fFalse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fFalse_1_1_T) ).
thf(2200,plain,
! [A: $i] :
( ( ( ti @ bool @ A )
= fTrue )
| ( ( ti @ bool @ A )
= fFalse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(889,axiom,
! [A: $i,B: $i,C: $i] :
( ( bot_bot @ ( fun @ A @ bool ) )
!= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_empty__not__insert) ).
thf(4187,plain,
! [A: $i,B: $i,C: $i] :
( ( bot_bot @ ( fun @ A @ bool ) )
!= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).
thf(321,axiom,
ordered_ring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__ring) ).
thf(2213,plain,
ordered_ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).
thf(41,axiom,
ring_char_0 @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Int_Oring__char__0) ).
thf(1281,plain,
ring_char_0 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(71,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_854_of__nat__mult) ).
thf(1365,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(442,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_order__antisym) ).
thf(2628,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).
thf(663,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_748_zmult__commute) ).
thf(3403,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).
thf(569,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_312_order__less__imp__not__less) ).
thf(3090,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).
thf(358,axiom,
( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ pls ) )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_954_numeral__1__eq__Suc__0) ).
thf(2332,plain,
( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ pls ) )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).
thf(501,axiom,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fequal @ A ) )
= ( fequal @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_fequal_res) ).
thf(2820,plain,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fequal @ A ) )
= ( fequal @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).
thf(720,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ B @ bool ) ) @ E ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ D ) ) )
=> ? [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ F ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ F ) )
& ( ( ti @ ( fun @ B @ bool ) @ E )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ F ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_finite__subset__image) ).
thf(3600,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ B @ bool ) ) @ E ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ D ) ) )
=> ? [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ F ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ F ) )
& ( ( ti @ ( fun @ B @ bool ) @ E )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).
thf(159,axiom,
ab_group_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oab__group__add) ).
thf(1687,plain,
ab_group_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(806,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fdisj @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fdisj_1_1_U) ).
thf(3894,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fdisj @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).
thf(276,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( one_one @ nat ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_diff__Suc__1) ).
thf(2068,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( one_one @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(200,axiom,
! [A: $i] :
( ( one @ A )
=> ( ( ti @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Oone__class_Oone_res) ).
thf(1819,plain,
! [A: $i] :
( ( one @ A )
=> ( ( ti @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(143,axiom,
( ( zero_zero @ int )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_863_transfer__int__nat__numerals_I1_J) ).
thf(1624,plain,
( ( zero_zero @ int )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(944,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_770_zadd__commute) ).
thf(4402,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).
thf(206,axiom,
mult_zero @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Omult__zero) ).
thf(1836,plain,
mult_zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(189,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_646_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) ).
thf(1787,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(1127,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B )
= ( ti @ ( fun @ A @ bool ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_Collect__def) ).
thf(5113,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B )
= ( ti @ ( fun @ A @ bool ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).
thf(309,axiom,
( ( ti @ ( fun @ int @ ( fun @ int @ int ) ) @ nat_tsub )
= nat_tsub ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Nat__Transfer_Otsub_res) ).
thf(2181,plain,
( ( ti @ ( fun @ int @ ( fun @ int @ int ) ) @ nat_tsub )
= nat_tsub ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(1090,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_765_le__imp__0__less) ).
thf(4979,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).
thf(9,axiom,
semiring_char_0 @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Nat_Osemiring__char__0) ).
thf(1210,plain,
semiring_char_0 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(149,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ pls ) @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_981_add__Pls) ).
thf(1656,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ pls ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(948,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_418_psubset__subset__trans) ).
thf(4417,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).
thf(556,axiom,
! [A: $i] :
( ( semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) @ C ) ) @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_591_combine__common__factor) ).
thf(3050,plain,
! [A: $i] :
( ( semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) @ C ) ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).
thf(422,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_727_less__add__iff2) ).
thf(2547,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ E ) @ B ) ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).
thf(363,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_310_less__imp__neq) ).
thf(2344,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).
thf(817,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_404_less__diff__conv) ).
thf(3931,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).
thf(171,axiom,
mult_zero @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Omult__zero) ).
thf(1740,plain,
mult_zero @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(227,axiom,
comm_semiring_1 @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ocomm__semiring__1) ).
thf(1910,plain,
comm_semiring_1 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(718,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) ).
thf(3595,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).
thf(1111,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_674_mult__pos__pos) ).
thf(5066,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1111]) ).
thf(78,axiom,
( ( zero_zero @ int )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ pls ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_967_zero__is__num__zero) ).
thf(1402,plain,
( ( zero_zero @ int )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ pls ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(303,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( one_one @ nat ) )
<=> ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_610_nat__mult__eq__1__iff) ).
thf(2157,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( one_one @ nat ) )
=> ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) )
& ( ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( one_one @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).
thf(1088,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_846_of__nat__less__imp__less) ).
thf(4974,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).
thf(510,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_112_subsetI) ).
thf(2841,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).
thf(1116,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_731_one__less__mult) ).
thf(5081,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).
thf(1190,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_639_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) ).
thf(5313,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1190]) ).
thf(873,axiom,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_504_le__iff__diff__le__0) ).
thf(4123,plain,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).
thf(1153,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B )
= A )
<=> ( C
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_268_le__imp__diff__is__add) ).
thf(5189,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B )
= A )
=> ( C
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) )
& ( ( C
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B )
= A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1153]) ).
thf(680,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_306_not__less__iff__gr__or__eq) ).
thf(3467,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).
thf(136,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ( ( ti @ ( fun @ A @ A ) @ ( abs_abs @ A ) )
= ( abs_abs @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Oabs__class_Oabs_res) ).
thf(1592,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ( ( ti @ ( fun @ A @ A ) @ ( abs_abs @ A ) )
= ( abs_abs @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(810,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ suc @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_275_diff__Suc__diff__eq1) ).
thf(3909,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ suc @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).
thf(122,axiom,
( pls
= ( zero_zero @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_976_Pls__def) ).
thf(1541,plain,
( pls
= ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(684,axiom,
! [A: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ fNot @ A ) )
| ~ ( hBOOL @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fNot_1_1_U) ).
thf(3482,plain,
! [A: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ fNot @ A ) )
| ~ ( hBOOL @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).
thf(1138,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_638_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) ).
thf(5145,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1138]) ).
thf(180,axiom,
linordered_idom @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__idom) ).
thf(1762,plain,
linordered_idom @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(620,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) )
<=> ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_794_nat__eq__add__iff2) ).
thf(3254,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) )
=> ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) )
& ( ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).
thf(184,axiom,
ordere216010020id_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__comm__monoid__add) ).
thf(1772,plain,
ordere216010020id_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(1122,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_496_add__increasing) ).
thf(5098,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1122]) ).
thf(371,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_339_linorder__antisym__conv2) ).
thf(2377,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) )
& ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).
thf(1010,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) )
<=> ? [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ E ) )
& ( ( ti @ A @ C )
= ( hAPP @ B @ A @ D @ F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_image__iff) ).
thf(4668,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) )
=> ? [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ E ) )
& ( ( ti @ A @ C )
= ( hAPP @ B @ A @ D @ F ) ) ) )
& ( ? [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ E ) )
& ( ( ti @ A @ C )
= ( hAPP @ B @ A @ D @ F ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).
thf(119,axiom,
! [A: $i] :
( A
!= ( hAPP @ nat @ nat @ suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_19_n__not__Suc__n) ).
thf(1531,plain,
! [A: $i] :
( A
!= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(1532,plain,
~ ? [A: $i] :
( A
= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(miniscope,[status(thm)],[1531]) ).
thf(1533,plain,
! [A: $i] :
( A
!= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(cnf,[status(esa)],[1532]) ).
thf(1534,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= A ),
inference(lifteq,[status(thm)],[1533]) ).
thf(241,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_934_abs__int__eq) ).
thf(1953,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(525,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
<=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_all__not__in__conv) ).
thf(2895,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ) )
& ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ! [C: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).
thf(908,axiom,
! [A: $i,B: $i] :
( ( A = B )
<=> ( ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_887_transfer__nat__int__set__relations_I3_J) ).
thf(4248,plain,
! [A: $i,B: $i] :
( ( ( A = B )
=> ( ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) )
& ( ( ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).
thf(541,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ E ) @ C ) ) ) ) )
=> ( ? [D: $i] :
! [E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ B @ E ) ) ) )
=> ( ? [D: $i] : ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ? [D: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_825_minusinfinity) ).
thf(2973,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ E ) @ C ) ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ E ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) ) ) )
=> ( ? [D: $i] :
! [E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ E ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ B @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) ) ) ) )
=> ( ? [D: $i] : ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ? [D: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).
thf(965,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_519_diff__less) ).
thf(4486,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).
thf(1046,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_43_diff__le__mono) ).
thf(4789,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).
thf(855,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_673_mult__less__cancel__left__pos) ).
thf(4048,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).
thf(980,axiom,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( bot_bot @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_354_not__less__bot) ).
thf(4544,plain,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( bot_bot @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).
thf(465,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_diff__card__le__card__Diff) ).
thf(2692,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).
thf(133,axiom,
ordered_ring_abs @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__ring__abs) ).
thf(1587,plain,
ordered_ring_abs @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(466,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_453_Collect__mono) ).
thf(2694,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).
thf(1056,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ C ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ C ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ C ) ) @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ C ) @ A ) ) ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_937_nat__number__of__add__left) ).
thf(4845,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ C ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ C ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ C ) ) @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ C ) @ A ) ) ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).
thf(476,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ? [B: $i] :
( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_553_gr0__implies__Suc) ).
thf(2726,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ? [B: $i] :
( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).
thf(443,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_492_zero__le__double__add__iff__zero__le__single__add) ).
thf(2631,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).
thf(475,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_502_add__neg__neg) ).
thf(2724,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).
thf(901,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_518_zero__less__diff) ).
thf(4227,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).
thf(417,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_718_mult__le__cancel__left__pos) ).
thf(2537,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).
thf(1055,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ti @ ( fun @ B @ bool ) @ E )
= ( ti @ ( fun @ B @ bool ) @ F ) )
=> ( ! [G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ G ) @ F ) )
=> ( ( hAPP @ B @ A @ C @ G )
= ( hAPP @ B @ A @ D @ G ) ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_445_image__cong) ).
thf(4840,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ti @ ( fun @ B @ bool ) @ E )
= ( ti @ ( fun @ B @ bool ) @ F ) )
=> ( ! [G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ G ) @ F ) )
=> ( ( hAPP @ B @ A @ C @ G )
= ( hAPP @ B @ A @ D @ G ) ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ F ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).
thf(565,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_216_card_Oinsert__remove) ).
thf(3078,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).
thf(1065,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( one_one @ int ) ) ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_783_zless__add1__eq) ).
thf(4883,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( A = B ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).
thf(735,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_381_less__imp__le__nat) ).
thf(3650,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).
thf(842,axiom,
! [A: $i] :
( ( linordered_field @ A )
=> ! [B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_570_field__le__epsilon) ).
thf(4012,plain,
! [A: $i] :
( ( linordered_field @ A )
=> ! [B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).
thf(675,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ B )
<=> ( hBOOL @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ B ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_818_imp__le__cong) ).
thf(3440,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( hBOOL @ B )
=> ( hBOOL @ A ) )
& ( ( hBOOL @ A )
=> ( hBOOL @ B ) ) ) )
=> ( ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ A ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).
thf(605,axiom,
! [A: $i] :
( ( mult_zero @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_580_mult__zero__right) ).
thf(3205,plain,
! [A: $i] :
( ( mult_zero @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).
thf(1011,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_768_zadd__assoc) ).
thf(4676,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).
thf(823,axiom,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_emptyE) ).
thf(3954,plain,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).
thf(1080,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ? [D: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_433_psubset__imp__ex__mem) ).
thf(4951,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ? [D: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1080]) ).
thf(974,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
& ( A != B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_379_nat__less__le) ).
thf(4518,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
& ( A != B ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
& ( A != B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).
thf(354,axiom,
( ( ti @ nat @ na )
= na ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_v_na_res) ).
thf(2323,plain,
( ( ti @ nat @ na )
= na ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).
thf(793,axiom,
! [A: $i,B: $i] :
( ( ab_semigroup_mult @ B )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ F ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ E ) @ F ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ F ) )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( times_times @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ D ) @ F ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_735_fold__image__insert) ).
thf(3846,plain,
! [A: $i,B: $i] :
( ( ab_semigroup_mult @ B )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ F ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ E ) @ F ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ F ) )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( times_times @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ D ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).
thf(607,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ B @ C ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) ) )
=> ( ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( collect @ int ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) ) @ ( combs @ int @ bool @ bool ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ int ) @ fconj ) @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) ) ) @ B ) )
= ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( collect @ int ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) ) @ ( combs @ int @ bool @ bool ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ int ) @ fconj ) @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) ) ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_815_transfer__nat__int__set__cong) ).
thf(3210,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ B @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ C ) ) ) ) )
=> ( ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( collect @ int ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) ) @ ( combs @ int @ bool @ bool ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ int ) @ fconj ) @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) ) ) @ B ) )
= ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( collect @ int ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) ) @ ( combs @ int @ bool @ bool ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ int ) @ fconj ) @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).
thf(571,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_535_Suc__pred) ).
thf(3094,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).
thf(753,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_933_not__neg__int) ).
thf(3714,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).
thf(280,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_649_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) ).
thf(2080,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(890,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ ( hAPP @ B @ ( fun @ A @ B ) @ ( combk @ B @ A ) @ C ) @ D )
= ( ti @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBK_1_1_U) ).
thf(4191,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ ( hAPP @ B @ ( fun @ A @ B ) @ ( combk @ B @ A ) @ C ) @ D )
= ( ti @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).
thf(6,axiom,
monoid_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Omonoid__add) ).
thf(1207,plain,
monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(139,axiom,
linordered_semiring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semiring) ).
thf(1612,plain,
linordered_semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(173,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ A @ ( fun @ B @ A ) ) @ ( combk @ A @ B ) )
= ( combk @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_COMBK_res) ).
thf(1744,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ A @ ( fun @ B @ A ) ) @ ( combk @ A @ B ) )
= ( combk @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(393,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_307_linorder__less__linear) ).
thf(2460,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).
thf(108,axiom,
! [A: $i] :
( ( monoid_add @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( plus_plus @ A ) )
= ( plus_plus @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_0_res) ).
thf(1493,plain,
! [A: $i] :
( ( monoid_add @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( plus_plus @ A ) )
= ( plus_plus @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(924,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( ti @ ( fun @ A @ bool ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_Diff__empty) ).
thf(4320,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( ti @ ( fun @ A @ bool ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).
thf(157,axiom,
comm_semiring @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ocomm__semiring) ).
thf(1683,plain,
comm_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(245,axiom,
bot @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Obot) ).
thf(1964,plain,
bot @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(150,axiom,
! [A: $i] :
~ ! [B: $i,C: $i] :
( A
!= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_896_int__diff__cases) ).
thf(1659,plain,
! [A: $i] :
~ ! [B: $i,C: $i] :
( A
!= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(46,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
<=> ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_695_mult__eq__1__iff) ).
thf(1292,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
=> ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) )
& ( ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(727,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_349_add__strict__right__mono) ).
thf(3622,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).
thf(1125,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_309_linorder__neqE) ).
thf(5107,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).
thf(505,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) @ C )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_194_Diff__idemp) ).
thf(2830,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) @ C )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).
thf(950,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_insert__absorb2) ).
thf(4421,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).
thf(89,axiom,
! [A: $i] :
( ( ( ti @ bool @ A )
= fTrue )
| ( ( ti @ bool @ A )
= fFalse ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_3_1_T) ).
thf(1424,plain,
! [A: $i] :
( ( ( ti @ bool @ A )
= fTrue )
| ( ( ti @ bool @ A )
= fFalse ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(351,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( zero_zero @ A ) )
<=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_470_right__minus__eq) ).
thf(2312,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) )
& ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).
thf(392,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_701_Suc__mult__less__cancel1) ).
thf(2455,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).
thf(1032,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_677_zero__less__mult__pos) ).
thf(4750,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).
thf(1160,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_351_add__strict__mono) ).
thf(5213,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1160]) ).
thf(439,axiom,
! [A: $i] :
( ( linord626643107strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B )
= ( one_one @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ E ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_577_convex__bound__lt) ).
thf(2613,plain,
! [A: $i] :
( ( linord626643107strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B )
= ( one_one @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ E ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).
thf(235,axiom,
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_838_int__0) ).
thf(1937,plain,
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(306,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) ) ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_630_add__number__of__left) ).
thf(2172,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(1004,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_card__Diff__subset) ).
thf(4648,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).
thf(13,axiom,
one @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oone) ).
thf(1214,plain,
one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(602,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_266_le__diff__conv2) ).
thf(3194,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).
thf(365,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_366_Suc__lessD) ).
thf(2349,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).
thf(733,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_757_zmult__zless__mono2) ).
thf(3643,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).
thf(4444,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ pls ) ),
inference(polarity_switch,[status(thm)],[4443]) ).
thf(494,axiom,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ F ) ) @ D ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ G ) @ H ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ C @ G ) ) @ ( hAPP @ A @ B @ C @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_443_order__less__le__subst2) ).
thf(2781,plain,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ F ) ) @ D ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ G ) @ H ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ C @ G ) ) @ ( hAPP @ A @ B @ C @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).
thf(976,axiom,
! [A: $i,B: $i] :
( ( bot @ B )
=> ( bot @ ( fun @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun___Orderings_Obot) ).
thf(4530,plain,
! [A: $i,B: $i] :
( ( bot @ B )
=> ( bot @ ( fun @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).
thf(86,axiom,
ordere236663937imp_le @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(1419,plain,
ordere236663937imp_le @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(134,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_972_add__numeral__0) ).
thf(1588,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(939,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ C ) ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_diff__commute) ).
thf(4384,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ C ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).
thf(461,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_diff__le__self) ).
thf(2681,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).
thf(1200,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ A )
= ( one_one @ int ) )
<=> ( ( B
= ( one_one @ int ) )
& ( A
= ( one_one @ int ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_758_pos__zmult__eq__1__iff) ).
thf(5374,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ A )
= ( one_one @ int ) )
=> ( ( B
= ( one_one @ int ) )
& ( A
= ( one_one @ int ) ) ) )
& ( ( ( B
= ( one_one @ int ) )
& ( A
= ( one_one @ int ) ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ A )
= ( one_one @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1200]) ).
thf(559,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_75_mem__def) ).
thf(3059,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).
thf(762,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_362_less__SucI) ).
thf(3743,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).
thf(1165,axiom,
! [A: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_516_card_Oempty) ).
thf(5234,plain,
! [A: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1165]) ).
thf(233,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_647_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) ).
thf(1932,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(395,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_eq__imp__le) ).
thf(2465,plain,
! [A: $i,B: $i] :
( ( A = B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).
thf(1052,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) )
<=> ( C = A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_eq__diff__iff) ).
thf(4829,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) )
=> ( C = A ) )
& ( ( C = A )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).
thf(260,axiom,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_583_divisors__zero) ).
thf(2006,plain,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(500,axiom,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] :
( ( ( ti @ A @ B )
!= ( bot_bot @ A ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( bot_bot @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_355_bot__less) ).
thf(2813,plain,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] :
( ( ( ( ti @ A @ B )
!= ( bot_bot @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( bot_bot @ A ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( bot_bot @ A ) ) @ B ) )
=> ( ( ti @ A @ B )
!= ( bot_bot @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).
thf(1081,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_insert__Diff__single) ).
thf(4953,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1081]) ).
thf(486,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_304_order__less__irrefl) ).
thf(2756,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).
thf(567,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_875_zero__less__int__conv) ).
thf(3083,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).
thf(539,axiom,
! [A: $i] :
( ( ab_semigroup_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_609_ab__semigroup__mult__class_Omult__ac_I1_J) ).
thf(2968,plain,
! [A: $i] :
( ( ab_semigroup_mult @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).
thf(1110,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ C )
!= ( ti @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_314_order__less__imp__not__eq2) ).
thf(5063,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ C )
!= ( ti @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).
thf(59,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_953_number__of__Bit1) ).
thf(1324,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(404,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fimplies ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ fNot ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_insert__Collect) ).
thf(2504,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fimplies ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ fNot ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).
thf(507,axiom,
hBOOL @ ( hAPP @ ( fun @ pname @ bool ) @ bool @ ( finite_finite @ pname ) @ u ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
thf(2836,plain,
hBOOL @ ( hAPP @ ( fun @ pname @ bool ) @ bool @ ( finite_finite @ pname ) @ u ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).
thf(96,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( ti @ ( fun @ nat @ A ) @ ( semiring_1_of_nat @ A ) )
= ( semiring_1_of_nat @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Nat_Osemiring__1__class_Oof__nat_res) ).
thf(1456,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( ti @ ( fun @ nat @ A ) @ ( semiring_1_of_nat @ A ) )
= ( semiring_1_of_nat @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(6134,plain,
! [B: $i,A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) ) @ A )
!= ( hAPP @ nat @ nat @ suc @ B ) ),
inference(paramod_ordered,[status(thm)],[1238,1243]) ).
thf(6242,plain,
! [B: $i,A: $i] :
( ( nat != nat )
| ( nat != nat )
| ( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= suc )
| ( A != B ) ),
inference(simp,[status(thm)],[6134]) ).
thf(6331,plain,
( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= suc ),
inference(simp,[status(thm)],[6242]) ).
thf(302,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_488_Suc__not__Zero) ).
thf(2153,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(513,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_789_nat__le__add__iff1) ).
thf(2851,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).
thf(331,axiom,
preorder @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Opreorder) ).
thf(2243,plain,
preorder @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).
thf(427,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( zero_zero @ int ) ) )
<=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_868_int__le__0__conv) ).
thf(2566,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( zero_zero @ int ) ) )
=> ( A
= ( zero_zero @ nat ) ) )
& ( ( A
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( zero_zero @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).
thf(1108,axiom,
( ( ti @ ( fun @ x_a @ bool ) @ g )
= g ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_v_G_res) ).
thf(5058,plain,
( ( ti @ ( fun @ x_a @ bool ) @ g )
= g ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).
thf(193,axiom,
! [A: $i] :
( ( bot @ A )
=> ( ( ti @ A @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Orderings_Obot__class_Obot_res) ).
thf(1799,plain,
! [A: $i] :
( ( bot @ A )
=> ( ( ti @ A @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(238,axiom,
( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ pls )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_974_nat__number__of__Pls) ).
thf(1947,plain,
( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ pls )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(872,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( hAPP @ C @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ ( fun @ C @ B ) ) @ ( fun @ C @ ( fun @ A @ B ) ) @ ( combc @ A @ C @ B ) @ D ) @ E ) @ F )
= ( hAPP @ C @ B @ ( hAPP @ A @ ( fun @ C @ B ) @ D @ F ) @ E ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBC_1_1_U) ).
thf(4120,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( hAPP @ C @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ ( fun @ C @ B ) ) @ ( fun @ C @ ( fun @ A @ B ) ) @ ( combc @ A @ C @ B ) @ D ) @ E ) @ F )
= ( hAPP @ C @ B @ ( hAPP @ A @ ( fun @ C @ B ) @ D @ F ) @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).
thf(218,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) ) ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_652_add__number__of__diff1) ).
thf(1874,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(577,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) )
<=> ( ( ti @ A @ C )
= ( ti @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_308_linorder__antisym__conv3) ).
thf(3115,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ B ) ) )
& ( ( ( ti @ A @ C )
= ( ti @ A @ B ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).
thf(1187,axiom,
! [A: $i] :
( ( ab_sem1668676832m_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_607_mult__idem) ).
thf(5305,plain,
! [A: $i] :
( ( ab_sem1668676832m_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1187]) ).
thf(826,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C ) ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_803_q__neg__lemma) ).
thf(3967,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C ) ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( zero_zero @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).
thf(617,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_686_mult__strict__left__mono__neg) ).
thf(3247,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).
thf(665,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_51_equalityI) ).
thf(3409,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).
thf(516,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_991_rel__simps_I17_J) ).
thf(2862,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).
thf(488,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_294_lessI) ).
thf(2766,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).
thf(467,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_348_add__less__cancel__left) ).
thf(2697,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).
thf(29,axiom,
! [A: $i] :
( ( minus @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( minus_minus @ A ) )
= ( minus_minus @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Ominus__class_Ominus_res) ).
