%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : ITP225_1 : TPTP v9.0.0. Released v8.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.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 : Sat Jun 21 05:12:07 AM UTC 2025
% Result : Theorem 58.60s 58.82s
% Output : Proof 59.37s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 18
% Syntax : Number of formulae : 49 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 462 ( 209 equ)
% Maximal formula atoms : 25 ( 9 avg)
% Number of connectives : 850 ( 455 ~; 305 |; 47 &)
% ( 43 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 4 ( 2 avg)
% Number of FOOLs : 18 ( 18 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 14 ( 11 usr; 7 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 5 con; 0-2 aty)
% Number of variables : 108 ( 96 !; 4 ?; 108 :)
% Comments :
%------------------------------------------------------------------------------
tff(pp_type,type,
pp: bool > $o ).
tff(aa_nat_bool_type,type,
aa_nat_bool: ( fun_nat_bool * nat ) > bool ).
tff(aa_nat_nat_type,type,
aa_nat_nat: ( fun_nat_nat * nat ) > nat ).
tff(na_type,type,
na: nat ).
tff(suc_type,type,
suc: fun_nat_nat ).
tff(aa_nat_fun_nat_bool_type,type,
aa_nat_fun_nat_bool: ( fun_nat_fun_nat_bool * nat ) > fun_nat_bool ).
tff(zero_zero_nat_type,type,
zero_zero_nat: nat ).
tff(ord_less_nat_type,type,
ord_less_nat: fun_nat_fun_nat_bool ).
tff(m_type,type,
m: nat ).
tff(tptp_fun_J3_30_type,type,
tptp_fun_J3_30: ( nat * nat ) > nat ).
tff(1,plain,
( ( m = aa_nat_nat(suc,na) )
<=> ( m = aa_nat_nat(suc,na) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
m = aa_nat_nat(suc,na),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4__C3_Ohyps_C_I3_J) ).
tff(3,plain,
m = aa_nat_nat(suc,na),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
aa_nat_nat(suc,na) = m,
inference(symmetry,[status(thm)],[3]) ).
tff(5,plain,
aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)) = aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m),
inference(monotonicity,[status(thm)],[4]) ).
tff(6,plain,
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m)) ),
inference(monotonicity,[status(thm)],[5]) ).
tff(7,plain,
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(symmetry,[status(thm)],[6]) ).
tff(8,plain,
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(monotonicity,[status(thm)],[7]) ).
tff(9,plain,
( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m))
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m)) ),
inference(rewrite,[status(thm)],[]) ).
tff(10,axiom,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
tff(11,plain,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),m)),
inference(modus_ponens,[status(thm)],[10,9]) ).
tff(12,plain,
~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))),
inference(modus_ponens,[status(thm)],[11,8]) ).
tff(13,plain,
^ [M: nat,N: nat] :
refl(( ~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(14,plain,
( ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,plain,
^ [M: nat,N: nat] :
rewrite(( ~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(16,plain,
( ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) ),
inference(quant_intro,[status(thm)],[15]) ).
tff(17,plain,
( ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) ),
inference(transitivity,[status(thm)],[16,14]) ).
tff(18,plain,
^ [M: nat,N: nat] :
trans(monotonicity(trans(monotonicity(rewrite(( ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) )
<=> ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )),
( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
<=> ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) ) )),
rewrite(( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
<=> ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) ) )),
( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
<=> ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) ) )),
rewrite(( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) )
<=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )),
( ( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) )),
rewrite(( ( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) )),
( ( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(19,plain,
( ! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) ),
inference(quant_intro,[status(thm)],[18]) ).
tff(20,plain,
^ [M: nat,N: nat] :
rewrite(( ( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( M = zero_zero_nat )
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(21,plain,
( ! [M: nat,N: nat] :
( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( M = zero_zero_nat )
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
<=> ! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ) ),
inference(quant_intro,[status(thm)],[20]) ).
tff(22,plain,
( ! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) )
<=> ! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,axiom,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_less__Suc__eq__0__disj) ).
tff(24,plain,
! [M: nat,N: nat] :
( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
<=> ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
! [M: nat,N: nat] :
( ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( M = zero_zero_nat )
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ),
inference(skolemize,[status(sab)],[24]) ).
