%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT153^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n029.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 : Wed Feb 25 08:44:57 AM UTC 2026
% Result : Theorem 2.94s 1.65s
% Output : Refutation 2.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : DAT153^1 : TPTP v9.2.1. Released v7.0.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n029.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 06:10:14 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : DAT153^1 : TPTP v9.2.1. Released v7.0.0.
% 0.28/0.46 % Domain : Data Structures
% 0.28/0.46 % Problem : Coinductive stream 491
% 0.28/0.46 % Version : [Bla16] axioms : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.28/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.28/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Bla16]
% 0.28/0.46 % Names : coinductive_stream__491.p [Bla16]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.1.0
% 0.28/0.46 % Syntax : Number of formulae : 351 ( 145 unt; 57 typ; 0 def)
% 0.28/0.46 % Number of atoms : 785 ( 302 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 24 ( 2 avg)
% 0.28/0.46 % Number of connectives : 3264 ( 100 ~; 21 |; 33 &;2811 @)
% 0.28/0.46 % ( 0 <=>; 299 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 15 ( 6 avg)
% 0.28/0.46 % Number of types : 5 ( 4 usr)
% 0.28/0.46 % Number of type conns : 180 ( 180 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 56 ( 53 usr; 3 con; 0-4 aty)
% 0.28/0.46 % Number of variables : 784 ( 63 ^; 649 !; 27 ?; 784 :)
% 0.28/0.46 % ( 45 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.46 % SPC : TH1_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.28/0.46 % 2016-07-13 15:15:11.527
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 %----Could-be-implicit typings (7)
% 0.28/0.46 thf(ty_t_Extended__Nat_Oenat,type,
% 0.28/0.46 extended_enat: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Stream_Ostream,type,
% 0.28/0.46 stream: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Set_Oset,type,
% 0.28/0.46 set: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Nat_Onat,type,
% 0.28/0.46 nat: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Int_Oint,type,
% 0.28/0.46 int: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_itself,type,
% 0.28/0.46 itself: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_tf_a,type,
% 0.28/0.46 a: $tType ).
% 0.28/0.46
% 0.28/0.46 %----Explicit typings (50)
% 0.28/0.46 thf(sy_cl_Groups_Ozero,type,
% 0.28/0.46 zero:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Fields_Ofield,type,
% 0.28/0.46 field:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Omult__zero,type,
% 0.28/0.46 mult_zero:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Osemiring__1,type,
% 0.28/0.46 semiring_1:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.28/0.46 semiring_char_0:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Fields_Odivision__ring,type,
% 0.28/0.46 division_ring:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.28/0.46 semigroup_mult:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.28/0.46 comm_semiring_1:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.28/0.46 linordered_idom:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__ring,type,
% 0.28/0.46 linordered_ring:
% 0.28/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Fields_Olinordered__field,type,
% 0.28/0.47 linordered_field:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.28/0.47 ab_semigroup_mult:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.28/0.47 linordered_semidom:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 0.28/0.47 linord581940658strict:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 0.28/0.47 archim1727834104eiling:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 0.28/0.47 semiri1193490041visors:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Rings_Olinordered__semiring__strict,type,
% 0.28/0.47 linord20386208strict:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 0.28/0.47 archim1804426504_field:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.28/0.47 canoni770627133id_add:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
% 0.28/0.47 linord893533164strict:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 0.28/0.47 semiri1923998003cancel:
% 0.28/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Archimedean__Field_Oceiling,type,
% 0.28/0.47 archimedean_ceiling:
% 0.28/0.47 !>[A: $tType] : ( A > int ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oscount,type,
% 0.28/0.47 coindu1365464361scount:
% 0.28/0.47 !>[S: $tType] : ( ( ( stream @ S ) > $o ) > ( stream @ S ) > extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Osfirst,type,
% 0.28/0.47 coindu76177514sfirst:
% 0.28/0.47 !>[S: $tType] : ( ( ( stream @ S ) > $o ) > ( stream @ S ) > extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Extended__Nat_Oenat,type,
% 0.28/0.47 extended_enat2: nat > extended_enat ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 0.28/0.47 extended_case_enat:
% 0.28/0.47 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 0.28/0.47 extended_rec_enat:
% 0.28/0.47 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
% 0.28/0.47 extended_size_enat: extended_enat > nat ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 0.28/0.47 extend1396239628finity:
% 0.28/0.47 !>[A: $tType] : A ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Extended__Nat_Othe__enat,type,
% 0.28/0.47 extended_the_enat: extended_enat > nat ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.28/0.47 inverse_inverse:
% 0.28/0.47 !>[A: $tType] : ( A > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Finite__Set_Ocard,type,
% 0.28/0.47 finite_card:
% 0.28/0.47 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.28/0.47 times_times:
% 0.28/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.28/0.47 zero_zero:
% 0.28/0.47 !>[A: $tType] : A ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_If,type,
% 0.28/0.47 if:
% 0.28/0.47 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Int_Onat,type,
% 0.28/0.47 nat2: int > nat ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Linear__Temporal__Logic__on__Streams_OUNTIL,type,
% 0.28/0.47 linear1341015445_UNTIL:
% 0.28/0.47 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw,type,
% 0.28/0.47 linear1386806755on_alw:
% 0.28/0.47 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev,type,
% 0.28/0.47 linear505997466_on_ev:
% 0.28/0.47 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Linear__Temporal__Logic__on__Streams_Owait,type,
% 0.28/0.47 linear1335279038n_wait:
% 0.28/0.47 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.28/0.47 semiring_1_of_nat:
% 0.28/0.47 !>[A: $tType] : ( nat > A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.28/0.47 size_size:
% 0.28/0.47 !>[A: $tType] : ( A > nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Nat__Transfer_Otransfer__morphism,type,
% 0.28/0.47 nat_tr1645093318rphism:
% 0.28/0.47 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.28/0.47 ord_less:
% 0.28/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Pure_Otype,type,
% 0.28/0.47 type:
% 0.28/0.47 !>[A: $tType] : ( itself @ A ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Set_OCollect,type,
% 0.28/0.47 collect:
% 0.28/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_Stream_Osdrop,type,
% 0.28/0.47 sdrop:
% 0.28/0.47 !>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_c_member,type,
% 0.28/0.47 member:
% 0.28/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v_P,type,
% 0.28/0.47 p: ( stream @ a ) > $o ).
% 0.28/0.47
% 0.28/0.47 thf(sy_v__092_060omega_062_H____,type,
% 0.28/0.47 omega: stream @ a ).
% 0.28/0.47
% 0.28/0.47 %----Relevant facts (253)
% 0.28/0.47 thf(fact_0__C0_Oprems_C,axiom,
% 0.28/0.47 ( linear1341015445_UNTIL @ a
% 0.28/0.47 @ ^ [Xs: stream @ a] :
% 0.28/0.47 ~ ( p @ Xs )
% 0.28/0.47 @ ( linear1386806755on_alw @ a @ p )
% 0.28/0.47 @ omega ) ).
% 0.28/0.47
% 0.28/0.47 % "0.prems"
% 0.28/0.47 thf(fact_1_less__sfirstD,axiom,
% 0.28/0.47 ! [A: $tType,N: nat,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu76177514sfirst @ A @ P @ Omega ) )
% 0.28/0.47 => ~ ( P @ ( sdrop @ A @ N @ Omega ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_sfirstD
% 0.28/0.47 thf(fact_2_not__gr__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.28/0.47 => ! [N: A] :
% 0.28/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.28/0.47 = ( N
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_gr_zero
% 0.28/0.47 thf(fact_3_enat_Oinject,axiom,
% 0.28/0.47 ! [Nat: nat,Nat2: nat] :
% 0.28/0.47 ( ( ( extended_enat2 @ Nat )
% 0.28/0.47 = ( extended_enat2 @ Nat2 ) )
% 0.28/0.47 = ( Nat = Nat2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.inject
% 0.28/0.47 thf(fact_4_sdrop_Osimps_I1_J,axiom,
% 0.28/0.47 ! [A: $tType,S2: stream @ A] :
% 0.28/0.47 ( ( sdrop @ A @ ( zero_zero @ nat ) @ S2 )
% 0.28/0.47 = S2 ) ).
% 0.28/0.47
% 0.28/0.47 % sdrop.simps(1)
% 0.28/0.47 thf(fact_5_chain__incr,axiom,
% 0.28/0.47 ! [A: $tType,Y: A > extended_enat,K: nat] :
% 0.28/0.47 ( ! [I: A] :
% 0.28/0.47 ? [J: A] : ( ord_less @ extended_enat @ ( Y @ I ) @ ( Y @ J ) )
% 0.28/0.47 => ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y @ J2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % chain_incr
% 0.28/0.47 thf(fact_6_enat__iless,axiom,
% 0.28/0.47 ! [N: extended_enat,M: nat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.28/0.47 => ? [K2: nat] :
% 0.28/0.47 ( N
% 0.28/0.47 = ( extended_enat2 @ K2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_iless
% 0.28/0.47 thf(fact_7_less__numeral__extra_I3_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.28/0.47 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_numeral_extra(3)
% 0.28/0.47 thf(fact_8_gr__zeroI,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.28/0.47 => ! [N: A] :
% 0.28/0.47 ( ( N
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % gr_zeroI
% 0.28/0.47 thf(fact_9_not__less__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.28/0.47 => ! [N: A] :
% 0.28/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_less_zero
% 0.28/0.47 thf(fact_10_gr__implies__not__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.28/0.47 => ! [M: A,N: A] :
% 0.28/0.47 ( ( ord_less @ A @ M @ N )
% 0.28/0.47 => ( N
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % gr_implies_not_zero
% 0.28/0.47 thf(fact_11_zero__less__iff__neq__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 0.28/0.47 => ! [N: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.28/0.47 = ( N
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_iff_neq_zero
% 0.28/0.47 thf(fact_12_fun__cong__unused__0,axiom,
% 0.28/0.47 ! [A: $tType,B: $tType,C: $tType] :
% 0.28/0.47 ( ( zero @ B @ ( type @ B ) )
% 0.28/0.47 => ! [F: ( A > B ) > C,G: C] :
% 0.28/0.47 ( ( F
% 0.28/0.47 = ( ^ [X: A > B] : G ) )
% 0.28/0.47 => ( ( F
% 0.28/0.47 @ ^ [X: A] : ( zero_zero @ B ) )
% 0.28/0.47 = G ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % fun_cong_unused_0
% 0.28/0.47 thf(fact_13_i0__less,axiom,
% 0.28/0.47 ! [N: extended_enat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 0.28/0.47 = ( N
% 0.28/0.47 != ( zero_zero @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % i0_less
% 0.28/0.47 thf(fact_14_sfirst__0,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( P @ Omega )
% 0.28/0.47 => ( ( coindu76177514sfirst @ A @ P @ Omega )
% 0.28/0.47 = ( zero_zero @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % sfirst_0
% 0.28/0.47 thf(fact_15_enat__ord__simps_I2_J,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.28/0.47 = ( ord_less @ nat @ M @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_ord_simps(2)
% 0.28/0.47 thf(fact_16_not__iless0,axiom,
% 0.28/0.47 ! [N: extended_enat] :
% 0.28/0.47 ~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_iless0
% 0.28/0.47 thf(fact_17_zero__enat__def,axiom,
% 0.28/0.47 ( ( zero_zero @ extended_enat )
% 0.28/0.47 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_enat_def
% 0.28/0.47 thf(fact_18_enat__0__iff_I1_J,axiom,
% 0.28/0.47 ! [X2: nat] :
% 0.28/0.47 ( ( ( extended_enat2 @ X2 )
% 0.28/0.47 = ( zero_zero @ extended_enat ) )
% 0.28/0.47 = ( X2
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_0_iff(1)
% 0.28/0.47 thf(fact_19_enat__0__iff_I2_J,axiom,
% 0.28/0.47 ! [X2: nat] :
% 0.28/0.47 ( ( ( zero_zero @ extended_enat )
% 0.28/0.47 = ( extended_enat2 @ X2 ) )
% 0.28/0.47 = ( X2
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_0_iff(2)
% 0.28/0.47 thf(fact_20_less__enatE,axiom,
% 0.28/0.47 ! [N: extended_enat,M: nat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 0.28/0.47 => ~ ! [K2: nat] :
% 0.28/0.47 ( ( N
% 0.28/0.47 = ( extended_enat2 @ K2 ) )
% 0.28/0.47 => ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_enatE
% 0.28/0.47 thf(fact_21_sfirst__eq__0,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( ( coindu76177514sfirst @ A @ P @ Omega )
% 0.28/0.47 = ( zero_zero @ extended_enat ) )
% 0.28/0.47 = ( P @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % sfirst_eq_0
% 0.28/0.47 thf(fact_22_zero__reorient,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( zero @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X2: A] :
% 0.28/0.47 ( ( ( zero_zero @ A )
% 0.28/0.47 = X2 )
% 0.28/0.47 = ( X2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_reorient
% 0.28/0.47 thf(fact_23_enat__less__induct,axiom,
% 0.28/0.47 ! [P: extended_enat > $o,N: extended_enat] :
% 0.28/0.47 ( ! [N2: extended_enat] :
% 0.28/0.47 ( ! [M2: extended_enat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ M2 @ N2 )
% 0.28/0.47 => ( P @ M2 ) )
% 0.28/0.47 => ( P @ N2 ) )
% 0.28/0.47 => ( P @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_less_induct
% 0.28/0.47 thf(fact_24_enat_Osize_I1_J,axiom,
% 0.28/0.47 ! [Nat: nat] :
% 0.28/0.47 ( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.size(1)
% 0.28/0.47 thf(fact_25_until__false,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1341015445_UNTIL @ A @ Phi
% 0.28/0.47 @ ^ [Xs: stream @ A] : $false )
% 0.28/0.47 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 0.28/0.47
% 0.28/0.47 % until_false
% 0.28/0.47 thf(fact_26_alw__alw,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ Phi ) )
% 0.28/0.47 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_alw
% 0.28/0.47 thf(fact_27_neq0__conv,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( N
% 0.28/0.47 != ( zero_zero @ nat ) )
% 0.28/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % neq0_conv
% 0.28/0.47 thf(fact_28_alw__iff__sdrop,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A )
% 0.28/0.47 = ( ^ [P2: ( stream @ A ) > $o,Omega2: stream @ A] :
% 0.28/0.47 ! [M3: nat] : ( P2 @ ( sdrop @ A @ M3 @ Omega2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_iff_sdrop
% 0.28/0.47 thf(fact_29_alw__sdrop,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,N: nat] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 0.28/0.47 => ( linear1386806755on_alw @ A @ Phi @ ( sdrop @ A @ N @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_sdrop
% 0.28/0.47 thf(fact_30_enat_Osize_I3_J,axiom,
% 0.28/0.47 ! [Nat: nat] :
% 0.28/0.47 ( ( size_size @ extended_enat @ ( extended_enat2 @ Nat ) )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.size(3)
% 0.28/0.47 thf(fact_31_enat_Osimps_I6_J,axiom,
% 0.28/0.47 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 0.28/0.47 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 0.28/0.47 = ( F1 @ Nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.simps(6)
% 0.28/0.47 thf(fact_32_alw__False,axiom,
% 0.28/0.47 ! [A: $tType,Omega: stream @ A] :
% 0.28/0.47 ~ ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [X: stream @ A] : $false
% 0.28/0.47 @ Omega ) ).
% 0.28/0.47
% 0.28/0.47 % alw_False
% 0.28/0.47 thf(fact_33_alw__aand,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs )
% 0.28/0.47 & ( Psi @ Xs ) ) )
% 0.28/0.47 = ( ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 0.28/0.47 & ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_aand
% 0.28/0.47 thf(fact_34_alw__mp,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 0.28/0.47 => ( ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs )
% 0.28/0.47 => ( Psi @ Xs ) )
% 0.28/0.47 @ Xs2 )
% 0.28/0.47 => ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_mp
% 0.28/0.47 thf(fact_35_nat__neq__iff,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( M != N )
% 0.28/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_neq_iff
% 0.28/0.47 thf(fact_36_less__not__refl,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.28/0.47
% 0.28/0.47 % less_not_refl
% 0.28/0.47 thf(fact_37_less__not__refl2,axiom,
% 0.28/0.47 ! [N: nat,M: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ N @ M )
% 0.28/0.47 => ( M != N ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_not_refl2
% 0.28/0.47 thf(fact_38_less__not__refl3,axiom,
% 0.28/0.47 ! [S2: nat,T2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ S2 @ T2 )
% 0.28/0.47 => ( S2 != T2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_not_refl3
% 0.28/0.47 thf(fact_39_measure__induct,axiom,
% 0.28/0.47 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ! [Y2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.28/0.47 => ( P @ Y2 ) )
% 0.28/0.47 => ( P @ X3 ) )
% 0.28/0.47 => ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % measure_induct
% 0.28/0.47 thf(fact_40_less__irrefl__nat,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.28/0.47
% 0.28/0.47 % less_irrefl_nat
% 0.28/0.47 thf(fact_41_nat__less__induct,axiom,
% 0.28/0.47 ! [P: nat > $o,N: nat] :
% 0.28/0.47 ( ! [N2: nat] :
% 0.28/0.47 ( ! [M2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/0.47 => ( P @ M2 ) )
% 0.28/0.47 => ( P @ N2 ) )
% 0.28/0.47 => ( P @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_less_induct
% 0.28/0.47 thf(fact_42_infinite__descent,axiom,
% 0.28/0.47 ! [P: nat > $o,N: nat] :
% 0.28/0.47 ( ! [N2: nat] :
% 0.28/0.47 ( ~ ( P @ N2 )
% 0.28/0.47 => ? [M2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/0.47 & ~ ( P @ M2 ) ) )
% 0.28/0.47 => ( P @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinite_descent
% 0.28/0.47 thf(fact_43_linorder__neqE__nat,axiom,
% 0.28/0.47 ! [X2: nat,Y3: nat] :
% 0.28/0.47 ( ( X2 != Y3 )
% 0.28/0.47 => ( ~ ( ord_less @ nat @ X2 @ Y3 )
% 0.28/0.47 => ( ord_less @ nat @ Y3 @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linorder_neqE_nat
% 0.28/0.47 thf(fact_44_mem__Collect__eq,axiom,
% 0.28/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.28/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.28/0.47 = ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % mem_Collect_eq
% 0.28/0.47 thf(fact_45_Collect__mem__eq,axiom,
% 0.28/0.47 ! [A: $tType,A3: set @ A] :
% 0.28/0.47 ( ( collect @ A
% 0.28/0.47 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 0.28/0.47 = A3 ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_mem_eq
% 0.28/0.47 thf(fact_46_Collect__cong,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( P @ X3 )
% 0.28/0.47 = ( Q @ X3 ) )
% 0.28/0.47 => ( ( collect @ A @ P )
% 0.28/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Collect_cong
% 0.28/0.47 thf(fact_47_ext,axiom,
% 0.28/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( F @ X3 )
% 0.28/0.47 = ( G @ X3 ) )
% 0.28/0.47 => ( F = G ) ) ).