thf(1244,plain,
! [A: $i] :
( ( minus @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( minus_minus @ A ) )
= ( minus_minus @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(1245,plain,
! [A: $i] :
( ~ ( minus @ A )
| ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( minus_minus @ A ) )
= ( minus_minus @ A ) ) ),
inference(cnf,[status(esa)],[1244]) ).
thf(1246,plain,
! [A: $i] :
( ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( minus_minus @ A ) )
= ( minus_minus @ A ) )
| ~ ( minus @ A ) ),
inference(lifteq,[status(thm)],[1245]) ).
thf(560,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_658_mult__right__mono__neg) ).
thf(3064,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).
thf(165,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( zero_zero @ A )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_467_double__zero__sym) ).
thf(1720,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( ( zero_zero @ A )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( zero_zero @ A )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(910,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_769_zadd__left__commute) ).
thf(4259,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).
thf(575,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_336_xt1_I12_J) ).
thf(3109,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).
thf(250,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_644_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) ).
thf(1979,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(1015,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ D ) @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ D ) @ E ) ) @ F ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ F ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_805_zdiv__mono2__lemma) ).
thf(4689,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ D ) @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ D ) @ E ) ) @ F ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ F ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).
thf(546,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_344_xt1_I7_J) ).
thf(3005,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).
thf(814,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_950_neg__number__of__Bit1) ).
thf(3920,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).
thf(650,axiom,
! [A: $i,B: $i] :
( ( bot @ B )
=> ! [C: $i] :
( ( hAPP @ A @ B @ ( bot_bot @ ( fun @ A @ B ) ) @ C )
= ( bot_bot @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_bot__apply) ).
thf(3364,plain,
! [A: $i,B: $i] :
( ( bot @ B )
=> ! [C: $i] :
( ( hAPP @ A @ B @ ( bot_bot @ ( fun @ A @ B ) ) @ C )
= ( bot_bot @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).
thf(935,axiom,
! [A: $i] :
( ( ordered_ring_abs @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_919_abs__eq__mult) ).
thf(4361,plain,
! [A: $i] :
( ( ordered_ring_abs @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).
thf(117,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_215_card_Oremove) ).
thf(1525,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(681,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ B ) ) ) @ ( one_one @ nat ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_211_card__Diff__insert) ).
thf(3476,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ B ) ) ) @ ( one_one @ nat ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).
thf(766,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ C ) @ D ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ C @ B ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_folding__one__idem_Oidem) ).
thf(3755,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ C ) @ D ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ C @ B ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).
thf(1189,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hAPP @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ nat @ ( finite_card @ ( sum_sum @ B @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ D ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ ( fun @ B @ bool ) @ nat @ ( finite_card @ B ) @ D ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_444_card__Plus) ).
thf(5310,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hAPP @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ nat @ ( finite_card @ ( sum_sum @ B @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ D ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ ( fun @ B @ bool ) @ nat @ ( finite_card @ B ) @ D ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1189]) ).
thf(1002,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_912_abs__triangle__ineq3) ).
thf(4644,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).
thf(468,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_folding__image__simple_Oinsert) ).
thf(2702,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).
thf(728,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_317_xt1_I9_J) ).
thf(3624,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).
thf(521,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) )
=> ~ ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_equalityE) ).
thf(2883,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) )
=> ~ ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).
thf(632,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_280_termination__basic__simps_I3_J) ).
thf(3306,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).
thf(44,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_829_number__of__int) ).
thf(1286,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ! [B: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(5395,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( ti @ A @ B ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
!= ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) ) ),
inference(paramod_ordered,[status(thm)],[1227,1203]) ).
thf(5404,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( ti @ A @ B ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( A != x_a )
| ( A
!= ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) )
| ( ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) )
!= ( insert @ x_a ) )
| ( B
!= ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) ),
inference(simp,[status(thm)],[5395]) ).
thf(5417,plain,
( ~ ( comm_semiring_1 @ x_a )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( ti @ x_a @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) )
!= x_a )
| ( ( hAPP @ x_a @ ( fun @ x_a @ x_a ) @ ( plus_plus @ x_a ) @ ( zero_zero @ x_a ) )
!= ( insert @ x_a ) ) ),
inference(simp,[status(thm)],[5404]) ).
thf(692,axiom,
! [A: $i] :
( ( linord1278240602ring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B )
= ( one_one @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ E ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) @ F ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_737_convex__bound__le) ).
thf(3508,plain,
! [A: $i] :
( ( linord1278240602ring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B )
= ( one_one @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ E ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) @ F ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).
thf(361,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_302_less__not__refl3) ).
thf(2338,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).
thf(780,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_601_mult__le__mono1) ).
thf(3807,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).
thf(1148,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_499_zero__less__double__add__iff__zero__less__single__add) ).
thf(5170,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1148]) ).
thf(589,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_432_psubset__trans) ).
thf(3147,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).
thf(678,axiom,
! [A: $i] :
( ( ring_n68954251visors @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_581_mult__eq__0__iff) ).
thf(3454,plain,
! [A: $i] :
( ( ring_n68954251visors @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).
thf(211,axiom,
ring_1 @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oring__1) ).
thf(1857,plain,
ring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(888,axiom,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ A ) ) @ ( partial_flat_lub @ A ) )
= ( partial_flat_lub @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Partial__Function_Oflat__lub_res) ).
thf(4184,plain,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ A ) ) @ ( partial_flat_lub @ A ) )
= ( partial_flat_lub @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).
thf(934,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
<=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_331_order__le__less) ).
thf(4352,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
| ( ( ti @ A @ B )
= ( ti @ A @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).
thf(316,axiom,
( ( one_one @ nat )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_958_Numeral1__eq1__nat) ).
thf(2203,plain,
( ( one_one @ nat )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).
thf(895,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the_elem @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( ti @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_187_the__elem__eq) ).
thf(4212,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the_elem @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( ti @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).
thf(715,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_insertCI) ).
thf(3587,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).
thf(1069,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_751_less__eq__number__of__int__code) ).
thf(4906,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).
thf(956,axiom,
! [A: $i] :
( ( hBOOL @ A )
| ( hBOOL @ ( hAPP @ bool @ bool @ fNot @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fNot_2_1_U) ).
thf(4448,plain,
! [A: $i] :
( ( hBOOL @ A )
| ( hBOOL @ ( hAPP @ bool @ bool @ fNot @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).
thf(498,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_card__insert) ).
thf(2793,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).
thf(151,axiom,
linordered_semidom @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__semidom) ).
thf(1663,plain,
linordered_semidom @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(283,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ D ) @ ( one_one @ nat ) ) ) ) @ ( hAPP @ nat @ int @ B @ D ) ) ) ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ B @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ nat @ int @ B @ C ) ) )
=> ? [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ C ) )
& ( ( hAPP @ nat @ int @ B @ D )
= A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_940_int__val__lemma) ).
thf(2089,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ D ) @ ( one_one @ nat ) ) ) ) @ ( hAPP @ nat @ int @ B @ D ) ) ) ) @ ( one_one @ int ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ B @ ( zero_zero @ nat ) ) ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ nat @ int @ B @ C ) ) )
=> ? [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ C ) )
& ( ( hAPP @ nat @ int @ B @ D )
= A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(1155,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_353_add__less__imp__less__left) ).
thf(5198,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1155]) ).
thf(141,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( hAPP @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) @ ( hAPP @ ( fun @ A @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ nat ) ) @ ( fun @ ( fun @ A @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) ) ) @ ( finite_fold_image @ nat @ A ) @ ( plus_plus @ nat ) ) @ ( hAPP @ nat @ ( fun @ A @ nat ) @ ( combk @ nat @ A ) @ ( one_one @ nat ) ) ) @ ( zero_zero @ nat ) ) @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_572_card__def) ).
thf(1616,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( hAPP @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) @ ( hAPP @ ( fun @ A @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ nat ) ) @ ( fun @ ( fun @ A @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) ) ) @ ( finite_fold_image @ nat @ A ) @ ( plus_plus @ nat ) ) @ ( hAPP @ nat @ ( fun @ A @ nat ) @ ( combk @ nat @ A ) @ ( one_one @ nat ) ) ) @ ( zero_zero @ nat ) ) @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(670,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_811_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) ).
thf(3422,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).
thf(293,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_935_neg__imp__number__of__eq__0) ).
thf(2121,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(39,axiom,
ordere453448008miring @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Oordered__cancel__semiring) ).
thf(1278,plain,
ordere453448008miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(477,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( hAPP @ ( fun @ A @ C ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ C @ B ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ A @ B ) ) @ ( combb @ C @ B @ A ) @ D ) @ E ) @ F )
= ( hAPP @ C @ B @ D @ ( hAPP @ A @ C @ E @ F ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_COMBB_1_1_U) ).
thf(2729,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ A @ B @ ( hAPP @ ( fun @ A @ C ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ C @ B ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ A @ B ) ) @ ( combb @ C @ B @ A ) @ D ) @ E ) @ F )
= ( hAPP @ C @ B @ D @ ( hAPP @ A @ C @ E @ F ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).
thf(258,axiom,
! [A: $i] :
( ( zero_zero @ nat )
!= ( hAPP @ nat @ nat @ suc @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_489_nat_Osimps_I2_J) ).
thf(2001,plain,
! [A: $i] :
( ( zero_zero @ nat )
!= ( hAPP @ nat @ nat @ suc @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(156,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_963_mult__numeral__1__right) ).
thf(1680,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(162,axiom,
! [A: $i] :
( ( ab_semigroup_mult @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_0_res) ).
thf(1698,plain,
! [A: $i] :
( ( ab_semigroup_mult @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(430,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ~ ! [B: $i] :
( A
!= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_893_nonneg__int__cases) ).
thf(2585,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ~ ! [B: $i] :
( A
!= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).
thf(126,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ A )
= B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_257_diff__add__inverse) ).
thf(1548,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ A )
= B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(436,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
<=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_360_not__less__less__Suc__eq) ).
thf(2602,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( A = B ) )
& ( ( A = B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).
thf(1100,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) )
<=> ? [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ B @ bool ) ) @ F ) @ E ) )
& ( ( ti @ ( fun @ A @ bool ) @ C )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_109_subset__image__iff) ).
thf(5016,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) )
=> ? [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ B @ bool ) ) @ F ) @ E ) )
& ( ( ti @ ( fun @ A @ bool ) @ C )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ F ) ) ) )
& ( ? [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ B @ bool ) ) @ F ) @ E ) )
& ( ( ti @ ( fun @ A @ bool ) @ C )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).
thf(958,axiom,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_480_gr0I) ).
thf(4453,plain,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).
thf(644,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_472_diff__self) ).
thf(3343,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).
thf(166,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ C )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_add__diff__cancel) ).
thf(1727,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ C )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(1195,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_716_mult__strict__mono) ).
thf(5327,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1195]) ).
thf(113,axiom,
! [A: $i,B: $i] :
( ( ( finite_finite_1 @ B )
& ( finite_finite_1 @ A ) )
=> ( finite_finite_1 @ ( sum_sum @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_sum___Finite__Set_Ofinite) ).
thf(1510,plain,
! [A: $i,B: $i] :
( ( ( finite_finite_1 @ B )
& ( finite_finite_1 @ A ) )
=> ( finite_finite_1 @ ( sum_sum @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(33,axiom,
ordered_ab_group_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__ab__group__add) ).
thf(1251,plain,
ordered_ab_group_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(1073,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
<=> ! [C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_Collect__empty__eq) ).
thf(4919,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ! [C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) ) )
& ( ! [C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).
thf(582,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite908156982e_idem @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_folding__image__simple__idem_Oin__idem) ).
thf(3130,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite908156982e_idem @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).
thf(962,axiom,
! [A: $i] :
( ? [B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) )
<=> ? [B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_866_transfer__int__nat__quantifiers_I2_J) ).
thf(4470,plain,
! [A: $i] :
( ( ? [B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) )
=> ? [B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) )
& ( ? [B: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
=> ? [B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
& ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).
thf(176,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ pls )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_980_diff__bin__simps_I1_J) ).
thf(1754,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ pls )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(998,axiom,
( ( ti @ ( fun @ pname @ ( option @ com ) ) @ body )
= body ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Com_Obody_res) ).
thf(4613,plain,
( ( ti @ ( fun @ pname @ ( option @ com ) ) @ body )
= body ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).
thf(341,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( one_one @ nat ) ) ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_208_diff__Suc__eq__diff__pred) ).
thf(2275,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( one_one @ nat ) ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).
thf(18,axiom,
zero @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ozero) ).
thf(1219,plain,
zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(722,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_908_abs__ge__zero) ).
thf(3607,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).
thf(688,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_666_mult__nonneg__nonneg) ).
thf(3491,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).
thf(105,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( one_one @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_273_Suc__eq__plus1) ).
thf(1483,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( one_one @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(909,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ D @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_folding__one_Osingleton) ).
thf(4256,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ D @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).
thf(267,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_585_mult__0__right) ).
thf(2028,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(47,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( zero_zero @ nat ) ) @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_479_plus__nat_Oadd__0) ).
thf(1301,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( zero_zero @ nat ) ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(653,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( C
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( one_one @ int ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_801_self__quotient__aux2) ).
thf(3375,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( C
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).
thf(137,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) )
=> ( ( hAPP @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ nat @ ( finite_card @ ( sum_sum @ B @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ D ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ ( fun @ B @ bool ) @ nat @ ( finite_card @ B ) @ D ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) )
& ( ~ ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) )
=> ( ( hAPP @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ nat @ ( finite_card @ ( sum_sum @ B @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ D ) @ C ) )
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_451_card__Plus__conv__if) ).
thf(1595,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) )
=> ( ( hAPP @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ nat @ ( finite_card @ ( sum_sum @ B @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ D ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ ( fun @ B @ bool ) @ nat @ ( finite_card @ B ) @ D ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) )
& ( ~ ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) )
=> ( ( hAPP @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ nat @ ( finite_card @ ( sum_sum @ B @ A ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ D ) @ C ) )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(804,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_20_le__antisym) ).
thf(3888,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).
thf(698,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_917_abs__le__D1) ).
thf(3523,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).
thf(479,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( hAPP @ int @ int @ bit1 @ A ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_994_rel__simps_I5_J) ).
thf(2734,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( hAPP @ int @ int @ bit1 @ A ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ pls ) @ ( hAPP @ int @ int @ bit1 @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).
thf(912,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i] :
? [C: $i] :
! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_822_minf_I6_J) ).
thf(4269,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i] :
? [C: $i] :
! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).
thf(5630,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ~ ( comm_semiring_1 @ ( fun @ x_a @ bool ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( ti @ ( fun @ x_a @ bool ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( fun @ x_a @ bool )
!= x_a )
| ( ( plus_plus @ ( fun @ x_a @ bool ) )
!= ( insert @ x_a ) )
| ( ( ti @ A @ B )
!= ( zero_zero @ ( fun @ x_a @ bool ) ) )
| ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
!= ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) ),
inference(paramod_ordered,[status(thm)],[1227,5625]) ).
thf(5637,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ~ ( comm_semiring_1 @ ( fun @ x_a @ bool ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( ti @ ( fun @ x_a @ bool ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( fun @ x_a @ bool )
!= x_a )
| ( ( plus_plus @ ( fun @ x_a @ bool ) )
!= ( insert @ x_a ) )
| ( ( ti @ A @ B )
!= ( zero_zero @ ( fun @ x_a @ bool ) ) )
| ( A != pname )
| ( A != x_a )
| ( ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) )
!= mgt_call )
| ( B != pn ) ),
inference(simp,[status(thm)],[5630]) ).
thf(5647,plain,
( ~ ( comm_semiring_1 @ pname )
| ~ ( comm_semiring_1 @ ( fun @ x_a @ bool ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( ti @ ( fun @ x_a @ bool ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( fun @ x_a @ bool )
!= x_a )
| ( ( plus_plus @ ( fun @ x_a @ bool ) )
!= ( insert @ x_a ) )
| ( ( ti @ pname @ pn )
!= ( zero_zero @ ( fun @ x_a @ bool ) ) )
| ( pname != x_a )
| ( ( hAPP @ pname @ ( fun @ pname @ pname ) @ ( plus_plus @ pname ) @ ( zero_zero @ pname ) )
!= mgt_call ) ),
inference(simp,[status(thm)],[5637]) ).
thf(1045,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_750_zmult__assoc) ).
thf(4786,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).
thf(562,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_diff__diff__cancel) ).
thf(3070,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).
thf(815,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite908156982e_idem @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_folding__image__simple__idem_Osubset__idem) ).
thf(3925,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite908156982e_idem @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).
thf(497,axiom,
! [A: $i] :
( ( ord @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( C = D )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_318_ord__eq__less__trans) ).
thf(2789,plain,
! [A: $i] :
( ( ord @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( C = D )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).
thf(566,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_663_mult__nonpos__nonneg) ).
thf(3081,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).
thf(626,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
& ! [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_539_nat__diff__split) ).
thf(3272,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
& ! [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) )
& ! [D: $i] :
( ( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).
thf(1021,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( bot_bot @ ( fun @ B @ bool ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_image__empty) ).
thf(4709,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( bot_bot @ ( fun @ B @ bool ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).
thf(809,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
& ( A != B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_777_zless__le) ).
thf(3900,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
& ( A != B ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
& ( A != B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).
thf(120,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_927_abs__one) ).
thf(1535,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(1121,axiom,
bot @ bool,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Obot) ).
thf(5097,plain,
bot @ bool,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).
thf(472,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_271_diff__add__assoc2) ).
thf(2716,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).
thf(246,axiom,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) )
<=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D )
= ( ti @ A @ F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_691_eq__add__iff1) ).
thf(1965,plain,
! [A: $i] :
( ( ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D )
= ( ti @ A @ F ) ) )
& ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D )
= ( ti @ A @ F ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(1079,axiom,
! [A: $i,B: $i] :
( ( comm_monoid_mult @ B )
=> ! [C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ! [E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ E ) @ D ) )
=> ( ( hAPP @ A @ B @ C @ E )
= ( one_one @ B ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ ( one_one @ B ) ) @ D )
= ( one_one @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_741_fold__image__1) ).
thf(4946,plain,
! [A: $i,B: $i] :
( ( comm_monoid_mult @ B )
=> ! [C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ! [E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ E ) @ D ) )
=> ( ( hAPP @ A @ B @ C @ E )
= ( one_one @ B ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ ( one_one @ B ) ) @ D )
= ( one_one @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1079]) ).
thf(504,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) ) )
| ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ C ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_finite__Collect__conjI) ).
thf(2827,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) ) )
| ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ C ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).
thf(657,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool @ ( finite_finite @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( collect @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( combc @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_finite__Collect__subsets) ).
thf(3387,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ bool ) @ bool @ ( finite_finite @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( collect @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( combc @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).
thf(1097,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_732_one__le__mult__iff) ).
thf(4998,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).
thf(761,axiom,
( ( ti @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ fimplies )
= fimplies ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_fimplies_res) ).
thf(3741,plain,
( ( ti @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ fimplies )
= fimplies ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).
thf(705,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ! [E: $i,F: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ C @ E ) @ F ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ F ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ D @ B ) ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_213_folding__one_Oclosed) ).
thf(3547,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ! [E: $i,F: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ C @ E ) @ F ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ F ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ D @ B ) ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).
thf(205,axiom,
! [A: $i] :
( ( ab_semigroup_add @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( plus_plus @ A ) )
= ( plus_plus @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_2_res) ).
thf(1833,plain,
! [A: $i] :
( ( ab_semigroup_add @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( plus_plus @ A ) )
= ( plus_plus @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(188,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) ) )
!= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_542_card__less) ).
thf(1784,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) ) )
!= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(75,axiom,
ord @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Oord) ).
thf(1389,plain,
ord @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(412,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_167_singleton__conv2) ).
thf(2525,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).
thf(723,axiom,
! [A: $i] :
( ( hBOOL @ ( ti @ bool @ A ) )
<=> ( hBOOL @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_hBOOL_arg1) ).
thf(3609,plain,
! [A: $i] :
( ( ( hBOOL @ ( ti @ bool @ A ) )
=> ( hBOOL @ A ) )
& ( ( hBOOL @ A )
=> ( hBOOL @ ( ti @ bool @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).
thf(452,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ B ) @ A )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_880_tsub__eq) ).
thf(2657,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ B ) @ A )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).
thf(512,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_card__insert__le) ).
thf(2849,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).
thf(1130,axiom,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_239_add__le__imp__le__left) ).
thf(5121,plain,
! [A: $i] :
( ( ordere236663937imp_le @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).
thf(400,axiom,
! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_insertI1) ).
thf(2489,plain,
! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).
thf(1074,axiom,
! [A: $i,B: $i] :
( ( minus @ B )
=> ( minus @ ( fun @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun___Groups_Ominus) ).
thf(4926,plain,
! [A: $i,B: $i] :
( ( minus @ B )
=> ( minus @ ( fun @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).
thf(408,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ B ) )
=> ( ! [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ suc @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_558_zero__induct) ).
thf(2514,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ B ) )
=> ( ! [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( hAPP @ nat @ nat @ suc @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ A @ ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).
thf(163,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) )
<=> ( ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( zero_zero @ nat ) ) )
| ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_513_one__is__add) ).
thf(1701,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) )
=> ( ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( zero_zero @ nat ) ) )
| ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( zero_zero @ nat ) ) )
| ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(927,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_642_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) ).
thf(4328,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).
thf(800,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_671_mult__less__cancel__right__disj) ).
thf(3868,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).
thf(1040,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ~ ( hBOOL @ A )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fconj_1_1_U) ).
thf(4769,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ~ ( hBOOL @ A )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).
thf(609,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_270_add__diff__assoc2) ).
thf(3223,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).
thf(978,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_327_linorder__not__le) ).
thf(4537,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).
thf(552,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ C ) ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( zero_zero @ int ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_879_zdiff__int__split) ).
thf(3023,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ C ) ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( zero_zero @ int ) ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ C ) ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( zero_zero @ int ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).
thf(865,axiom,
! [A: $i,B: $i] :
( ( ( finite_finite_1 @ B )
& ( finite_finite_1 @ A ) )
=> ( finite_finite_1 @ ( fun @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun___Finite__Set_Ofinite) ).
thf(4096,plain,
! [A: $i,B: $i] :
( ( ( finite_finite_1 @ B )
& ( finite_finite_1 @ A ) )
=> ( finite_finite_1 @ ( fun @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).
thf(758,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ C ) ) ) @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ C ) ) ) @ B ) )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_Collect__conv__if2) ).
thf(3726,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ C ) ) ) @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ C ) ) ) @ B ) )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).
thf(545,axiom,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( bot_bot @ A ) ) )
=> ( ( ti @ A @ B )
= ( bot_bot @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_161_le__bot) ).
thf(3002,plain,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( bot_bot @ A ) ) )
=> ( ( ti @ A @ B )
= ( bot_bot @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).
thf(751,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ E ) @ C ) ) ) ) )
=> ( ? [D: $i] :
! [E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ B @ E ) ) ) )
=> ( ? [D: $i] : ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ? [D: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_826_plusinfinity) ).
thf(3700,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i,E: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ E ) @ C ) ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ E ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) ) ) )
=> ( ? [D: $i] :
! [E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ E ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ B @ E ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ E ) ) ) ) )
=> ( ? [D: $i] : ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ? [D: $i] : ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).
thf(425,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_680_mult__neg__pos) ).
thf(2557,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).
thf(383,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_171_finite__Diff__insert) ).
thf(2425,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).
thf(1105,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_772_zle__linear) ).
thf(5033,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).
thf(325,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_272_Suc__eq__plus1__left) ).
thf(2220,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).
thf(1129,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_335_order__neq__le__trans) ).
thf(5118,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).
thf(347,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ A ) ) ) @ ( one_one @ int ) ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_939_incr__lemma) ).
thf(2293,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ A ) ) ) @ ( one_one @ int ) ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).
thf(53,axiom,
cancel146912293up_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) ).
thf(1312,plain,
cancel146912293up_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(869,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ B ) ) @ ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_764_less__bin__lemma) ).
thf(4112,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ B ) ) @ ( zero_zero @ int ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ B ) ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).
thf(606,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_373_add__less__mono1) ).
thf(3208,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).
thf(215,axiom,
! [A: $i] :
( ( semiring @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_2_res) ).
thf(1865,plain,
! [A: $i] :
( ( semiring @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(988,axiom,
! [A: $i] :
( ( mult_zero @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_579_mult__zero__left) ).
thf(4573,plain,
! [A: $i] :
( ( mult_zero @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( zero_zero @ A ) ) @ B )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).
thf(27,axiom,
linord893533164strict @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__comm__semiring__strict) ).