tff(26,plain,
! [M: nat,N: nat] :
( ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M = aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
& ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ( ( M != zero_zero_nat )
& ! [J3: nat] :
~ ( ( M = aa_nat_nat(suc,J3) )
& pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ),
inference(modus_ponens,[status(thm)],[26,19]) ).
tff(28,plain,
! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) ),
inference(modus_ponens,[status(thm)],[27,17]) ).
tff(29,plain,
( ( ~ ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) )
<=> ( ~ ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,plain,
( ~ ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(rewrite,[status(thm)],[]) ).
tff(31,plain,
( ( $false
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) )
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,plain,
( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ ! [J3: nat] :
( ( zero_zero_nat != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),na)) ) ) )
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,plain,
( ~ $true
<=> $false ),
inference(rewrite,[status(thm)],[]) ).
tff(34,plain,
( ( $true
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
<=> $true ),
inference(rewrite,[status(thm)],[]) ).
tff(35,plain,
( ( zero_zero_nat = zero_zero_nat )
<=> $true ),
inference(rewrite,[status(thm)],[]) ).
tff(36,plain,
( ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
<=> ( $true
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) ) ),
inference(monotonicity,[status(thm)],[35]) ).
tff(37,plain,
( ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
<=> $true ),
inference(transitivity,[status(thm)],[36,34]) ).
tff(38,plain,
( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
<=> ~ $true ),
inference(monotonicity,[status(thm)],[37]) ).
tff(39,plain,
( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
<=> $false ),
inference(transitivity,[status(thm)],[38,33]) ).
tff(40,plain,
( ( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ ! [J3: nat] :
( ( zero_zero_nat != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),na)) ) ) ) )
<=> ( $false
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ) ),
inference(monotonicity,[status(thm)],[39,32]) ).
tff(41,plain,
( ( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ ! [J3: nat] :
( ( zero_zero_nat != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),na)) ) ) ) )
<=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(transitivity,[status(thm)],[40,31]) ).
tff(42,plain,
( ~ ( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ ! [J3: nat] :
( ( zero_zero_nat != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),na)) ) ) ) )
<=> ~ ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(monotonicity,[status(thm)],[41]) ).
tff(43,plain,
( ~ ( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ ! [J3: nat] :
( ( zero_zero_nat != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),na)) ) ) ) )
<=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(transitivity,[status(thm)],[42,30]) ).
tff(44,plain,
( ( ~ ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
| ~ ( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ ! [J3: nat] :
( ( zero_zero_nat != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),na)) ) ) ) ) )
<=> ( ~ ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ) ),
inference(monotonicity,[status(thm)],[43]) ).
tff(45,plain,
( ( ~ ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
| ~ ( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ ! [J3: nat] :
( ( zero_zero_nat != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),na)) ) ) ) ) )
<=> ( ~ ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ) ),
inference(transitivity,[status(thm)],[44,29]) ).
tff(46,plain,
( ~ ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
| ~ ( ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat != aa_nat_nat(suc,tptp_fun_J3_30(na,zero_zero_nat)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(na,zero_zero_nat)),na)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na)))
| ~ ( ( zero_zero_nat = zero_zero_nat )
| ~ ! [J3: nat] :
( ( zero_zero_nat != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),na)) ) ) ) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
( ~ ! [M: nat,N: nat] :
~ ( ~ ( ( M = zero_zero_nat )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M != aa_nat_nat(suc,tptp_fun_J3_30(N,M)) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,tptp_fun_J3_30(N,M)),N)) ) )
| ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),aa_nat_nat(suc,N)))
| ~ ( ( M = zero_zero_nat )
| ~ ! [J3: nat] :
( ( M != aa_nat_nat(suc,J3) )
| ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),N)) ) ) ) )
| pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))) ),
inference(modus_ponens,[status(thm)],[46,45]) ).
tff(48,plain,
pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,na))),
inference(unit_resolution,[status(thm)],[47,28]) ).
tff(49,plain,
$false,
inference(unit_resolution,[status(thm)],[48,12]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : ITP225_1 : TPTP v9.0.0. Released v8.0.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n005.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 Jun 19 18:32:05 EDT 2025
% 0.12/0.33 % CPUTime :
% 58.60/58.82 % SZS status Theorem
% 58.60/58.82 % SZS output start Proof
% See solution above
% 59.47/59.69 % E exiting
%------------------------------------------------------------------------------