% 0.28/0.47
% 0.28/0.47 % ext
% 0.28/0.47 thf(fact_48_measure__induct__rule,axiom,
% 0.28/0.47 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ! [Y2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.28/0.47 => ( P @ Y2 ) )
% 0.28/0.47 => ( P @ X3 ) )
% 0.28/0.47 => ( P @ A2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % measure_induct_rule
% 0.28/0.47 thf(fact_49_infinite__descent__measure,axiom,
% 0.28/0.47 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ~ ( P @ X3 )
% 0.28/0.47 => ? [Y2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
% 0.28/0.47 & ~ ( P @ Y2 ) ) )
% 0.28/0.47 => ( P @ X2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinite_descent_measure
% 0.28/0.47 thf(fact_50_size__enat__overloaded__def,axiom,
% 0.28/0.47 ( ( size_size @ extended_enat )
% 0.28/0.47 = ( extended_rec_enat @ nat
% 0.28/0.47 @ ^ [X: nat] : ( zero_zero @ nat )
% 0.28/0.47 @ ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % size_enat_overloaded_def
% 0.28/0.47 thf(fact_51_size__enat__def,axiom,
% 0.28/0.47 ( extended_size_enat
% 0.28/0.47 = ( extended_rec_enat @ nat
% 0.28/0.47 @ ^ [X: nat] : ( zero_zero @ nat )
% 0.28/0.47 @ ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % size_enat_def
% 0.28/0.47 thf(fact_52_gr0I,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( N
% 0.28/0.47 != ( zero_zero @ nat ) )
% 0.28/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.28/0.47
% 0.28/0.47 % gr0I
% 0.28/0.47 thf(fact_53_not__gr0,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.28/0.47 = ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_gr0
% 0.28/0.47 thf(fact_54_not__less0,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_less0
% 0.28/0.47 thf(fact_55_less__zeroE,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_zeroE
% 0.28/0.47 thf(fact_56_gr__implies__not0,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 => ( N
% 0.28/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % gr_implies_not0
% 0.28/0.47 thf(fact_57_infinite__descent0,axiom,
% 0.28/0.47 ! [P: nat > $o,N: nat] :
% 0.28/0.47 ( ( P @ ( zero_zero @ nat ) )
% 0.28/0.47 => ( ! [N2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.28/0.47 => ( ~ ( P @ N2 )
% 0.28/0.47 => ? [M2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.28/0.47 & ~ ( P @ M2 ) ) ) )
% 0.28/0.47 => ( P @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinite_descent0
% 0.28/0.47 thf(fact_58_less__nat__zero__code,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_nat_zero_code
% 0.28/0.47 thf(fact_59_infinite__descent0__measure,axiom,
% 0.28/0.47 ! [A: $tType,V: A > nat,P: A > $o,X2: A] :
% 0.28/0.47 ( ! [X3: A] :
% 0.28/0.47 ( ( ( V @ X3 )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 => ( P @ X3 ) )
% 0.28/0.47 => ( ! [X3: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
% 0.28/0.47 => ( ~ ( P @ X3 )
% 0.28/0.47 => ? [Y2: A] :
% 0.28/0.47 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
% 0.28/0.47 & ~ ( P @ Y2 ) ) ) )
% 0.28/0.47 => ( P @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinite_descent0_measure
% 0.28/0.47 thf(fact_60_all__imp__alw,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.28/0.47 ( ! [X1: stream @ A] : ( Phi @ X1 )
% 0.28/0.47 => ( linear1386806755on_alw @ A @ Phi @ Xs2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % all_imp_alw
% 0.28/0.47 thf(fact_61_alw__mono,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 0.28/0.47 => ( ! [Xs3: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs3 )
% 0.28/0.47 => ( Psi @ Xs3 ) )
% 0.28/0.47 => ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_mono
% 0.28/0.47 thf(fact_62_alw__cong,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 0.28/0.47 => ( ! [Omega3: stream @ A] :
% 0.28/0.47 ( ( P @ Omega3 )
% 0.28/0.47 => ( ( Q1 @ Omega3 )
% 0.28/0.47 = ( Q2 @ Omega3 ) ) )
% 0.28/0.47 => ( ( linear1386806755on_alw @ A @ Q1 @ Omega )
% 0.28/0.47 = ( linear1386806755on_alw @ A @ Q2 @ Omega ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_cong
% 0.28/0.47 thf(fact_63_alw__alwD,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 0.28/0.47 => ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ P ) @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_alwD
% 0.28/0.47 thf(fact_64_alwD,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,X2: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ P @ X2 )
% 0.28/0.47 => ( P @ X2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % alwD
% 0.28/0.47 thf(fact_65_until__monoR,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi_1: ( stream @ A ) > $o,Xs2: stream @ A,Psi_2: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1341015445_UNTIL @ A @ Phi @ Psi_1 @ Xs2 )
% 0.28/0.47 => ( ! [Xs3: stream @ A] :
% 0.28/0.47 ( ( Psi_1 @ Xs3 )
% 0.28/0.47 => ( Psi_2 @ Xs3 ) )
% 0.28/0.47 => ( linear1341015445_UNTIL @ A @ Phi @ Psi_2 @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % until_monoR
% 0.28/0.47 thf(fact_66_until__monoL,axiom,
% 0.28/0.47 ! [A: $tType,Phi_1: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs2: stream @ A,Phi_2: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1341015445_UNTIL @ A @ Phi_1 @ Psi @ Xs2 )
% 0.28/0.47 => ( ! [Xs3: stream @ A] :
% 0.28/0.47 ( ( Phi_1 @ Xs3 )
% 0.28/0.47 => ( Phi_2 @ Xs3 ) )
% 0.28/0.47 => ( linear1341015445_UNTIL @ A @ Phi_2 @ Psi @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % until_monoL
% 0.28/0.47 thf(fact_67_until__mono,axiom,
% 0.28/0.47 ! [A: $tType,Phi_1: ( stream @ A ) > $o,Psi_1: ( stream @ A ) > $o,Xs2: stream @ A,Phi_2: ( stream @ A ) > $o,Psi_2: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1341015445_UNTIL @ A @ Phi_1 @ Psi_1 @ Xs2 )
% 0.28/0.47 => ( ! [Xs3: stream @ A] :
% 0.28/0.47 ( ( Phi_1 @ Xs3 )
% 0.28/0.47 => ( Phi_2 @ Xs3 ) )
% 0.28/0.47 => ( ! [Xs3: stream @ A] :
% 0.28/0.47 ( ( Psi_1 @ Xs3 )
% 0.28/0.47 => ( Psi_2 @ Xs3 ) )
% 0.28/0.47 => ( linear1341015445_UNTIL @ A @ Phi_2 @ Psi_2 @ Xs2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % until_mono
% 0.28/0.47 thf(fact_68_UNTIL_Obase,axiom,
% 0.28/0.47 ! [A: $tType,Psi: ( stream @ A ) > $o,Xs2: stream @ A,Phi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( Psi @ Xs2 )
% 0.28/0.47 => ( linear1341015445_UNTIL @ A @ Phi @ Psi @ Xs2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % UNTIL.base
% 0.28/0.47 thf(fact_69_of__nat__0__less__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.28/0.47 => ! [N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_0_less_iff
% 0.28/0.47 thf(fact_70_less__enat__def,axiom,
% 0.28/0.47 ( ( ord_less @ extended_enat )
% 0.28/0.47 = ( ^ [M3: extended_enat,N3: extended_enat] :
% 0.28/0.47 ( extended_case_enat @ $o
% 0.28/0.47 @ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N3 )
% 0.28/0.47 @ $false
% 0.28/0.47 @ M3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_enat_def
% 0.28/0.47 thf(fact_71_case__enat__def,axiom,
% 0.28/0.47 ! [T: $tType] :
% 0.28/0.47 ( ( extended_case_enat @ T )
% 0.28/0.47 = ( extended_rec_enat @ T ) ) ).
% 0.28/0.47
% 0.28/0.47 % case_enat_def
% 0.28/0.47 thf(fact_72_enat_Osimps_I4_J,axiom,
% 0.28/0.47 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 0.28/0.47 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 0.28/0.47 = ( F1 @ Nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.simps(4)
% 0.28/0.47 thf(fact_73_scount__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( ( coindu1365464361scount @ A @ P @ Omega )
% 0.28/0.47 = ( zero_zero @ extended_enat ) )
% 0.28/0.47 = ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( P @ Xs )
% 0.28/0.47 @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % scount_eq_0_iff
% 0.28/0.47 thf(fact_74_scount__eq__0I,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( P @ Xs )
% 0.28/0.47 @ Omega )
% 0.28/0.47 => ( ( coindu1365464361scount @ A @ P @ Omega )
% 0.28/0.47 = ( zero_zero @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % scount_eq_0I
% 0.28/0.47 thf(fact_75_scount__eq__0D,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( ( coindu1365464361scount @ A @ P @ Omega )
% 0.28/0.47 = ( zero_zero @ extended_enat ) )
% 0.28/0.47 => ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( P @ Xs )
% 0.28/0.47 @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % scount_eq_0D
% 0.28/0.47 thf(fact_76_enat_Osize_I4_J,axiom,
% 0.28/0.47 ( ( size_size @ extended_enat @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.size(4)
% 0.28/0.47 thf(fact_77_of__nat__eq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.28/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( M = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_eq_iff
% 0.28/0.47 thf(fact_78_not__infinity__eq,axiom,
% 0.28/0.47 ! [X2: extended_enat] :
% 0.28/0.47 ( ( X2
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( ? [I2: nat] :
% 0.28/0.47 ( X2
% 0.28/0.47 = ( extended_enat2 @ I2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_infinity_eq
% 0.28/0.47 thf(fact_79_not__enat__eq,axiom,
% 0.28/0.47 ! [X2: extended_enat] :
% 0.28/0.47 ( ( ! [Y4: nat] :
% 0.28/0.47 ( X2
% 0.28/0.47 != ( extended_enat2 @ Y4 ) ) )
% 0.28/0.47 = ( X2
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_enat_eq
% 0.28/0.47 thf(fact_80_enat__ord__simps_I4_J,axiom,
% 0.28/0.47 ! [Q3: extended_enat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ Q3 @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( Q3
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_ord_simps(4)
% 0.28/0.47 thf(fact_81_enat__ord__simps_I6_J,axiom,
% 0.28/0.47 ! [Q3: extended_enat] :
% 0.28/0.47 ~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q3 ) ).
% 0.28/0.47
% 0.28/0.47 % enat_ord_simps(6)
% 0.28/0.47 thf(fact_82_enat_Osimps_I5_J,axiom,
% 0.28/0.47 ! [T: $tType,F1: nat > T,F2: T] :
% 0.28/0.47 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = F2 ) ).
% 0.28/0.47
% 0.28/0.47 % enat.simps(5)
% 0.28/0.47 thf(fact_83_enat_Osimps_I7_J,axiom,
% 0.28/0.47 ! [T: $tType,F1: nat > T,F2: T] :
% 0.28/0.47 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = F2 ) ).
% 0.28/0.47
% 0.28/0.47 % enat.simps(7)
% 0.28/0.47 thf(fact_84_of__nat__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [M: nat] :
% 0.28/0.47 ( ( ( semiring_1_of_nat @ A @ M )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( M
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_eq_0_iff
% 0.28/0.47 thf(fact_85_of__nat__0__eq__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [N: nat] :
% 0.28/0.47 ( ( ( zero_zero @ A )
% 0.28/0.47 = ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( ( zero_zero @ nat )
% 0.28/0.47 = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_0_eq_iff
% 0.28/0.47 thf(fact_86_of__nat__0,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_0
% 0.28/0.47 thf(fact_87_of__nat__less__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_less_iff
% 0.28/0.47 thf(fact_88_case__enat__0,axiom,
% 0.28/0.47 ! [A: $tType,F: nat > A,I3: A] :
% 0.28/0.47 ( ( extended_case_enat @ A @ F @ I3 @ ( zero_zero @ extended_enat ) )
% 0.28/0.47 = ( F @ ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % case_enat_0
% 0.28/0.47 thf(fact_89_enat_Osize_I2_J,axiom,
% 0.28/0.47 ( ( extended_size_enat @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.size(2)
% 0.28/0.47 thf(fact_90_of__nat__eq__enat,axiom,
% 0.28/0.47 ( ( semiring_1_of_nat @ extended_enat )
% 0.28/0.47 = extended_enat2 ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_eq_enat
% 0.28/0.47 thf(fact_91_enat__ex__split,axiom,
% 0.28/0.47 ( ( ^ [P3: extended_enat > $o] :
% 0.28/0.47 ? [X4: extended_enat] : ( P3 @ X4 ) )
% 0.28/0.47 = ( ^ [P2: extended_enat > $o] :
% 0.28/0.47 ( ( P2 @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 | ? [X: nat] : ( P2 @ ( extended_enat2 @ X ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_ex_split
% 0.28/0.47 thf(fact_92_enat_Oinducts,axiom,
% 0.28/0.47 ! [P: extended_enat > $o,Enat: extended_enat] :
% 0.28/0.47 ( ! [Nat3: nat] : ( P @ ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ( ( P @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( P @ Enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.inducts
% 0.28/0.47 thf(fact_93_enat_Oexhaust,axiom,
% 0.28/0.47 ! [Y3: extended_enat] :
% 0.28/0.47 ( ! [Nat3: nat] :
% 0.28/0.47 ( Y3
% 0.28/0.47 != ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ( Y3
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.exhaust
% 0.28/0.47 thf(fact_94_enat3__cases,axiom,
% 0.28/0.47 ! [Y3: extended_enat,Ya: extended_enat,Yb: extended_enat] :
% 0.28/0.47 ( ( ? [Nat3: nat] :
% 0.28/0.47 ( Y3
% 0.28/0.47 = ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ( ? [Nata: nat] :
% 0.28/0.47 ( Ya
% 0.28/0.47 = ( extended_enat2 @ Nata ) )
% 0.28/0.47 => ! [Natb: nat] :
% 0.28/0.47 ( Yb
% 0.28/0.47 != ( extended_enat2 @ Natb ) ) ) )
% 0.28/0.47 => ( ( ? [Nat3: nat] :
% 0.28/0.47 ( Y3
% 0.28/0.47 = ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ( ? [Nata: nat] :
% 0.28/0.47 ( Ya
% 0.28/0.47 = ( extended_enat2 @ Nata ) )
% 0.28/0.47 => ( Yb
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.28/0.47 => ( ( ? [Nat3: nat] :
% 0.28/0.47 ( Y3
% 0.28/0.47 = ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ( ( Ya
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ! [Nata: nat] :
% 0.28/0.47 ( Yb
% 0.28/0.47 != ( extended_enat2 @ Nata ) ) ) )
% 0.28/0.47 => ( ( ? [Nat3: nat] :
% 0.28/0.47 ( Y3
% 0.28/0.47 = ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ( ( Ya
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( Yb
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.28/0.47 => ( ( ( Y3
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( ? [Nat3: nat] :
% 0.28/0.47 ( Ya
% 0.28/0.47 = ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ! [Nata: nat] :
% 0.28/0.47 ( Yb
% 0.28/0.47 != ( extended_enat2 @ Nata ) ) ) )
% 0.28/0.47 => ( ( ( Y3
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( ? [Nat3: nat] :
% 0.28/0.47 ( Ya
% 0.28/0.47 = ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ( Yb
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) ) ) )
% 0.28/0.47 => ( ( ( Y3
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( ( Ya
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ! [Nat3: nat] :
% 0.28/0.47 ( Yb
% 0.28/0.47 != ( extended_enat2 @ Nat3 ) ) ) )
% 0.28/0.47 => ~ ( ( Y3
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( ( Ya
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( Yb
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat3_cases
% 0.28/0.47 thf(fact_95_enat2__cases,axiom,
% 0.28/0.47 ! [Y3: extended_enat,Ya: extended_enat] :
% 0.28/0.47 ( ( ? [Nat3: nat] :
% 0.28/0.47 ( Y3
% 0.28/0.47 = ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ! [Nata: nat] :
% 0.28/0.47 ( Ya
% 0.28/0.47 != ( extended_enat2 @ Nata ) ) )
% 0.28/0.47 => ( ( ? [Nat3: nat] :
% 0.28/0.47 ( Y3
% 0.28/0.47 = ( extended_enat2 @ Nat3 ) )
% 0.28/0.47 => ( Ya
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) ) )
% 0.28/0.47 => ( ( ( Y3
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ! [Nat3: nat] :
% 0.28/0.47 ( Ya
% 0.28/0.47 != ( extended_enat2 @ Nat3 ) ) )
% 0.28/0.47 => ~ ( ( Y3
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( Ya
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat2_cases
% 0.28/0.47 thf(fact_96_enat_Odistinct_I1_J,axiom,
% 0.28/0.47 ! [Nat: nat] :
% 0.28/0.47 ( ( extended_enat2 @ Nat )
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.distinct(1)
% 0.28/0.47 thf(fact_97_enat_Odistinct_I2_J,axiom,
% 0.28/0.47 ! [Nat4: nat] :
% 0.28/0.47 ( ( extend1396239628finity @ extended_enat )
% 0.28/0.47 != ( extended_enat2 @ Nat4 ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat.distinct(2)
% 0.28/0.47 thf(fact_98_infinity__ne__i0,axiom,
% 0.28/0.47 ( ( extend1396239628finity @ extended_enat )
% 0.28/0.47 != ( zero_zero @ extended_enat ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinity_ne_i0
% 0.28/0.47 thf(fact_99_of__nat__less__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.28/0.47 => ! [M: nat] :
% 0.28/0.47 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_less_0_iff
% 0.28/0.47 thf(fact_100_of__nat__less__imp__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_less_imp_less
% 0.28/0.47 thf(fact_101_less__imp__of__nat__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_semidom @ A @ ( type @ A ) )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ N )
% 0.28/0.47 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_of_nat_less
% 0.28/0.47 thf(fact_102_infinity__ilessE,axiom,
% 0.28/0.47 ! [M: nat] :
% 0.28/0.47 ~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% 0.28/0.47
% 0.28/0.47 % infinity_ilessE
% 0.28/0.47 thf(fact_103_less__infinityE,axiom,
% 0.28/0.47 ! [N: extended_enat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ N @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ~ ! [K2: nat] :
% 0.28/0.47 ( N
% 0.28/0.47 != ( extended_enat2 @ K2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_infinityE
% 0.28/0.47 thf(fact_104_enat__ord__code_I4_J,axiom,
% 0.28/0.47 ! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_ord_code(4)
% 0.28/0.47 thf(fact_105_zero__less__imp__eq__int,axiom,
% 0.28/0.47 ! [K: int] :
% 0.28/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 => ? [N2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.28/0.47 & ( K
% 0.28/0.47 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_imp_eq_int
% 0.28/0.47 thf(fact_106_reals__Archimedean2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1804426504_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X2: A] :
% 0.28/0.47 ? [N2: nat] : ( ord_less @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % reals_Archimedean2
% 0.28/0.47 thf(fact_107_times__enat__simps_I3_J,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( ( N
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
% 0.28/0.47 = ( zero_zero @ extended_enat ) ) )
% 0.28/0.47 & ( ( N
% 0.28/0.47 != ( zero_zero @ nat ) )
% 0.28/0.47 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % times_enat_simps(3)
% 0.28/0.47 thf(fact_108_times__enat__simps_I4_J,axiom,
% 0.28/0.47 ! [M: nat] :
% 0.28/0.47 ( ( ( M
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 => ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( zero_zero @ extended_enat ) ) )
% 0.28/0.47 & ( ( M
% 0.28/0.47 != ( zero_zero @ nat ) )
% 0.28/0.47 => ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % times_enat_simps(4)
% 0.28/0.47 thf(fact_109_the__enat__0,axiom,
% 0.28/0.47 ( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % the_enat_0
% 0.28/0.47 thf(fact_110_scount__finite,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A
% 0.28/0.47 @ ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( P @ Xs ) )
% 0.28/0.47 @ Omega )
% 0.28/0.47 => ( ord_less @ extended_enat @ ( coindu1365464361scount @ A @ P @ Omega ) @ ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % scount_finite
% 0.28/0.47 thf(fact_111_ev__ev,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ ( linear505997466_on_ev @ A @ Phi ) )
% 0.28/0.47 = ( linear505997466_on_ev @ A @ Phi ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_ev
% 0.28/0.47 thf(fact_112_of__nat__mult,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [M: nat,N: nat] :
% 0.28/0.47 ( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
% 0.28/0.47 = ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % of_nat_mult
% 0.28/0.47 thf(fact_113_times__enat__simps_I2_J,axiom,
% 0.28/0.47 ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) ) ).