thf(1239,plain,
linord893533164strict @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(1139,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_621_Ints__mult) ).
thf(5148,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1139]) ).
thf(579,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_333_not__leE) ).
thf(3123,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).
thf(388,axiom,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( bot_bot @ A ) ) )
<=> ( ( ti @ A @ B )
= ( bot_bot @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_bot__unique) ).
thf(2442,plain,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( bot_bot @ A ) ) )
=> ( ( ti @ A @ B )
= ( bot_bot @ A ) ) )
& ( ( ( ti @ A @ B )
= ( bot_bot @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( bot_bot @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).
thf(487,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ A ) ) @ ( one_one @ int ) ) )
<=> ( A
= ( zero_zero @ int ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_903_zabs__less__one__iff) ).
thf(2759,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ A ) ) @ ( one_one @ int ) ) )
=> ( A
= ( zero_zero @ int ) ) )
& ( ( A
= ( zero_zero @ int ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ A ) ) @ ( one_one @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).
thf(686,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_675_mult__pos__neg) ).
thf(3487,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).
thf(273,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_930_abs__zero) ).
thf(2055,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(304,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_702_mult__Suc__right) ).
thf(2166,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(713,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_190_DiffE) ).
thf(3579,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
=> ~ ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).
thf(914,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_345_order__le__less__trans) ).
thf(4273,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).
thf(36,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
<=> ( ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( zero_zero @ nat ) ) )
| ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_514_add__is__1) ).
thf(1258,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
=> ( ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( zero_zero @ nat ) ) )
| ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ) )
& ( ( ( ( A
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
& ( B
= ( zero_zero @ nat ) ) )
| ( ( A
= ( zero_zero @ nat ) )
& ( B
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(647,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_503_pos__add__strict) ).
thf(3355,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).
thf(23,axiom,
zero_neq_one @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Ozero__neq__one) ).
thf(1224,plain,
zero_neq_one @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(367,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_494_add__nonneg__nonneg) ).
thf(2358,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).
thf(352,axiom,
linordered_semidom @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semidom) ).
thf(2319,plain,
linordered_semidom @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).
thf(68,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( one_one @ A ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_965_add__special_I3_J) ).
thf(1356,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( one_one @ A ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(428,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ pls ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_993_rel__simps_I29_J) ).
thf(2573,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).
thf(300,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) @ A )
!= ( zero_zero @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_759_odd__nonzero) ).
thf(2148,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) @ A )
!= ( zero_zero @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(406,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_card__Diff1__le) ).
thf(2510,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).
thf(745,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) )
=> ~ ! [F: $i] :
( ( ( ti @ A @ C )
= ( hAPP @ B @ A @ D @ F ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_imageE) ).
thf(3678,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) )
=> ~ ! [F: $i] :
( ( ( ti @ A @ C )
= ( hAPP @ B @ A @ D @ F ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).
thf(410,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ E ) ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ D ) ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ E ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_902_abs__diff__triangle__ineq) ).
thf(2521,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ E ) ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ D ) ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).
thf(460,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( zero_zero @ int ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_928_not__neg__0) ).
thf(2679,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( zero_zero @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).
thf(42,axiom,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
=> ? [B: $i] :
( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_559_not0__implies__Suc) ).
thf(1282,plain,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
=> ? [B: $i] :
( A
= ( hAPP @ nat @ nat @ suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(920,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) )
<=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_511_less__Suc0) ).
thf(4305,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) )
=> ( A
= ( zero_zero @ nat ) ) )
& ( ( A
= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).
thf(773,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_676_mult__pos__neg2) ).
thf(3779,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).
thf(344,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_970_number__of__Pls) ).
thf(2284,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).
thf(536,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_530_add__nonpos__neg) ).
thf(2955,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).
thf(603,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) )
<=> ( B = C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_nat__add__left__cancel) ).
thf(3197,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) )
=> ( B = C ) )
& ( ( B = C )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).
thf(918,axiom,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) ) @ ( finite2073411215e_idem @ A ) )
= ( finite2073411215e_idem @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofolding__one__idem_res) ).
thf(4300,plain,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) ) @ ( finite2073411215e_idem @ A ) )
= ( finite2073411215e_idem @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).
thf(561,axiom,
! [A: $i] :
( ( ord @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( D = B )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_320_ord__less__eq__trans) ).
thf(3066,plain,
! [A: $i] :
( ( ord @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( D = B )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).
thf(656,axiom,
! [A: $i] :
( ( ord @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( ord_less @ A ) )
= ( ord_less @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Orderings_Oord__class_Oless_res) ).
thf(3384,plain,
! [A: $i] :
( ( ord @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( ord_less @ A ) )
= ( ord_less @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).
thf(129,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( ring_char_0 @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ B )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) )
<=> ( B = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_619_eq__number__of) ).
thf(1571,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( ring_char_0 @ A ) )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ B )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) )
=> ( B = C ) )
& ( ( B = C )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ B )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(275,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( zero_zero @ int ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_760_zadd__0__right) ).
thf(2065,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( zero_zero @ int ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(8,axiom,
no_zero_divisors @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ono__zero__divisors) ).
thf(1209,plain,
no_zero_divisors @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(474,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) )
=> ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_insertE) ).
thf(2721,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) )
=> ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).
thf(1113,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_396_less__eq__Suc__le) ).
thf(5071,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).
thf(836,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_326_linorder__not__less) ).
thf(3996,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).
thf(730,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_361_less__antisym) ).
thf(3631,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( B = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).
thf(81,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( hAPP @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) ) @ ( combb @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ nat ) @ ( member @ nat ) ) @ suc ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ A ) ) ) ) )
= ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_541_card__less__Suc) ).
thf(1408,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( hAPP @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) ) @ ( combb @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ nat ) @ ( member @ nat ) ) @ suc ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ A ) ) ) ) )
= ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ ( fun @ nat @ bool ) @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ ( fun @ nat @ bool ) @ bool ) @ ( member @ nat ) ) @ B ) ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(591,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_trans__le__add1) ).
thf(3155,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).
thf(328,axiom,
! [A: $i] :
( ( comm_monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_465_add_Ocomm__neutral) ).
thf(2229,plain,
! [A: $i] :
( ( comm_monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).
thf(322,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ B ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ fNot ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_196_set__diff__eq) ).
thf(2214,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ B ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ bool @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combb @ bool @ bool @ A ) @ fNot ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).
thf(1169,axiom,
( ( ti @ ( fun @ pname @ bool ) @ u )
= u ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_v_U_res) ).
thf(5244,plain,
( ( ti @ ( fun @ pname @ bool ) @ u )
= u ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1169]) ).
thf(1124,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( B = C )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_order__eq__refl) ).
thf(5103,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( B = C )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1124]) ).
thf(402,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_719_sum__squares__le__zero__iff) ).
thf(2493,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).
thf(824,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_equalityD2) ).
thf(3957,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).
thf(695,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_918_abs__ge__self) ).
thf(3515,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).
thf(1071,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_332_leI) ).
thf(4914,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).
thf(625,axiom,
~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( member @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_5) ).
thf(3270,plain,
~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( member @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).
thf(456,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_order__refl) ).
thf(2669,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).
thf(1162,axiom,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_670_not__square__less__zero) ).
thf(5222,plain,
! [A: $i] :
( ( linordered_ring @ A )
=> ! [B: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1162]) ).
thf(291,axiom,
( ( ti @ ( fun @ int @ int ) @ succ )
= succ ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Int_Osucc_res) ).
thf(2116,plain,
( ( ti @ ( fun @ int @ int ) @ succ )
= succ ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(509,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29_rev__finite__subset) ).
thf(2839,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).
thf(885,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ nat @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) @ ( semiri532925092at_aux @ A ) @ B ) @ ( hAPP @ nat @ nat @ suc @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ nat @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) @ ( semiri532925092at_aux @ A ) @ B ) @ C ) @ ( hAPP @ A @ A @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_892_of__nat__aux_Osimps_I2_J) ).
thf(4176,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ nat @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) @ ( semiri532925092at_aux @ A ) @ B ) @ ( hAPP @ nat @ nat @ suc @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ nat @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) @ ( semiri532925092at_aux @ A ) @ B ) @ C ) @ ( hAPP @ A @ A @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).
thf(585,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ A ) @ B ) )
| ~ ( hBOOL @ A )
| ( hBOOL @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fimplies_3_1_U) ).
thf(3138,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ A ) @ B ) )
| ~ ( hBOOL @ A )
| ( hBOOL @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).
thf(314,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ C )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_diff__add__cancel) ).
thf(2197,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ C )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).
thf(963,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ D ) @ E ) @ F ) @ G ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ G @ C )
= ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ D ) @ F ) @ E ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_574_folding__image__simple_Oeq__fold__g) ).
thf(4477,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ D ) @ E ) @ F ) @ G ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ G @ C )
= ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ D ) @ F ) @ E ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).
thf(593,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_383_less__or__eq__imp__le) ).
thf(3160,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).
thf(51,axiom,
( ( ti @ bool @ fFalse )
= fFalse ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_fFalse_res) ).
thf(1307,plain,
( ( ti @ bool @ fFalse )
= fFalse ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(153,axiom,
! [A: $i] :
( ( zero_neq_one @ A )
=> ( ( one_one @ A )
!= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_463_one__neq__zero) ).
thf(1667,plain,
! [A: $i] :
( ( zero_neq_one @ A )
=> ( ( one_one @ A )
!= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(266,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) ).
thf(2025,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(481,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ C ) )
=> ( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ E ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_finite__empty__induct) ).
thf(2742,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ C ) )
=> ( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ E ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).
thf(1149,axiom,
! [A: $i,B: $i,C: $i] :
( ? [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) )
& ! [E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ C @ E ) )
=> ( ( ti @ A @ E )
= ( ti @ A @ D ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ C )
= ( ti @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_290_the1__equality) ).
thf(5175,plain,
! [A: $i,B: $i,C: $i] :
( ? [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) )
& ! [E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ C @ E ) )
=> ( ( ti @ A @ E )
= ( ti @ A @ D ) ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ C @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ C )
= ( ti @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1149]) ).
thf(724,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ A ) @ B ) )
| ( hBOOL @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fconj_2_1_U) ).
thf(3614,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ A ) @ B ) )
| ( hBOOL @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).
thf(917,axiom,
! [A: $i,B: $i] :
( ? [C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) )
=> ( ( ti @ A @ D )
= ( ti @ A @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_289_theI_H) ).
thf(4296,plain,
! [A: $i,B: $i] :
( ? [C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) )
=> ( ( ti @ A @ D )
= ( ti @ A @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).
thf(1025,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_367_Suc__less__SucD) ).
thf(4726,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).
thf(342,axiom,
! [A: $i] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_464_zero__neq__one) ).
thf(2278,plain,
! [A: $i] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).
thf(1013,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_788_nat__mult__le__cancel1) ).
thf(4684,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).
thf(1017,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_723_mult__left__le__one__le) ).
thf(4694,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).
thf(893,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_531_zero__less__two) ).
thf(4203,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).
thf(764,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ ( one_one @ int ) ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_775_zle__diff1__eq) ).
thf(3747,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ ( one_one @ int ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ ( one_one @ int ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).
thf(1106,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
<=> ( ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
& ( ( ti @ A @ C )
= ( ti @ A @ E ) ) )
| ( ( ( ti @ A @ B )
= ( ti @ A @ E ) )
& ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153_doubleton__eq__iff) ).
thf(5035,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
& ( ( ti @ A @ C )
= ( ti @ A @ E ) ) )
| ( ( ( ti @ A @ B )
= ( ti @ A @ E ) )
& ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ) )
& ( ( ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
& ( ( ti @ A @ C )
= ( ti @ A @ E ) ) )
| ( ( ( ti @ A @ B )
= ( ti @ A @ E ) )
& ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1106]) ).
thf(219,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_618_arith__simps_I32_J) ).
thf(1877,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(62,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_628_add__number__of__eq) ).
thf(1339,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(955,axiom,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ A ) @ ( the @ A ) )
= ( the @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_HOL_OThe_res) ).
thf(4445,plain,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ A ) @ ( the @ A ) )
= ( the @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).
thf(583,axiom,
! [A: $i] :
( ( dense_linorder @ A )
=> ! [B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_438_dense__le) ).
thf(3133,plain,
! [A: $i] :
( ( dense_linorder @ A )
=> ! [B: $i,C: $i] :
( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).
thf(868,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) )
<=> ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
| ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_insert__iff) ).
thf(4103,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) )
=> ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
| ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
| ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).
thf(362,axiom,
~ ( hBOOL @ fFalse ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fFalse_1_1_U) ).
thf(2342,plain,
~ ( hBOOL @ fFalse ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).
thf(2343,plain,
~ ( hBOOL @ fFalse ),
inference(polarity_switch,[status(thm)],[2342]) ).
thf(740,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_le__add2) ).
thf(3668,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).
thf(1070,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_193_DiffD1) ).
thf(4911,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).
thf(289,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ A ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_390_finite__M__bounded__by__nat) ).
thf(2111,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( combs @ nat @ bool @ bool ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ nat ) @ fconj ) @ A ) ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(563,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_254_add__leD1) ).
thf(3073,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).
thf(337,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ ( one_one @ A ) ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_693_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) ).
thf(2261,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ ( one_one @ A ) ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).
thf(168,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ C ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_753_zdiff__zmult__distrib2) ).
thf(1731,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ C ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(778,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( finite_finite @ int ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_861_transfer__nat__int__set__relations_I1_J) ).
thf(3800,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( finite_finite @ int ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( finite_finite @ int ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).
thf(712,axiom,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ B ) @ A )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ A ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ B ) @ A )
= ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_827_tsub__def) ).
thf(3572,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ B ) @ A )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ B ) @ A ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ A ) @ B ) )
=> ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ B ) @ A )
= ( zero_zero @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).
thf(1183,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ A ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_763_int__one__le__iff__zero__less) ).
thf(5286,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1183]) ).
thf(594,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ D ) ) @ E ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_797_unique__quotient__lemma__neg) ).
thf(3165,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ D ) ) @ E ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).
thf(214,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_460_diff__self__eq__0) ).
thf(1862,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(887,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_341_order__le__neq__trans) ).
thf(4181,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).
thf(6183,plain,
! [A: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( zero_zero @ nat ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) ) @ A )
!= ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ) ) ),
inference(paramod_ordered,[status(thm)],[1238,1203]) ).
thf(6230,plain,
! [A: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( zero_zero @ nat ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( fun @ x_a @ bool )
!= nat )
| ( ( fun @ ( fun @ x_a @ bool ) @ bool )
!= nat )
| ( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= ( ord_less_eq @ ( fun @ x_a @ bool ) ) )
| ( A
!= ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) @ g ) ) ),
inference(simp,[status(thm)],[6183]) ).
thf(6319,plain,
( ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( zero_zero @ nat ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( fun @ x_a @ bool )
!= nat )
| ( ( fun @ ( fun @ x_a @ bool ) @ bool )
!= nat )
| ( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= ( ord_less_eq @ ( fun @ x_a @ bool ) ) ) ),
inference(simp,[status(thm)],[6230]) ).
thf(630,axiom,
! [A: $i] :
( ( ti @ ( fun @ A @ A ) @ ( combi @ A ) )
= ( combi @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_COMBI_res) ).
thf(3298,plain,
! [A: $i] :
( ( ti @ ( fun @ A @ A ) @ ( combi @ A ) )
= ( combi @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).
thf(519,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ C )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( hAPP @ bool @ ( fun @ nat @ ( fun @ nat @ nat ) ) @ ( if @ nat ) @ C ) @ A ) @ B ) ) ) )
& ( ~ ( hBOOL @ C )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( hAPP @ bool @ ( fun @ nat @ ( fun @ nat @ nat ) ) @ ( if @ nat ) @ C ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_889_int__if__cong) ).
thf(2874,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ C )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( hAPP @ bool @ ( fun @ nat @ ( fun @ nat @ nat ) ) @ ( if @ nat ) @ C ) @ A ) @ B ) ) ) )
& ( ~ ( hBOOL @ C )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( hAPP @ bool @ ( fun @ nat @ ( fun @ nat @ nat ) ) @ ( if @ nat ) @ C ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).
thf(420,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ bool @ ( finite_comp_fun_idem @ A @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ ( combb @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ A ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( combc @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_282_comp__fun__idem__remove) ).
thf(2542,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ bool @ ( finite_comp_fun_idem @ A @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) ) @ ( combb @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ A ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( combc @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).
thf(1150,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C )
!= ( bot_bot @ ( fun @ A @ bool ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_insert__not__empty) ).
thf(5180,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C )
!= ( bot_bot @ ( fun @ A @ bool ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1150]) ).
thf(429,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
<=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_395_less__iff__Suc__add) ).
thf(2578,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) )
& ( ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).
thf(160,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ B )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_256_diff__add__inverse2) ).
thf(1688,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ B )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(490,axiom,
( ( ti @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) ) @ p )
= p ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_v_P_res) ).
thf(2770,plain,
( ( ti @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) ) @ p )
= p ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).
thf(1173,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ pls ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_951_not__neg__number__of__Pls) ).
thf(5254,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ pls ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1173]) ).
thf(902,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_linorder__le__cases) ).
thf(4232,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).
thf(118,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( number_number_of @ int ) @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_798_number__of__is__id) ).
thf(1528,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( number_number_of @ int ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(282,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ suc @ A ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_956_Suc3__eq__add__3) ).
thf(2086,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ suc @ A ) ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(147,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_593_mult__1__right) ).
thf(1643,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(813,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_564_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) ).
thf(3917,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).
thf(1171,axiom,
! [A: $i] :
( ( dense_linorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ! [E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ E ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_439_dense__le__bounded) ).
thf(5248,plain,
! [A: $i] :
( ( dense_linorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ! [E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ E ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ E ) @ B ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1171]) ).
thf(743,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_105_subset__insertI2) ).
thf(3674,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).
thf(1051,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ D ) ) @ E ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_804_unique__quotient__lemma) ).
thf(4827,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ D ) ) @ E ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ E ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).
thf(614,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_923_abs__minus__commute) ).
thf(3241,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).
thf(634,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ C )
= ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_920_abs__mult__pos) ).
thf(3310,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ C )
= ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).
thf(1136,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) @ ( hAPP @ A @ ( fun @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) ) @ ( nat_case @ A ) @ B ) @ C ) @ ( hAPP @ nat @ nat @ suc @ D ) )
= ( hAPP @ nat @ A @ C @ D ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_nat__case__Suc) ).
thf(5139,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ A @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) @ ( hAPP @ A @ ( fun @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) ) @ ( nat_case @ A ) @ B ) @ C ) @ ( hAPP @ nat @ nat @ suc @ D ) )
= ( hAPP @ nat @ A @ C @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1136]) ).
thf(517,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_844_of__nat__less__iff) ).
thf(2867,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).
thf(12,axiom,
number_semiring @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Int_Onumber__semiring) ).
thf(1213,plain,
number_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(767,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_563_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) ).
thf(3759,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).
thf(789,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_order__trans) ).
thf(3838,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).
thf(271,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_964_mult__numeral__1) ).
thf(2051,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(1120,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_562_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) ).
thf(5094,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1120]) ).
thf(332,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_849_zadd__int) ).
thf(2244,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).
thf(484,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_506_not__one__le__zero) ).
thf(2752,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).
thf(186,axiom,
( ( ti @ int @ pls )
= pls ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Int_OPls_res) ).
thf(1775,plain,
( ( ti @ int @ pls )
= pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(432,axiom,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( hAPP @ B @ A @ D @ E ) ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ H ) @ G ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ B @ A @ D @ H ) ) @ ( hAPP @ B @ A @ D @ G ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( hAPP @ B @ A @ D @ F ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_440_xt5) ).
thf(2592,plain,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( hAPP @ B @ A @ D @ E ) ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less @ B ) @ H ) @ G ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ B @ A @ D @ H ) ) @ ( hAPP @ B @ A @ D @ G ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( hAPP @ B @ A @ D @ F ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).
thf(640,axiom,
! [A: $i] :
( ( ti @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ ( if @ A ) )
= ( if @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_If_res) ).
thf(3325,plain,
! [A: $i] :
( ( ti @ ( fun @ bool @ ( fun @ A @ ( fun @ A @ A ) ) ) @ ( if @ A ) )
= ( if @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).
thf(587,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_717_mult__le__cancel__left__neg) ).
thf(3142,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( zero_zero @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).
thf(1019,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ C )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_212_folding__one_Oinsert) ).
thf(4703,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite_folding_one @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ C )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).
thf(416,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_689_less__1__mult) ).
thf(2535,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).
thf(1363,plain,
( ( zero_zero @ int )
!= ( one_one @ int ) ),
inference(polarity_switch,[status(thm)],[1362]) ).
thf(1364,plain,
( ( one_one @ int )
!= ( zero_zero @ int ) ),
inference(lifteq,[status(thm)],[1363]) ).
thf(32,axiom,
linord1278240602ring_1 @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Olinordered__semiring__1) ).
thf(1250,plain,
linord1278240602ring_1 @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(109,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A )
= ( zero_zero @ nat ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_459_diffs0__imp__equal) ).
thf(1496,plain,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A )
= ( zero_zero @ nat ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(892,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_insert__Diff__if) ).
thf(4196,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).
thf(542,axiom,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_622_le__number__of) ).
thf(2984,plain,
! [A: $i] :
( ( ( number_ring @ A )
& ( linordered_idom @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).
thf(248,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_859_Nat__Transfer_Otransfer__int__nat__functions_I1_J) ).
thf(1973,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(880,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_807_int__le__induct) ).
thf(4158,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ A @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( one_one @ int ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).
thf(195,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ suc @ B ) )
=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_Suc__inject) ).
thf(1805,plain,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ suc @ B ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(259,axiom,
cancel146912293up_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) ).
thf(2005,plain,
cancel146912293up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(272,axiom,
ordered_semiring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__semiring) ).
thf(2054,plain,
ordered_semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(945,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_83_insert__absorb) ).
thf(4405,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).
thf(284,axiom,
! [A: $i] :
( ( number @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ B )
= ( ti @ A @ C ) )
<=> ( ( ti @ A @ C )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_617_number__of__reorient) ).
thf(2096,plain,
! [A: $i] :
( ( number @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ B )
= ( ti @ A @ C ) )
=> ( ( ti @ A @ C )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) )
& ( ( ( ti @ A @ C )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ B )
= ( ti @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(48,axiom,
order @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Oorder) ).
thf(1304,plain,
order @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(454,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_878_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) ).
thf(2665,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).
thf(294,axiom,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_3_res) ).
thf(2124,plain,
! [A: $i] :
( ( no_zero_divisors @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(1063,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_536_diff__Suc__less) ).
thf(4879,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).
thf(389,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_trans__le__add2) ).
thf(2449,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).
thf(1135,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fdisj ) @ B ) ) @ C ) ) ) )
<=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_finite__Collect__disjI) ).
thf(5133,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fdisj ) @ B ) ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fdisj ) @ B ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1135]) ).
thf(794,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ti @ A @ C )
= ( ti @ A @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_xt1_I3_J) ).
thf(3849,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ti @ A @ C )
= ( ti @ A @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).
thf(144,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_616_number__of__mult) ).
thf(1626,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(928,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_217_folding__one__idem_Osubset__idem) ).
thf(4331,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) )
= ( hAPP @ ( fun @ A @ bool ) @ A @ E @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).
thf(548,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) )
<=> ( ( ti @ ( fun @ A @ bool ) @ D )
= ( ti @ ( fun @ A @ bool ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_insert__ident) ).
thf(3009,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) )
=> ( ( ti @ ( fun @ A @ bool ) @ D )
= ( ti @ ( fun @ A @ bool ) @ B ) ) )
& ( ( ( ti @ ( fun @ A @ bool ) @ D )
= ( ti @ ( fun @ A @ bool ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).
thf(818,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_391_add__less__le__mono) ).
thf(3936,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).
thf(178,axiom,
! [A: $i,B: $i] :
( ( A
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) )
=> ( ( B
= ( one_one @ nat ) )
| ( A
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_705_mult__eq__self__implies__10) ).
thf(1758,plain,
! [A: $i,B: $i] :
( ( A
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) )
=> ( ( B
= ( one_one @ nat ) )
| ( A
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(287,axiom,
( ( ti @ ( fun @ nat @ nat ) @ suc )
= suc ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Nat_OSuc_res) ).
thf(2107,plain,
( ( ti @ ( fun @ nat @ nat ) @ suc )
= suc ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(7,axiom,
monoid_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Omonoid__add) ).