% 0.28/0.47
% 0.28/0.47 % times_enat_simps(2)
% 0.28/0.47 thf(fact_114_not__alw__not,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xt: stream @ A] :
% 0.28/0.47 ~ ( Phi @ Xt )
% 0.28/0.47 @ Xs ) )
% 0.28/0.47 = ( linear505997466_on_ev @ A @ Phi ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_alw_not
% 0.28/0.47 thf(fact_115_not__ev__not,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( linear505997466_on_ev @ A
% 0.28/0.47 @ ^ [Xt: stream @ A] :
% 0.28/0.47 ~ ( Phi @ Xt )
% 0.28/0.47 @ Xs ) )
% 0.28/0.47 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_ev_not
% 0.28/0.47 thf(fact_116_int__int__eq,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( semiring_1_of_nat @ int @ M )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ N ) )
% 0.28/0.47 = ( M = N ) ) ).
% 0.28/0.47
% 0.28/0.47 % int_int_eq
% 0.28/0.47 thf(fact_117_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_mult @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 0.28/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ab_semigroup_mult_class.mult_ac(1)
% 0.28/0.47 thf(fact_118_mult_Oassoc,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semigroup_mult @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 0.28/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.assoc
% 0.28/0.47 thf(fact_119_mult_Ocommute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_mult @ A @ ( type @ A ) )
% 0.28/0.47 => ( ( times_times @ A )
% 0.28/0.47 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.commute
% 0.28/0.47 thf(fact_120_mult_Oleft__commute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( ab_semigroup_mult @ A @ ( type @ A ) )
% 0.28/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.28/0.47 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
% 0.28/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult.left_commute
% 0.28/0.47 thf(fact_121_ev__or,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs )
% 0.28/0.47 | ( Psi @ Xs ) ) )
% 0.28/0.47 = ( ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ Phi @ Xs )
% 0.28/0.47 | ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_or
% 0.28/0.47 thf(fact_122_ev_Obase,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs2 )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Phi @ Xs2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev.base
% 0.28/0.47 thf(fact_123_ev__mono,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.28/0.47 => ( ! [Xs3: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs3 )
% 0.28/0.47 => ( Psi @ Xs3 ) )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_mono
% 0.28/0.47 thf(fact_124_ev__False,axiom,
% 0.28/0.47 ! [A: $tType,Omega: stream @ A] :
% 0.28/0.47 ~ ( linear505997466_on_ev @ A
% 0.28/0.47 @ ^ [X: stream @ A] : $false
% 0.28/0.47 @ Omega ) ).
% 0.28/0.47
% 0.28/0.47 % ev_False
% 0.28/0.47 thf(fact_125_less__int__code_I1_J,axiom,
% 0.28/0.47 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_int_code(1)
% 0.28/0.47 thf(fact_126_ev__alw__imp__alw__ev,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 0.28/0.47 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_alw_imp_alw_ev
% 0.28/0.47 thf(fact_127_ev__cong,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 0.28/0.47 => ( ! [Omega3: stream @ A] :
% 0.28/0.47 ( ( P @ Omega3 )
% 0.28/0.47 => ( ( Q1 @ Omega3 )
% 0.28/0.47 = ( Q2 @ Omega3 ) ) )
% 0.28/0.47 => ( ( linear505997466_on_ev @ A @ Q1 @ Omega )
% 0.28/0.47 = ( linear505997466_on_ev @ A @ Q2 @ Omega ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_cong
% 0.28/0.47 thf(fact_128_ev__iff__sdrop,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A )
% 0.28/0.47 = ( ^ [P2: ( stream @ A ) > $o,Omega2: stream @ A] :
% 0.28/0.47 ? [M3: nat] : ( P2 @ ( sdrop @ A @ M3 @ Omega2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_iff_sdrop
% 0.28/0.47 thf(fact_129_mult__of__nat__commute,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X2: nat,Y3: A] :
% 0.28/0.47 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ X2 ) @ Y3 )
% 0.28/0.47 = ( times_times @ A @ Y3 @ ( semiring_1_of_nat @ A @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_of_nat_commute
% 0.28/0.47 thf(fact_130_imult__is__0,axiom,
% 0.28/0.47 ! [M: extended_enat,N: extended_enat] :
% 0.28/0.47 ( ( ( times_times @ extended_enat @ M @ N )
% 0.28/0.47 = ( zero_zero @ extended_enat ) )
% 0.28/0.47 = ( ( M
% 0.28/0.47 = ( zero_zero @ extended_enat ) )
% 0.28/0.47 | ( N
% 0.28/0.47 = ( zero_zero @ extended_enat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imult_is_0
% 0.28/0.47 thf(fact_131_alw__ev__imp__ev__alw,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega )
% 0.28/0.47 => ( linear505997466_on_ev @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( P @ Xs )
% 0.28/0.47 & ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Xs ) )
% 0.28/0.47 @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_ev_imp_ev_alw
% 0.28/0.47 thf(fact_132_ev__alw__alw__impl,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 0.28/0.47 => ( ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 0.28/0.47 @ Xs2 )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_alw_alw_impl
% 0.28/0.47 thf(fact_133_alw__impl__ev__alw,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 0.28/0.47 @ Xs2 )
% 0.28/0.47 => ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ Phi @ Xs )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 0.28/0.47 @ Xs2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_impl_ev_alw
% 0.28/0.47 thf(fact_134_alw__alw__impl__ev,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs ) ) )
% 0.28/0.47 = ( ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs )
% 0.28/0.47 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Psi ) @ Xs ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_alw_impl_ev
% 0.28/0.47 thf(fact_135_ev__alw__impl__ev,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.28/0.47 => ( ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 0.28/0.47 @ Xs2 )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_alw_impl_ev
% 0.28/0.47 thf(fact_136_not__alw__iff,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( ~ ( linear1386806755on_alw @ A @ P @ Omega ) )
% 0.28/0.47 = ( linear505997466_on_ev @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( P @ Xs )
% 0.28/0.47 @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_alw_iff
% 0.28/0.47 thf(fact_137_ev__alw__impl,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.28/0.47 => ( ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs )
% 0.28/0.47 => ( Psi @ Xs ) )
% 0.28/0.47 @ Xs2 )
% 0.28/0.47 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_alw_impl
% 0.28/0.47 thf(fact_138_ev__alw__aand,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 0.28/0.47 => ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Psi ) @ Xs2 )
% 0.28/0.47 => ( linear505997466_on_ev @ A
% 0.28/0.47 @ ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ( ( Phi @ Xs )
% 0.28/0.47 & ( Psi @ Xs ) ) )
% 0.28/0.47 @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ev_alw_aand
% 0.28/0.47 thf(fact_139_not__ev__iff,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( ~ ( linear505997466_on_ev @ A @ P @ Omega ) )
% 0.28/0.47 = ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( P @ Xs )
% 0.28/0.47 @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_ev_iff
% 0.28/0.47 thf(fact_140_not__alw,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( linear1386806755on_alw @ A @ Phi @ Xs ) )
% 0.28/0.47 = ( linear505997466_on_ev @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( Phi @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_alw
% 0.28/0.47 thf(fact_141_not__ev,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 0.28/0.47 ( ( ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( linear505997466_on_ev @ A @ Phi @ Xs ) )
% 0.28/0.47 = ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( Phi @ Xs ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_ev
% 0.28/0.47 thf(fact_142_the__enat_Osimps,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 0.28/0.47 = N ) ).
% 0.28/0.47
% 0.28/0.47 % the_enat.simps
% 0.28/0.47 thf(fact_143_alw__ev__sdrop,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,M: nat,Xs2: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ ( sdrop @ A @ M @ Xs2 ) )
% 0.28/0.47 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% 0.28/0.47
% 0.28/0.47 % alw_ev_sdrop
% 0.28/0.47 thf(fact_144_ex__less__of__nat__mult,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1804426504_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X2: A,Y3: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
% 0.28/0.47 => ? [N2: nat] : ( ord_less @ A @ Y3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ex_less_of_nat_mult
% 0.28/0.47 thf(fact_145_imult__is__infinity,axiom,
% 0.28/0.47 ! [A2: extended_enat,B2: extended_enat] :
% 0.28/0.47 ( ( ( times_times @ extended_enat @ A2 @ B2 )
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( ( ( A2
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 & ( B2
% 0.28/0.47 != ( zero_zero @ extended_enat ) ) )
% 0.28/0.47 | ( ( B2
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 & ( A2
% 0.28/0.47 != ( zero_zero @ extended_enat ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imult_is_infinity
% 0.28/0.47 thf(fact_146_enat__0__less__mult__iff,axiom,
% 0.28/0.47 ! [M: extended_enat,N: extended_enat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N ) )
% 0.28/0.47 = ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M )
% 0.28/0.47 & ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_0_less_mult_iff
% 0.28/0.47 thf(fact_147_scount__infinite__iff,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( ( coindu1365464361scount @ A @ P @ Omega )
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % scount_infinite_iff
% 0.28/0.47 thf(fact_148_scount__infinite,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega )
% 0.28/0.47 => ( ( coindu1365464361scount @ A @ P @ Omega )
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % scount_infinite
% 0.28/0.47 thf(fact_149_sfirst__finite,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ ( coindu76177514sfirst @ A @ P @ Omega ) @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( linear505997466_on_ev @ A @ P @ Omega ) ) ).
% 0.28/0.47
% 0.28/0.47 % sfirst_finite
% 0.28/0.47 thf(fact_150_imult__infinity__right,axiom,
% 0.28/0.47 ! [N: extended_enat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 0.28/0.47 => ( ( times_times @ extended_enat @ N @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imult_infinity_right
% 0.28/0.47 thf(fact_151_imult__infinity,axiom,
% 0.28/0.47 ! [N: extended_enat] :
% 0.28/0.47 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 0.28/0.47 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ N )
% 0.28/0.47 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % imult_infinity
% 0.28/0.47 thf(fact_152_enat__the__enat,axiom,
% 0.28/0.47 ! [N: extended_enat] :
% 0.28/0.47 ( ( N
% 0.28/0.47 != ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 => ( ( extended_enat2 @ ( extended_the_enat @ N ) )
% 0.28/0.47 = N ) ) ).
% 0.28/0.47
% 0.28/0.47 % enat_the_enat
% 0.28/0.47 thf(fact_153_mult__cancel__right,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1923998003cancel @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ C2 )
% 0.28/0.47 = ( times_times @ A @ B2 @ C2 ) )
% 0.28/0.47 = ( ( C2
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel_right
% 0.28/0.47 thf(fact_154_mult__cancel__left,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1923998003cancel @ A @ ( type @ A ) )
% 0.28/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.28/0.47 ( ( ( times_times @ A @ C2 @ A2 )
% 0.28/0.47 = ( times_times @ A @ C2 @ B2 ) )
% 0.28/0.47 = ( ( C2
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( A2 = B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel_left
% 0.28/0.47 thf(fact_155_mult__eq__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1193490041visors @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ B2 )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( ( A2
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( B2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_eq_0_iff
% 0.28/0.47 thf(fact_156_mult__zero__right,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( mult_zero @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_zero_right
% 0.28/0.47 thf(fact_157_mult__zero__left,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( mult_zero @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_zero_left
% 0.28/0.47 thf(fact_158_mult__cancel2,axiom,
% 0.28/0.47 ! [M: nat,K: nat,N: nat] :
% 0.28/0.47 ( ( ( times_times @ nat @ M @ K )
% 0.28/0.47 = ( times_times @ nat @ N @ K ) )
% 0.28/0.47 = ( ( M = N )
% 0.28/0.47 | ( K
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel2
% 0.28/0.47 thf(fact_159_mult__cancel1,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ( times_times @ nat @ K @ M )
% 0.28/0.47 = ( times_times @ nat @ K @ N ) )
% 0.28/0.47 = ( ( M = N )
% 0.28/0.47 | ( K
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_cancel1
% 0.28/0.47 thf(fact_160_mult__0__right,axiom,
% 0.28/0.47 ! [M: nat] :
% 0.28/0.47 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_0_right
% 0.28/0.47 thf(fact_161_mult__is__0,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ( times_times @ nat @ M @ N )
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 = ( ( M
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 | ( N
% 0.28/0.47 = ( zero_zero @ nat ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_is_0
% 0.28/0.47 thf(fact_162_mult__less__cancel2,axiom,
% 0.28/0.47 ! [M: nat,K: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 0.28/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.28/0.47 & ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_less_cancel2
% 0.28/0.47 thf(fact_163_nat__0__less__mult__iff,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
% 0.28/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 0.28/0.47 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_0_less_mult_iff
% 0.28/0.47 thf(fact_164_times__enat__simps_I1_J,axiom,
% 0.28/0.47 ! [M: nat,N: nat] :
% 0.28/0.47 ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.28/0.47 = ( extended_enat2 @ ( times_times @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % times_enat_simps(1)
% 0.28/0.47 thf(fact_165_times__int__code_I1_J,axiom,
% 0.28/0.47 ! [K: int] :
% 0.28/0.47 ( ( times_times @ int @ K @ ( zero_zero @ int ) )
% 0.28/0.47 = ( zero_zero @ int ) ) ).
% 0.28/0.47
% 0.28/0.47 % times_int_code(1)
% 0.28/0.47 thf(fact_166_times__int__code_I2_J,axiom,
% 0.28/0.47 ! [L: int] :
% 0.28/0.47 ( ( times_times @ int @ ( zero_zero @ int ) @ L )
% 0.28/0.47 = ( zero_zero @ int ) ) ).
% 0.28/0.47
% 0.28/0.47 % times_int_code(2)
% 0.28/0.47 thf(fact_167_mult__0,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_0
% 0.28/0.47 thf(fact_168_mult__less__mono1,axiom,
% 0.28/0.47 ! [I3: nat,J3: nat,K: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I3 @ J3 )
% 0.28/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.28/0.47 => ( ord_less @ nat @ ( times_times @ nat @ I3 @ K ) @ ( times_times @ nat @ J3 @ K ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_less_mono1
% 0.28/0.47 thf(fact_169_mult__less__mono2,axiom,
% 0.28/0.47 ! [I3: nat,J3: nat,K: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ I3 @ J3 )
% 0.28/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.28/0.47 => ( ord_less @ nat @ ( times_times @ nat @ K @ I3 ) @ ( times_times @ nat @ K @ J3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_less_mono2
% 0.28/0.47 thf(fact_170_zmult__zless__mono2,axiom,
% 0.28/0.47 ! [I3: int,J3: int,K: int] :
% 0.28/0.47 ( ( ord_less @ int @ I3 @ J3 )
% 0.28/0.47 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.28/0.47 => ( ord_less @ int @ ( times_times @ int @ K @ I3 ) @ ( times_times @ int @ K @ J3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zmult_zless_mono2
% 0.28/0.47 thf(fact_171_linorder__neqE__linordered__idom,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_idom @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X2: A,Y3: A] :
% 0.28/0.47 ( ( X2 != Y3 )
% 0.28/0.47 => ( ~ ( ord_less @ A @ X2 @ Y3 )
% 0.28/0.47 => ( ord_less @ A @ Y3 @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linorder_neqE_linordered_idom
% 0.28/0.47 thf(fact_172_zmult__zless__mono2__lemma,axiom,
% 0.28/0.47 ! [I3: int,J3: int,K: nat] :
% 0.28/0.47 ( ( ord_less @ int @ I3 @ J3 )
% 0.28/0.47 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.28/0.47 => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I3 ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J3 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zmult_zless_mono2_lemma
% 0.28/0.47 thf(fact_173_times__enat__def,axiom,
% 0.28/0.47 ( ( times_times @ extended_enat )
% 0.28/0.47 = ( ^ [M3: extended_enat,N3: extended_enat] :
% 0.28/0.47 ( extended_case_enat @ extended_enat
% 0.28/0.47 @ ^ [O: nat] :
% 0.28/0.47 ( extended_case_enat @ extended_enat
% 0.28/0.47 @ ^ [P4: nat] : ( extended_enat2 @ ( times_times @ nat @ O @ P4 ) )
% 0.28/0.47 @ ( if @ extended_enat
% 0.28/0.47 @ ( O
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 @ ( zero_zero @ extended_enat )
% 0.28/0.47 @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 @ N3 )
% 0.28/0.47 @ ( if @ extended_enat
% 0.28/0.47 @ ( N3
% 0.28/0.47 = ( zero_zero @ extended_enat ) )
% 0.28/0.47 @ ( zero_zero @ extended_enat )
% 0.28/0.47 @ ( extend1396239628finity @ extended_enat ) )
% 0.28/0.47 @ M3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % times_enat_def
% 0.28/0.47 thf(fact_174_mult__not__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( mult_zero @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ B2 )
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 & ( B2
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_not_zero
% 0.28/0.47 thf(fact_175_divisors__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1193490041visors @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ( times_times @ A @ A2 @ B2 )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 | ( B2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % divisors_zero
% 0.28/0.47 thf(fact_176_no__zero__divisors,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1193490041visors @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( B2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( times_times @ A @ A2 @ B2 )
% 0.28/0.47 != ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % no_zero_divisors
% 0.28/0.47 thf(fact_177_mult__left__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1923998003cancel @ A @ ( type @ A ) )
% 0.28/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.28/0.47 ( ( C2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ( times_times @ A @ C2 @ A2 )
% 0.28/0.47 = ( times_times @ A @ C2 @ B2 ) )
% 0.28/0.47 = ( A2 = B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_left_cancel
% 0.28/0.47 thf(fact_178_mult__right__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( semiri1923998003cancel @ A @ ( type @ A ) )
% 0.28/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.28/0.47 ( ( C2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ( times_times @ A @ A2 @ C2 )
% 0.28/0.47 = ( times_times @ A @ B2 @ C2 ) )
% 0.28/0.47 = ( A2 = B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_right_cancel
% 0.28/0.47 thf(fact_179_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord893533164strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.28/0.47 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.28/0.47 thf(fact_180_mult__less__cancel__right__disj,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,C2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.28/0.47 & ( ord_less @ A @ A2 @ B2 ) )
% 0.28/0.47 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_less_cancel_right_disj
% 0.28/0.47 thf(fact_181_mult__strict__right__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord20386208strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.28/0.47 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_strict_right_mono
% 0.28/0.47 thf(fact_182_mult__strict__right__mono__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_strict_right_mono_neg
% 0.28/0.47 thf(fact_183_mult__less__cancel__left__disj,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.28/0.47 & ( ord_less @ A @ A2 @ B2 ) )
% 0.28/0.47 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_less_cancel_left_disj
% 0.28/0.47 thf(fact_184_mult__strict__left__mono,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord20386208strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.28/0.47 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_strict_left_mono
% 0.28/0.47 thf(fact_185_mult__strict__left__mono__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.28/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_strict_left_mono_neg
% 0.28/0.47 thf(fact_186_mult__less__cancel__left__pos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.28/0.47 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_less_cancel_left_pos
% 0.28/0.47 thf(fact_187_mult__less__cancel__left__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [C2: A,A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.28/0.47 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_less_cancel_left_neg
% 0.28/0.47 thf(fact_188_zero__less__mult__pos2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord20386208strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [B2: A,A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_mult_pos2
% 0.28/0.47 thf(fact_189_zero__less__mult__pos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord20386208strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_mult_pos
% 0.28/0.47 thf(fact_190_zero__less__mult__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.28/0.47 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_mult_iff
% 0.28/0.47 thf(fact_191_mult__pos__neg2,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord20386208strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_pos_neg2
% 0.28/0.47 thf(fact_192_mult__pos__pos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord20386208strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_pos_pos
% 0.28/0.47 thf(fact_193_mult__pos__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord20386208strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_pos_neg
% 0.28/0.47 thf(fact_194_mult__neg__pos,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord20386208strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_neg_pos
% 0.28/0.47 thf(fact_195_mult__less__0__iff,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.28/0.47 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_less_0_iff
% 0.28/0.47 thf(fact_196_not__square__less__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_ring @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % not_square_less_zero
% 0.28/0.47 thf(fact_197_mult__neg__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linord581940658strict @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % mult_neg_neg
% 0.28/0.47 thf(fact_198_nat__mult__less__cancel__disj,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.28/0.47 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.28/0.47 & ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_mult_less_cancel_disj
% 0.28/0.47 thf(fact_199_transfer__int__nat__relations_I2_J,axiom,
% 0.28/0.47 ! [X2: nat,Y3: nat] :
% 0.28/0.47 ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ X2 ) @ ( semiring_1_of_nat @ int @ Y3 ) )
% 0.28/0.47 = ( ord_less @ nat @ X2 @ Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % transfer_int_nat_relations(2)
% 0.28/0.47 thf(fact_200_transfer__int__nat__numerals_I1_J,axiom,
% 0.28/0.47 ( ( zero_zero @ int )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % transfer_int_nat_numerals(1)
% 0.28/0.47 thf(fact_201_transfer__int__nat__relations_I1_J,axiom,
% 0.28/0.47 ! [X2: nat,Y3: nat] :
% 0.28/0.47 ( ( ( semiring_1_of_nat @ int @ X2 )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ Y3 ) )
% 0.28/0.47 = ( X2 = Y3 ) ) ).
% 0.28/0.47
% 0.28/0.47 % transfer_int_nat_relations(1)
% 0.28/0.47 thf(fact_202_int__if__cong,axiom,
% 0.28/0.47 ! [P: $o,X2: nat,Y3: nat] :
% 0.28/0.47 ( ( P
% 0.28/0.47 => ( ( semiring_1_of_nat @ int @ X2 )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X2 @ Y3 ) ) ) )
% 0.28/0.47 & ( ~ P
% 0.28/0.47 => ( ( semiring_1_of_nat @ int @ Y3 )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X2 @ Y3 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % int_if_cong
% 0.28/0.47 thf(fact_203_nat__mult__eq__cancel__disj,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ( times_times @ nat @ K @ M )
% 0.28/0.47 = ( times_times @ nat @ K @ N ) )
% 0.28/0.47 = ( ( K
% 0.28/0.47 = ( zero_zero @ nat ) )
% 0.28/0.47 | ( M = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_mult_eq_cancel_disj
% 0.28/0.47 thf(fact_204_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
% 0.28/0.47 ! [X2: nat,Y3: nat] :
% 0.28/0.47 ( ( times_times @ int @ ( semiring_1_of_nat @ int @ X2 ) @ ( semiring_1_of_nat @ int @ Y3 ) )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ ( times_times @ nat @ X2 @ Y3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % Nat_Transfer.transfer_int_nat_functions(2)
% 0.28/0.47 thf(fact_205_nat__mult__less__cancel1,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.28/0.47 => ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 0.28/0.47 = ( ord_less @ nat @ M @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_mult_less_cancel1
% 0.28/0.47 thf(fact_206_nat__mult__eq__cancel1,axiom,
% 0.28/0.47 ! [K: nat,M: nat,N: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.28/0.47 => ( ( ( times_times @ nat @ K @ M )
% 0.28/0.47 = ( times_times @ nat @ K @ N ) )
% 0.28/0.47 = ( M = N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_mult_eq_cancel1
% 0.28/0.47 thf(fact_207_linordered__field__class_Osign__simps_I45_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 0.28/0.47 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_class.sign_simps(45)
% 0.28/0.47 thf(fact_208_linordered__field__class_Osign__simps_I44_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
% 0.28/0.47 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 0.28/0.47 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_class.sign_simps(44)
% 0.28/0.47 thf(fact_209_linordered__field__no__ub,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X5: A] :
% 0.28/0.47 ? [X1: A] : ( ord_less @ A @ X5 @ X1 ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_no_ub
% 0.28/0.47 thf(fact_210_linordered__field__no__lb,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X5: A] :
% 0.28/0.47 ? [Y5: A] : ( ord_less @ A @ Y5 @ X5 ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_no_lb
% 0.28/0.47 thf(fact_211_linordered__field__class_Osign__simps_I25_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [B2: A,A2: A,C2: A] :
% 0.28/0.47 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
% 0.28/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_class.sign_simps(25)
% 0.28/0.47 thf(fact_212_linordered__field__class_Osign__simps_I24_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ( ( times_times @ A )
% 0.28/0.47 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_class.sign_simps(24)
% 0.28/0.47 thf(fact_213_linordered__field__class_Osign__simps_I23_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A,C2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 0.28/0.47 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % linordered_field_class.sign_simps(23)
% 0.28/0.47 thf(fact_214_scount__eq,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( coindu1365464361scount @ A )
% 0.28/0.47 = ( ^ [P2: ( stream @ A ) > $o,Omega2: stream @ A] :
% 0.28/0.47 ( if @ extended_enat @ ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P2 ) @ Omega2 ) @ ( extend1396239628finity @ extended_enat )
% 0.28/0.47 @ ( extended_enat2
% 0.28/0.47 @ ( finite_card @ nat
% 0.28/0.47 @ ( collect @ nat
% 0.28/0.47 @ ^ [I2: nat] : ( P2 @ ( sdrop @ A @ I2 @ Omega2 ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % scount_eq
% 0.28/0.47 thf(fact_215_semiring__normalization__rules_I10_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(10)
% 0.28/0.47 thf(fact_216_semiring__normalization__rules_I7_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ( ( times_times @ A )
% 0.28/0.47 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(7)
% 0.28/0.47 thf(fact_217_semiring__normalization__rules_I13_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.28/0.47 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ ( times_times @ A @ Ly @ Ry ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(13)
% 0.28/0.47 thf(fact_218_semiring__normalization__rules_I14_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.28/0.47 = ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ ( times_times @ A @ Rx @ Ry ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(14)
% 0.28/0.47 thf(fact_219_semiring__normalization__rules_I15_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.28/0.47 = ( times_times @ A @ Rx @ ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Ry ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(15)
% 0.28/0.47 thf(fact_220_semiring__normalization__rules_I16_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [Lx: A,Ly: A,Rx: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
% 0.28/0.47 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ly ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(16)
% 0.28/0.47 thf(fact_221_semiring__normalization__rules_I17_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [Lx: A,Ly: A,Rx: A] :
% 0.28/0.47 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
% 0.28/0.47 = ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ Rx ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(17)
% 0.28/0.47 thf(fact_222_semiring__normalization__rules_I18_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [Lx: A,Rx: A,Ry: A] :
% 0.28/0.47 ( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
% 0.28/0.47 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ry ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(18)
% 0.28/0.47 thf(fact_223_semiring__normalization__rules_I19_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [Lx: A,Rx: A,Ry: A] :
% 0.28/0.47 ( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
% 0.28/0.47 = ( times_times @ A @ Rx @ ( times_times @ A @ Lx @ Ry ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(19)
% 0.28/0.47 thf(fact_224_scount__eq__card,axiom,
% 0.28/0.47 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A
% 0.28/0.47 @ ( linear1386806755on_alw @ A
% 0.28/0.47 @ ^ [Xs: stream @ A] :
% 0.28/0.47 ~ ( P @ Xs ) )
% 0.28/0.47 @ Omega )
% 0.28/0.47 => ( ( coindu1365464361scount @ A @ P @ Omega )
% 0.28/0.47 = ( extended_enat2
% 0.28/0.47 @ ( finite_card @ nat
% 0.28/0.47 @ ( collect @ nat
% 0.28/0.47 @ ^ [I2: nat] : ( P @ ( sdrop @ A @ I2 @ Omega ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % scount_eq_card
% 0.28/0.47 thf(fact_225_semiring__normalization__rules_I9_J,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % semiring_normalization_rules(9)
% 0.28/0.47 thf(fact_226_card__Collect__less__nat,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( finite_card @ nat
% 0.28/0.47 @ ( collect @ nat
% 0.28/0.47 @ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) )
% 0.28/0.47 = N ) ).
% 0.28/0.47
% 0.28/0.47 % card_Collect_less_nat
% 0.28/0.47 thf(fact_227_sdrop__wait,axiom,
% 0.28/0.47 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 0.28/0.47 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 0.28/0.47 => ( Phi @ ( sdrop @ A @ ( linear1335279038n_wait @ A @ Phi @ Xs2 ) @ Xs2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % sdrop_wait
% 0.28/0.47 thf(fact_228_transfer__morphism__int__nat,axiom,
% 0.28/0.47 ( nat_tr1645093318rphism @ nat @ int @ ( semiring_1_of_nat @ int )
% 0.28/0.47 @ ^ [N3: nat] : $true ) ).
% 0.28/0.47
% 0.28/0.47 % transfer_morphism_int_nat
% 0.28/0.47 thf(fact_229_zero__less__ceiling,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1727834104eiling @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X2: A] :
% 0.28/0.47 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
% 0.28/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_ceiling
% 0.28/0.47 thf(fact_230_ceiling__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1727834104eiling @ A @ ( type @ A ) )
% 0.28/0.47 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ int ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ceiling_zero
% 0.28/0.47 thf(fact_231_ceiling__of__nat,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1727834104eiling @ A @ ( type @ A ) )
% 0.28/0.47 => ! [N: nat] :
% 0.28/0.47 ( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.28/0.47 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ceiling_of_nat
% 0.28/0.47 thf(fact_232_ceiling__less__cancel,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1727834104eiling @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X2: A,Y3: A] :
% 0.28/0.47 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y3 ) )
% 0.28/0.47 => ( ord_less @ A @ X2 @ Y3 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ceiling_less_cancel
% 0.28/0.47 thf(fact_233_zero__less__nat__eq,axiom,
% 0.28/0.47 ! [Z: int] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z ) )
% 0.28/0.47 = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% 0.28/0.47
% 0.28/0.47 % zero_less_nat_eq
% 0.28/0.47 thf(fact_234_ex__inverse__of__nat__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( archim1804426504_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [X2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
% 0.28/0.47 => ? [N2: nat] :
% 0.28/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 0.28/0.47 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % ex_inverse_of_nat_less
% 0.28/0.47 thf(fact_235_inverse__zero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A @ ( type @ A ) )
% 0.28/0.47 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 0.28/0.47 = ( zero_zero @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_zero
% 0.28/0.47 thf(fact_236_inverse__nonzero__iff__nonzero,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( division_ring @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ( inverse_inverse @ A @ A2 )
% 0.28/0.47 = ( zero_zero @ A ) )
% 0.28/0.47 = ( A2
% 0.28/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_nonzero_iff_nonzero
% 0.28/0.47 thf(fact_237_inverse__mult__distrib,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
% 0.28/0.47 = ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_mult_distrib
% 0.28/0.47 thf(fact_238_nat__int,axiom,
% 0.28/0.47 ! [N: nat] :
% 0.28/0.47 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 0.28/0.47 = N ) ).
% 0.28/0.47
% 0.28/0.47 % nat_int
% 0.28/0.47 thf(fact_239_inverse__positive__iff__positive,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.28/0.47 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_positive_iff_positive
% 0.28/0.47 thf(fact_240_inverse__negative__iff__negative,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_negative_iff_negative
% 0.28/0.47 thf(fact_241_inverse__less__iff__less__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_less_iff_less_neg
% 0.28/0.47 thf(fact_242_inverse__less__iff__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_less_iff_less
% 0.28/0.47 thf(fact_243_nat__0,axiom,
% 0.28/0.47 ( ( nat2 @ ( zero_zero @ int ) )
% 0.28/0.47 = ( zero_zero @ nat ) ) ).
% 0.28/0.47
% 0.28/0.47 % nat_0
% 0.28/0.47 thf(fact_244_zless__nat__conj,axiom,
% 0.28/0.47 ! [W: int,Z: int] :
% 0.28/0.47 ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.28/0.47 = ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
% 0.28/0.47 & ( ord_less @ int @ W @ Z ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % zless_nat_conj
% 0.28/0.47 thf(fact_245_zless__nat__eq__int__zless,axiom,
% 0.28/0.47 ! [M: nat,Z: int] :
% 0.28/0.47 ( ( ord_less @ nat @ M @ ( nat2 @ Z ) )
% 0.28/0.47 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z ) ) ).
% 0.28/0.47
% 0.28/0.47 % zless_nat_eq_int_zless
% 0.28/0.47 thf(fact_246_positive__imp__inverse__positive,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % positive_imp_inverse_positive
% 0.28/0.47 thf(fact_247_negative__imp__inverse__negative,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % negative_imp_inverse_negative
% 0.28/0.47 thf(fact_248_inverse__positive__imp__positive,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_positive_imp_positive
% 0.28/0.47 thf(fact_249_inverse__negative__imp__negative,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ( A2
% 0.28/0.47 != ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_negative_imp_negative
% 0.28/0.47 thf(fact_250_less__imp__inverse__less__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_inverse_less_neg
% 0.28/0.47 thf(fact_251_inverse__less__imp__less__neg,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.28/0.47 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.28/0.47 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % inverse_less_imp_less_neg
% 0.28/0.47 thf(fact_252_less__imp__inverse__less,axiom,
% 0.28/0.47 ! [A: $tType] :
% 0.28/0.47 ( ( linordered_field @ A @ ( type @ A ) )
% 0.28/0.47 => ! [A2: A,B2: A] :
% 0.28/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.47 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 0.28/0.47 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 % less_imp_inverse_less
% 0.28/0.47
% 0.28/0.47 %----Type constructors (37)
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 0.28/0.47 semiri1923998003cancel @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
% 0.28/0.47 linord893533164strict @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
% 0.28/0.47 linord20386208strict @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 0.28/0.47 semiri1193490041visors @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
% 0.28/0.47 linord581940658strict @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 0.28/0.47 linordered_semidom @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
% 0.28/0.47 ab_semigroup_mult @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
% 0.28/0.47 linordered_ring @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 0.28/0.47 linordered_idom @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 0.28/0.47 comm_semiring_1 @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
% 0.28/0.47 semigroup_mult @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 0.28/0.47 semiring_char_0 @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 0.28/0.47 semiring_1 @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
% 0.28/0.47 mult_zero @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 0.28/0.47 zero @ int @ ( type @ int ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_1,axiom,
% 0.28/0.47 semiri1923998003cancel @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_2,axiom,
% 0.28/0.47 linord893533164strict @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.28/0.47 canoni770627133id_add @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_3,axiom,
% 0.28/0.47 linord20386208strict @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_4,axiom,
% 0.28/0.47 semiri1193490041visors @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_5,axiom,
% 0.28/0.47 linordered_semidom @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_6,axiom,
% 0.28/0.47 ab_semigroup_mult @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_7,axiom,
% 0.28/0.47 comm_semiring_1 @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_8,axiom,
% 0.28/0.47 semigroup_mult @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_9,axiom,
% 0.28/0.47 semiring_char_0 @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Osemiring__1_10,axiom,
% 0.28/0.47 semiring_1 @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Rings_Omult__zero_11,axiom,
% 0.28/0.47 mult_zero @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Nat_Onat___Groups_Ozero_12,axiom,
% 0.28/0.47 zero @ nat @ ( type @ nat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_13,axiom,
% 0.28/0.47 canoni770627133id_add @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_14,axiom,
% 0.28/0.47 semiri1193490041visors @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_15,axiom,
% 0.28/0.47 ab_semigroup_mult @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_16,axiom,
% 0.28/0.47 comm_semiring_1 @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_17,axiom,
% 0.28/0.47 semigroup_mult @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_18,axiom,
% 0.28/0.47 semiring_char_0 @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_19,axiom,
% 0.28/0.47 semiring_1 @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_20,axiom,
% 0.28/0.47 mult_zero @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_21,axiom,
% 0.28/0.47 zero @ extended_enat @ ( type @ extended_enat ) ).