thf(1208,plain,
monoid_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(495,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_399_le__imp__less__Suc) ).
thf(2784,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).
thf(959,axiom,
! [A: $i] :
( ( ab_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_221_ab__semigroup__add__class_Oadd__ac_I1_J) ).
thf(4456,plain,
! [A: $i] :
( ( ab_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).
thf(450,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_323_xt1_I10_J) ).
thf(2652,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).
thf(628,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_790_nat__diff__add__eq1) ).
thf(3293,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).
thf(742,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_337_leD) ).
thf(3672,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).
thf(87,axiom,
minus @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ominus) ).
thf(1420,plain,
minus @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(832,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ti @ A @ ( hAPP @ B @ A @ C @ D ) )
= ( hAPP @ B @ A @ C @ D ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_hAPP_res) ).
thf(3981,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ti @ A @ ( hAPP @ B @ A @ C @ D ) )
= ( hAPP @ B @ A @ C @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).
thf(158,axiom,
! [A: $i] :
( ( ( number @ A )
& ( ring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_624_left__diff__distrib__number__of) ).
thf(1684,plain,
! [A: $i] :
( ( ( number @ A )
& ( ring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(555,axiom,
! [A: $i,B: $i] :
( ( ord @ B )
=> ( ord @ ( fun @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun___Orderings_Oord) ).
thf(3048,plain,
! [A: $i,B: $i] :
( ( ord @ B )
=> ( ord @ ( fun @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).
thf(986,axiom,
! [A: $i,B: $i] :
( ( ab_semigroup_mult @ B )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ F ) )
=> ( ! [G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ G ) @ F ) )
=> ( ( hAPP @ A @ B @ D @ G )
= ( hAPP @ A @ B @ E @ G ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ D ) @ C ) @ F )
= ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ E ) @ C ) @ F ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_808_fold__image__cong) ).
thf(4566,plain,
! [A: $i,B: $i] :
( ( ab_semigroup_mult @ B )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ F ) )
=> ( ! [G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ G ) @ F ) )
=> ( ( hAPP @ A @ B @ D @ G )
= ( hAPP @ A @ B @ E @ G ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ D ) @ C ) @ F )
= ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ E ) @ C ) @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).
thf(1146,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ bool @ ( finite_comp_fun_idem @ A @ ( fun @ A @ bool ) ) @ ( insert @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_286_comp__fun__idem__insert) ).
thf(5165,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ bool @ ( finite_comp_fun_idem @ A @ ( fun @ A @ bool ) ) @ ( insert @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1146]) ).
thf(434,axiom,
! [A: $i] :
( ( ord @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( ord_less_eq @ A ) )
= ( ord_less_eq @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Orderings_Oord__class_Oless__eq_res) ).
thf(2597,plain,
! [A: $i] :
( ( ord @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( ord_less_eq @ A ) )
= ( ord_less_eq @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).
thf(990,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_599_mult__le__mono) ).
thf(4581,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).
thf(930,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ D ) @ ( hAPP @ ( fun @ C @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ C @ A ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ C @ A ) @ E ) @ F ) )
= ( hAPP @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ C @ B ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ C @ B ) @ ( hAPP @ ( fun @ C @ A ) @ ( fun @ C @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ C @ B ) ) @ ( combb @ A @ B @ C ) @ D ) @ E ) ) @ F ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_image__image) ).
thf(4336,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ A @ B ) @ D ) @ ( hAPP @ ( fun @ C @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ C @ A ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ C @ A ) @ E ) @ F ) )
= ( hAPP @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ C @ B ) @ ( fun @ ( fun @ C @ bool ) @ ( fun @ B @ bool ) ) @ ( image @ C @ B ) @ ( hAPP @ ( fun @ C @ A ) @ ( fun @ C @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ C @ B ) ) @ ( combb @ A @ B @ C ) @ D ) @ E ) ) @ F ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).
thf(127,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) )
<=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_722_sum__squares__gt__zero__iff) ).
thf(1551,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) )
=> ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
!= ( zero_zero @ A ) )
| ( ( ti @ A @ C )
!= ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(1006,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ C ) @ D ) @ E ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( ti @ B @ E ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_573_fold__image__empty) ).
thf(4658,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ C ) @ D ) @ E ) @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( ti @ B @ E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).
thf(125,axiom,
ordere453448008miring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__cancel__semiring) ).
thf(1547,plain,
ordere453448008miring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(183,axiom,
( ( one_one @ nat )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_523_One__nat__def) ).
thf(1770,plain,
( ( one_one @ nat )
= ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(116,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_688_sum__squares__eq__zero__iff) ).
thf(1516,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
& ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ C ) )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(1059,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_493_double__add__le__zero__iff__single__add__le__zero) ).
thf(4864,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).
thf(576,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_finite__Diff2) ).
thf(3112,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).
thf(1132,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_finite__surj) ).
thf(5125,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ E ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1132]) ).
thf(1076,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_325_linorder__cases) ).
thf(4937,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).
thf(821,axiom,
! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_Diff__subset) ).
thf(3949,plain,
! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).
thf(409,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_nat__add__assoc) ).
thf(2518,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).
thf(1229,plain,
! [B: $i,A: $i] :
( ~ ( number_ring @ A )
| ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) )
= ( ti @ A @ B ) ) ),
inference(cnf,[status(esa)],[1228]) ).
thf(1230,plain,
! [B: $i,A: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) )
= ( ti @ A @ B ) )
| ~ ( number_ring @ A ) ),
inference(lifteq,[status(thm)],[1229]) ).
thf(464,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ! [E: $i] :
( ( hAPP @ A @ B @ C @ E )
= ( hAPP @ A @ B @ D @ E ) )
=> ( ( ti @ ( fun @ A @ B ) @ C )
= ( ti @ ( fun @ A @ B ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_ext) ).
thf(2689,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ! [E: $i] :
( ( hAPP @ A @ B @ C @ E )
= ( hAPP @ A @ B @ D @ E ) )
=> ( ( ti @ ( fun @ A @ B ) @ C )
= ( ti @ ( fun @ A @ B ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).
thf(244,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_597_add__mult__distrib) ).
thf(1961,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(851,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_311_order__less__not__sym) ).
thf(4035,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).
thf(6419,plain,
! [A: $i] :
( ~ ( hBOOL @ ( zero_zero @ nat ) )
| ( ( fun @ x_a @ bool )
!= nat )
| ( ( fun @ ( fun @ x_a @ bool ) @ bool )
!= nat )
| ( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= ( ord_less_eq @ ( fun @ x_a @ bool ) ) )
| ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) ) @ A )
!= ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( zero_zero @ nat ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) ) ),
inference(paramod_ordered,[status(thm)],[1238,6319]) ).
thf(6441,plain,
! [A: $i] :
( ~ ( hBOOL @ ( zero_zero @ nat ) )
| ( ( fun @ x_a @ bool )
!= nat )
| ( ( fun @ ( fun @ x_a @ bool ) @ bool )
!= nat )
| ( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= ( ord_less_eq @ ( fun @ x_a @ bool ) ) )
| ( ( fun @ x_a @ bool )
!= nat )
| ( nat != bool )
| ( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= ( zero_zero @ nat ) )
| ( A
!= ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) ),
inference(simp,[status(thm)],[6419]) ).
thf(6453,plain,
( ~ ( hBOOL @ ( zero_zero @ nat ) )
| ( ( fun @ x_a @ bool )
!= nat )
| ( ( fun @ ( fun @ x_a @ bool ) @ bool )
!= nat )
| ( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= ( ord_less_eq @ ( fun @ x_a @ bool ) ) )
| ( nat != bool )
| ( ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) )
!= ( zero_zero @ nat ) ) ),
inference(simp,[status(thm)],[6441]) ).
thf(138,axiom,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( one_one @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ succ @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_946_Suc__nat__number__of) ).
thf(1605,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( one_one @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ succ @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(621,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( ( ti @ A @ C )
= ( ti @ A @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_xt1_I4_J) ).
thf(3260,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( ( ti @ A @ C )
= ( ti @ A @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).
thf(787,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_528_add__strict__increasing2) ).
thf(3833,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).
thf(737,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_454_less__zeroE) ).
thf(3659,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).
thf(478,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_403_diff__less__Suc) ).
thf(2732,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).
thf(34,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) @ ( hAPP @ A @ ( fun @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) ) @ ( nat_case @ A ) @ B ) @ C ) @ ( zero_zero @ nat ) )
= ( ti @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_491_nat__case__0) ).
thf(1252,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) @ ( hAPP @ A @ ( fun @ ( fun @ nat @ A ) @ ( fun @ nat @ A ) ) @ ( nat_case @ A ) @ B ) @ C ) @ ( zero_zero @ nat ) )
= ( ti @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(72,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) )
<=> ( ( A
= ( zero_zero @ nat ) )
| ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_586_mult__is__0) ).
thf(1368,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( A
= ( zero_zero @ nat ) )
| ( B
= ( zero_zero @ nat ) ) ) )
& ( ( ( A
= ( zero_zero @ nat ) )
| ( B
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(307,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_961_numeral__1__eq__1) ).
thf(2175,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
thf(916,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ C ) @ D )
= ( bot_bot @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) )
<=> ( ( ( ti @ ( fun @ B @ bool ) @ C )
= ( bot_bot @ ( fun @ B @ bool ) ) )
& ( ( ti @ ( fun @ A @ bool ) @ D )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_450_Plus__eq__empty__conv) ).
thf(4287,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ C ) @ D )
= ( bot_bot @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) )
=> ( ( ( ti @ ( fun @ B @ bool ) @ C )
= ( bot_bot @ ( fun @ B @ bool ) ) )
& ( ( ti @ ( fun @ A @ bool ) @ D )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
& ( ( ( ( ti @ ( fun @ B @ bool ) @ C )
= ( bot_bot @ ( fun @ B @ bool ) ) )
& ( ( ti @ ( fun @ A @ bool ) @ D )
= ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) @ ( sum_Plus @ B @ A ) @ C ) @ D )
= ( bot_bot @ ( fun @ ( sum_sum @ B @ A ) @ bool ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).
thf(961,axiom,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( one_one @ int ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_926_not__neg__1) ).
thf(4468,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( one_one @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).
thf(4469,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( one_one @ int ) ) ),
inference(polarity_switch,[status(thm)],[4468]) ).
thf(1034,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_191_DiffI) ).
thf(4754,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).
thf(374,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) )
=> ? [E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ E ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( collect @ B ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ B @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( combs @ B @ bool @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ B ) @ fconj ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( combc @ B @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) ) @ D ) ) ) @ ( hAPP @ A @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( combc @ B @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( combb @ A @ ( fun @ A @ bool ) @ B ) @ ( fequal @ A ) ) @ C ) ) @ ( hAPP @ B @ A @ C @ E ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_pigeonhole__infinite) ).
thf(2386,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) )
=> ? [E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ E ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( collect @ B ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ B @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( combs @ B @ bool @ bool ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ B ) @ fconj ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ B @ bool ) ) @ ( combc @ B @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) ) @ D ) ) ) @ ( hAPP @ A @ ( fun @ B @ bool ) @ ( hAPP @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ B @ bool ) ) @ ( combc @ B @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ ( fun @ B @ A ) @ ( fun @ B @ ( fun @ A @ bool ) ) ) @ ( combb @ A @ ( fun @ A @ bool ) @ B ) @ ( fequal @ A ) ) @ C ) ) @ ( hAPP @ B @ A @ C @ E ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).
thf(378,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( B
!= ( hAPP @ nat @ nat @ suc @ A ) )
=> ~ ! [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) )
=> ( B
!= ( hAPP @ nat @ nat @ suc @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_429_lessE) ).
thf(2407,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( B
!= ( hAPP @ nat @ nat @ suc @ A ) )
=> ~ ! [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) )
=> ( B
!= ( hAPP @ nat @ nat @ suc @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).
thf(775,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_857_transfer__int__nat__relations_I3_J) ).
thf(3786,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).
thf(1067,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_357_not__less__eq) ).
thf(4894,plain,
! [A: $i,B: $i] :
( ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ suc @ A ) ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).
thf(1084,axiom,
! [A: $i,B: $i] :
( ( ( bot_bot @ ( fun @ A @ bool ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) )
<=> ! [C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_empty__Collect__eq) ).
thf(4961,plain,
! [A: $i,B: $i] :
( ( ( ( bot_bot @ ( fun @ A @ bool ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) )
=> ! [C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) ) )
& ( ! [C: $i] :
~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( bot_bot @ ( fun @ A @ bool ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).
thf(1086,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ B )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fimplies_1_1_U) ).
thf(4971,plain,
! [A: $i,B: $i] :
( ( hBOOL @ B )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1086]) ).
thf(937,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_Suc__le__mono) ).
thf(4377,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).
thf(993,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) ) @ ( sum_Plus @ A @ B ) )
= ( sum_Plus @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Sum__Type_OPlus_res) ).
thf(4593,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) ) @ ( sum_Plus @ A @ B ) )
= ( sum_Plus @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).
thf(522,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_le__add1) ).
thf(2888,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).
thf(616,axiom,
order @ bool,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Oorder) ).
thf(3246,plain,
order @ bool,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).
thf(922,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fequal_2_1_T) ).
thf(4314,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ A @ B )
!= ( ti @ A @ C ) )
| ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( fequal @ A ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).
thf(984,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_372_trans__less__add2) ).
thf(4561,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).
thf(529,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ? [B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
& ( A
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_881_zero__less__imp__eq__int) ).
thf(2920,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ? [B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
& ( A
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).
thf(838,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_525_add__pos__nonneg) ).
thf(4003,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).
thf(982,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_509_not__one__less__zero) ).
thf(4549,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).
thf(899,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_394_less__add__Suc2) ).
thf(4222,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).
thf(1141,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_537_rev__predicate1D) ).
thf(5152,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1141]) ).
thf(666,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_792_nat__le__add__iff2) ).
thf(3412,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).
thf(270,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ int @ int @ bit1 @ A )
= ( hAPP @ int @ int @ bit1 @ B ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_977_rel__simps_I51_J) ).
thf(2044,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ int @ int @ bit1 @ A )
= ( hAPP @ int @ int @ bit1 @ B ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hAPP @ int @ int @ bit1 @ A )
= ( hAPP @ int @ int @ bit1 @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(870,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ pls ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_985_rel__simps_I19_J) ).
thf(4117,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ pls ) @ pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).
thf(61,axiom,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_969_semiring__numeral__0__eq__0) ).
thf(1336,plain,
! [A: $i] :
( ( number_semiring @ A )
=> ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(482,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ A @ A ) @ ( combi @ A ) ) @ B )
= ( ti @ ( fun @ A @ bool ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_image__ident) ).
thf(2747,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ A @ A ) @ ( combi @ A ) ) @ B )
= ( ti @ ( fun @ A @ bool ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).
thf(110,axiom,
ab_semigroup_mult @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oab__semigroup__mult) ).
thf(1499,plain,
ab_semigroup_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(190,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) ).
thf(1790,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(820,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_380_le__eq__less__or__eq) ).
thf(3940,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( A = B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).
thf(5392,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( ti @ A @ B ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) @ B )
!= ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) ),
inference(paramod_ordered,[status(thm)],[1227,1203]) ).
thf(5405,plain,
! [B: $i,A: $i] :
( ~ ( comm_semiring_1 @ A )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( ti @ A @ B ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( A != pname )
| ( A != x_a )
| ( ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( zero_zero @ A ) )
!= mgt_call )
| ( B != pn ) ),
inference(simp,[status(thm)],[5392]) ).
thf(5418,plain,
( ~ ( comm_semiring_1 @ pname )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( ti @ pname @ pn ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( pname != x_a )
| ( ( hAPP @ pname @ ( fun @ pname @ pname ) @ ( plus_plus @ pname ) @ ( zero_zero @ pname ) )
!= mgt_call ) ),
inference(simp,[status(thm)],[5405]) ).
thf(198,axiom,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_592_mult_Ocomm__neutral) ).
thf(1813,plain,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(94,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B )
= ( hAPP @ A @ A @ ( hAPP @ nat @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) @ ( semiri532925092at_aux @ A ) @ ( hAPP @ A @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ ( fun @ A @ A ) ) @ ( combc @ A @ A @ A ) @ ( plus_plus @ A ) ) @ ( one_one @ A ) ) ) @ B ) @ ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_882_semiring__1__class_Oof__nat__code) ).
thf(1452,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B )
= ( hAPP @ A @ A @ ( hAPP @ nat @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) @ ( semiri532925092at_aux @ A ) @ ( hAPP @ A @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ ( fun @ A @ A ) ) @ ( combc @ A @ A @ A ) @ ( plus_plus @ A ) ) @ ( one_one @ A ) ) ) @ B ) @ ( zero_zero @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(748,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i] :
? [C: $i] :
! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_823_pinf_I8_J) ).
thf(3691,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i] :
? [C: $i] :
! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).
thf(776,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_907_abs__le__zero__iff) ).
thf(3791,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( zero_zero @ A ) ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).
thf(785,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_734_mult__le__cancel2) ).
thf(3823,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).
thf(360,axiom,
ordere779506340up_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).
thf(2337,plain,
ordere779506340up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).
thf(1103,axiom,
! [A: $i] :
( ( ab_sem1668676832m_mult @ A )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ bool @ ( finite_comp_fun_idem @ A @ A ) @ ( times_times @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_589_comp__fun__idem) ).
thf(5030,plain,
! [A: $i] :
( ( ab_sem1668676832m_mult @ A )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ bool @ ( finite_comp_fun_idem @ A @ A ) @ ( times_times @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).
thf(1142,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( A != B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_382_le__neq__implies__less) ).
thf(5154,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( A != B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1142]) ).
thf(1157,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( ord_less @ ( fun @ nat @ bool ) ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( ord_less @ ( fun @ int @ bool ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_884_transfer__nat__int__set__relations_I4_J) ).
thf(5203,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( ord_less @ ( fun @ nat @ bool ) ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( ord_less @ ( fun @ int @ bool ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ int @ bool ) @ bool @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ ( fun @ int @ bool ) @ bool ) @ ( ord_less @ ( fun @ int @ bool ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ A ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ ( fun @ nat @ bool ) @ bool ) @ ( ord_less @ ( fun @ nat @ bool ) ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1157]) ).
thf(771,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_795_nat__less__add__iff1) ).
thf(3774,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ E ) @ D ) ) @ A ) ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ E ) @ A ) ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ D ) @ A ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).
thf(204,axiom,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_473_diff__0__right) ).
thf(1830,plain,
! [A: $i] :
( ( group_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(181,axiom,
! [A: $i,B: $i,C: $i] :
( ( ti @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ C @ B ) ) ) @ ( combb @ A @ B @ C ) )
= ( combb @ A @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_COMBB_res) ).
thf(1763,plain,
! [A: $i,B: $i,C: $i] :
( ( ti @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ C @ A ) @ ( fun @ C @ B ) ) ) @ ( combb @ A @ B @ C ) )
= ( combb @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(121,axiom,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_595_mult__1__left) ).
thf(1538,plain,
! [A: $i] :
( ( monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(906,axiom,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_356_diff__eq__diff__less) ).
thf(4241,plain,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ D ) @ E ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).
thf(43,axiom,
linorder @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Orderings_Olinorder) ).
thf(1285,plain,
linorder @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(17,axiom,
zero @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ozero) ).
thf(1218,plain,
zero @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(1152,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_insert__Diff) ).
thf(5186,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1152]) ).
thf(19,axiom,
monoid_mult @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Omonoid__mult) ).
thf(1220,plain,
monoid_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(1158,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_104_subset__insert) ).
thf(5208,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1158]) ).
thf(736,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
= ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_828_of__nat__number__of__lemma) ).
thf(3652,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ B ) )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).
thf(1083,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_665_mult__nonneg__nonpos) ).
thf(4959,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1083]) ).
thf(544,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_672_mult__less__cancel__left__disj) ).
thf(2992,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).
thf(90,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) )
<=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_533_card__eq__0__iff) ).
thf(1427,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) )
& ( ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( bot_bot @ ( fun @ A @ bool ) ) )
| ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(154,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_add__Suc) ).
thf(1670,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(164,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_842_of__nat__add) ).
thf(1717,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(534,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_subset__insert__iff) ).
thf(2939,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).
thf(697,axiom,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_368_not__add__less1) ).
thf(3520,plain,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).
thf(79,axiom,
ab_semigroup_mult @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oab__semigroup__mult) ).
thf(1404,plain,
ab_semigroup_mult @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(1192,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_386_diff__less__mono2) ).
thf(5319,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1192]) ).
thf(333,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ ( fun @ nat @ nat ) @ ( fun @ nat @ nat ) @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ nat ) @ ( fun @ nat @ nat ) ) @ ( nat_case @ nat ) @ ( zero_zero @ nat ) ) @ ( combi @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_524_diff__Suc) ).
thf(2247,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ ( fun @ nat @ nat ) @ ( fun @ nat @ nat ) @ ( hAPP @ nat @ ( fun @ ( fun @ nat @ nat ) @ ( fun @ nat @ nat ) ) @ ( nat_case @ nat ) @ ( zero_zero @ nat ) ) @ ( combi @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).
thf(123,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( A
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( one_one @ nat ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_546_Suc__pred_H) ).
thf(1543,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( A
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( one_one @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(182,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_Suc__n__not__n) ).
thf(1766,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ suc @ A )
!= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(290,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( zero_zero @ nat ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_478_Nat_Oadd__0__right) ).
thf(2113,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( zero_zero @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(667,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ D @ E ) ) @ ( hAPP @ nat @ A @ D @ ( hAPP @ nat @ nat @ suc @ E ) ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ D @ B ) ) @ ( hAPP @ nat @ A @ D @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_48_lift__Suc__mono__le) ).
thf(3415,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ D @ E ) ) @ ( hAPP @ nat @ A @ D @ ( hAPP @ nat @ nat @ suc @ E ) ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ nat @ A @ D @ B ) ) @ ( hAPP @ nat @ A @ D @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).
thf(194,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( plus_plus @ A ) )
= ( plus_plus @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_1_res) ).
thf(1802,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( plus_plus @ A ) )
= ( plus_plus @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(689,axiom,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ B @ A @ E @ F ) ) )
=> ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ F ) @ C ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ G ) @ H ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ B @ A @ E @ G ) ) @ ( hAPP @ B @ A @ E @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ B @ A @ E @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_437_order__less__le__subst1) ).
thf(3493,plain,
! [A: $i,B: $i] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ B @ A @ E @ F ) ) )
=> ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ F ) @ C ) )
=> ( ! [G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ G ) @ H ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ B @ A @ E @ G ) ) @ ( hAPP @ B @ A @ E @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ ( hAPP @ B @ A @ E @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).
thf(997,axiom,
( ( ti @ bool @ fTrue )
= fTrue ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_fTrue_res) ).
thf(4611,plain,
( ( ti @ bool @ fTrue )
= fTrue ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).
thf(1089,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( hAPP @ bool @ ( fun @ A @ ( fun @ A @ A ) ) @ ( if @ A ) @ fTrue ) @ B ) @ C )
= ( ti @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_If_1_1_T) ).
thf(4976,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( hAPP @ bool @ ( fun @ A @ ( fun @ A @ A ) ) @ ( if @ A ) @ fTrue ) @ B ) @ C )
= ( ti @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).
thf(907,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_finite__subset) ).
thf(4246,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).
thf(1167,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_279_termination__basic__simps_I4_J) ).
thf(5240,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1167]) ).
thf(20,axiom,
monoid_mult @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Omonoid__mult) ).
thf(1221,plain,
monoid_mult @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(966,axiom,
! [A: $i,B: $i] :
( ( minus @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hAPP @ A @ B @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( minus_minus @ ( fun @ A @ B ) ) @ C ) @ D ) @ E )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( minus_minus @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_minus__apply) ).
thf(4488,plain,
! [A: $i,B: $i] :
( ( minus @ B )
=> ! [C: $i,D: $i,E: $i] :
( ( hAPP @ A @ B @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ A @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ A @ B ) ) @ ( minus_minus @ ( fun @ A @ B ) ) @ C ) @ D ) @ E )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( minus_minus @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).
thf(67,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( hAPP @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) @ ( hAPP @ ( fun @ A @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ nat ) ) @ ( fun @ ( fun @ A @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) ) ) @ ( finite_fold_image @ nat @ A ) @ ( plus_plus @ nat ) ) @ ( hAPP @ nat @ ( fun @ A @ nat ) @ ( combk @ nat @ A ) @ ( one_one @ nat ) ) ) @ ( zero_zero @ nat ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_571_card_Oeq__fold__g) ).
thf(1353,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ nat @ ( hAPP @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) @ ( hAPP @ ( fun @ A @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ nat ) ) @ ( fun @ ( fun @ A @ nat ) @ ( fun @ nat @ ( fun @ ( fun @ A @ bool ) @ nat ) ) ) @ ( finite_fold_image @ nat @ A ) @ ( plus_plus @ nat ) ) @ ( hAPP @ nat @ ( fun @ A @ nat ) @ ( combk @ nat @ A ) @ ( one_one @ nat ) ) ) @ ( zero_zero @ nat ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(1031,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_253_add__leD2) ).
thf(4748,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).
thf(299,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) @ D ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B ) ) @ D ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_785_left__add__mult__distrib) ).
thf(2145,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) @ D ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) @ B ) ) @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(247,axiom,
ordered_semiring @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Oordered__semiring) ).
thf(1972,plain,
ordered_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(353,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_852_zmult__int) ).
thf(2320,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).
thf(1094,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_654_split__mult__pos__le) ).
thf(4991,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i] :
( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1094]) ).
thf(904,axiom,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_empty__iff) ).
thf(4235,plain,
! [A: $i,B: $i] :
~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).
thf(131,axiom,
ordere1490568538miring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oordered__comm__semiring) ).
thf(1585,plain,
ordere1490568538miring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(1001,axiom,
! [A: $i,B: $i] :
( ( A != B )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_298_nat__neq__iff) ).
thf(4634,plain,
! [A: $i,B: $i] :
( ( ( A != B )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).
thf(1054,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_914_abs__triangle__ineq2__sym) ).
thf(4838,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).
thf(1117,axiom,
! [A: $i] :
( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ A ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_905_not__neg__eq__ge__0) ).
thf(5083,plain,
! [A: $i] :
( ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).
thf(812,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_152_singletonE) ).
thf(3914,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).
thf(1123,axiom,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) ) @ ( finite908156982e_idem @ A @ B ) )
= ( finite908156982e_idem @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofolding__image__simple__idem_res) ).
thf(5100,plain,
! [A: $i,B: $i] :
( ( ti @ ( fun @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ A @ ( fun @ ( fun @ B @ A ) @ ( fun @ ( fun @ ( fun @ B @ bool ) @ A ) @ bool ) ) ) ) @ ( finite908156982e_idem @ A @ B ) )
= ( finite908156982e_idem @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).
thf(547,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_916_abs__triangle__ineq) ).
thf(3007,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).
thf(339,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) )
<=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_544_card__gt__0__iff) ).
thf(2266,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) )
& ( ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).
thf(973,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_714_mult__less__le__imp__less) ).
thf(4516,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).
thf(878,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ E ) @ F ) )
=> ( ( ( ti @ A @ C )
= ( hAPP @ B @ A @ D @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ F ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_rev__image__eqI) ).
thf(4150,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ E ) @ F ) )
=> ( ( ( ti @ A @ C )
= ( hAPP @ B @ A @ D @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ D ) @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).
thf(941,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_insert__commute) ).
thf(4392,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).
thf(627,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_745_less__number__of__int__code) ).
thf(3288,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).
thf(897,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_xt1_I5_J) ).
thf(4217,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).
thf(677,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_add__le__mono) ).
thf(3452,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).
thf(208,axiom,
! [A: $i] :
( ( ( number @ A )
& ( ring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_625_right__diff__distrib__number__of) ).
thf(1844,plain,
! [A: $i] :
( ( ( number @ A )
& ( ring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(313,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_755_plus__numeral__code_I9_J) ).
thf(2194,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ B ) )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(1028,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_397_less__Suc__eq__le) ).
thf(4738,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).
thf(739,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ B )
= ( ti @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_911_abs__of__pos) ).
thf(3665,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hAPP @ A @ A @ ( abs_abs @ A ) @ B )
= ( ti @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).
thf(88,axiom,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ nat ) @ ( finite_card @ A ) )
= ( finite_card @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ocard_res) ).
thf(1421,plain,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ nat ) @ ( finite_card @ A ) )
= ( finite_card @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(530,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ B ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_not__less__eq__eq) ).
thf(2924,plain,
! [A: $i,B: $i] :
( ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ B ) ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ B ) ) @ A ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).
thf(1005,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B )
= ( zero_zero @ A ) )
<=> ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
& ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_495_add__nonneg__eq__0__iff) ).
thf(4651,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B )
= ( zero_zero @ A ) )
=> ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
& ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) )
& ( ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
& ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B )
= ( zero_zero @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).
thf(726,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_106_insert__mono) ).
thf(3620,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).
thf(229,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_635_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) ).
thf(1918,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( one_one @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(951,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ D ) @ E ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ ( hAPP @ A @ ( fun @ B @ A ) @ ( combk @ A @ B ) @ C ) ) @ E )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_image__constant) ).
thf(4424,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ D ) @ E ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ ( hAPP @ A @ ( fun @ B @ A ) @ ( combk @ A @ B ) @ C ) ) @ E )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).
thf(55,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( one_one @ A )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_957_semiring__norm_I110_J) ).
thf(1316,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( one_one @ A )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(709,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_diff__le__mono2) ).
thf(3565,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).
thf(662,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_192_DiffD2) ).
thf(3401,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
=> ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).
thf(570,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ E ) ) @ ( hAPP @ nat @ A @ D @ ( hAPP @ nat @ nat @ suc @ E ) ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ B ) ) @ ( hAPP @ nat @ A @ D @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_413_lift__Suc__mono__less) ).
thf(3092,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ E ) ) @ ( hAPP @ nat @ A @ D @ ( hAPP @ nat @ nat @ suc @ E ) ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ nat @ A @ D @ B ) ) @ ( hAPP @ nat @ A @ D @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).
thf(97,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_853_int__mult) ).
thf(1459,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(926,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_874_of__nat__diff) ).
thf(4325,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ C ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).
thf(239,axiom,
group_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Ogroup__add) ).
thf(1949,plain,
group_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(882,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
<=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_334_linorder__antisym__conv1) ).
thf(4165,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) )
& ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).
thf(949,axiom,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_661_mult__right__mono) ).
thf(4419,plain,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).
thf(1202,axiom,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_664_mult__nonneg__nonpos2) ).
thf(5386,plain,
! [A: $i] :
( ( ordere453448008miring @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1202]) ).
thf(1078,axiom,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( bot_bot @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_163_bot__least) ).
thf(4944,plain,
! [A: $i] :
( ( bot @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( bot_bot @ A ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1078]) ).
thf(364,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_376_add__lessD1) ).
thf(2347,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).
thf(83,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ B ) ) @ B ) ) @ ( zero_zero @ A ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_575_Ints__odd__less__0) ).
thf(1412,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ B ) ) @ B ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( one_one @ A ) ) @ B ) ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(438,axiom,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_393_less__add__Suc1) ).
thf(2611,plain,
! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).
thf(355,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( one_one @ int ) ) @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_746_zmult__1) ).
thf(2325,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( one_one @ int ) ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).
thf(816,axiom,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( insert @ A ) )
= ( insert @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Set_Oinsert_res) ).
thf(3928,plain,
! [A: $i] :
( ( ti @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( insert @ A ) )
= ( insert @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).
thf(15,axiom,
one @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oone) ).
thf(1216,plain,
one @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(319,axiom,
ordere236663937imp_le @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(2209,plain,
ordere236663937imp_le @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).
thf(1003,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( zero_zero @ nat ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_458_less__eq__nat_Osimps_I1_J) ).
thf(4646,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( zero_zero @ nat ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).
thf(1188,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fimplies_2_1_U) ).
thf(5308,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ B )
| ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fimplies @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1188]) ).
thf(312,axiom,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( one_one @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ succ @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_948_nat__1__add__number__of) ).
thf(2187,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( one_one @ nat ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( one_one @ nat ) ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ succ @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).
thf(645,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) )
=> ( ( ti @ A @ D )
= ( ti @ A @ C ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ B )
= ( ti @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_288_the__equality) ).
thf(3346,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) )
=> ( ( ti @ A @ D )
= ( ti @ A @ C ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ B )
= ( ti @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).
thf(638,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_322_order__less__trans) ).
thf(3320,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).
thf(76,axiom,
! [A: $i,B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A )
= A ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_547_add__eq__if) ).
thf(1390,plain,
! [A: $i,B: $i] :
( ( ( B
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A )
= A ) )
& ( ( B
!= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(350,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_890_transfer__int__nat__relations_I1_J) ).
thf(2305,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).
thf(883,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ ( ti @ ( fun @ A @ B ) @ C ) @ D )
= ( hAPP @ A @ B @ C @ D ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_hAPP_arg1) ).
thf(4170,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hAPP @ A @ B @ ( ti @ ( fun @ A @ B ) @ C ) @ D )
= ( hAPP @ A @ B @ C @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).
thf(45,axiom,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_594_mult__1) ).
thf(1289,plain,
! [A: $i] :
( ( comm_monoid_mult @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(558,axiom,
! [A: $i] :
( ( comm_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_590_comm__semiring__class_Odistrib) ).
thf(3056,plain,
! [A: $i] :
( ( comm_semiring @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) @ D )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).
thf(327,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ ( one_one @ int ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_747_zmult__1__right) ).
thf(2226,plain,
! [A: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ ( one_one @ int ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).
thf(1197,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( ( ( B = C )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_303_nat__less__cases) ).
thf(5332,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( ( ( B = C )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ A @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1197]) ).
thf(277,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_add__Suc__right) ).
thf(2071,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(492,axiom,
! [A: $i] :
( ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( collect @ int ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) ) @ ( combs @ int @ bool @ bool ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ int ) @ fconj ) @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) ) ) @ A ) )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ ( fun @ nat @ int ) @ ( fun @ nat @ bool ) ) @ ( combb @ int @ bool @ nat ) @ A ) @ ( semiring_1_of_nat @ int ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_864_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J) ).
thf(2775,plain,
! [A: $i] :
( ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( collect @ int ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ bool ) ) @ ( combs @ int @ bool @ bool ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ int @ bool ) @ ( fun @ int @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ int ) @ fconj ) @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) ) ) @ A ) )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ int @ bool ) ) @ ( image @ nat @ int ) @ ( semiring_1_of_nat @ int ) ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ ( fun @ nat @ int ) @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ int @ bool ) @ ( fun @ ( fun @ nat @ int ) @ ( fun @ nat @ bool ) ) @ ( combb @ int @ bool @ nat ) @ A ) @ ( semiring_1_of_nat @ int ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).
thf(169,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( one_one @ nat ) ) )
= A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_545_Suc__diff__1) ).
thf(1734,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( one_one @ nat ) ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(1112,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ E ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_643_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) ).
thf(5068,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).
thf(1151,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_529_add__neg__nonpos) ).
thf(5184,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ ( zero_zero @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( zero_zero @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1151]) ).
thf(257,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ bit1 @ A )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_968_Bit1__def) ).
thf(1998,plain,
! [A: $i] :
( ( hAPP @ int @ int @ bit1 @ A )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ A ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(717,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_251_add__le__mono1) ).
thf(3593,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).
thf(148,axiom,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ E ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
<=> ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( ( ti @ A @ C )
= ( ti @ A @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_648_crossproduct__eq) ).
thf(1646,plain,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ E ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( ( ti @ A @ C )
= ( ti @ A @ E ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( ( ti @ A @ C )
= ( ti @ A @ E ) ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ E ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ E ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(470,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_515_add__gr__0) ).
thf(2707,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).
thf(74,axiom,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
=> ~ ! [B: $i] :
( A
!= ( hAPP @ nat @ nat @ suc @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_557_nat_Oexhaust) ).
thf(1385,plain,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
=> ~ ! [B: $i] :
( A
!= ( hAPP @ nat @ nat @ suc @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(685,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_le__diff__iff) ).
thf(3484,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).
thf(415,axiom,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( collect @ A ) )
= ( collect @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Set_OCollect_res) ).
thf(2532,plain,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( collect @ A ) )
= ( collect @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).
thf(1164,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_330_less__le__not__le) ).
thf(5228,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1164]) ).
thf(540,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) )
=> ( hBOOL @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_315_order__less__imp__triv) ).
thf(2971,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) )
=> ( hBOOL @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).
thf(619,axiom,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i] :
? [C: $i] :
! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_820_minf_I8_J) ).
thf(3252,plain,
! [A: $i] :
( ( linorder @ A )
=> ! [B: $i] :
? [C: $i] :
! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).
thf(375,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ C ) )
=> ? [D: $i,E: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
& ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ E )
= C )
& ( ( C
= ( zero_zero @ nat ) )
=> ( ( ti @ ( fun @ A @ bool ) @ E )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_556_card__eq__SucD) ).
thf(2389,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ C ) )
=> ? [D: $i,E: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
& ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ E )
= C )
& ( ( C
= ( zero_zero @ nat ) )
=> ( ( ti @ ( fun @ A @ bool ) @ E )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).
thf(102,axiom,
! [A: $i] :
( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ A ) ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_389_card__Collect__less__nat) ).
thf(1476,plain,
! [A: $i] :
( ( hAPP @ ( fun @ nat @ bool ) @ nat @ ( finite_card @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ A ) ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(1179,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_709_mult__less__imp__less__left) ).
thf(5272,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1179]) ).
thf(346,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( one_one @ A ) ) ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) ).
thf(2290,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( one_one @ A ) ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).
thf(999,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( C
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_800_self__quotient__aux1) ).
thf(4615,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( C
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ C ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( one_one @ int ) ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).
thf(335,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) )
= ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_932_abs__of__nat) ).
thf(2251,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) )
= ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).
thf(224,axiom,
ab_semigroup_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oab__semigroup__add) ).
thf(1903,plain,
ab_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(1009,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_833_zero__le__imp__of__nat) ).
thf(4666,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).
thf(297,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_634_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) ).
thf(2135,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( one_one @ A ) ) @ B )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(1201,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ~ ! [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) )
=> ( B
!= ( hAPP @ nat @ nat @ suc @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_430_Suc__lessE) ).
thf(5382,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ~ ! [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ C ) )
=> ( B
!= ( hAPP @ nat @ nat @ suc @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1201]) ).
thf(424,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ D ) @ E ) )
=> ( ! [F: $i,G: $i] :
( ( hAPP @ A @ A @ C @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ F ) @ G ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ ( hAPP @ A @ A @ C @ F ) ) @ ( hAPP @ A @ A @ C @ G ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ A @ A @ C @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ A @ A ) @ C ) @ B ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_278_folding__one__idem_Ohom__commute) ).
thf(2554,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool @ ( hAPP @ ( fun @ A @ ( fun @ A @ A ) ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ A ) @ bool ) @ ( finite2073411215e_idem @ A ) @ D ) @ E ) )
=> ( ! [F: $i,G: $i] :
( ( hAPP @ A @ A @ C @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ F ) @ G ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ D @ ( hAPP @ A @ A @ C @ F ) ) @ ( hAPP @ A @ A @ C @ G ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ( ( hAPP @ A @ A @ C @ ( hAPP @ ( fun @ A @ bool ) @ A @ E @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ A @ E @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ A @ A ) @ C ) @ B ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).
thf(502,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_subsetD) ).
thf(2823,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).
thf(979,axiom,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_901_abs__triangle__ineq4) ).
thf(4542,plain,
! [A: $i] :
( ( ordere142940540dd_abs @ A )
=> ! [B: $i,C: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).
thf(1023,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) )
<=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_set__eq__subset) ).
thf(4715,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ B ) ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).
thf(1047,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_424_psubset__insert__iff) ).
thf(4791,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ D ) ) )
& ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ D ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).
thf(145,axiom,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B )
= ( zero_zero @ A ) )
<=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_555_double__eq__0__iff) ).
thf(1629,plain,
! [A: $i] :
( ( linord219039673up_add @ A )
=> ! [B: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ B )
= ( zero_zero @ A ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ B )
= ( zero_zero @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(574,axiom,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_728_less__add__iff1) ).
thf(3104,plain,
! [A: $i] :
( ( ordered_ring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ E ) ) @ C ) ) @ D ) ) @ F ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ D ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).
thf(437,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_276_diff__Suc__diff__eq2) ).
thf(2608,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ suc @ C ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).
thf(613,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) )
<=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_786_nat__mult__eq__cancel1) ).
thf(3235,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
=> ( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).
thf(201,axiom,
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
= ( one_one @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_869_int__Suc0__eq__1) ).
thf(1822,plain,
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) )
= ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(841,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24_le__refl) ).
thf(4010,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).
thf(968,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) )
=> ( ( ti @ A @ D )
= ( ti @ A @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_291_theI) ).
thf(4495,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ B @ D ) )
=> ( ( ti @ A @ D )
= ( ti @ A @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).
thf(584,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_402_Suc__le__lessD) ).
thf(3136,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).
thf(1064,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less_eq @ nat ) ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_finite__Collect__le__nat) ).
thf(4881,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less_eq @ nat ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).
thf(738,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6_card__seteq) ).
thf(3662,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).
thf(975,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_484_not__less0) ).
thf(4527,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).
thf(532,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( A = B )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_776_zless__linear) ).
thf(2934,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
| ( A = B )
| ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).
thf(301,axiom,
preorder @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Opreorder) ).
thf(2152,plain,
preorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(1030,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite908156982e_idem @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_folding__image__simple__idem_Oinsert__idem) ).
thf(4745,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite908156982e_idem @ B @ A ) @ E ) @ F ) @ G ) @ H ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ H @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
= ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ E @ ( hAPP @ A @ B @ G @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ H @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).
thf(261,axiom,
ordere223160158up_add @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(2009,plain,
ordere223160158up_add @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(879,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ A ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_904_neg__def) ).
thf(4153,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ ( zero_zero @ int ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).
thf(187,axiom,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
<=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_471_eq__iff__diff__eq__0) ).
thf(1777,plain,
! [A: $i] :
( ( ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( zero_zero @ A ) ) )
& ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(1036,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_767_zadd__left__mono) ).
thf(4759,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).
thf(590,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ suc @ A ) ) )
<=> ( B = A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_401_le__less__Suc__eq) ).
thf(3149,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ suc @ A ) ) )
=> ( B = A ) )
& ( ( B = A )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).
thf(807,axiom,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( one_one @ int ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_814_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J) ).
thf(3896,plain,
hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( one_one @ int ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).
thf(940,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_370_nat__add__left__cancel__less) ).
thf(4387,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).
thf(701,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fdisj @ A ) @ B ) )
| ( hBOOL @ A )
| ( hBOOL @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fdisj_3_1_U) ).
thf(3529,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fdisj @ A ) @ B ) )
| ( hBOOL @ A )
| ( hBOOL @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).
thf(646,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( B
!= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_481_gr__implies__not0) ).
thf(3351,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( B
!= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).
thf(1060,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( one_one @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_508_zero__less__one) ).
thf(4869,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ ( one_one @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).
thf(669,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_292_card__Diff2__less) ).
thf(3420,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).
thf(373,axiom,
( ( ti @ ( fun @ int @ bool ) @ nat_neg )
= nat_neg ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Nat__Numeral_Oneg_res) ).
thf(2384,plain,
( ( ti @ ( fun @ int @ bool ) @ nat_neg )
= nat_neg ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).
thf(397,axiom,
! [A: $i] :
( ( ( number @ A )
& ( linorder @ A ) )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) )
<=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_651_le__number__of__eq__not__less) ).
thf(2475,plain,
! [A: $i] :
( ( ( number @ A )
& ( linorder @ A ) )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) )
=> ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).
thf(1057,axiom,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) )
<=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_add__left__cancel) ).
thf(4855,plain,
! [A: $i] :
( ( cancel_semigroup_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) )
=> ( ( ti @ A @ C )
= ( ti @ A @ D ) ) )
& ( ( ( ti @ A @ C )
= ( ti @ A @ D ) )
=> ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).
thf(496,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_517_card__infinite) ).
thf(2786,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).
thf(499,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ C ) )
<=> ? [D: $i,E: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
& ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ E )
= C )
& ( ( C
= ( zero_zero @ nat ) )
=> ( ( ti @ ( fun @ A @ bool ) @ E )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_549_card__Suc__eq) ).
thf(2796,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ C ) )
=> ? [D: $i,E: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
& ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ E )
= C )
& ( ( C
= ( zero_zero @ nat ) )
=> ( ( ti @ ( fun @ A @ bool ) @ E )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) )
& ( ? [D: $i,E: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ D ) @ E ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ D ) @ E ) )
& ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ E )
= C )
& ( ( C
= ( zero_zero @ nat ) )
=> ( ( ti @ ( fun @ A @ bool ) @ E )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( hAPP @ nat @ nat @ suc @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).
thf(1145,axiom,
! [A: $i,B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_Diff__insert__absorb) ).
thf(5162,plain,
! [A: $i,B: $i,C: $i] :
( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1145]) ).
thf(5483,plain,
! [B: $i,A: $i] :
( ~ ( number_ring @ A )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( ti @ A @ B ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) )
!= ( hAPP @ pname @ x_a @ mgt_call @ pn ) ) ),
inference(paramod_ordered,[status(thm)],[1230,1203]) ).
thf(5506,plain,
! [B: $i,A: $i] :
( ~ ( number_ring @ A )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( ti @ A @ B ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( A != pname )
| ( A != x_a )
| ( ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B )
!= mgt_call )
| ( ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls )
!= pn ) ),
inference(simp,[status(thm)],[5483]) ).
thf(5532,plain,
! [A: $i] :
( ~ ( number_ring @ pname )
| ~ ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ x_a @ bool ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( ti @ pname @ A ) ) @ g ) ) @ ( hAPP @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ ( fun @ pname @ x_a ) @ ( fun @ ( fun @ pname @ bool ) @ ( fun @ x_a @ bool ) ) @ ( image @ pname @ x_a ) @ mgt_call ) @ u ) ) )
| ( pname != x_a )
| ( ( hAPP @ pname @ ( fun @ pname @ pname ) @ ( plus_plus @ pname ) @ A )
!= mgt_call )
| ( ( hAPP @ int @ pname @ ( number_number_of @ pname ) @ pls )
!= pn ) ),
inference(simp,[status(thm)],[5506]) ).
thf(376,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) @ D ) )
<=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_103_insert__subset) ).
thf(2399,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ D ) )
& ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).
thf(269,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ( ( zero_zero @ A )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_971_semiring__norm_I112_J) ).
thf(2041,plain,
! [A: $i] :
( ( number_ring @ A )
=> ( ( zero_zero @ A )
= ( hAPP @ int @ A @ ( number_number_of @ A ) @ pls ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(624,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_Suc__n__not__le__n) ).
thf(3267,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).
thf(1042,axiom,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the_elem @ A ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ ( combb @ ( fun @ A @ bool ) @ bool @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fequal @ ( fun @ A @ bool ) ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_the__elem__def) ).
thf(4774,plain,
! [A: $i,B: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ A @ ( the_elem @ A ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ A @ ( the @ A ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fun @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ bool ) ) @ ( combb @ ( fun @ A @ bool ) @ bool @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( fequal @ ( fun @ A @ bool ) ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ A @ bool ) ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).
thf(719,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_nat__le__linear) ).
thf(3598,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
| ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).
thf(2680,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( zero_zero @ int ) ) ),
inference(polarity_switch,[status(thm)],[2679]) ).
thf(310,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_835_of__nat__0) ).
thf(2183,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(11,axiom,
comm_monoid_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocomm__monoid__add) ).
thf(1212,plain,
comm_monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(526,axiom,
! [A: $i,B: $i,C: $i] :
( ( comm_monoid_mult @ C )
=> ! [D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ J ) )
=> ( ! [K: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ K ) @ I ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ ( hAPP @ A @ B @ H @ K ) ) @ J ) )
& ( ( hAPP @ B @ A @ G @ ( hAPP @ A @ B @ H @ K ) )
= ( ti @ A @ K ) ) ) )
=> ( ! [K: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ K ) @ J ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ B @ A @ G @ K ) ) @ I ) )
& ( ( hAPP @ A @ B @ H @ ( hAPP @ B @ A @ G @ K ) )
= ( ti @ B @ K ) )
& ( ( hAPP @ A @ C @ E @ ( hAPP @ B @ A @ G @ K ) )
= ( hAPP @ B @ C @ F @ K ) ) ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ C @ ( hAPP @ C @ ( fun @ ( fun @ B @ bool ) @ C ) @ ( hAPP @ ( fun @ B @ C ) @ ( fun @ C @ ( fun @ ( fun @ B @ bool ) @ C ) ) @ ( hAPP @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ C @ ( fun @ ( fun @ B @ bool ) @ C ) ) ) @ ( finite_fold_image @ C @ B ) @ ( times_times @ C ) ) @ F ) @ D ) @ J )
= ( hAPP @ ( fun @ A @ bool ) @ C @ ( hAPP @ C @ ( fun @ ( fun @ A @ bool ) @ C ) @ ( hAPP @ ( fun @ A @ C ) @ ( fun @ C @ ( fun @ ( fun @ A @ bool ) @ C ) ) @ ( hAPP @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ C @ ( fun @ ( fun @ A @ bool ) @ C ) ) ) @ ( finite_fold_image @ C @ A ) @ ( times_times @ C ) ) @ E ) @ D ) @ I ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_898_fold__image__eq__general__inverses) ).