% 0.28/0.47
% 0.28/0.47 %----Helper facts (3)
% 0.28/0.47 thf(help_If_3_1_T,axiom,
% 0.28/0.47 ! [P: $o] :
% 0.28/0.47 ( ( P = $true )
% 0.28/0.47 | ( P = $false ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(help_If_2_1_T,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Y3: A] :
% 0.28/0.47 ( ( if @ A @ $false @ X2 @ Y3 )
% 0.28/0.47 = Y3 ) ).
% 0.28/0.47
% 0.28/0.47 thf(help_If_1_1_T,axiom,
% 0.28/0.47 ! [A: $tType,X2: A,Y3: A] :
% 0.28/0.47 ( ( if @ A @ $true @ X2 @ Y3 )
% 0.28/0.47 = X2 ) ).
% 0.28/0.47
% 0.28/0.47 %----Conjectures (1)
% 0.28/0.47 thf(conj_0,conjecture,
% 0.28/0.47 ( ( ord_less @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ ( coindu76177514sfirst @ a @ p @ omega ) )
% 0.28/0.47 = ( ~ ( p @ ( sdrop @ a @ ( zero_zero @ nat ) @ omega ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 ------------------------
% 2.85/1.54 ---- Embedded in THF ---
% 2.85/1.54 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.85/1.54 %%% Generated on Tue Feb 24 06:10:15 EST 2026
% 2.85/1.54 %%% using '$$dhol' embedding, version 1.3.0.
% 2.85/1.54 %%% Logic specification used:
% 2.85/1.54 %%% thf(spec, logic, $$dhol).
% 2.85/1.54
% 2.85/1.54 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.plLMjpaOMh/DTF2DTF20370.p
% 2.85/1.54 thf(ty_t_Extended__Nat_Oenat, type, extended_enat: $tType).
% 2.85/1.54 thf(ty_t_Stream_Ostream, type, stream: ($tType > $tType)).
% 2.85/1.54 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.85/1.54 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.85/1.54 thf(ty_t_Int_Oint, type, int: $tType).
% 2.85/1.54 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.85/1.54 thf(ty_tf_a, type, a: $tType).
% 2.85/1.54 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Omult__zero, type, mult_zero: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Groups_Osemigroup__mult, type, semigroup_mult: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Olinordered__ring, type, linordered_ring: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Groups_Oab__semigroup__mult, type, ab_semigroup_mult: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Olinordered__ring__strict, type, linord581940658strict: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Archimedean__Field_Ofloor__ceiling, type, archim1727834104eiling: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri1193490041visors: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Olinordered__semiring__strict, type, linord20386208strict: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim1804426504_field: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Olinordered__comm__semiring__strict, type, linord893533164strict: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel, type, semiri1923998003cancel: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.85/1.54 thf(sy_c_Archimedean__Field_Oceiling, type, archimedean_ceiling: (!> [A:$tType]: ((A > int)))).
% 2.85/1.54 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oscount, type, coindu1365464361scount: (!> [S:$tType]: ((((stream @ S) > $o) > ((stream @ S) > extended_enat))))).
% 2.85/1.54 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Osfirst, type, coindu76177514sfirst: (!> [S:$tType]: ((((stream @ S) > $o) > ((stream @ S) > extended_enat))))).
% 2.85/1.54 thf(sy_c_Extended__Nat_Oenat, type, extended_enat2: (nat > extended_enat)).
% 2.85/1.54 thf(sy_c_Extended__Nat_Oenat_Ocase__enat, type, extended_case_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.85/1.54 thf(sy_c_Extended__Nat_Oenat_Orec__enat, type, extended_rec_enat: (!> [T:$tType]: (((nat > T) > (T > (extended_enat > T)))))).
% 2.85/1.54 thf(sy_c_Extended__Nat_Oenat_Osize__enat, type, extended_size_enat: (extended_enat > nat)).
% 2.85/1.54 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity, type, extend1396239628finity: (!> [A:$tType]: (A))).
% 2.85/1.54 thf(sy_c_Extended__Nat_Othe__enat, type, extended_the_enat: (extended_enat > nat)).
% 2.85/1.54 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))).
% 2.85/1.54 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))).
% 2.85/1.54 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))).
% 2.85/1.54 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.85/1.54 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.85/1.54 thf(sy_c_Int_Onat, type, nat2: (int > nat)).
% 2.85/1.54 thf(sy_c_Linear__Temporal__Logic__on__Streams_OUNTIL, type, linear1341015445_UNTIL: (!> [A:$tType]: ((((stream @ A) > $o) > (((stream @ A) > $o) > ((stream @ A) > $o)))))).
% 2.85/1.54 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw, type, linear1386806755on_alw: (!> [A:$tType]: ((((stream @ A) > $o) > ((stream @ A) > $o))))).
% 2.85/1.54 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev, type, linear505997466_on_ev: (!> [A:$tType]: ((((stream @ A) > $o) > ((stream @ A) > $o))))).
% 2.85/1.54 thf(sy_c_Linear__Temporal__Logic__on__Streams_Owait, type, linear1335279038n_wait: (!> [A:$tType]: ((((stream @ A) > $o) > ((stream @ A) > nat))))).
% 2.85/1.54 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 2.85/1.54 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.85/1.54 thf(sy_c_Nat__Transfer_Otransfer__morphism, type, nat_tr1645093318rphism: (!> [B:$tType,A:$tType]: (((B > A) > ((B > $o) > $o))))).
% 2.85/1.54 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.85/1.54 thf(sy_c_Pure_Otype, type, type: (!> [A:$tType]: ((itself @ A)))).
% 2.85/1.54 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.85/1.54 thf(sy_c_Stream_Osdrop, type, sdrop: (!> [A:$tType]: ((nat > ((stream @ A) > (stream @ A)))))).
% 2.85/1.54 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.85/1.54 thf(sy_v_P, type, p: ((stream @ a) > $o)).
% 2.85/1.54 thf(sy_v__092_060omega_062_H____, type, omega: (stream @ a)).
% 2.85/1.54 thf(fact_0__C0_Oprems_C, axiom, ((((linear1341015445_UNTIL @ a) @ (^ [Xs:(stream @ a)]: ((~ ((p @ Xs)))))) @ ((linear1386806755on_alw @ a) @ p)) @ omega)).
% 2.85/1.54 thf(fact_1_less__sfirstD, axiom, (! [A:$tType,N:nat,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((ord_less @ extended_enat) @ (extended_enat2 @ N)) @ (((coindu76177514sfirst @ A) @ P) @ Omega)) => (~ ((P @ (((sdrop @ A) @ N) @ Omega)))))))).
% 2.85/1.54 thf(fact_2_not__gr__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.85/1.54 thf(fact_3_enat_Oinject, axiom, (! [Nat:nat,Nat2:nat]: ((((extended_enat2 @ Nat) = (extended_enat2 @ Nat2)) = (Nat = Nat2))))).
% 2.85/1.54 thf(fact_4_sdrop_Osimps_I1_J, axiom, (! [A:$tType,S2:(stream @ A)]: (((((sdrop @ A) @ (zero_zero @ nat)) @ S2) = S2)))).
% 2.85/1.54 thf(fact_5_chain__incr, axiom, (! [A:$tType,Y:(A > extended_enat),K:nat]: (((! [I:A]: ((? [J:A]: ((((ord_less @ extended_enat) @ (Y @ I)) @ (Y @ J)))))) => (? [J2:A]: ((((ord_less @ extended_enat) @ (extended_enat2 @ K)) @ (Y @ J2)))))))).
% 2.85/1.54 thf(fact_6_enat__iless, axiom, (! [N:extended_enat,M:nat]: (((((ord_less @ extended_enat) @ N) @ (extended_enat2 @ M)) => (? [K2:nat]: ((N = (extended_enat2 @ K2)))))))).
% 2.85/1.54 thf(fact_7_less__numeral__extra_I3_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ (zero_zero @ A)))))))).
% 2.85/1.54 thf(fact_8_gr__zeroI, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.85/1.54 thf(fact_9_not__less__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_10_gr__implies__not__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.85/1.54 thf(fact_11_zero__less__iff__neq__zero, axiom, (! [A:$tType]: ((((canoni770627133id_add @ A) @ (type @ A)) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.85/1.54 thf(fact_12_fun__cong__unused__0, axiom, (! [A:$tType,B:$tType,C:$tType]: ((((zero @ B) @ (type @ B)) => (! [F:((A > B) > C),G:C]: (((F = (^ [X:(A > B)]: (G))) => ((F @ (^ [X:A]: ((zero_zero @ B)))) = G)))))))).
% 2.85/1.54 thf(fact_13_i0__less, axiom, (! [N:extended_enat]: (((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ N) = (N != (zero_zero @ extended_enat)))))).
% 2.85/1.54 thf(fact_14_sfirst__0, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((P @ Omega) => ((((coindu76177514sfirst @ A) @ P) @ Omega) = (zero_zero @ extended_enat)))))).
% 2.85/1.54 thf(fact_15_enat__ord__simps_I2_J, axiom, (! [M:nat,N:nat]: (((((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 2.85/1.54 thf(fact_16_not__iless0, axiom, (! [N:extended_enat]: ((~ ((((ord_less @ extended_enat) @ N) @ (zero_zero @ extended_enat))))))).
% 2.85/1.54 thf(fact_17_zero__enat__def, axiom, ((zero_zero @ extended_enat) = (extended_enat2 @ (zero_zero @ nat)))).
% 2.85/1.54 thf(fact_18_enat__0__iff_I1_J, axiom, (! [X2:nat]: ((((extended_enat2 @ X2) = (zero_zero @ extended_enat)) = (X2 = (zero_zero @ nat)))))).
% 2.85/1.54 thf(fact_19_enat__0__iff_I2_J, axiom, (! [X2:nat]: ((((zero_zero @ extended_enat) = (extended_enat2 @ X2)) = (X2 = (zero_zero @ nat)))))).
% 2.85/1.54 thf(fact_20_less__enatE, axiom, (! [N:extended_enat,M:nat]: (((((ord_less @ extended_enat) @ N) @ (extended_enat2 @ M)) => (~ ((! [K2:nat]: (((N = (extended_enat2 @ K2)) => (~ ((((ord_less @ nat) @ K2) @ M)))))))))))).
% 2.85/1.54 thf(fact_21_sfirst__eq__0, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: ((((((coindu76177514sfirst @ A) @ P) @ Omega) = (zero_zero @ extended_enat)) = (P @ Omega))))).
% 2.85/1.54 thf(fact_22_zero__reorient, axiom, (! [A:$tType]: ((((zero @ A) @ (type @ A)) => (! [X2:A]: ((((zero_zero @ A) = X2) = (X2 = (zero_zero @ A))))))))).
% 2.85/1.54 thf(fact_23_enat__less__induct, axiom, (! [P:(extended_enat > $o),N:extended_enat]: (((! [N2:extended_enat]: (((! [M2:extended_enat]: (((((ord_less @ extended_enat) @ M2) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 2.85/1.54 thf(fact_24_enat_Osize_I1_J, axiom, (! [Nat:nat]: (((extended_size_enat @ (extended_enat2 @ Nat)) = (zero_zero @ nat))))).
% 2.85/1.54 thf(fact_25_until__false, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((((linear1341015445_UNTIL @ A) @ Phi) @ (^ [Xs:(stream @ A)]: ($false))) = ((linear1386806755on_alw @ A) @ Phi))))).
% 2.85/1.54 thf(fact_26_alw__alw, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: ((((linear1386806755on_alw @ A) @ ((linear1386806755on_alw @ A) @ Phi)) = ((linear1386806755on_alw @ A) @ Phi))))).
% 2.85/1.54 thf(fact_27_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.85/1.54 thf(fact_28_alw__iff__sdrop, axiom, (! [A:$tType]: (((linear1386806755on_alw @ A) = (^ [P2:((stream @ A) > $o),Omega2:(stream @ A)]: ((! [M3:nat]: ((P2 @ (((sdrop @ A) @ M3) @ Omega2)))))))))).
% 2.85/1.54 thf(fact_29_alw__sdrop, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),N:nat]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs2) => (((linear1386806755on_alw @ A) @ Phi) @ (((sdrop @ A) @ N) @ Xs2)))))).
% 2.85/1.54 thf(fact_30_enat_Osize_I3_J, axiom, (! [Nat:nat]: ((((size_size @ extended_enat) @ (extended_enat2 @ Nat)) = (zero_zero @ nat))))).
% 2.85/1.54 thf(fact_31_enat_Osimps_I6_J, axiom, (! [T:$tType,F1:(nat > T),F2:T,Nat:nat]: ((((((extended_rec_enat @ T) @ F1) @ F2) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.85/1.54 thf(fact_32_alw__False, axiom, (! [A:$tType,Omega:(stream @ A)]: ((~ ((((linear1386806755on_alw @ A) @ (^ [X:(stream @ A)]: ($false))) @ Omega)))))).
% 2.85/1.54 thf(fact_33_alw__aand, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi:((stream @ A) > $o)]: ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) & (Psi @ Xs))))) = (^ [Xs:(stream @ A)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs) & (((linear1386806755on_alw @ A) @ Psi) @ Xs)))))))).
% 2.85/1.54 thf(fact_34_alw__mp, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => (Psi @ Xs))))) @ Xs2) => (((linear1386806755on_alw @ A) @ Psi) @ Xs2)))))).
% 2.85/1.54 thf(fact_35_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.85/1.54 thf(fact_36_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.85/1.54 thf(fact_37_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.85/1.54 thf(fact_38_less__not__refl3, axiom, (! [S2:nat,T2:nat]: (((((ord_less @ nat) @ S2) @ T2) => (S2 != T2))))).
% 2.85/1.54 thf(fact_39_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ nat) @ (F @ Y2)) @ (F @ X3)) => (P @ Y2)))) => (P @ X3)))) => (P @ A2))))).
% 2.85/1.54 thf(fact_40_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.85/1.54 thf(fact_41_nat__less__induct, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) => (P @ M2)))) => (P @ N2)))) => (P @ N))))).
% 2.85/1.54 thf(fact_42_infinite__descent, axiom, (! [P:(nat > $o),N:nat]: (((! [N2:nat]: (((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2))))))))) => (P @ N))))).
% 2.85/1.54 thf(fact_43_linorder__neqE__nat, axiom, (! [X2:nat,Y3:nat]: (((X2 != Y3) => ((~ ((((ord_less @ nat) @ X2) @ Y3))) => (((ord_less @ nat) @ Y3) @ X2)))))).
% 2.85/1.54 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.85/1.54 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))).
% 2.85/1.54 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.85/1.54 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.85/1.54 thf(fact_48_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ nat) @ (F @ Y2)) @ (F @ X3)) => (P @ Y2)))) => (P @ X3)))) => (P @ A2))))).
% 2.85/1.54 thf(fact_49_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X2:A]: (((! [X3:A]: (((~ ((P @ X3))) => (? [Y2:A]: (((((ord_less @ nat) @ (V @ Y2)) @ (V @ X3)) & (~ ((P @ Y2))))))))) => (P @ X2))))).
% 2.85/1.54 thf(fact_50_size__enat__overloaded__def, axiom, ((size_size @ extended_enat) = (((extended_rec_enat @ nat) @ (^ [X:nat]: ((zero_zero @ nat)))) @ (zero_zero @ nat)))).
% 2.85/1.54 thf(fact_51_size__enat__def, axiom, (extended_size_enat = (((extended_rec_enat @ nat) @ (^ [X:nat]: ((zero_zero @ nat)))) @ (zero_zero @ nat)))).
% 2.85/1.54 thf(fact_52_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.85/1.54 thf(fact_53_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.85/1.54 thf(fact_54_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.85/1.54 thf(fact_55_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.85/1.54 thf(fact_56_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 2.85/1.54 thf(fact_57_infinite__descent0, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) => ((~ ((P @ N2))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N2) & (~ ((P @ M2)))))))))) => (P @ N)))))).
% 2.85/1.54 thf(fact_58_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.85/1.54 thf(fact_59_infinite__descent0__measure, axiom, (! [A:$tType,V:(A > nat),P:(A > $o),X2:A]: (((! [X3:A]: ((((V @ X3) = (zero_zero @ nat)) => (P @ X3)))) => ((! [X3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V @ X3)) => ((~ ((P @ X3))) => (? [Y2:A]: (((((ord_less @ nat) @ (V @ Y2)) @ (V @ X3)) & (~ ((P @ Y2)))))))))) => (P @ X2)))))).
% 2.85/1.54 thf(fact_60_all__imp__alw, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((! [X1:(stream @ A)]: ((Phi @ X1))) => (((linear1386806755on_alw @ A) @ Phi) @ Xs2))))).
% 2.85/1.54 thf(fact_61_alw__mono, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs2) => ((! [Xs3:(stream @ A)]: (((Phi @ Xs3) => (Psi @ Xs3)))) => (((linear1386806755on_alw @ A) @ Psi) @ Xs2)))))).
% 2.85/1.54 thf(fact_62_alw__cong, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A),Q1:((stream @ A) > $o),Q2:((stream @ A) > $o)]: (((((linear1386806755on_alw @ A) @ P) @ Omega) => ((! [Omega3:(stream @ A)]: (((P @ Omega3) => ((Q1 @ Omega3) = (Q2 @ Omega3))))) => ((((linear1386806755on_alw @ A) @ Q1) @ Omega) = (((linear1386806755on_alw @ A) @ Q2) @ Omega))))))).
% 2.85/1.54 thf(fact_63_alw__alwD, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear1386806755on_alw @ A) @ P) @ Omega) => (((linear1386806755on_alw @ A) @ ((linear1386806755on_alw @ A) @ P)) @ Omega))))).
% 2.85/1.54 thf(fact_64_alwD, axiom, (! [A:$tType,P:((stream @ A) > $o),X2:(stream @ A)]: (((((linear1386806755on_alw @ A) @ P) @ X2) => (P @ X2))))).
% 2.85/1.54 thf(fact_65_until__monoR, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi_1:((stream @ A) > $o),Xs2:(stream @ A),Psi_2:((stream @ A) > $o)]: ((((((linear1341015445_UNTIL @ A) @ Phi) @ Psi_1) @ Xs2) => ((! [Xs3:(stream @ A)]: (((Psi_1 @ Xs3) => (Psi_2 @ Xs3)))) => ((((linear1341015445_UNTIL @ A) @ Phi) @ Psi_2) @ Xs2)))))).