thf(2902,plain,
! [A: $i,B: $i,C: $i] :
( ( comm_monoid_mult @ C )
=> ! [D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ J ) )
=> ( ! [K: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ K ) @ I ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ ( hAPP @ A @ B @ H @ K ) ) @ J ) )
& ( ( hAPP @ B @ A @ G @ ( hAPP @ A @ B @ H @ K ) )
= ( ti @ A @ K ) ) ) )
=> ( ! [K: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ K ) @ J ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ B @ A @ G @ K ) ) @ I ) )
& ( ( hAPP @ A @ B @ H @ ( hAPP @ B @ A @ G @ K ) )
= ( ti @ B @ K ) )
& ( ( hAPP @ A @ C @ E @ ( hAPP @ B @ A @ G @ K ) )
= ( hAPP @ B @ C @ F @ K ) ) ) )
=> ( ( hAPP @ ( fun @ B @ bool ) @ C @ ( hAPP @ C @ ( fun @ ( fun @ B @ bool ) @ C ) @ ( hAPP @ ( fun @ B @ C ) @ ( fun @ C @ ( fun @ ( fun @ B @ bool ) @ C ) ) @ ( hAPP @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ B @ C ) @ ( fun @ C @ ( fun @ ( fun @ B @ bool ) @ C ) ) ) @ ( finite_fold_image @ C @ B ) @ ( times_times @ C ) ) @ F ) @ D ) @ J )
= ( hAPP @ ( fun @ A @ bool ) @ C @ ( hAPP @ C @ ( fun @ ( fun @ A @ bool ) @ C ) @ ( hAPP @ ( fun @ A @ C ) @ ( fun @ C @ ( fun @ ( fun @ A @ bool ) @ C ) ) @ ( hAPP @ ( fun @ C @ ( fun @ C @ C ) ) @ ( fun @ ( fun @ A @ C ) @ ( fun @ C @ ( fun @ ( fun @ A @ bool ) @ C ) ) ) @ ( finite_fold_image @ C @ A ) @ ( times_times @ C ) ) @ E ) @ D ) @ I ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).
thf(1159,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_21_le__trans) ).
thf(5211,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1159]) ).
thf(223,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) )
<=> ( ( A
= ( zero_zero @ nat ) )
| ( B = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_784_nat__mult__eq__cancel__disj) ).
thf(1893,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) )
=> ( ( A
= ( zero_zero @ nat ) )
| ( B = C ) ) )
& ( ( ( A
= ( zero_zero @ nat ) )
| ( B = C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(52,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_605_diff__mult__distrib2) ).
thf(1309,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(734,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ bool @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( nat_case @ bool ) @ fFalse ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_less__eq__nat_Osimps_I2_J) ).
thf(3645,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ bool @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( nat_case @ bool ) @ fFalse ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( hAPP @ bool @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) ) @ ( nat_case @ bool ) @ fFalse ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).
thf(480,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_921_abs__mult) ).
thf(2739,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( abs_abs @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).
thf(866,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_385_less__imp__diff__less) ).
thf(4098,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ A ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).
thf(92,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( ? [C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( ( one_one @ nat )
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_550_card_Oneutral) ).
thf(1444,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ( ( ? [C: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ B ) )
=> ( ( one_one @ nat )
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B )
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(1154,axiom,
! [A: $i,B: $i] :
( ! [C: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ ( hAPP @ nat @ nat @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) ) @ ( combb @ nat @ ( fun @ nat @ bool ) @ nat ) @ ( ord_less_eq @ nat ) ) @ B ) ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_finite__less__ub) ).
thf(5196,plain,
! [A: $i,B: $i] :
( ! [C: $i] : ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ C ) @ ( hAPP @ nat @ nat @ B @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ ( fun @ nat @ nat ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) ) @ ( combb @ nat @ ( fun @ nat @ bool ) @ nat ) @ ( ord_less_eq @ nat ) ) @ B ) ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1154]) ).
thf(65,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_462_diff__0__eq__0) ).
thf(1348,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(345,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_584_mult__0) ).
thf(2287,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( zero_zero @ nat ) ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).
thf(114,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ B ) ) ) @ ( one_one @ int ) ) ) @ C ) ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_900_decr__lemma) ).
thf(1512,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( abs_abs @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ A ) @ B ) ) ) @ ( one_one @ int ) ) ) @ C ) ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(612,axiom,
! [A: $i] :
( ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A )
= ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_408_less__eq__Suc__le__raw) ).
thf(3232,plain,
! [A: $i] :
( ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A )
= ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).
thf(1178,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite908156982e_idem @ B @ A ) @ D ) @ E ) @ F ) @ G ) )
=> ( ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ D @ C ) @ C )
= ( ti @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_folding__image__simple__idem_Oidem) ).
thf(5268,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite908156982e_idem @ B @ A ) @ D ) @ E ) @ F ) @ G ) )
=> ( ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ D @ C ) @ C )
= ( ti @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1178]) ).
thf(994,axiom,
! [A: $i] :
( ( A
!= ( zero_zero @ nat ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_483_neq0__conv) ).
thf(4596,plain,
! [A: $i] :
( ( ( A
!= ( zero_zero @ nat ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( A
!= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).
thf(124,axiom,
linordered_semiring @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__semiring) ).
thf(1546,plain,
linordered_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(679,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( ( hAPP @ nat @ nat @ suc @ A )
!= B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_363_Suc__lessI) ).
thf(3464,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( ( ( hAPP @ nat @ nat @ suc @ A )
!= B )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).
thf(253,axiom,
! [A: $i] :
( ( ( number @ A )
& ( semiring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_629_right__distrib__number__of) ).
thf(1986,plain,
! [A: $i] :
( ( ( number @ A )
& ( semiring @ A ) )
=> ! [B: $i,C: $i,D: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(155,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_883_int__int__eq) ).
thf(1673,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(324,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_475_bot__nat__def) ).
thf(2218,plain,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).
thf(1107,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ C ) ) ) @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ C ) ) ) @ B ) )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_Collect__conv__if) ).
thf(5051,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ C ) ) ) @ B ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
& ( ~ ( hBOOL @ ( hAPP @ A @ bool @ B @ C ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fconj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ C ) ) ) @ B ) )
= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).
thf(349,axiom,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( one_one @ nat ) ) @ A )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_613_nat__mult__1) ).
thf(2302,plain,
! [A: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( one_one @ nat ) ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).
thf(1099,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
<=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_le__iff__add) ).
thf(5009,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) )
& ( ? [C: $i] :
( B
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1099]) ).
thf(462,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ C ) @ D ) @ E ) @ F ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ F @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( ti @ B @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_folding__image__simple_Oempty) ).
thf(2683,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) @ ( hAPP @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ B @ ( fun @ ( fun @ A @ B ) @ ( fun @ ( fun @ ( fun @ A @ bool ) @ B ) @ bool ) ) ) @ ( finite1357897459simple @ B @ A ) @ C ) @ D ) @ E ) @ F ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ B @ F @ ( bot_bot @ ( fun @ A @ bool ) ) )
= ( ti @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).
thf(130,axiom,
! [A: $i] :
( ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( zero_zero @ int ) )
<=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_839_int__eq__0__conv) ).
thf(1578,plain,
! [A: $i] :
( ( ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( zero_zero @ int ) )
=> ( A
= ( zero_zero @ nat ) ) )
& ( ( A
= ( zero_zero @ nat ) )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A )
= ( zero_zero @ int ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(943,axiom,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_743_nat__mult__commute) ).
thf(4399,plain,
! [A: $i,B: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).
thf(1014,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_371_trans__less__add1) ).
thf(4687,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).
thf(1020,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_equalityD1) ).
thf(4706,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( ti @ ( fun @ A @ bool ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).
thf(798,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_992_less__int__code_I16_J) ).
thf(3860,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ ( hAPP @ int @ int @ bit1 @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).
thf(693,axiom,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_343_order__less__le__trans) ).
thf(3511,plain,
! [A: $i] :
( ( preorder @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).
thf(799,axiom,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ A ) @ ( the_elem @ A ) )
= ( the_elem @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Set_Othe__elem_res) ).
thf(3865,plain,
! [A: $i] :
( ( ti @ ( fun @ ( fun @ A @ bool ) @ A ) @ ( the_elem @ A ) )
= ( the_elem @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).
thf(473,axiom,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_655_mult__mono) ).
thf(2719,plain,
! [A: $i] :
( ( ordered_semiring @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).
thf(1098,axiom,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ C ) @ D ) )
<=> ! [E: $i] : ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_le__fun__def) ).
thf(5004,plain,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ C ) @ D ) )
=> ! [E: $i] : ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) ) )
& ( ! [E: $i] : ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).
thf(759,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_699_mult__less__cancel1) ).
thf(3733,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).
thf(535,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ~ ! [B: $i] :
( A
!= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_894_nonneg__eq__int) ).
thf(2951,plain,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ~ ! [B: $i] :
( A
!= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).
thf(1115,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ D ) @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ D ) @ E ) ) @ F ) ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_796_zdiv__mono2__neg__lemma) ).
thf(5078,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ C )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ D ) @ E ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ D ) @ E ) ) @ F ) ) @ ( zero_zero @ int ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ C ) @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ D ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ E ) @ B ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1115]) ).
thf(1199,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ B @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ suc @ D ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_zero__induct__lemma) ).
thf(5371,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ B @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ suc @ D ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1199]) ).
thf(31,axiom,
( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_955_numeral__3__eq__3) ).
thf(1248,plain,
( ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ ( hAPP @ int @ int @ bit1 @ ( hAPP @ int @ int @ bit1 @ pls ) ) )
= ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ suc @ ( hAPP @ nat @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(1008,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fdisj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_insert__compr__raw) ).
thf(4663,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combs @ A @ bool @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) @ ( hAPP @ ( fun @ bool @ ( fun @ bool @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ ( fun @ bool @ bool ) ) ) @ ( combb @ bool @ ( fun @ bool @ bool ) @ A ) @ fdisj ) @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( fun @ A @ ( fun @ A @ bool ) ) @ ( combc @ A @ A @ bool ) @ ( fequal @ A ) ) @ B ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( combc @ A @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).
thf(690,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) )
<=> ( A = C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_nat__add__right__cancel) ).
thf(3496,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) )
=> ( A = C ) )
& ( ( A = C )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).
thf(967,axiom,
! [A: $i] :
( ( ord @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( D = B )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_ord__le__eq__trans) ).
thf(4491,plain,
! [A: $i] :
( ( ord @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( ( D = B )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).
thf(177,axiom,
semiring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Osemiring) ).
thf(1757,plain,
semiring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(489,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_417_subset__psubset__trans) ).
thf(2768,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).
thf(104,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_858_Nat__Transfer_Otransfer__int__nat__functions_I3_J) ).
thf(1480,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ nat_tsub @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(213,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_754_zadd__zmult__distrib2) ).
thf(1859,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ B ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(199,axiom,
! [A: $i] :
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ suc @ A ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_871_int__Suc) ).
thf(1816,plain,
! [A: $i] :
( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ suc @ A ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( one_one @ int ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(311,axiom,
ring_n68954251visors @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oring__no__zero__divisors) ).
thf(2186,plain,
ring_n68954251visors @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).
thf(952,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
<=> ? [C: $i,D: $i] :
( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ nat @ A ) @ D ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_426_finite__conv__nat__seg__image) ).
thf(4427,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) )
=> ? [C: $i,D: $i] :
( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ nat @ A ) @ D ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ C ) ) ) ) )
& ( ? [C: $i,D: $i] :
( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ nat @ A ) @ ( fun @ ( fun @ nat @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ nat @ A ) @ D ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ C ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).
thf(846,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_377_termination__basic__simps_I1_J) ).
thf(4024,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).
thf(128,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) )
<=> ( ( A = C )
| ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_588_mult__cancel2) ).
thf(1561,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) )
=> ( ( A = C )
| ( B
= ( zero_zero @ nat ) ) ) )
& ( ( ( A = C )
| ( B
= ( zero_zero @ nat ) ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ C ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(528,axiom,
! [A: $i,B: $i] :
( ( comm_monoid_mult @ B )
=> ! [C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ G ) @ G ) )
=> ( ! [H: $i,I: $i,J: $i,K: $i] :
( ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ H ) @ J ) )
& ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ I ) @ K ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( times_times @ B ) @ H ) @ I ) ) @ ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( times_times @ B ) @ J ) @ K ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ! [H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ H ) @ E ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ ( hAPP @ A @ B @ C @ H ) ) @ ( hAPP @ A @ B @ D @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ G ) @ E ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ D ) @ G ) @ E ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_897_fold__image__related) ).
thf(2913,plain,
! [A: $i,B: $i] :
( ( comm_monoid_mult @ B )
=> ! [C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ G ) @ G ) )
=> ( ! [H: $i,I: $i,J: $i,K: $i] :
( ( ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ H ) @ J ) )
& ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ I ) @ K ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( times_times @ B ) @ H ) @ I ) ) @ ( hAPP @ B @ B @ ( hAPP @ B @ ( fun @ B @ B ) @ ( times_times @ B ) @ J ) @ K ) ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ E ) )
=> ( ! [H: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ H ) @ E ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ ( hAPP @ A @ B @ C @ H ) ) @ ( hAPP @ A @ B @ D @ H ) ) ) )
=> ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ F @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ C ) @ G ) @ E ) ) @ ( hAPP @ ( fun @ A @ bool ) @ B @ ( hAPP @ B @ ( fun @ ( fun @ A @ bool ) @ B ) @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) @ ( hAPP @ ( fun @ B @ ( fun @ B @ B ) ) @ ( fun @ ( fun @ A @ B ) @ ( fun @ B @ ( fun @ ( fun @ A @ bool ) @ B ) ) ) @ ( finite_fold_image @ B @ A ) @ ( times_times @ B ) ) @ D ) @ G ) @ E ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).
thf(264,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( one_one @ nat ) )
= ( one_one @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_836_of__nat__1) ).
thf(2021,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ( ( hAPP @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( one_one @ nat ) )
= ( one_one @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(1194,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_819_incr__mult__lemma) ).
thf(5324,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1194]) ).
thf(588,axiom,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_497_add__increasing2) ).
thf(3145,plain,
! [A: $i] :
( ( ordere216010020id_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).
thf(444,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ! [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ B @ A @ C @ F ) ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_image__subsetI) ).
thf(2636,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ! [F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( hAPP @ B @ ( fun @ ( fun @ B @ bool ) @ bool ) @ ( member @ B ) @ F ) @ E ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ B @ A @ C @ F ) ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ E ) ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).
thf(848,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_626_Ints__add) ).
thf(4029,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ ( ring_1_Ints @ A ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( ring_1_Ints @ A ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( ring_1_Ints @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).
thf(641,axiom,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) )
<=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_668_zero__le__mult__iff) ).
thf(3328,plain,
! [A: $i] :
( ( linord581940658strict @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) )
=> ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) ) ) )
& ( ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) ) )
| ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( zero_zero @ A ) ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ ( zero_zero @ A ) ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).
thf(919,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ A ) ) @ B ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ com @ x_a @ mgt @ ( hAPP @ ( option @ com ) @ com @ ( the_1 @ com ) @ ( hAPP @ pname @ ( option @ com ) @ body @ A ) ) ) ) @ ( bot_bot @ ( fun @ x_a @ bool ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ B ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ A ) ) @ ( bot_bot @ ( fun @ x_a @ bool ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_182_assms_I2_J) ).
thf(4303,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ A ) ) @ B ) ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ com @ x_a @ mgt @ ( hAPP @ ( option @ com ) @ com @ ( the_1 @ com ) @ ( hAPP @ pname @ ( option @ com ) @ body @ A ) ) ) ) @ ( bot_bot @ ( fun @ x_a @ bool ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ B ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ A ) ) @ ( bot_bot @ ( fun @ x_a @ bool ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).
thf(808,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_816_decr__mult__lemma) ).
thf(3897,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ C ) ) ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ! [D: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ B @ D ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ B @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( minus_minus @ int ) @ D ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ A ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).
thf(1043,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( ! [E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ E ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ F ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ F ) ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_181_finite__subset__induct) ).
thf(4777,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( ! [E: $i,F: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ E ) @ C ) )
=> ( ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ E ) @ F ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ F ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ E ) @ F ) ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).
thf(254,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ C ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_850_zadd__int__left) ).
thf(1989,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) @ C ) )
= ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(856,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_329_order__less__le) ).
thf(4051,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ( ( ti @ A @ B )
!= ( ti @ A @ C ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).
thf(1091,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ A @ B ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) @ ( sum_Plus @ A @ B ) @ C ) @ D ) ) )
<=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_449_finite__Plus__iff) ).
thf(4981,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ A @ B ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) @ ( sum_Plus @ A @ B ) @ C ) @ D ) ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
& ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ bool @ ( finite_finite @ ( sum_sum @ A @ B ) ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ ( sum_sum @ A @ B ) @ bool ) ) @ ( sum_Plus @ A @ B ) @ C ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1091]) ).
thf(359,axiom,
! [A: $i] :
( ( monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_466_add__0__right) ).
thf(2334,plain,
! [A: $i] :
( ( monoid_add @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).
thf(708,axiom,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
<=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_505_less__iff__diff__less__0) ).
thf(3560,plain,
! [A: $i] :
( ( ordered_ab_group_add @ A )
=> ! [B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) ) )
& ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( minus_minus @ A ) @ B ) @ C ) ) @ ( zero_zero @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).
thf(1085,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_365_less__SucE) ).
thf(4968,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) )
=> ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).
thf(840,axiom,
! [A: $i] :
( ( ring_1 @ A )
=> ( ( ti @ ( fun @ A @ bool ) @ ( ring_1_Ints @ A ) )
= ( ring_1_Ints @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Int_Oring__1__class_OInts_res) ).
thf(4007,plain,
! [A: $i] :
( ( ring_1 @ A )
=> ( ( ti @ ( fun @ A @ bool ) @ ( ring_1_Ints @ A ) )
= ( ring_1_Ints @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).
thf(682,axiom,
! [A: $i] :
( ( finite_finite_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_finite) ).
thf(3479,plain,
! [A: $i] :
( ( finite_finite_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).
thf(381,axiom,
! [A: $i,B: $i] :
( ( A != B )
=> ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_299_linorder__neqE__nat) ).
thf(2417,plain,
! [A: $i,B: $i] :
( ( A != B )
=> ( ~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).
thf(471,axiom,
! [A: $i,B: $i] :
( ( preorder @ B )
=> ( preorder @ ( fun @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_fun___Orderings_Opreorder) ).
thf(2714,plain,
! [A: $i,B: $i] :
( ( preorder @ B )
=> ( preorder @ ( fun @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).
thf(672,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
<=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_order__eq__iff) ).
thf(3426,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i] :
( ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
& ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ B ) ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).
thf(326,axiom,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_627_number__of__add) ).
thf(2223,plain,
! [A: $i] :
( ( number_ring @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ int @ A @ ( number_number_of @ A ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ B ) ) @ ( hAPP @ int @ A @ ( number_number_of @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).
thf(100,axiom,
( ( one_one @ int )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_960_one__is__num__one) ).
thf(1473,plain,
( ( one_one @ int )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ ( hAPP @ int @ int @ bit1 @ pls ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(1161,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( bot_bot @ ( fun @ A @ bool ) )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) )
<=> ( ( ti @ ( fun @ B @ bool ) @ D )
= ( bot_bot @ ( fun @ B @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_empty__is__image) ).
thf(5215,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( bot_bot @ ( fun @ A @ bool ) )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) )
=> ( ( ti @ ( fun @ B @ bool ) @ D )
= ( bot_bot @ ( fun @ B @ bool ) ) ) )
& ( ( ( ti @ ( fun @ B @ bool ) @ D )
= ( bot_bot @ ( fun @ B @ bool ) ) )
=> ( ( bot_bot @ ( fun @ A @ bool ) )
= ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1161]) ).
thf(441,axiom,
! [A: $i,B: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
<=> ? [C: $i,D: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_188_nonempty__iff) ).
thf(2619,plain,
! [A: $i,B: $i] :
( ( ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) )
=> ? [C: $i,D: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) ) ) )
& ( ? [C: $i,D: $i] :
( ( ( ti @ ( fun @ A @ bool ) @ B )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ D ) )
& ~ ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ C ) @ D ) ) )
=> ( ( ti @ ( fun @ A @ bool ) @ B )
!= ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).
thf(671,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_711_mult__less__imp__less__right) ).
thf(3424,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).
thf(554,axiom,
! [A: $i] :
( ( bot_bot @ ( fun @ A @ bool ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ bool @ ( fun @ A @ bool ) @ ( combk @ bool @ A ) @ fFalse ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_empty__def) ).
thf(3045,plain,
! [A: $i] :
( ( bot_bot @ ( fun @ A @ bool ) )
= ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( collect @ A ) @ ( hAPP @ bool @ ( fun @ A @ bool ) @ ( combk @ bool @ A ) @ fFalse ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).
thf(791,axiom,
! [A: $i] :
( ( finite_finite_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_finite__code) ).
thf(3843,plain,
! [A: $i] :
( ( finite_finite_1 @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).
thf(1156,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_double__diff) ).
thf(5200,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ D ) @ B ) )
=> ( ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ D ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ B ) @ C ) )
= ( ti @ ( fun @ A @ bool ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1156]) ).
thf(1126,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ C @ D ) ) @ ( hAPP @ nat @ nat @ C @ E ) ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ C @ A ) ) @ ( hAPP @ nat @ nat @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_428_less__mono__imp__le__mono) ).
thf(5110,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ D ) @ E ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ nat @ nat @ C @ D ) ) @ ( hAPP @ nat @ nat @ C @ E ) ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ C @ A ) ) @ ( hAPP @ nat @ nat @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1126]) ).
thf(779,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_finite_OemptyI) ).
thf(3805,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).
thf(615,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_le__SucI) ).
thf(3244,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).
thf(649,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_293_card__Diff1__less) ).
thf(3362,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( minus_minus @ ( fun @ A @ bool ) ) @ C ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).
thf(581,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( one_one @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_414_less__add__one) ).
thf(3128,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ! [B: $i] : ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( one_one @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).
thf(384,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_378_termination__basic__simps_I2_J) ).
thf(2430,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ B ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).
thf(73,axiom,
! [A: $i,B: $i] :
( ( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ suc @ B ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_nat_Oinject) ).
thf(1378,plain,
! [A: $i,B: $i] :
( ( ( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ suc @ B ) )
=> ( A = B ) )
& ( ( A = B )
=> ( ( hAPP @ nat @ nat @ suc @ A )
= ( hAPP @ nat @ nat @ suc @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(871,axiom,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_add__mono) ).
thf(4118,plain,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).
thf(687,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_682_mult__strict__right__mono) ).
thf(3489,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).
thf(405,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_Nat_Odiff__diff__eq) ).
thf(2507,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ A ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).
thf(659,axiom,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_296_finite__Collect__less__nat) ).
thf(3392,plain,
! [A: $i] : ( hBOOL @ ( hAPP @ ( fun @ nat @ bool ) @ bool @ ( finite_finite @ nat ) @ ( hAPP @ ( fun @ nat @ bool ) @ ( fun @ nat @ bool ) @ ( collect @ nat ) @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( hAPP @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( fun @ nat @ ( fun @ nat @ bool ) ) @ ( combc @ nat @ nat @ bool ) @ ( ord_less @ nat ) ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).
thf(942,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ ( one_one @ nat ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_548_ex__least__nat__less) ).
thf(4395,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( zero_zero @ nat ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ B @ A ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ C ) @ A ) )
& ! [D: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ D ) @ C ) )
=> ~ ( hBOOL @ ( hAPP @ nat @ bool @ B @ D ) ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ B @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ ( one_one @ nat ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).
thf(220,axiom,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i] :
( ( ( ti @ A @ B )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) )
<=> ( ( ti @ A @ C )
= ( zero_zero @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_568_add__0__iff) ).
thf(1880,plain,
! [A: $i] :
( ( semiri456707255roduct @ A )
=> ! [B: $i,C: $i] :
( ( ( ( ti @ A @ B )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) )
=> ( ( ti @ A @ C )
= ( zero_zero @ A ) ) )
& ( ( ( ti @ A @ C )
= ( zero_zero @ A ) )
=> ( ( ti @ A @ B )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(660,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_848_zle__int) ).
thf(3394,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).
thf(782,axiom,
! [A: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ pls ) )
<=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_990_rel__simps_I12_J) ).
thf(3812,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) ) )
& ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ pls ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ int @ int @ bit1 @ A ) ) @ pls ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).
thf(970,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_261_diff__diff__right) ).
thf(4501,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ C ) @ B ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).
thf(370,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C ) )
<=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_262_le__diff__conv) ).
thf(2372,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) ) ) )
& ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ C ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).
thf(1233,plain,
! [B: $i,A: $i] :
( ( B
= ( zero_zero @ nat ) )
| ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ),
inference(cnf,[status(esa)],[1232]) ).
thf(1235,plain,
! [B: $i,A: $i] :
( ( B
= ( zero_zero @ nat ) )
| ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ),
inference(lifteq,[status(thm)],[1233]) ).
thf(1236,plain,
! [B: $i,A: $i] :
( ( B
= ( zero_zero @ nat ) )
| ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ B ) @ A )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ ( one_one @ nat ) ) ) @ A ) ) ) ),
inference(simp,[status(thm)],[1235]) ).
thf(403,axiom,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( one_one @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_507_zero__le__one) ).
thf(2502,plain,
! [A: $i] :
( ( linordered_semidom @ A )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ ( one_one @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).
thf(330,axiom,
( ( zero_zero @ nat )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ pls ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_975_semiring__norm_I113_J) ).
thf(2241,plain,
( ( zero_zero @ nat )
= ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ pls ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).
thf(295,axiom,
! [A: $i] :
( ( hAPP @ int @ int @ bit1 @ A )
!= pls ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_978_rel__simps_I46_J) ).
thf(2127,plain,
! [A: $i] :
( ( hAPP @ int @ int @ bit1 @ A )
!= pls ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(829,axiom,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_add__right__mono) ).
thf(3974,plain,
! [A: $i] :
( ( ordere779506340up_add @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ C ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ C ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).
thf(203,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_258_diff__diff__left) ).
thf(1827,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ B ) ) @ C )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ A ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(435,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_813_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) ).
thf(2600,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).
thf(938,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ D ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ D ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_915_abs__mult__less) ).
thf(4382,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ D ) ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ D ) ) @ ( hAPP @ A @ A @ ( abs_abs @ A ) @ B ) ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).
thf(721,axiom,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_724_mult__right__le__one__le) ).
thf(3605,plain,
! [A: $i] :
( ( linordered_idom @ A )
=> ! [B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ B ) @ ( one_one @ A ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ C ) @ B ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).
thf(770,axiom,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ A ) @ B ) )
| ( hBOOL @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_fconj_3_1_U) ).
thf(3772,plain,
! [A: $i,B: $i] :
( ~ ( hBOOL @ ( hAPP @ bool @ bool @ ( hAPP @ bool @ ( fun @ bool @ bool ) @ fconj @ A ) @ B ) )
| ( hBOOL @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).
thf(1062,axiom,
! [A: $i] :
( ( ab_sem1668676832m_mult @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_608_mult__left__idem) ).
thf(4876,plain,
! [A: $i] :
( ( ab_sem1668676832m_mult @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) )
= ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).
thf(249,axiom,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_259_diff__cancel) ).
thf(1976,plain,
! [A: $i,B: $i,C: $i] :
( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C ) )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( minus_minus @ nat ) @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(1092,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_card__mono) ).
thf(4987,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).
thf(783,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_300_less__irrefl__nat) ).
thf(3817,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).
thf(111,axiom,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( zero_zero @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_938_int__nat__number__of) ).
thf(1500,plain,
! [A: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( zero_zero @ int ) ) )
& ( ~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) )
=> ( ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ int @ nat @ ( number_number_of @ nat ) @ A ) )
= ( hAPP @ int @ int @ ( number_number_of @ int ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(1109,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ com @ bool @ wt @ B ) )
=> ( ! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ pname @ bool ) @ bool @ ( hAPP @ pname @ ( fun @ ( fun @ pname @ bool ) @ bool ) @ ( member @ pname ) @ C ) @ u ) )
=> ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ A ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ C ) ) @ ( bot_bot @ ( fun @ x_a @ bool ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ A ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ com @ x_a @ mgt @ B ) ) @ ( bot_bot @ ( fun @ x_a @ bool ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_assms_I3_J) ).
thf(5060,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ com @ bool @ wt @ B ) )
=> ( ! [C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ pname @ bool ) @ bool @ ( hAPP @ pname @ ( fun @ ( fun @ pname @ bool ) @ bool ) @ ( member @ pname ) @ C ) @ u ) )
=> ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ A ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ pname @ x_a @ mgt_call @ C ) ) @ ( bot_bot @ ( fun @ x_a @ bool ) ) ) ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ x_a @ bool ) @ bool @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ ( fun @ x_a @ bool ) @ bool ) @ p @ A ) @ ( hAPP @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) @ ( hAPP @ x_a @ ( fun @ ( fun @ x_a @ bool ) @ ( fun @ x_a @ bool ) ) @ ( insert @ x_a ) @ ( hAPP @ com @ x_a @ mgt @ B ) ) @ ( bot_bot @ ( fun @ x_a @ bool ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1109]) ).
thf(913,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_425_card__psubset) ).
thf(4271,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).
thf(863,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_700_nat__0__less__mult__iff) ).
thf(4088,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
& ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).
thf(298,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) )
<=> ( B = C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_596_Suc__mult__cancel1) ).
thf(2138,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) )
=> ( B = C ) )
& ( ( B = C )
=> ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ B )
= ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ ( hAPP @ nat @ nat @ suc @ A ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(643,axiom,
! [A: $i] :
( ( linord893533164strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_684_comm__mult__strict__left__mono) ).
thf(3341,plain,
! [A: $i] :
( ( linord893533164strict @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ C ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ C ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ B ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).
thf(825,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
<=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_singleton__iff) ).
thf(3960,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
=> ( ( ti @ A @ B )
= ( ti @ A @ C ) ) )
& ( ( ( ti @ A @ B )
= ( ti @ A @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ C ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).
thf(485,axiom,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_715_mult__strict__mono_H) ).
thf(2754,plain,
! [A: $i] :
( ( linord20386208strict @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ ( zero_zero @ A ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( times_times @ A ) @ E ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).
thf(278,axiom,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_569_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) ).
thf(2074,plain,
! [A: $i] :
( ( comm_semiring_1 @ A )
=> ! [B: $i] :
( ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ B ) @ ( zero_zero @ A ) )
= ( ti @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(391,axiom,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i] :
( ! [E: $i] : ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ C ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_le__funI) ).
thf(2453,plain,
! [A: $i,B: $i] :
( ( ord @ B )
=> ! [C: $i,D: $i] :
( ! [E: $i] : ( hBOOL @ ( hAPP @ B @ bool @ ( hAPP @ B @ ( fun @ B @ bool ) @ ( ord_less_eq @ B ) @ ( hAPP @ A @ B @ C @ E ) ) @ ( hAPP @ A @ B @ D @ E ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ B ) @ bool @ ( hAPP @ ( fun @ A @ B ) @ ( fun @ ( fun @ A @ B ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ B ) ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).
thf(960,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) @ D ) )
<=> ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_insert__code) ).
thf(4459,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) @ D ) )
=> ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) ) ) )
& ( ( ( ( ti @ A @ B )
= ( ti @ A @ D ) )
| ( hBOOL @ ( hAPP @ A @ bool @ C @ D ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ ( fun @ A @ bool ) ) @ ( insert @ A ) @ B ) @ C ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).
thf(756,axiom,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ti @ A @ C )
= ( ti @ A @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_319_xt1_I1_J) ).
thf(3721,plain,
! [A: $i] :
( ( order @ A )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ti @ A @ C )
= ( ti @ A @ D ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).
thf(629,axiom,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_392_add__le__less__mono) ).
thf(3296,plain,
! [A: $i] :
( ( ordere223160158up_add @ A )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less_eq @ A ) @ D ) @ E ) )
=> ( ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( hAPP @ A @ ( fun @ A @ bool ) @ ( ord_less @ A ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ D ) @ B ) ) @ ( hAPP @ A @ A @ ( hAPP @ A @ ( fun @ A @ A ) @ ( plus_plus @ A ) @ E ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).
thf(1027,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( B != A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_301_less__not__refl2) ).
thf(4734,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).
thf(520,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( one_one @ int ) ) ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_782_zless__imp__add1__zle) ).
thf(2881,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ A ) @ B ) )
=> ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( plus_plus @ int ) @ A ) @ ( one_one @ int ) ) ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).
thf(729,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ A @ bool @ ( bot_bot @ ( fun @ A @ bool ) ) @ B ) )
<=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_532_bot__empty__eq) ).
thf(3626,plain,
! [A: $i,B: $i] :
( ( ( hBOOL @ ( hAPP @ A @ bool @ ( bot_bot @ ( fun @ A @ bool ) ) @ B ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) ) )
& ( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ A @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( member @ A ) @ B ) @ ( bot_bot @ ( fun @ A @ bool ) ) ) )
=> ( hBOOL @ ( hAPP @ A @ bool @ ( bot_bot @ ( fun @ A @ bool ) ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).
thf(318,axiom,
! [A: $i] :
( ( mult_zero @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tsy_c_Groups_Otimes__class_Otimes_4_res) ).
thf(2206,plain,
! [A: $i] :
( ( mult_zero @ A )
=> ( ( ti @ ( fun @ A @ ( fun @ A @ A ) ) @ ( times_times @ A ) )
= ( times_times @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).
thf(170,axiom,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_855_Nat__Transfer_Otransfer__int__nat__functions_I2_J) ).
thf(1737,plain,
! [A: $i,B: $i] :
( ( hAPP @ int @ int @ ( hAPP @ int @ ( fun @ int @ int ) @ ( times_times @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ B ) )
= ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(1175,axiom,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( zero_zero @ int ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_831_int__less__0__conv) ).
thf(5261,plain,
! [A: $i] :
~ ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less @ int ) @ ( hAPP @ nat @ int @ ( semiring_1_of_nat @ int ) @ A ) ) @ ( zero_zero @ int ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1175]) ).
thf(981,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) ) @ ( hAPP @ ( fun @ B @ bool ) @ nat @ ( finite_card @ B ) @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_card__image__le) ).
thf(4547,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ B @ bool ) @ bool @ ( finite_finite @ B ) @ D ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ ( hAPP @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) @ ( hAPP @ ( fun @ B @ A ) @ ( fun @ ( fun @ B @ bool ) @ ( fun @ A @ bool ) ) @ ( image @ B @ A ) @ C ) @ D ) ) ) @ ( hAPP @ ( fun @ B @ bool ) @ nat @ ( finite_card @ B ) @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).
thf(1033,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_419_psubset__imp__subset) ).
thf(4752,plain,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less @ ( fun @ A @ bool ) ) @ B ) @ C ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( hAPP @ ( fun @ A @ bool ) @ ( fun @ ( fun @ A @ bool ) @ bool ) @ ( ord_less_eq @ ( fun @ A @ bool ) ) @ B ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).
thf(5255,plain,
~ ( hBOOL @ ( hAPP @ int @ bool @ nat_neg @ ( hAPP @ int @ int @ ( number_number_of @ int ) @ pls ) ) ),
inference(polarity_switch,[status(thm)],[5254]) ).
thf(636,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_534_card__ge__0__finite) ).
thf(3316,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ ( hAPP @ ( fun @ A @ bool ) @ nat @ ( finite_card @ A ) @ B ) ) )
=> ( hBOOL @ ( hAPP @ ( fun @ A @ bool ) @ bool @ ( finite_finite @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).
thf(366,axiom,
! [A: $i,B: $i,C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) )
<=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_733_mult__le__cancel1) ).
thf(2351,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) )
=> ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) ) )
& ( ( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ A ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ B ) @ C ) ) )
=> ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less_eq @ nat ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ B ) ) @ ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( times_times @ nat ) @ A ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).
thf(3,axiom,
ring @ int,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oring) ).
thf(1204,plain,
ring @ int,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(93,axiom,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ nat @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) @ ( semiri532925092at_aux @ A ) @ B ) @ ( zero_zero @ nat ) ) @ C )
= ( ti @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_891_of__nat__aux_Osimps_I1_J) ).
thf(1449,plain,
! [A: $i] :
( ( semiring_1 @ A )
=> ! [B: $i,C: $i] :
( ( hAPP @ A @ A @ ( hAPP @ nat @ ( fun @ A @ A ) @ ( hAPP @ ( fun @ A @ A ) @ ( fun @ nat @ ( fun @ A @ A ) ) @ ( semiri532925092at_aux @ A ) @ B ) @ ( zero_zero @ nat ) ) @ C )
= ( ti @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(1077,axiom,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
& ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_552_less__imp__add__positive) ).
thf(4940,plain,
! [A: $i,B: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ A ) @ B ) )
=> ? [C: $i] :
( ( hBOOL @ ( hAPP @ nat @ bool @ ( hAPP @ nat @ ( fun @ nat @ bool ) @ ( ord_less @ nat ) @ ( zero_zero @ nat ) ) @ C ) )
& ( ( hAPP @ nat @ nat @ ( hAPP @ nat @ ( fun @ nat @ nat ) @ ( plus_plus @ nat ) @ A ) @ C )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).
thf(553,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( hBOOL @ B )
<=> ( hBOOL @ A ) ) )
=> ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ B ) )
<=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_817_conj__le__cong) ).
thf(3035,plain,
! [A: $i,B: $i,C: $i] :
( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
=> ( ( ( hBOOL @ B )
=> ( hBOOL @ A ) )
& ( ( hBOOL @ A )
=> ( hBOOL @ B ) ) ) )
=> ( ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ B ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ A ) ) )
& ( ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ A ) )
=> ( ( hBOOL @ ( hAPP @ int @ bool @ ( hAPP @ int @ ( fun @ int @ bool ) @ ( ord_less_eq @ int ) @ ( zero_zero @ int ) ) @ C ) )
& ( hBOOL @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).
thf(41687,plain,
$false,
inference(cvc4,[status(thm)],[1586,1411,5116,4094,2778,4022,3167,2412,2463,2544,2331,4728,3172,2250,3021,1322,5088,4835,2295,4450,4005,4571,3053,2109,2131,2031,4262,5329,4062,3399,1995,4899,3798,3698,3979,3830,2595,1850,2480,3969,1982,2559,4015,1956,1950,1750,1211,4339,4075,3858,5242,1939,1591,2837,4215,1305,1437,3382,2077,4043,3534,4037,2512,1924,2210,2104,5225,3762,2644,1808,3417,3912,5625,2671,2676,4603,1205,1793,2854,5237,4408,4973,1486,4532,3017,4234,3120,3745,2686,3367,4026,3338,2434,3781,3313,2844,4762,4220,3175,1275,5316,1327,3693,1761,1342,3481,1507,1222,3717,2360,3370,5123,3249,1613,1921,3809,2825,3880,2405,2957,3634,4142,5252,2432,3994,2491,1243,2020,3513,4679,4618,3088,3777,3308,2010,1935,4916,1359,3140,1475,3878,1904,1228,1623,5024,5342,1887,4956,4415,4761,1323,2274,3845,1992,3820,3891,2654,4238,3204,3263,3221,3976,2328,3938,4743,4563,1985,4070,2667,4692,5284,3689,4224,1871,1907,5274,3617,4334,2217,3125,5150,1313,4349,1279,4275,5130,5160,1455,4766,1462,3855,2420,3099,3517,4179,1479,4928,2929,3014,3951,1773,5091,4771,1847,4756,3840,5256,2857,4862,2169,3638,1514,4194,1247,5127,2552,1319,1890,1741,1398,4317,4551,1931,2024,3835,2989,5076,1362,4507,3557,2178,1664,4583,4173,3724,1911,2650,3527,5032,1960,4783,1489,4696,4205,1415,5264,3322,2616,1946,1465,4443,3075,1811,4018,5027,2893,2469,3670,4001,1418,2451,1858,3438,4147,3525,4815,4724,3357,2254,3569,2540,4871,4810,4162,3389,2058,1215,2483,4504,1351,4825,4040,3226,2965,4892,4588,3378,2367,2437,3537,4033,2833,3189,3739,2281,2530,2523,1796,4996,5296,2772,3157,5246,5321,3676,1345,2639,2103,2750,1405,3406,2205,4082,4065,3097,1206,5157,3301,3531,3186,5142,5291,3318,4370,2641,1730,5298,3435,3582,3229,3450,3503,2910,4323,3591,3567,1691,3984,3169,4480,3265,2646,3852,1306,5167,2118,3767,2232,4576,2083,4031,4100,4312,2660,4059,1868,2382,3550,5281,1636,2589,2541,4989,1751,4344,4499,1824,3682,1238,1274,2673,2705,2264,1223,4712,3711,3972,1255,4434,1217,5266,3752,3989,4128,3884,2937,2890,2414,4661,2528,3191,1837,4140,4608,4994,2872,3718,2200,4187,2213,1281,1365,2628,3403,3090,2332,2820,3600,1687,3894,2068,1819,1624,4402,1836,1787,5113,2181,4979,1210,1656,4417,3050,2547,2344,3931,1740,1910,3595,5066,1402,2157,4974,2841,1227,5081,5313,4123,5189,3467,1592,3909,1541,3482,5145,1762,3254,1772,5098,2377,4668,1534,1953,2895,4248,2973,4486,4789,4048,4544,2692,1587,2694,4845,2726,2631,2724,4227,2537,4840,3078,4883,3650,4012,3440,3205,4676,3954,4951,4518,2323,3846,3210,3094,3714,2080,4191,1207,1612,1744,2460,1493,4320,1683,1964,1659,1292,3622,5107,2830,4421,1424,2312,2455,4750,5213,2613,1937,2172,4648,1214,3194,2349,3643,4444,2781,4530,1419,1588,4384,2681,5374,3059,3743,5234,1932,2465,4829,2006,2813,4953,2756,3083,2968,5063,1324,2504,2836,1456,6331,2153,1231,2851,2243,2566,5058,1799,1947,4120,1874,3115,5305,3967,3247,3409,2862,2766,2697,1246,3064,1720,4259,3109,1979,4689,3005,3920,3364,4361,1525,3476,3755,5310,4644,2702,3624,2883,5415,3306,1286,5417,3508,2338,3807,5170,3147,3454,1857,4184,4352,2203,4212,3587,4906,4448,2793,1663,2089,5198,1616,3422,2121,1278,2729,2001,1680,1698,2585,1548,2602,5016,4453,3343,1727,5327,1510,1251,4919,3130,4470,1754,4613,2275,1219,3607,3491,1483,4256,1747,2028,1301,3375,1595,3888,3523,2734,4269,5647,4786,3070,3925,2789,3081,3272,4709,3900,1535,5097,2716,1965,4946,2827,3387,4998,3741,3547,1833,1784,1389,2525,3609,2657,2849,5121,2489,4926,2514,1701,4328,3868,4769,3223,4537,3023,4096,3726,3002,3700,2557,2425,5033,2220,5118,2293,1312,4112,3208,1865,4573,1239,5148,3123,2442,2759,3487,2055,2166,3579,4273,1258,3355,1224,2358,2319,1356,2573,2148,2510,3678,2521,2679,1282,4305,3779,2284,2955,3197,4300,3066,3384,1571,2065,1209,2721,5071,3996,3631,1408,3155,2229,2214,5244,5103,2493,3957,3515,4914,3270,2669,5222,2116,2839,4176,3138,2197,4477,3160,1307,1667,2025,2742,5175,3614,4296,4726,2278,4684,4694,4203,3747,5035,1877,1339,4445,3133,4103,2343,3668,4911,2111,3073,2261,1731,3800,3572,5286,3165,1862,4181,6319,3298,2874,2542,5180,2578,1688,2770,5254,4232,1528,2086,1643,3917,5248,3674,4827,3241,3310,5139,2867,1213,3759,3838,2051,5094,2244,2752,1775,2592,3325,3142,4703,2535,1364,1250,1496,4196,2984,1973,4158,1805,2005,2054,4405,2096,1304,2665,2124,4879,2449,5133,3849,1626,4331,3009,3936,1758,2107,1203,1208,2784,4456,2652,3293,3672,1420,3981,1684,3048,4566,5165,2597,4581,4336,1551,4658,1547,1770,1516,4864,3112,5125,4937,1225,3949,2518,1531,1230,2689,1961,4035,6453,1605,3260,3833,3659,2732,1252,1368,2175,4287,4469,4754,2386,2407,3786,4894,4961,4971,4377,4593,2888,3246,4314,4561,2920,4003,4549,4222,5152,3412,2044,4117,1336,2747,1499,1790,3940,5418,1813,1452,3691,3791,3823,2337,5030,5154,5203,3774,1830,1763,1538,4241,1285,1218,5186,1220,5208,3652,4959,2992,1427,1670,1717,2939,3520,1404,5319,2247,1543,1766,2113,3415,1802,3493,4611,4976,4246,5240,1221,4488,1353,4748,2145,1972,2320,4991,4235,1585,4634,4838,5083,3914,5100,3007,2266,4516,4150,4392,3288,1240,4217,3452,1844,2194,4738,3665,1421,2924,4651,3620,1918,4424,1316,3565,3401,3092,1459,4325,1949,4165,4419,5386,4944,2347,1412,2611,2325,3928,1216,2209,4646,5308,2187,3346,3320,1390,2305,4170,2342,1289,3056,2226,5332,2071,2775,1734,5068,5184,1998,3593,1646,2707,1385,3484,2532,5228,2971,3252,2389,1476,5272,2290,4615,2251,1903,4666,2135,5382,2554,4468,2823,4542,4715,4791,1629,3104,2608,3235,1244,1822,4010,4495,3136,4881,3662,4527,2934,2152,4745,2009,4153,1777,4759,3149,3896,4387,3529,3351,4869,3420,2384,2475,4855,2786,2796,5162,5532,2399,2041,3267,4774,3598,2680,2183,1212,2902,5211,1893,1309,3645,2739,4098,1444,5196,1348,2287,1512,3232,5268,4596,1546,3464,1986,1673,2218,5051,2302,5009,2683,1578,4399,4687,4706,3860,3511,3865,2719,5004,3733,2951,5078,5371,1248,4663,3496,4491,1757,2768,1480,1859,1816,2186,4427,4024,1561,2913,2021,5324,3145,2636,4029,3328,4303,3897,4777,1989,4051,4981,2334,3560,4968,4007,3479,2417,2714,3426,2223,1473,5215,2619,3424,3045,3843,5200,5110,3805,3244,3362,3128,2430,1378,4118,3489,2507,3392,4395,1880,3394,3812,4501,2372,1236,2502,2241,2127,3974,1827,2600,4382,3605,3772,4876,1976,4987,3817,1500,5060,4271,4088,2138,3341,3960,2754,2074,2453,4459,3721,3296,4734,2881,3626,2206,1737,5261,4547,4752,5255,3316,2351,1204,1449,4940,3035]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.10 % Problem : SWW473+7 : TPTP v8.1.2. Released v5.3.0.
% 0.10/0.12 % Command : run_Leo-III %s %d
% 0.12/0.33 % Computer : n016.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu May 18 23:11:12 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.92/0.87 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.03/1.20 % [INFO] Parsing done (325ms).
% 2.03/1.21 % [INFO] Running in sequential loop mode.
% 2.46/1.44 % [INFO] eprover registered as external prover.
% 2.46/1.44 % [INFO] cvc4 registered as external prover.
% 2.46/1.45 % [INFO] Scanning for conjecture ...
% 3.50/1.69 % [INFO] Found a conjecture and 1200 axioms. Running axiom selection ...
% 4.87/2.04 % [INFO] Axiom selection finished. Selected 1200 axioms (removed 0 axioms).
% 7.33/2.95 % [INFO] Problem is first-order (TPTP FOF).
% 7.33/2.99 % [INFO] Type checking passed.
% 7.33/2.99 % [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 ...
% 201.42/52.60 % External prover 'cvc4' found a proof!
% 201.42/52.60 % [INFO] Killing All external provers ...