% 2.85/1.54 thf(fact_66_until__monoL, axiom, (! [A:$tType,Phi_1:((stream @ A) > $o),Psi:((stream @ A) > $o),Xs2:(stream @ A),Phi_2:((stream @ A) > $o)]: ((((((linear1341015445_UNTIL @ A) @ Phi_1) @ Psi) @ Xs2) => ((! [Xs3:(stream @ A)]: (((Phi_1 @ Xs3) => (Phi_2 @ Xs3)))) => ((((linear1341015445_UNTIL @ A) @ Phi_2) @ Psi) @ Xs2)))))).
% 2.85/1.54 thf(fact_67_until__mono, axiom, (! [A:$tType,Phi_1:((stream @ A) > $o),Psi_1:((stream @ A) > $o),Xs2:(stream @ A),Phi_2:((stream @ A) > $o),Psi_2:((stream @ A) > $o)]: ((((((linear1341015445_UNTIL @ A) @ Phi_1) @ Psi_1) @ Xs2) => ((! [Xs3:(stream @ A)]: (((Phi_1 @ Xs3) => (Phi_2 @ Xs3)))) => ((! [Xs3:(stream @ A)]: (((Psi_1 @ Xs3) => (Psi_2 @ Xs3)))) => ((((linear1341015445_UNTIL @ A) @ Phi_2) @ Psi_2) @ Xs2))))))).
% 2.85/1.54 thf(fact_68_UNTIL_Obase, axiom, (! [A:$tType,Psi:((stream @ A) > $o),Xs2:(stream @ A),Phi:((stream @ A) > $o)]: (((Psi @ Xs2) => ((((linear1341015445_UNTIL @ A) @ Phi) @ Psi) @ Xs2))))).
% 2.85/1.54 thf(fact_69_of__nat__0__less__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 2.85/1.54 thf(fact_70_less__enat__def, axiom, ((ord_less @ extended_enat) = (^ [M3:extended_enat,N3:extended_enat]: (((((extended_case_enat @ $o) @ (^ [M1:nat]: (((((extended_case_enat @ $o) @ ((ord_less @ nat) @ M1)) @ $true) @ N3)))) @ $false) @ M3))))).
% 2.85/1.54 thf(fact_71_case__enat__def, axiom, (! [T:$tType]: (((extended_case_enat @ T) = (extended_rec_enat @ T))))).
% 2.85/1.54 thf(fact_72_enat_Osimps_I4_J, axiom, (! [T:$tType,F1:(nat > T),F2:T,Nat:nat]: ((((((extended_case_enat @ T) @ F1) @ F2) @ (extended_enat2 @ Nat)) = (F1 @ Nat))))).
% 2.85/1.54 thf(fact_73_scount__eq__0__iff, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: ((((((coindu1365464361scount @ A) @ P) @ Omega) = (zero_zero @ extended_enat)) = (((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs)))))) @ Omega))))).
% 2.85/1.54 thf(fact_74_scount__eq__0I, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs)))))) @ Omega) => ((((coindu1365464361scount @ A) @ P) @ Omega) = (zero_zero @ extended_enat)))))).
% 2.85/1.54 thf(fact_75_scount__eq__0D, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: ((((((coindu1365464361scount @ A) @ P) @ Omega) = (zero_zero @ extended_enat)) => (((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs)))))) @ Omega))))).
% 2.85/1.54 thf(fact_76_enat_Osize_I4_J, axiom, (((size_size @ extended_enat) @ (extend1396239628finity @ extended_enat)) = (zero_zero @ nat))).
% 2.85/1.54 thf(fact_77_of__nat__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type @ A)) => (! [M:nat,N:nat]: (((((semiring_1_of_nat @ A) @ M) = ((semiring_1_of_nat @ A) @ N)) = (M = N)))))))).
% 2.85/1.54 thf(fact_78_not__infinity__eq, axiom, (! [X2:extended_enat]: (((X2 != (extend1396239628finity @ extended_enat)) = (? [I2:nat]: ((X2 = (extended_enat2 @ I2)))))))).
% 2.85/1.54 thf(fact_79_not__enat__eq, axiom, (! [X2:extended_enat]: (((! [Y4:nat]: ((X2 != (extended_enat2 @ Y4)))) = (X2 = (extend1396239628finity @ extended_enat)))))).
% 2.85/1.54 thf(fact_80_enat__ord__simps_I4_J, axiom, (! [Q3:extended_enat]: (((((ord_less @ extended_enat) @ Q3) @ (extend1396239628finity @ extended_enat)) = (Q3 != (extend1396239628finity @ extended_enat)))))).
% 2.85/1.54 thf(fact_81_enat__ord__simps_I6_J, axiom, (! [Q3:extended_enat]: ((~ ((((ord_less @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ Q3)))))).
% 2.85/1.54 thf(fact_82_enat_Osimps_I5_J, axiom, (! [T:$tType,F1:(nat > T),F2:T]: ((((((extended_case_enat @ T) @ F1) @ F2) @ (extend1396239628finity @ extended_enat)) = F2)))).
% 2.85/1.54 thf(fact_83_enat_Osimps_I7_J, axiom, (! [T:$tType,F1:(nat > T),F2:T]: ((((((extended_rec_enat @ T) @ F1) @ F2) @ (extend1396239628finity @ extended_enat)) = F2)))).
% 2.85/1.54 thf(fact_84_of__nat__eq__0__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type @ A)) => (! [M:nat]: (((((semiring_1_of_nat @ A) @ M) = (zero_zero @ A)) = (M = (zero_zero @ nat))))))))).
% 2.85/1.54 thf(fact_85_of__nat__0__eq__iff, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type @ A)) => (! [N:nat]: ((((zero_zero @ A) = ((semiring_1_of_nat @ A) @ N)) = ((zero_zero @ nat) = N)))))))).
% 2.85/1.54 thf(fact_86_of__nat__0, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type @ A)) => (((semiring_1_of_nat @ A) @ (zero_zero @ nat)) = (zero_zero @ A)))))).
% 2.85/1.54 thf(fact_87_of__nat__less__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) = (((ord_less @ nat) @ M) @ N)))))))).
% 2.85/1.54 thf(fact_88_case__enat__0, axiom, (! [A:$tType,F:(nat > A),I3:A]: ((((((extended_case_enat @ A) @ F) @ I3) @ (zero_zero @ extended_enat)) = (F @ (zero_zero @ nat)))))).
% 2.85/1.54 thf(fact_89_enat_Osize_I2_J, axiom, ((extended_size_enat @ (extend1396239628finity @ extended_enat)) = (zero_zero @ nat))).
% 2.85/1.54 thf(fact_90_of__nat__eq__enat, axiom, ((semiring_1_of_nat @ extended_enat) = extended_enat2)).
% 2.85/1.54 thf(fact_91_enat__ex__split, axiom, ((^ [P3:(extended_enat > $o)]: ((? [X4:extended_enat]: ((P3 @ X4))))) = (^ [P2:(extended_enat > $o)]: (((P2 @ (extend1396239628finity @ extended_enat)) | (? [X:nat]: ((P2 @ (extended_enat2 @ X))))))))).
% 2.85/1.54 thf(fact_92_enat_Oinducts, axiom, (! [P:(extended_enat > $o),Enat:extended_enat]: (((! [Nat3:nat]: ((P @ (extended_enat2 @ Nat3)))) => ((P @ (extend1396239628finity @ extended_enat)) => (P @ Enat)))))).
% 2.85/1.54 thf(fact_93_enat_Oexhaust, axiom, (! [Y3:extended_enat]: (((! [Nat3:nat]: ((Y3 != (extended_enat2 @ Nat3)))) => (Y3 = (extend1396239628finity @ extended_enat)))))).
% 2.85/1.54 thf(fact_94_enat3__cases, axiom, (! [Y3:extended_enat,Ya:extended_enat,Yb:extended_enat]: ((((? [Nat3:nat]: ((Y3 = (extended_enat2 @ Nat3)))) => ((? [Nata:nat]: ((Ya = (extended_enat2 @ Nata)))) => (! [Natb:nat]: ((Yb != (extended_enat2 @ Natb)))))) => (((? [Nat3:nat]: ((Y3 = (extended_enat2 @ Nat3)))) => ((? [Nata:nat]: ((Ya = (extended_enat2 @ Nata)))) => (Yb != (extend1396239628finity @ extended_enat)))) => (((? [Nat3:nat]: ((Y3 = (extended_enat2 @ Nat3)))) => ((Ya = (extend1396239628finity @ extended_enat)) => (! [Nata:nat]: ((Yb != (extended_enat2 @ Nata)))))) => (((? [Nat3:nat]: ((Y3 = (extended_enat2 @ Nat3)))) => ((Ya = (extend1396239628finity @ extended_enat)) => (Yb != (extend1396239628finity @ extended_enat)))) => (((Y3 = (extend1396239628finity @ extended_enat)) => ((? [Nat3:nat]: ((Ya = (extended_enat2 @ Nat3)))) => (! [Nata:nat]: ((Yb != (extended_enat2 @ Nata)))))) => (((Y3 = (extend1396239628finity @ extended_enat)) => ((? [Nat3:nat]: ((Ya = (extended_enat2 @ Nat3)))) => (Yb != (extend1396239628finity @ extended_enat)))) => (((Y3 = (extend1396239628finity @ extended_enat)) => ((Ya = (extend1396239628finity @ extended_enat)) => (! [Nat3:nat]: ((Yb != (extended_enat2 @ Nat3)))))) => (~ (((Y3 = (extend1396239628finity @ extended_enat)) => ((Ya = (extend1396239628finity @ extended_enat)) => (Yb != (extend1396239628finity @ extended_enat)))))))))))))))).
% 2.85/1.54 thf(fact_95_enat2__cases, axiom, (! [Y3:extended_enat,Ya:extended_enat]: ((((? [Nat3:nat]: ((Y3 = (extended_enat2 @ Nat3)))) => (! [Nata:nat]: ((Ya != (extended_enat2 @ Nata))))) => (((? [Nat3:nat]: ((Y3 = (extended_enat2 @ Nat3)))) => (Ya != (extend1396239628finity @ extended_enat))) => (((Y3 = (extend1396239628finity @ extended_enat)) => (! [Nat3:nat]: ((Ya != (extended_enat2 @ Nat3))))) => (~ (((Y3 = (extend1396239628finity @ extended_enat)) => (Ya != (extend1396239628finity @ extended_enat))))))))))).
% 2.85/1.54 thf(fact_96_enat_Odistinct_I1_J, axiom, (! [Nat:nat]: (((extended_enat2 @ Nat) != (extend1396239628finity @ extended_enat))))).
% 2.85/1.54 thf(fact_97_enat_Odistinct_I2_J, axiom, (! [Nat4:nat]: (((extend1396239628finity @ extended_enat) != (extended_enat2 @ Nat4))))).
% 2.85/1.54 thf(fact_98_infinity__ne__i0, axiom, ((extend1396239628finity @ extended_enat) != (zero_zero @ extended_enat))).
% 2.85/1.54 thf(fact_99_of__nat__less__0__iff, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [M:nat]: ((~ ((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_100_of__nat__less__imp__less, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [M:nat,N:nat]: (((((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N)) => (((ord_less @ nat) @ M) @ N)))))))).
% 2.85/1.54 thf(fact_101_less__imp__of__nat__less, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type @ A)) => (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 2.85/1.54 thf(fact_102_infinity__ilessE, axiom, (! [M:nat]: ((~ ((((ord_less @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ M))))))).
% 2.85/1.54 thf(fact_103_less__infinityE, axiom, (! [N:extended_enat]: (((((ord_less @ extended_enat) @ N) @ (extend1396239628finity @ extended_enat)) => (~ ((! [K2:nat]: ((N != (extended_enat2 @ K2)))))))))).
% 2.85/1.54 thf(fact_104_enat__ord__code_I4_J, axiom, (! [M:nat]: ((((ord_less @ extended_enat) @ (extended_enat2 @ M)) @ (extend1396239628finity @ extended_enat))))).
% 2.85/1.54 thf(fact_105_zero__less__imp__eq__int, axiom, (! [K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (? [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (K = ((semiring_1_of_nat @ int) @ N2))))))))).
% 2.85/1.54 thf(fact_106_reals__Archimedean2, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type @ A)) => (! [X2:A]: ((? [N2:nat]: ((((ord_less @ A) @ X2) @ ((semiring_1_of_nat @ A) @ N2)))))))))).
% 2.85/1.54 thf(fact_107_times__enat__simps_I3_J, axiom, (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((times_times @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ N)) = (zero_zero @ extended_enat))) & ((N != (zero_zero @ nat)) => ((((times_times @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ N)) = (extend1396239628finity @ extended_enat))))))).
% 2.85/1.54 thf(fact_108_times__enat__simps_I4_J, axiom, (! [M:nat]: ((((M = (zero_zero @ nat)) => ((((times_times @ extended_enat) @ (extended_enat2 @ M)) @ (extend1396239628finity @ extended_enat)) = (zero_zero @ extended_enat))) & ((M != (zero_zero @ nat)) => ((((times_times @ extended_enat) @ (extended_enat2 @ M)) @ (extend1396239628finity @ extended_enat)) = (extend1396239628finity @ extended_enat))))))).
% 2.85/1.54 thf(fact_109_the__enat__0, axiom, ((extended_the_enat @ (zero_zero @ extended_enat)) = (zero_zero @ nat))).
% 2.85/1.54 thf(fact_110_scount__finite, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs))))))) @ Omega) => (((ord_less @ extended_enat) @ (((coindu1365464361scount @ A) @ P) @ Omega)) @ (extend1396239628finity @ extended_enat)))))).
% 2.85/1.54 thf(fact_111_ev__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: ((((linear505997466_on_ev @ A) @ ((linear505997466_on_ev @ A) @ Phi)) = ((linear505997466_on_ev @ A) @ Phi))))).
% 2.85/1.54 thf(fact_112_of__nat__mult, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type @ A)) => (! [M:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((times_times @ nat) @ M) @ N)) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 2.85/1.54 thf(fact_113_times__enat__simps_I2_J, axiom, ((((times_times @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ (extend1396239628finity @ extended_enat)) = (extend1396239628finity @ extended_enat))).
% 2.85/1.54 thf(fact_114_not__alw__not, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((^ [Xs:(stream @ A)]: ((~ ((((linear1386806755on_alw @ A) @ (^ [Xt:(stream @ A)]: ((~ ((Phi @ Xt)))))) @ Xs))))) = ((linear505997466_on_ev @ A) @ Phi))))).
% 2.85/1.54 thf(fact_115_not__ev__not, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((^ [Xs:(stream @ A)]: ((~ ((((linear505997466_on_ev @ A) @ (^ [Xt:(stream @ A)]: ((~ ((Phi @ Xt)))))) @ Xs))))) = ((linear1386806755on_alw @ A) @ Phi))))).
% 2.85/1.54 thf(fact_116_int__int__eq, axiom, (! [M:nat,N:nat]: (((((semiring_1_of_nat @ int) @ M) = ((semiring_1_of_nat @ int) @ N)) = (M = N))))).
% 2.85/1.54 thf(fact_117_ab__semigroup__mult__class_Omult__ac_I1_J, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type @ A)) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.85/1.54 thf(fact_118_mult_Oassoc, axiom, (! [A:$tType]: ((((semigroup_mult @ A) @ (type @ A)) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.85/1.54 thf(fact_119_mult_Ocommute, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type @ A)) => ((times_times @ A) = (^ [A4:A,B3:A]: ((((times_times @ A) @ B3) @ A4)))))))).
% 2.85/1.54 thf(fact_120_mult_Oleft__commute, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type @ A)) => (! [B2:A,A2:A,C2:A]: (((((times_times @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.85/1.54 thf(fact_121_ev__or, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi:((stream @ A) > $o)]: ((((linear505997466_on_ev @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) | (Psi @ Xs))))) = (^ [Xs:(stream @ A)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs) | (((linear505997466_on_ev @ A) @ Psi) @ Xs)))))))).
% 2.85/1.54 thf(fact_122_ev_Obase, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((Phi @ Xs2) => (((linear505997466_on_ev @ A) @ Phi) @ Xs2))))).
% 2.85/1.54 thf(fact_123_ev__mono, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => ((! [Xs3:(stream @ A)]: (((Phi @ Xs3) => (Psi @ Xs3)))) => (((linear505997466_on_ev @ A) @ Psi) @ Xs2)))))).
% 2.85/1.54 thf(fact_124_ev__False, axiom, (! [A:$tType,Omega:(stream @ A)]: ((~ ((((linear505997466_on_ev @ A) @ (^ [X:(stream @ A)]: ($false))) @ Omega)))))).
% 2.85/1.54 thf(fact_125_less__int__code_I1_J, axiom, (~ ((((ord_less @ int) @ (zero_zero @ int)) @ (zero_zero @ int))))).
% 2.85/1.54 thf(fact_126_ev__alw__imp__alw__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs2) => (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ Phi)) @ Xs2))))).
% 2.85/1.54 thf(fact_127_ev__cong, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A),Q1:((stream @ A) > $o),Q2:((stream @ A) > $o)]: (((((linear1386806755on_alw @ A) @ P) @ Omega) => ((! [Omega3:(stream @ A)]: (((P @ Omega3) => ((Q1 @ Omega3) = (Q2 @ Omega3))))) => ((((linear505997466_on_ev @ A) @ Q1) @ Omega) = (((linear505997466_on_ev @ A) @ Q2) @ Omega))))))).
% 2.85/1.54 thf(fact_128_ev__iff__sdrop, axiom, (! [A:$tType]: (((linear505997466_on_ev @ A) = (^ [P2:((stream @ A) > $o),Omega2:(stream @ A)]: ((? [M3:nat]: ((P2 @ (((sdrop @ A) @ M3) @ Omega2)))))))))).
% 2.85/1.54 thf(fact_129_mult__of__nat__commute, axiom, (! [A:$tType]: ((((semiring_1 @ A) @ (type @ A)) => (! [X2:nat,Y3:A]: (((((times_times @ A) @ ((semiring_1_of_nat @ A) @ X2)) @ Y3) = (((times_times @ A) @ Y3) @ ((semiring_1_of_nat @ A) @ X2))))))))).
% 2.85/1.54 thf(fact_130_imult__is__0, axiom, (! [M:extended_enat,N:extended_enat]: ((((((times_times @ extended_enat) @ M) @ N) = (zero_zero @ extended_enat)) = ((M = (zero_zero @ extended_enat)) | (N = (zero_zero @ extended_enat))))))).
% 2.85/1.54 thf(fact_131_alw__ev__imp__ev__alw, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ P)) @ Omega) => (((linear505997466_on_ev @ A) @ (^ [Xs:(stream @ A)]: (((P @ Xs) & (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ P)) @ Xs))))) @ Omega))))).
% 2.85/1.54 thf(fact_132_ev__alw__alw__impl, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) @ Xs2) => (((linear505997466_on_ev @ A) @ Psi) @ Xs2)))))).
% 2.85/1.54 thf(fact_133_alw__impl__ev__alw, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi:((stream @ A) > $o),Xs2:(stream @ A)]: (((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) @ Xs2) => (((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) @ Xs2))))).
% 2.85/1.54 thf(fact_134_alw__alw__impl__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Psi:((stream @ A) > $o)]: ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((((linear1386806755on_alw @ A) @ Phi) @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) = (^ [Xs:(stream @ A)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs) => (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ Psi)) @ Xs)))))))).
% 2.85/1.54 thf(fact_135_ev__alw__impl__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => (((linear505997466_on_ev @ A) @ Psi) @ Xs))))) @ Xs2) => (((linear505997466_on_ev @ A) @ Psi) @ Xs2)))))).
% 2.85/1.54 thf(fact_136_not__alw__iff, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((~ ((((linear1386806755on_alw @ A) @ P) @ Omega))) = (((linear505997466_on_ev @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs)))))) @ Omega))))).
% 2.85/1.54 thf(fact_137_ev__alw__impl, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => ((((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) => (Psi @ Xs))))) @ Xs2) => (((linear505997466_on_ev @ A) @ Psi) @ Xs2)))))).
% 2.85/1.54 thf(fact_138_ev__alw__aand, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A),Psi:((stream @ A) > $o)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Phi)) @ Xs2) => ((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ Psi)) @ Xs2) => (((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: (((Phi @ Xs) & (Psi @ Xs)))))) @ Xs2)))))).
% 2.85/1.54 thf(fact_139_not__ev__iff, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((~ ((((linear505997466_on_ev @ A) @ P) @ Omega))) = (((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs)))))) @ Omega))))).
% 2.85/1.54 thf(fact_140_not__alw, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((^ [Xs:(stream @ A)]: ((~ ((((linear1386806755on_alw @ A) @ Phi) @ Xs))))) = ((linear505997466_on_ev @ A) @ (^ [Xs:(stream @ A)]: ((~ ((Phi @ Xs)))))))))).
% 2.85/1.54 thf(fact_141_not__ev, axiom, (! [A:$tType,Phi:((stream @ A) > $o)]: (((^ [Xs:(stream @ A)]: ((~ ((((linear505997466_on_ev @ A) @ Phi) @ Xs))))) = ((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((Phi @ Xs)))))))))).
% 2.85/1.54 thf(fact_142_the__enat_Osimps, axiom, (! [N:nat]: (((extended_the_enat @ (extended_enat2 @ N)) = N)))).
% 2.85/1.54 thf(fact_143_alw__ev__sdrop, axiom, (! [A:$tType,Phi:((stream @ A) > $o),M:nat,Xs2:(stream @ A)]: (((((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ Phi)) @ (((sdrop @ A) @ M) @ Xs2)) => (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ Phi)) @ Xs2))))).
% 2.85/1.54 thf(fact_144_ex__less__of__nat__mult, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type @ A)) => (! [X2:A,Y3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X2) => (? [N2:nat]: ((((ord_less @ A) @ Y3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N2)) @ X2))))))))))).
% 2.85/1.54 thf(fact_145_imult__is__infinity, axiom, (! [A2:extended_enat,B2:extended_enat]: ((((((times_times @ extended_enat) @ A2) @ B2) = (extend1396239628finity @ extended_enat)) = (((A2 = (extend1396239628finity @ extended_enat)) & (B2 != (zero_zero @ extended_enat))) | ((B2 = (extend1396239628finity @ extended_enat)) & (A2 != (zero_zero @ extended_enat)))))))).
% 2.85/1.54 thf(fact_146_enat__0__less__mult__iff, axiom, (! [M:extended_enat,N:extended_enat]: (((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ (((times_times @ extended_enat) @ M) @ N)) = ((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ M) & (((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ N)))))).
% 2.85/1.54 thf(fact_147_scount__infinite__iff, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: ((((((coindu1365464361scount @ A) @ P) @ Omega) = (extend1396239628finity @ extended_enat)) = (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ P)) @ Omega))))).
% 2.85/1.54 thf(fact_148_scount__infinite, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ P)) @ Omega) => ((((coindu1365464361scount @ A) @ P) @ Omega) = (extend1396239628finity @ extended_enat)))))).
% 2.85/1.54 thf(fact_149_sfirst__finite, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((ord_less @ extended_enat) @ (((coindu76177514sfirst @ A) @ P) @ Omega)) @ (extend1396239628finity @ extended_enat)) = (((linear505997466_on_ev @ A) @ P) @ Omega))))).
% 2.85/1.54 thf(fact_150_imult__infinity__right, axiom, (! [N:extended_enat]: (((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ N) => ((((times_times @ extended_enat) @ N) @ (extend1396239628finity @ extended_enat)) = (extend1396239628finity @ extended_enat)))))).
% 2.85/1.54 thf(fact_151_imult__infinity, axiom, (! [N:extended_enat]: (((((ord_less @ extended_enat) @ (zero_zero @ extended_enat)) @ N) => ((((times_times @ extended_enat) @ (extend1396239628finity @ extended_enat)) @ N) = (extend1396239628finity @ extended_enat)))))).
% 2.85/1.54 thf(fact_152_enat__the__enat, axiom, (! [N:extended_enat]: (((N != (extend1396239628finity @ extended_enat)) => ((extended_enat2 @ (extended_the_enat @ N)) = N))))).
% 2.85/1.54 thf(fact_153_mult__cancel__right, axiom, (! [A:$tType]: ((((semiri1923998003cancel @ A) @ (type @ A)) => (! [A2:A,C2:A,B2:A]: ((((((times_times @ A) @ A2) @ C2) = (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (A2 = B2))))))))).
% 2.85/1.54 thf(fact_154_mult__cancel__left, axiom, (! [A:$tType]: ((((semiri1923998003cancel @ A) @ (type @ A)) => (! [C2:A,A2:A,B2:A]: ((((((times_times @ A) @ C2) @ A2) = (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (A2 = B2))))))))).
% 2.85/1.54 thf(fact_155_mult__eq__0__iff, axiom, (! [A:$tType]: ((((semiri1193490041visors @ A) @ (type @ A)) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) = (zero_zero @ A)) = ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_156_mult__zero__right, axiom, (! [A:$tType]: ((((mult_zero @ A) @ (type @ A)) => (! [A2:A]: (((((times_times @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.85/1.54 thf(fact_157_mult__zero__left, axiom, (! [A:$tType]: ((((mult_zero @ A) @ (type @ A)) => (! [A2:A]: (((((times_times @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.85/1.54 thf(fact_158_mult__cancel2, axiom, (! [M:nat,K:nat,N:nat]: ((((((times_times @ nat) @ M) @ K) = (((times_times @ nat) @ N) @ K)) = ((M = N) | (K = (zero_zero @ nat))))))).
% 2.85/1.54 thf(fact_159_mult__cancel1, axiom, (! [K:nat,M:nat,N:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N)) = ((M = N) | (K = (zero_zero @ nat))))))).
% 2.85/1.54 thf(fact_160_mult__0__right, axiom, (! [M:nat]: (((((times_times @ nat) @ M) @ (zero_zero @ nat)) = (zero_zero @ nat))))).
% 2.85/1.54 thf(fact_161_mult__is__0, axiom, (! [M:nat,N:nat]: ((((((times_times @ nat) @ M) @ N) = (zero_zero @ nat)) = ((M = (zero_zero @ nat)) | (N = (zero_zero @ nat))))))).
% 2.85/1.54 thf(fact_162_mult__less__cancel2, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((times_times @ nat) @ N) @ K)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) & (((ord_less @ nat) @ M) @ N)))))).
% 2.85/1.54 thf(fact_163_nat__0__less__mult__iff, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((times_times @ nat) @ M) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 2.85/1.54 thf(fact_164_times__enat__simps_I1_J, axiom, (! [M:nat,N:nat]: (((((times_times @ extended_enat) @ (extended_enat2 @ M)) @ (extended_enat2 @ N)) = (extended_enat2 @ (((times_times @ nat) @ M) @ N)))))).
% 2.85/1.54 thf(fact_165_times__int__code_I1_J, axiom, (! [K:int]: (((((times_times @ int) @ K) @ (zero_zero @ int)) = (zero_zero @ int))))).
% 2.85/1.54 thf(fact_166_times__int__code_I2_J, axiom, (! [L:int]: (((((times_times @ int) @ (zero_zero @ int)) @ L) = (zero_zero @ int))))).
% 2.85/1.54 thf(fact_167_mult__0, axiom, (! [N:nat]: (((((times_times @ nat) @ (zero_zero @ nat)) @ N) = (zero_zero @ nat))))).
% 2.85/1.54 thf(fact_168_mult__less__mono1, axiom, (! [I3:nat,J3:nat,K:nat]: (((((ord_less @ nat) @ I3) @ J3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ nat) @ (((times_times @ nat) @ I3) @ K)) @ (((times_times @ nat) @ J3) @ K))))))).
% 2.85/1.54 thf(fact_169_mult__less__mono2, axiom, (! [I3:nat,J3:nat,K:nat]: (((((ord_less @ nat) @ I3) @ J3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ nat) @ (((times_times @ nat) @ K) @ I3)) @ (((times_times @ nat) @ K) @ J3))))))).
% 2.85/1.54 thf(fact_170_zmult__zless__mono2, axiom, (! [I3:int,J3:int,K:int]: (((((ord_less @ int) @ I3) @ J3) => ((((ord_less @ int) @ (zero_zero @ int)) @ K) => (((ord_less @ int) @ (((times_times @ int) @ K) @ I3)) @ (((times_times @ int) @ K) @ J3))))))).
% 2.85/1.54 thf(fact_171_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type @ A)) => (! [X2:A,Y3:A]: (((X2 != Y3) => ((~ ((((ord_less @ A) @ X2) @ Y3))) => (((ord_less @ A) @ Y3) @ X2))))))))).
% 2.85/1.54 thf(fact_172_zmult__zless__mono2__lemma, axiom, (! [I3:int,J3:int,K:nat]: (((((ord_less @ int) @ I3) @ J3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((ord_less @ int) @ (((times_times @ int) @ ((semiring_1_of_nat @ int) @ K)) @ I3)) @ (((times_times @ int) @ ((semiring_1_of_nat @ int) @ K)) @ J3))))))).
% 2.85/1.54 thf(fact_173_times__enat__def, axiom, ((times_times @ extended_enat) = (^ [M3:extended_enat,N3:extended_enat]: (((((extended_case_enat @ extended_enat) @ (^ [O:nat]: (((((extended_case_enat @ extended_enat) @ (^ [P4:nat]: ((extended_enat2 @ (((times_times @ nat) @ O) @ P4))))) @ ((((if @ extended_enat) @ (O = (zero_zero @ nat))) @ (zero_zero @ extended_enat)) @ (extend1396239628finity @ extended_enat))) @ N3)))) @ ((((if @ extended_enat) @ (N3 = (zero_zero @ extended_enat))) @ (zero_zero @ extended_enat)) @ (extend1396239628finity @ extended_enat))) @ M3))))).
% 2.85/1.54 thf(fact_174_mult__not__zero, axiom, (! [A:$tType]: ((((mult_zero @ A) @ (type @ A)) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) != (zero_zero @ A)) => ((A2 != (zero_zero @ A)) & (B2 != (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_175_divisors__zero, axiom, (! [A:$tType]: ((((semiri1193490041visors @ A) @ (type @ A)) => (! [A2:A,B2:A]: ((((((times_times @ A) @ A2) @ B2) = (zero_zero @ A)) => ((A2 = (zero_zero @ A)) | (B2 = (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_176_no__zero__divisors, axiom, (! [A:$tType]: ((((semiri1193490041visors @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((A2 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((times_times @ A) @ A2) @ B2) != (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_177_mult__left__cancel, axiom, (! [A:$tType]: ((((semiri1923998003cancel @ A) @ (type @ A)) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ C2) @ A2) = (((times_times @ A) @ C2) @ B2)) = (A2 = B2))))))))).
% 2.85/1.54 thf(fact_178_mult__right__cancel, axiom, (! [A:$tType]: ((((semiri1923998003cancel @ A) @ (type @ A)) => (! [C2:A,A2:A,B2:A]: (((C2 != (zero_zero @ A)) => (((((times_times @ A) @ A2) @ C2) = (((times_times @ A) @ B2) @ C2)) = (A2 = B2))))))))).
% 2.85/1.54 thf(fact_179_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom, (! [A:$tType]: ((((linord893533164strict @ A) @ (type @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 2.85/1.54 thf(fact_180_mult__less__cancel__right__disj, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [A2:A,C2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) & (((ord_less @ A) @ A2) @ B2)) | ((((ord_less @ A) @ C2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.85/1.54 thf(fact_181_mult__strict__right__mono, axiom, (! [A:$tType]: ((((linord20386208strict @ A) @ (type @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 2.85/1.54 thf(fact_182_mult__strict__right__mono__neg, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 2.85/1.54 thf(fact_183_mult__less__cancel__left__disj, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) & (((ord_less @ A) @ A2) @ B2)) | ((((ord_less @ A) @ C2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.85/1.54 thf(fact_184_mult__strict__left__mono, axiom, (! [A:$tType]: ((((linord20386208strict @ A) @ (type @ A)) => (! [A2:A,B2:A,C2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 2.85/1.54 thf(fact_185_mult__strict__left__mono__neg, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [B2:A,A2:A,C2:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)))))))))).
% 2.85/1.54 thf(fact_186_mult__less__cancel__left__pos, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less @ A) @ A2) @ B2))))))))).
% 2.85/1.54 thf(fact_187_mult__less__cancel__left__neg, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [C2:A,A2:A,B2:A]: (((((ord_less @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((times_times @ A) @ C2) @ A2)) @ (((times_times @ A) @ C2) @ B2)) = (((ord_less @ A) @ B2) @ A2))))))))).
% 2.85/1.54 thf(fact_188_zero__less__mult__pos2, axiom, (! [A:$tType]: ((((linord20386208strict @ A) @ (type @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ B2) @ A2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ B2))))))))).
% 2.85/1.54 thf(fact_189_zero__less__mult__pos, axiom, (! [A:$tType]: ((((linord20386208strict @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ B2))))))))).
% 2.85/1.54 thf(fact_190_zero__less__mult__iff, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ (zero_zero @ A))))))))))).
% 2.85/1.54 thf(fact_191_mult__pos__neg2, axiom, (! [A:$tType]: ((((linord20386208strict @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ B2) @ A2)) @ (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_192_mult__pos__pos, axiom, (! [A:$tType]: ((((linord20386208strict @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)))))))))).
% 2.85/1.54 thf(fact_193_mult__pos__neg, axiom, (! [A:$tType]: ((((linord20386208strict @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_194_mult__neg__pos, axiom, (! [A:$tType]: ((((linord20386208strict @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_195_mult__less__0__iff, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 2.85/1.54 thf(fact_196_not__square__less__zero, axiom, (! [A:$tType]: ((((linordered_ring @ A) @ (type @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ (((times_times @ A) @ A2) @ A2)) @ (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_197_mult__neg__neg, axiom, (! [A:$tType]: ((((linord581940658strict @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)))))))))).
% 2.85/1.54 thf(fact_198_nat__mult__less__cancel__disj, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) & (((ord_less @ nat) @ M) @ N)))))).
% 2.85/1.54 thf(fact_199_transfer__int__nat__relations_I2_J, axiom, (! [X2:nat,Y3:nat]: (((((ord_less @ int) @ ((semiring_1_of_nat @ int) @ X2)) @ ((semiring_1_of_nat @ int) @ Y3)) = (((ord_less @ nat) @ X2) @ Y3))))).
% 2.85/1.54 thf(fact_200_transfer__int__nat__numerals_I1_J, axiom, ((zero_zero @ int) = ((semiring_1_of_nat @ int) @ (zero_zero @ nat)))).
% 2.85/1.54 thf(fact_201_transfer__int__nat__relations_I1_J, axiom, (! [X2:nat,Y3:nat]: (((((semiring_1_of_nat @ int) @ X2) = ((semiring_1_of_nat @ int) @ Y3)) = (X2 = Y3))))).
% 2.85/1.54 thf(fact_202_int__if__cong, axiom, (! [P:$o,X2:nat,Y3:nat]: (((P => (((semiring_1_of_nat @ int) @ X2) = ((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ X2) @ Y3)))) & ((~ (P)) => (((semiring_1_of_nat @ int) @ Y3) = ((semiring_1_of_nat @ int) @ ((((if @ nat) @ P) @ X2) @ Y3)))))))).
% 2.85/1.54 thf(fact_203_nat__mult__eq__cancel__disj, axiom, (! [K:nat,M:nat,N:nat]: ((((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N)) = ((K = (zero_zero @ nat)) | (M = N)))))).
% 2.85/1.54 thf(fact_204_Nat__Transfer_Otransfer__int__nat__functions_I2_J, axiom, (! [X2:nat,Y3:nat]: (((((times_times @ int) @ ((semiring_1_of_nat @ int) @ X2)) @ ((semiring_1_of_nat @ int) @ Y3)) = ((semiring_1_of_nat @ int) @ (((times_times @ nat) @ X2) @ Y3)))))).
% 2.85/1.54 thf(fact_205_nat__mult__less__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((ord_less @ nat) @ (((times_times @ nat) @ K) @ M)) @ (((times_times @ nat) @ K) @ N)) = (((ord_less @ nat) @ M) @ N)))))).
% 2.85/1.54 thf(fact_206_nat__mult__eq__cancel1, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((((times_times @ nat) @ K) @ M) = (((times_times @ nat) @ K) @ N)) = (M = N)))))).
% 2.85/1.54 thf(fact_207_linordered__field__class_Osign__simps_I45_J, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A2) @ B2)) @ (zero_zero @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ B2) @ (zero_zero @ A))) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))))).
% 2.85/1.54 thf(fact_208_linordered__field__class_Osign__simps_I44_J, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((times_times @ A) @ A2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A2) & (((ord_less @ A) @ (zero_zero @ A)) @ B2)) | ((((ord_less @ A) @ A2) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ (zero_zero @ A))))))))))).
% 2.85/1.54 thf(fact_209_linordered__field__no__ub, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X5:A]: ((? [X1:A]: ((((ord_less @ A) @ X5) @ X1))))))))).
% 2.85/1.54 thf(fact_210_linordered__field__no__lb, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [X5:A]: ((? [Y5:A]: ((((ord_less @ A) @ Y5) @ X5))))))))).