% 201.42/52.60 % Time passed: 52131ms (effective reasoning time: 51386ms)
% 201.42/52.60 % 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)>
% 201.42/52.60 % Axioms used in derivation (1200): fact_790_nat__diff__add__eq1, fact_224_add__left__imp__eq, fact_78_insert__commute, arity_HOL_Obool___Finite__Set_Ofinite, fact_464_zero__neq__one, fact_675_mult__pos__neg, arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add, fact_933_not__neg__int, fact_178_diff__eq__diff__eq, arity_Int_Oint___Rings_Oordered__semiring, fact_122_finite__Diff, fact_44_diff__le__mono2, fact_837_int__1, fact_426_finite__conv__nat__seg__image, fact_55_zero__induct__lemma, help_fimplies_2_1_U, fact_400_Suc__leI, fact_195_Diff__iff, fact_905_not__neg__eq__ge__0, fact_665_mult__nonneg__nonpos, fact_186_finite__less__ub, fact_762_int__0__less__1, fact_688_sum__squares__eq__zero__iff, fact_507_zero__le__one, fact_561_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J, fact_482_less__nat__zero__code, fact_579_mult__zero__left, fact_928_not__neg__0, fact_48_lift__Suc__mono__le, arity_Nat_Onat___Orderings_Obot, fact_610_nat__mult__eq__1__iff, tsy_c_fconj_res, fact_908_abs__ge__zero, fact_406_diff__less__mono, fact_898_fold__image__eq__general__inverses, fact_659_comm__mult__left__mono, fact_422_psubset__card__mono, fact_458_less__eq__nat_Osimps_I1_J, fact_831_int__less__0__conv, fact_360_not__less__less__Suc__eq, arity_HOL_Obool___Orderings_Oorder, arity_Nat_Onat___Groups_Omonoid__mult, fact_896_int__diff__cases, fact_232_nat__add__right__cancel, fact_657_mult__left__mono__neg, fact_705_mult__eq__self__implies__10, fact_772_zle__linear, fact_164_Collect__conv__if, fact_901_abs__triangle__ineq4, fact_149_Diff__mono, fact_596_Suc__mult__cancel1, fact_798_number__of__is__id, fact_845_less__imp__of__nat__less, arity_Int_Oint___Groups_Oone, fact_810_int__ge__induct, tsy_c_Orderings_Obot__class_Obot_res, arity_Nat_Onat___Rings_Oordered__semiring, tsy_c_Nat_OSuc_res, fact_723_mult__left__le__one__le, fact_228_nat__add__commute, fact_71_insertI1, fact_608_mult__left__idem, fact_965_add__special_I3_J, fact_257_diff__add__inverse, help_COMBS_1_1_U, fact_984_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J, fact_667_mult__le__0__iff, fact_631_left__distrib__number__of, fact_542_card__less, arity_Int_Oint___Groups_Oordered__comm__monoid__add, fact_783_zless__add1__eq, fact_263_le__add__diff, fact_523_One__nat__def, fact_47_finite__subset__image, fact_941_nat0__intermed__int__val, fact_518_zero__less__diff, fact_63_xt1_I3_J, fact_88_in__mono, tsy_c_Nat_Osemiring__1__class_Oof__nat__aux_res, fact_264_le__add__diff__inverse, fact_225_add__imp__eq, fact_104_subset__insert, fact_591_combine__common__factor, arity_Nat_Onat___Groups_Oab__semigroup__mult, fact_876_zmult__zless__mono2__lemma, fact_498_add__nonpos__nonpos, fact_180_diff__card__le__card__Diff, fact_381_less__imp__le__nat, fact_395_less__iff__Suc__add, fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J, fact_954_numeral__1__eq__Suc__0, tsy_v_na_res, arity_Int_Oint___Rings_Olinordered__ring__strict, arity_HOL_Obool___Orderings_Oord, fact_208_diff__Suc__eq__diff__pred, fact_678_zero__less__mult__pos2, fact_899_of__nat__number__of__eq, fact_990_rel__simps_I12_J, fact_243_add__Suc, arity_Int_Oint___Groups_Oordered__ab__semigroup__add, arity_Int_Oint___Rings_Olinordered__idom, fact_304_order__less__irrefl, fact_13_card__Collect__le__nat, fact_295_Suc__mono, arity_Int_Oint___Int_Oring__char__0, fact_563_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J, fact_590_comm__semiring__class_Odistrib, fact_633_Ints__1, arity_Int_Oint___Rings_Olinordered__semiring, fact_983_mult__Pls, fact_174_nat__case__Suc, arity_Int_Oint___Rings_Oordered__cancel__semiring, arity_HOL_Obool___Orderings_Opreorder, fact_700_nat__0__less__mult__iff, fact_645_crossproduct__noteq, fact_328_linorder__le__less__linear, fact_325_linorder__cases, fact_679_mult__less__cancel__left__neg, tsy_c_COMBK_res, help_fconj_3_1_U, fact_437_order__less__le__subst1, fact_725_le__add__iff2, fact_750_zmult__assoc, arity_Nat_Onat___Rings_Osemiring__1, arity_fun___Orderings_Obot, fact_985_rel__simps_I19_J, arity_Int_Oint___Orderings_Oorder, fact_492_zero__le__double__add__iff__zero__le__single__add, fact_255_add__leE, fact_930_abs__zero, fact_163_bot__least, fact_757_zmult__zless__mono2, fact_465_add_Ocomm__neutral, fact_524_diff__Suc, fact_616_number__of__mult, fact_58_linorder__linear, fact_417_subset__psubset__trans, fact_72_insert__compr, fact_607_mult__idem, tsy_c_Int_OPls_res, fact_520_diff__add__0, fact_485_Suc__neq__Zero, arity_Nat_Onat___Rings_Olinordered__semiring, fact_445_image__cong, fact_935_neg__imp__number__of__eq__0, fact_654_split__mult__pos__le, fact_724_mult__right__le__one__le, fact_727_less__add__iff2, fact_135_empty__def, fact_598_add__mult__distrib2, fact_179_card__Diff__subset, fact_922_abs__mult__self, fact_853_int__mult, fact_676_mult__pos__neg2, fact_555_double__eq__0__iff, fact_499_zero__less__double__add__iff__zero__less__single__add, fact_721_not__sum__squares__lt__zero, arity_Int_Oint___Groups_Ominus, fact_207_diff__Suc__1, fact_211_card__Diff__insert, fact_432_psubset__trans, fact_38_diff__Suc__Suc, fact_272_Suc__eq__plus1__left, fact_536_diff__Suc__less, fact_599_mult__le__mono, fact_326_linorder__not__less, fact_522_diff__is__0__eq_H, fact_283_If__def, fact_15_finite, tsy_c_fimplies_res, fact_808_fold__image__cong, fact_261_diff__diff__right, fact_652_add__number__of__diff1, fact_367_Suc__less__SucD, fact_832_of__nat__0__le__iff, fact_175_image__constant, fact_770_zadd__commute, fact_538_predicate1D, tsy_c_member_res, fact_427_bounded__nat__set__is__finite, fact_521_diff__is__0__eq, conj_3, fact_142_folding__image__simple__idem_Oidem, fact_978_rel__simps_I46_J, tsy_c_Groups_Otimes__class_Otimes_0_res, fact_62_ord__eq__le__trans, fact_493_double__add__le__zero__iff__single__add__le__zero, fact_565_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J, fact_866_transfer__int__nat__quantifiers_I2_J, fact_371_trans__less__add1, arity_Int_Oint___Rings_Olinordered__ring, fact_868_int__le__0__conv, fact_5_card__mono, fact_93_image__iff, fact_341_order__le__neq__trans, arity_Int_Oint___Groups_Ocomm__monoid__add, fact_37_Suc__diff__diff, fact_695_mult__eq__1__iff, help_COMBI_1_1_U, fact_483_neq0__conv, tsy_c_Groups_Oplus__class_Oplus_0_res, fact_497_add__increasing2, fact_247_le__iff__add, fact_793_nat__diff__add__eq2, fact_187_the__elem__eq, fact_218_folding__one__idem_Oinsert__idem, fact_613_nat__mult__1, fact_571_card_Oeq__fold__g, fact_39_le__diff__iff, fact_128_Diff__empty, arity_fun___Orderings_Oord, fact_227_folding__one__idem_Oidem, fact_53_insertCI, fact_704_Suc__mult__le__cancel1, arity_Int_Oint___Orderings_Oord, fact_950_neg__number__of__Bit1, fact_252_add__le__mono, fact_572_card__def, fact_552_less__imp__add__positive, fact_936_eq__nat__number__of, fact_820_minf_I8_J, fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J, fact_409_mono__nat__linear__lb, fact_2_finite__imageI, fact_556_card__eq__SucD, fact_327_linorder__not__le, fact_886_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J, fact_666_mult__nonneg__nonneg, fact_18_Suc__n__not__n, fact_527_add__strict__increasing, fact_859_Nat__Transfer_Otransfer__int__nat__functions_I1_J, fact_292_card__Diff2__less, arity_Int_Oint___Orderings_Opreorder, fact_336_xt1_I12_J, fact_333_not__leE, arity_sum___Finite__Set_Ofinite, fact_918_abs__ge__self, arity_Nat_Onat___Rings_Oordered__comm__semiring, fact_192_DiffD2, fact_239_add__le__imp__le__left, arity_Nat_Onat___Int_Onumber__semiring, arity_Nat_Onat___Orderings_Opreorder, fact_844_of__nat__less__iff, arity_Int_Oint___Rings_Oordered__ring, arity_HOL_Obool___Groups_Ominus, fact_294_lessI, tsy_c_Nat_Onat_Onat__case_res, fact_236_add__left__mono, fact_977_rel__simps_I51_J, fact_685_mult__strict__right__mono__neg, fact_440_xt5, fact_902_abs__diff__triangle__ineq, fact_350_add__strict__left__mono, tsy_c_Finite__Set_Ofold__image_res, fact_19_n__not__Suc__n, fact_564_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J, fact_972_add__numeral__0, fact_330_less__le__not__le, fact_146_insert__Diff1, fact_29_rev__finite__subset, fact_618_arith__simps_I32_J, fact_729_n__less__m__mult__n, fact_425_card__psubset, fact_238_add__le__imp__le__right, fact_73_insert__Collect, fact_987_rel__simps_I34_J, fact_75_mem__def, fact_320_ord__less__eq__trans, arity_Int_Oint___Rings_Oring, fact_535_Suc__pred, arity_Int_Oint___Rings_Ono__zero__divisors, fact_506_not__one__le__zero, fact_517_card__infinite, fact_960_one__is__num__one, fact_971_semiring__norm_I112_J, fact_992_less__int__code_I16_J, fact_914_abs__triangle__ineq2__sym, fact_118_less__eq__nat_Osimps_I2_J, arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, fact_284_the__eq__trivial, fact_852_zmult__int, fact_648_crossproduct__eq, fact_495_add__nonneg__eq__0__iff, tsy_c_Set_OCollect_res, fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J, fact_30_Suc__leD, fact_582_no__zero__divisors, fact_906_abs__of__nonneg, fact_474_zero__reorient, fact_796_zdiv__mono2__neg__lemma, fact_305_linorder__neq__iff, tsy_c_Int_Onumber__class_Onumber__of_res, tsy_c_hAPP_res, fact_768_zadd__assoc, fact_714_mult__less__le__imp__less, fact_203_card__Diff__singleton__if, fact_467_double__zero__sym, fact_411_less__imp__Suc__add, fact_880_tsub__eq, fact_680_mult__neg__pos, tsy_c_Groups_Oplus__class_Oplus_1_res, fact_979_rel__simps_I39_J, fact_256_diff__add__inverse2, arity_Int_Oint___Groups_Oab__group__add, fact_686_mult__strict__left__mono__neg, tsy_c_hAPP_arg1, arity_Int_Oint___Groups_Oordered__ab__group__add__abs, fact_449_finite__Plus__iff, arity_Int_Oint___Int_Onumber__semiring, fact_230_nat__add__assoc, fact_741_fold__image__1, fact_94_imageI, arity_Int_Oint___Int_Onumber__ring, fact_575_Ints__odd__less__0, fact_46_finite__surj, arity_Int_Oint___Rings_Ocomm__semiring, fact_68_xt1_I5_J, fact_298_nat__neq__iff, fact_540_nat__diff__split__asm, fact_205_one__reorient, fact_861_transfer__nat__int__set__relations_I1_J, fact_576_expand__Suc, tsy_c_Orderings_Oord__class_Oless__eq_res, fact_182_assms_I2_J, fact_393_less__add__Suc1, fact_601_mult__le__mono1, fact_515_add__gr__0, fact_718_mult__le__cancel__left__pos, fact_847_zless__int, fact_594_mult__1, fact_223_add__right__cancel, fact_703_mult__Suc, arity_Int_Oint___Groups_Ocancel__ab__semigroup__add, fact_379_nat__less__le, fact_920_abs__mult__pos, fact_944_succ__def, help_fNot_2_1_U, fact_31_le__SucE, fact_354_not__less__bot, fact_749_times__numeral__code_I5_J, fact_771_zle__refl, fact_22_eq__imp__le, tsy_c_COMBC_res, fact_508_zero__less__one, fact_696_mult__less__mono2, fact_279_termination__basic__simps_I4_J, fact_89_set__rev__mp, fact_171_finite__Diff__insert, fact_549_card__Suc__eq, fact_606_times_Oidem, fact_969_semiring__numeral__0__eq__0, fact_240_diff__add__cancel, tsy_v_mgt_res, arity_Nat_Onat___Groups_Ocomm__monoid__add, fact_794_nat__eq__add__iff2, fact_115_finite__nat__set__iff__bounded__le, fact_144_subset__insert__iff, fact_973_add__numeral__0__right, fact_924_abs__add__abs, fact_376_add__lessD1, fact_568_add__0__iff, fact_728_less__add__iff1, fact_926_not__neg__1, fact_909_abs__not__less__zero, fact_414_less__add__one, help_fNot_1_1_U, arity_Int_Oint___Rings_Osemiring__1, fact_702_mult__Suc__right, fact_588_mult__cancel2, fact_428_less__mono__imp__le__mono, fact_689_less__1__mult, fact_840_Ints__of__nat, fact_869_int__Suc0__eq__1, tsy_c_COMBS_res, fact_383_less__or__eq__imp__le, fact_894_nonneg__eq__int, arity_Nat_Onat___Rings_Omult__zero, fact_373_add__less__mono1, fact_929_abs__eq__0, fact_873_of__nat__Suc, fact_448_finite__Plus, fact_270_add__diff__assoc2, fact_967_zero__is__num__zero, tsy_v_wt_res, fact_849_zadd__int, fact_699_mult__less__cancel1, fact_442_order__le__less__subst2, arity_Nat_Onat___Rings_Olinordered__semiring__strict, fact_229_nat__add__left__commute, fact_962_semiring__numeral__1__eq__1, fact_244_add__Suc__shift, tsy_v_U_res, fact_530_add__nonpos__neg, fact_644_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J, fact_893_nonneg__int__cases, help_fFalse_1_1_T, fact_41_eq__diff__iff, fact_347_add__less__cancel__right, fact_355_bot__less, fact_166_singleton__conv, fact_377_termination__basic__simps_I1_J, fact_181_finite__subset__induct, fact_54_insertE, arity_Int_Oint___Rings_Olinordered__semiring__strict, fact_529_add__neg__nonpos, fact_194_Diff__idemp, tsy_c_fFalse_res, fact_711_mult__less__imp__less__right, fact_74_ext, fact_697_mult__less__mono1, fact_732_one__le__mult__iff, fact_253_add__leD2, fact_717_mult__le__cancel__left__neg, fact_310_less__imp__neq, fact_532_bot__empty__eq, fact_165_Collect__conv__if2, fact_130_empty__Collect__eq, fact_469_add__0__left, fact_36_Suc__n__not__le__n, fact_365_less__SucE, fact_258_diff__diff__left, fact_865_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J, fact_807_int__le__induct, fact_656_mult__mono_H, fact_147_insert__Diff__if, fact_478_Nat_Oadd__0__right, fact_60_order__eq__refl, fact_266_le__diff__conv2, fact_450_Plus__eq__empty__conv, fact_855_Nat__Transfer_Otransfer__int__nat__functions_I2_J, fact_335_order__neq__le__trans, tsy_c_hBOOL_arg1, fact_351_add__strict__mono, fact_934_abs__int__eq, fact_402_Suc__le__lessD, fact_940_int__val__lemma, tsy_c_Groups_Otimes__class_Otimes_1_res, fact_214_folding__image__simple_Oneutral, arity_Int_Oint___Rings_Olinordered__semiring__1, fact_10_finite__Collect__conjI, fact_943_Suc__nat__number__of__add, fact_303_nat__less__cases, fact_470_right__minus__eq, fact_775_zle__diff1__eq, fact_134_bot__apply, fact_267_le__add__diff__inverse2, fact_825_minusinfinity, fact_858_Nat__Transfer_Otransfer__int__nat__functions_I3_J, fact_923_abs__minus__commute, fact_454_less__zeroE, fact_156_empty__not__insert, fact_569_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J, tsy_c_Groups_Otimes__class_Otimes_4_res, fact_138_Diff__insert__absorb, fact_879_zdiff__int__split, fact_262_le__diff__conv, fact_226_add__right__imp__eq, tsy_c_Finite__Set_Ofinite_res, fact_359_Suc__less__eq, fact_349_add__strict__right__mono, help_COMBB_1_1_U, fact_111_imageE, help_If_1_1_T, fact_334_linorder__antisym__conv1, fact_910_zero__less__abs__iff, fact_447_finite__PlusD_I1_J, fact_562_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J, fact_82_insertI2, fact_155_insert__not__empty, fact_951_not__neg__number__of__Pls, arity_Int_Oint___Rings_Oring__1, fact_366_Suc__lessD, conj_2, fact_787_nat__mult__less__cancel1, fact_634_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J, fact_7_card__insert__le, fact_791_nat__eq__add__iff1, tsy_c_Finite__Set_Ocomp__fun__idem_res, tsy_c_Groups_Otimes__class_Otimes_2_res, fact_986_rel__simps_I22_J, fact_500_double__add__less__zero__iff__single__add__less__zero, fact_99_le__fun__def, fact_248_nat__add__left__cancel__le, fact_778_zadd__strict__right__mono, fact_352_add__less__imp__less__right, fact_388_nat__seg__image__imp__finite, fact_611_nat__mult__1__right, fact_321_xt1_I2_J, tsy_c_hAPP_arg2, fact_151_singleton__inject, fact_710_mult__left__less__imp__less, arity_fun___Orderings_Oorder, fact_584_mult__0, fact_823_pinf_I8_J, fact_526_add__nonneg__pos, fact_23_nat__le__linear, fact_153_doubleton__eq__iff, arity_Nat_Onat___Orderings_Olinorder, tsy_c_Set_Othe__elem_res, fact_274_folding__one__idem_Oin__idem, help_fdisj_1_1_U, fact_461_minus__nat_Odiff__0, fact_774_zle__antisym, fact_854_of__nat__mult, fact_484_not__less0, fact_746_zmult__1, fact_475_bot__nat__def, fact_504_le__iff__diff__le__0, fact_795_nat__less__add__iff1, fact_534_card__ge__0__finite, fact_76_Collect__def, arity_fun___Orderings_Opreorder, fact_505_less__iff__diff__less__0, fact_57_order__refl, fact_415_less__fun__def, fact_490_Zero__not__Suc, fact_501_add__pos__pos, help_fconj_2_1_U, fact_441_xt7, fact_557_nat_Oexhaust, fact_733_mult__le__cancel1, fact_748_zmult__commute, fact_116_assms_I3_J, fact_150_Diff__subset, fact_554_even__less__0__iff, tsy_c_COMBB_res, tsy_c_fNot_res, tsy_c_HOL_OThe_res, tsy_c_Set_Oimage_res, fact_559_not0__implies__Suc, fact_867_transfer__int__nat__quantifiers_I1_J, fact_345_order__le__less__trans, fact_817_conj__le__cong, fact_533_card__eq__0__iff, fact_581_mult__eq__0__iff, tsy_c_Finite__Set_Ofolding__one__idem_res, fact_916_abs__triangle__ineq, fact_780_zle__add1__eq__le, fact_801_self__quotient__aux2, fact_736_fold__image__distrib, fact_196_set__diff__eq, tsy_c_Int_OBit1_res, fact_12_finite__Collect__le__nat, fact_816_decr__mult__lemma, fact_904_neg__def, fact_516_card_Oempty, fact_209_folding__one_Osingleton, fact_348_add__less__cancel__left, fact_826_plusinfinity, fact_201_fun__diff__def, fact_586_mult__is__0, fact_963_mult__numeral__1__right, fact_595_mult__1__left, fact_617_number__of__reorient, fact_585_mult__0__right, fact_636_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J, fact_42_diff__diff__cancel, fact_510_gr0__conv__Suc, fact_237_add__mono, fact_126_Collect__empty__eq, fact_570_field__le__epsilon, fact_839_int__eq__0__conv, fact_254_add__leD1, fact_112_subsetI, fact_344_xt1_I7_J, fact_621_Ints__mult, fact_761_zadd__0, fact_191_DiffI, arity_Int_Oint___Groups_Omonoid__mult, fact_84_subset__refl, fact_340_order__le__imp__less__or__eq, fact_460_diff__self__eq__0, fact_27_finite__insert, fact_769_zadd__left__commute, fact_763_int__one__le__iff__zero__less, fact_519_diff__less, fact_628_add__number__of__eq, fact_514_add__is__1, fact_184_finite__induct, fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J, fact_877_of__nat__0__less__iff, fact_766_odd__less__0, fact_404_less__diff__conv, fact_299_linorder__neqE__nat, fact_806_int__less__induct, fact_293_card__Diff1__less, arity_Int_Oint___Rings_Oring__no__zero__divisors, fact_913_abs__triangle__ineq2, fact_833_zero__le__imp__of__nat, fact_593_mult__1__right, fact_297_less__not__refl, fact_453_Collect__mono, fact_198_folding__one_Oremove, fact_439_dense__le__bounded, fact_300_less__irrefl__nat, fact_966_add__special_I2_J, fact_989_rel__simps_I2_J, fact_829_number__of__int, fact_994_rel__simps_I5_J, fact_961_numeral__1__eq__1, fact_619_eq__number__of, fact_313_order__less__imp__not__eq, fact_968_Bit1__def, fact_281_flat__lub__def, fact_959_nat__numeral__1__eq__1, tsy_c_HOL_Oundefined_res, fact_889_int__if__cong, fact_712_mult__right__less__imp__less, fact_407_less__diff__iff, tsy_c_Finite__Set_Ofolding__one_res, fact_433_psubset__imp__ex__mem, fact_213_folding__one_Oclosed, fact_673_mult__less__cancel__left__pos, fact_324_order__less__asym, fact_129_empty__iff, fact_21_le__trans, fact_846_of__nat__less__imp__less, fact_802_q__pos__lemma, fact_337_leD, fact_684_comm__mult__strict__left__mono, fact_834_of__nat__less__0__iff, arity_Nat_Onat___Rings_Ocomm__semiring, fact_332_leI, fact_200_folding__image__simple_Oremove, help_ti_idem, fact_672_mult__less__cancel__left__disj, arity_Nat_Onat___Rings_Olinordered__semidom, 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fact_412_lift__Suc__mono__less__iff, arity_fun___Groups_Ominus, arity_Nat_Onat___Groups_Ocancel__semigroup__add, fact_841_of__nat__eq__iff, fact_664_mult__nonneg__nonpos2, fact_278_folding__one__idem_Ohom__commute, fact_285_comp__fun__idem_Ofun__left__idem, fact_577_convex__bound__lt, fact_102_subset__insertI, fact_670_not__square__less__zero, fact_605_diff__mult__distrib2, fact_250_trans__le__add2, fact_895_zero__le__imp__eq__int, fact_812_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J, fact_110_image__mono, fact_83_insert__absorb, fact_315_order__less__imp__triv, tsy_c_fTrue_res, fact_370_nat__add__left__cancel__less, fact_604_diff__mult__distrib, fact_647_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J, fact_384_termination__basic__simps_I5_J, fact_318_ord__eq__less__trans, fact_630_add__number__of__left
% 201.42/52.60 % No. of inferences in proof: 2459
% 201.42/52.60 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 52131 ms resp. 51386 ms w/o parsing
% 203.44/53.41 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 203.44/53.42 % [INFO] Killing All external provers ...
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