% 2.85/1.54 thf(fact_211_linordered__field__class_Osign__simps_I25_J, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [B2:A,A2:A,C2:A]: (((((times_times @ A) @ B2) @ (((times_times @ A) @ A2) @ C2)) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.85/1.54 thf(fact_212_linordered__field__class_Osign__simps_I24_J, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => ((times_times @ A) = (^ [A4:A,B3:A]: ((((times_times @ A) @ B3) @ A4)))))))).
% 2.85/1.54 thf(fact_213_linordered__field__class_Osign__simps_I23_J, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C2) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C2))))))))).
% 2.85/1.54 thf(fact_214_scount__eq, axiom, (! [A:$tType]: (((coindu1365464361scount @ A) = (^ [P2:((stream @ A) > $o),Omega2:(stream @ A)]: (((((if @ extended_enat) @ (((linear1386806755on_alw @ A) @ ((linear505997466_on_ev @ A) @ P2)) @ Omega2)) @ (extend1396239628finity @ extended_enat)) @ (extended_enat2 @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [I2:nat]: ((P2 @ (((sdrop @ A) @ I2) @ Omega2)))))))))))))).
% 2.85/1.54 thf(fact_215_semiring__normalization__rules_I10_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [A2:A]: (((((times_times @ A) @ A2) @ (zero_zero @ A)) = (zero_zero @ A)))))))).
% 2.85/1.54 thf(fact_216_semiring__normalization__rules_I7_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => ((times_times @ A) = (^ [A4:A,B3:A]: ((((times_times @ A) @ B3) @ A4)))))))).
% 2.85/1.54 thf(fact_217_semiring__normalization__rules_I13_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [Lx:A,Ly:A,Rx:A,Ry:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ (((times_times @ A) @ Lx) @ Rx)) @ (((times_times @ A) @ Ly) @ Ry))))))))).
% 2.85/1.54 thf(fact_218_semiring__normalization__rules_I14_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [Lx:A,Ly:A,Rx:A,Ry:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ Lx) @ (((times_times @ A) @ Ly) @ (((times_times @ A) @ Rx) @ Ry)))))))))).
% 2.85/1.54 thf(fact_219_semiring__normalization__rules_I15_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [Lx:A,Ly:A,Rx:A,Ry:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ Rx) @ (((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ Ry))))))))).
% 2.85/1.54 thf(fact_220_semiring__normalization__rules_I16_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [Lx:A,Ly:A,Rx:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ Rx) = (((times_times @ A) @ (((times_times @ A) @ Lx) @ Rx)) @ Ly)))))))).
% 2.85/1.54 thf(fact_221_semiring__normalization__rules_I17_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [Lx:A,Ly:A,Rx:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ Rx) = (((times_times @ A) @ Lx) @ (((times_times @ A) @ Ly) @ Rx))))))))).
% 2.85/1.54 thf(fact_222_semiring__normalization__rules_I18_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [Lx:A,Rx:A,Ry:A]: (((((times_times @ A) @ Lx) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ (((times_times @ A) @ Lx) @ Rx)) @ Ry)))))))).
% 2.85/1.54 thf(fact_223_semiring__normalization__rules_I19_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [Lx:A,Rx:A,Ry:A]: (((((times_times @ A) @ Lx) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ Rx) @ (((times_times @ A) @ Lx) @ Ry))))))))).
% 2.85/1.54 thf(fact_224_scount__eq__card, axiom, (! [A:$tType,P:((stream @ A) > $o),Omega:(stream @ A)]: (((((linear505997466_on_ev @ A) @ ((linear1386806755on_alw @ A) @ (^ [Xs:(stream @ A)]: ((~ ((P @ Xs))))))) @ Omega) => ((((coindu1365464361scount @ A) @ P) @ Omega) = (extended_enat2 @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [I2:nat]: ((P @ (((sdrop @ A) @ I2) @ Omega)))))))))))).
% 2.85/1.54 thf(fact_225_semiring__normalization__rules_I9_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type @ A)) => (! [A2:A]: (((((times_times @ A) @ (zero_zero @ A)) @ A2) = (zero_zero @ A)))))))).
% 2.85/1.54 thf(fact_226_card__Collect__less__nat, axiom, (! [N:nat]: ((((finite_card @ nat) @ ((collect @ nat) @ (^ [I2:nat]: ((((ord_less @ nat) @ I2) @ N))))) = N)))).
% 2.85/1.54 thf(fact_227_sdrop__wait, axiom, (! [A:$tType,Phi:((stream @ A) > $o),Xs2:(stream @ A)]: (((((linear505997466_on_ev @ A) @ Phi) @ Xs2) => (Phi @ (((sdrop @ A) @ (((linear1335279038n_wait @ A) @ Phi) @ Xs2)) @ Xs2)))))).
% 2.85/1.54 thf(fact_228_transfer__morphism__int__nat, axiom, ((((nat_tr1645093318rphism @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (^ [N3:nat]: ($true)))).
% 2.85/1.54 thf(fact_229_zero__less__ceiling, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type @ A)) => (! [X2:A]: (((((ord_less @ int) @ (zero_zero @ int)) @ ((archimedean_ceiling @ A) @ X2)) = (((ord_less @ A) @ (zero_zero @ A)) @ X2)))))))).
% 2.85/1.54 thf(fact_230_ceiling__zero, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type @ A)) => (((archimedean_ceiling @ A) @ (zero_zero @ A)) = (zero_zero @ int)))))).
% 2.85/1.54 thf(fact_231_ceiling__of__nat, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type @ A)) => (! [N:nat]: ((((archimedean_ceiling @ A) @ ((semiring_1_of_nat @ A) @ N)) = ((semiring_1_of_nat @ int) @ N)))))))).
% 2.85/1.54 thf(fact_232_ceiling__less__cancel, axiom, (! [A:$tType]: ((((archim1727834104eiling @ A) @ (type @ A)) => (! [X2:A,Y3:A]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X2)) @ ((archimedean_ceiling @ A) @ Y3)) => (((ord_less @ A) @ X2) @ Y3)))))))).
% 2.85/1.54 thf(fact_233_zero__less__nat__eq, axiom, (! [Z:int]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (nat2 @ Z)) = (((ord_less @ int) @ (zero_zero @ int)) @ Z))))).
% 2.85/1.54 thf(fact_234_ex__inverse__of__nat__less, axiom, (! [A:$tType]: ((((archim1804426504_field @ A) @ (type @ A)) => (! [X2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ X2) => (? [N2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N2) & (((ord_less @ A) @ ((inverse_inverse @ A) @ ((semiring_1_of_nat @ A) @ N2))) @ X2))))))))))).
% 2.85/1.54 thf(fact_235_inverse__zero, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type @ A)) => (((inverse_inverse @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 2.85/1.54 thf(fact_236_inverse__nonzero__iff__nonzero, axiom, (! [A:$tType]: ((((division_ring @ A) @ (type @ A)) => (! [A2:A]: (((((inverse_inverse @ A) @ A2) = (zero_zero @ A)) = (A2 = (zero_zero @ A))))))))).
% 2.85/1.54 thf(fact_237_inverse__mult__distrib, axiom, (! [A:$tType]: ((((field @ A) @ (type @ A)) => (! [A2:A,B2:A]: ((((inverse_inverse @ A) @ (((times_times @ A) @ A2) @ B2)) = (((times_times @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2))))))))).
% 2.85/1.54 thf(fact_238_nat__int, axiom, (! [N:nat]: (((nat2 @ ((semiring_1_of_nat @ int) @ N)) = N)))).
% 2.85/1.54 thf(fact_239_inverse__positive__iff__positive, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) = (((ord_less @ A) @ (zero_zero @ A)) @ A2)))))))).
% 2.85/1.54 thf(fact_240_inverse__negative__iff__negative, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) = (((ord_less @ A) @ A2) @ (zero_zero @ A))))))))).
% 2.85/1.54 thf(fact_241_inverse__less__iff__less__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.85/1.54 thf(fact_242_inverse__less__iff__less, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) = (((ord_less @ A) @ B2) @ A2)))))))))).
% 2.85/1.54 thf(fact_243_nat__0, axiom, ((nat2 @ (zero_zero @ int)) = (zero_zero @ nat))).
% 2.85/1.54 thf(fact_244_zless__nat__conj, axiom, (! [W:int,Z:int]: (((((ord_less @ nat) @ (nat2 @ W)) @ (nat2 @ Z)) = ((((ord_less @ int) @ (zero_zero @ int)) @ Z) & (((ord_less @ int) @ W) @ Z)))))).
% 2.85/1.54 thf(fact_245_zless__nat__eq__int__zless, axiom, (! [M:nat,Z:int]: (((((ord_less @ nat) @ M) @ (nat2 @ Z)) = (((ord_less @ int) @ ((semiring_1_of_nat @ int) @ M)) @ Z))))).
% 2.85/1.54 thf(fact_246_positive__imp__inverse__positive, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2))))))))).
% 2.85/1.54 thf(fact_247_negative__imp__inverse__negative, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ A2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A))))))))).
% 2.85/1.54 thf(fact_248_inverse__positive__imp__positive, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((inverse_inverse @ A) @ A2)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ A2))))))))).
% 2.85/1.54 thf(fact_249_inverse__negative__imp__negative, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ (zero_zero @ A)) => ((A2 != (zero_zero @ A)) => (((ord_less @ A) @ A2) @ (zero_zero @ A)))))))))).
% 2.85/1.54 thf(fact_250_less__imp__inverse__less__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))).
% 2.85/1.54 thf(fact_251_inverse__less__imp__less__neg, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ A2)) @ ((inverse_inverse @ A) @ B2)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ A2))))))))).
% 2.85/1.54 thf(fact_252_less__imp__inverse__less, axiom, (! [A:$tType]: ((((linordered_field @ A) @ (type @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ (zero_zero @ A)) @ A2) => (((ord_less @ A) @ ((inverse_inverse @ A) @ B2)) @ ((inverse_inverse @ A) @ A2)))))))))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel, axiom, ((semiri1923998003cancel @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict, axiom, ((linord893533164strict @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict, axiom, ((linord20386208strict @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors, axiom, ((semiri1193490041visors @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict, axiom, ((linord581940658strict @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult, axiom, ((ab_semigroup_mult @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Olinordered__ring, axiom, ((linordered_ring @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, ((linordered_idom @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1, axiom, ((comm_semiring_1 @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Groups_Osemigroup__mult, axiom, ((semigroup_mult @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, ((semiring_char_0 @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, ((semiring_1 @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Rings_Omult__zero, axiom, ((mult_zero @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Int_Oint___Groups_Ozero, axiom, ((zero @ int) @ (type @ int))).
% 2.85/1.54 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_1, axiom, ((semiri1923998003cancel @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_2, axiom, ((linord893533164strict @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, ((canoni770627133id_add @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_3, axiom, ((linord20386208strict @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_4, axiom, ((semiri1193490041visors @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_5, axiom, ((linordered_semidom @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_6, axiom, ((ab_semigroup_mult @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_7, axiom, ((comm_semiring_1 @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_8, axiom, ((semigroup_mult @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_9, axiom, ((semiring_char_0 @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Rings_Osemiring__1_10, axiom, ((semiring_1 @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Rings_Omult__zero_11, axiom, ((mult_zero @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Nat_Onat___Groups_Ozero_12, axiom, ((zero @ nat) @ (type @ nat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_13, axiom, ((canoni770627133id_add @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_14, axiom, ((semiri1193490041visors @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_15, axiom, ((ab_semigroup_mult @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_16, axiom, ((comm_semiring_1 @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_17, axiom, ((semigroup_mult @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_18, axiom, ((semiring_char_0 @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_19, axiom, ((semiring_1 @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_20, axiom, ((mult_zero @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_21, axiom, ((zero @ extended_enat) @ (type @ extended_enat))).
% 2.85/1.54 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.85/1.54 thf(help_If_2_1_T, axiom, (! [A:$tType,X2:A,Y3:A]: ((((((if @ A) @ $false) @ X2) @ Y3) = Y3)))).
% 2.85/1.54 thf(help_If_1_1_T, axiom, (! [A:$tType,X2:A,Y3:A]: ((((((if @ A) @ $true) @ X2) @ Y3) = X2)))).
% 2.85/1.54 thf(conj_0, conjecture, ((((ord_less @ extended_enat) @ (extended_enat2 @ (zero_zero @ nat))) @ (((coindu76177514sfirst @ a) @ p) @ omega)) = (~ ((p @ (((sdrop @ a) @ (zero_zero @ nat)) @ omega)))))).
% 2.85/1.54 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.plLMjpaOMh/DTF2DTF20370.p
% 2.85/1.54 ------------------------
% 2.85/1.57 ---- Cleaned THF ---
% 2.85/1.57 thf(ty_t_Extended__Nat_Oenat,type,
% 2.85/1.57 extended_enat: $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_t_Stream_Ostream,type,
% 2.85/1.57 stream: $tType > $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_t_Set_Oset,type,
% 2.85/1.57 set: $tType > $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_t_Nat_Onat,type,
% 2.85/1.57 nat: $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_t_Int_Oint,type,
% 2.85/1.57 int: $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_t_itself,type,
% 2.85/1.57 itself: $tType > $tType ).
% 2.85/1.57
% 2.85/1.57 thf(ty_tf_a,type,
% 2.85/1.57 a: $tType ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Groups_Ozero,type,
% 2.85/1.57 zero:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Fields_Ofield,type,
% 2.85/1.57 field:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Omult__zero,type,
% 2.85/1.57 mult_zero:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Osemiring__1,type,
% 2.85/1.57 semiring_1:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.85/1.57 semiring_char_0:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Fields_Odivision__ring,type,
% 2.85/1.57 division_ring:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Groups_Osemigroup__mult,type,
% 2.85/1.57 semigroup_mult:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 2.85/1.57 comm_semiring_1:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.85/1.57 linordered_idom:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Olinordered__ring,type,
% 2.85/1.57 linordered_ring:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Fields_Olinordered__field,type,
% 2.85/1.57 linordered_field:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 2.85/1.57 ab_semigroup_mult:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.85/1.57 linordered_semidom:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 2.85/1.57 linord581940658strict:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 2.85/1.57 archim1727834104eiling:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 2.85/1.57 semiri1193490041visors:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Olinordered__semiring__strict,type,
% 2.85/1.57 linord20386208strict:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 2.85/1.57 archim1804426504_field:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.85/1.57 canoni770627133id_add:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
% 2.85/1.57 linord893533164strict:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
% 2.85/1.57 semiri1923998003cancel:
% 2.85/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Archimedean__Field_Oceiling,type,
% 2.85/1.57 archimedean_ceiling:
% 2.85/1.57 !>[A: $tType] : ( A > int ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oscount,type,
% 2.85/1.57 coindu1365464361scount:
% 2.85/1.57 !>[S: $tType] : ( ( ( stream @ S ) > $o ) > ( stream @ S ) > extended_enat ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Osfirst,type,
% 2.85/1.57 coindu76177514sfirst:
% 2.85/1.57 !>[S: $tType] : ( ( ( stream @ S ) > $o ) > ( stream @ S ) > extended_enat ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Extended__Nat_Oenat,type,
% 2.85/1.57 extended_enat2: nat > extended_enat ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
% 2.85/1.57 extended_case_enat:
% 2.85/1.57 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
% 2.85/1.57 extended_rec_enat:
% 2.85/1.57 !>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
% 2.85/1.57 extended_size_enat: extended_enat > nat ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
% 2.85/1.57 extend1396239628finity:
% 2.85/1.57 !>[A: $tType] : A ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Extended__Nat_Othe__enat,type,
% 2.85/1.57 extended_the_enat: extended_enat > nat ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 2.85/1.57 inverse_inverse:
% 2.85/1.57 !>[A: $tType] : ( A > A ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Finite__Set_Ocard,type,
% 2.85/1.57 finite_card:
% 2.85/1.57 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 2.85/1.57 times_times:
% 2.85/1.57 !>[A: $tType] : ( A > A > A ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.85/1.57 zero_zero:
% 2.85/1.57 !>[A: $tType] : A ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_If,type,
% 2.85/1.57 if:
% 2.85/1.57 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Int_Onat,type,
% 2.85/1.57 nat2: int > nat ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Linear__Temporal__Logic__on__Streams_OUNTIL,type,
% 2.85/1.57 linear1341015445_UNTIL:
% 2.85/1.57 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw,type,
% 2.85/1.57 linear1386806755on_alw:
% 2.85/1.57 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev,type,
% 2.85/1.57 linear505997466_on_ev:
% 2.85/1.57 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Linear__Temporal__Logic__on__Streams_Owait,type,
% 2.85/1.57 linear1335279038n_wait:
% 2.85/1.57 !>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > nat ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 2.85/1.57 semiring_1_of_nat:
% 2.85/1.57 !>[A: $tType] : ( nat > A ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.85/1.57 size_size:
% 2.85/1.57 !>[A: $tType] : ( A > nat ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Nat__Transfer_Otransfer__morphism,type,
% 2.85/1.57 nat_tr1645093318rphism:
% 2.85/1.57 !>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.85/1.57 ord_less:
% 2.85/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Pure_Otype,type,
% 2.85/1.57 type:
% 2.85/1.57 !>[A: $tType] : ( itself @ A ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Set_OCollect,type,
% 2.85/1.57 collect:
% 2.85/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_Stream_Osdrop,type,
% 2.85/1.57 sdrop:
% 2.85/1.57 !>[A: $tType] : ( nat > ( stream @ A ) > ( stream @ A ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_c_member,type,
% 2.85/1.57 member:
% 2.85/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.85/1.57
% 2.85/1.57 thf(sy_v_P,type,
% 2.85/1.57 p: ( stream @ a ) > $o ).
% 2.85/1.57
% 2.85/1.57 thf(sy_v__092_060omega_062_H____,type,
% 2.85/1.57 omega: stream @ a ).
% 2.85/1.57
% 2.85/1.57 thf(fact_0__C0_Oprems_C,axiom,
% 2.85/1.57 ( linear1341015445_UNTIL @ a
% 2.85/1.57 @ ^ [Xs: stream @ a] :
% 2.85/1.57 ~ ( p @ Xs )
% 2.85/1.57 @ ( linear1386806755on_alw @ a @ p )
% 2.85/1.57 @ omega ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_1_less__sfirstD,axiom,
% 2.85/1.57 ! [A: $tType,N: nat,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.57 ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu76177514sfirst @ A @ P @ Omega ) )
% 2.85/1.57 => ~ ( P @ ( sdrop @ A @ N @ Omega ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_2_not__gr__zero,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 2.85/1.57 => ! [N: A] :
% 2.85/1.57 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.85/1.57 = ( N
% 2.85/1.57 = ( zero_zero @ A ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_3_enat_Oinject,axiom,
% 2.85/1.57 ! [Nat: nat,Nat2: nat] :
% 2.85/1.57 ( ( ( extended_enat2 @ Nat )
% 2.85/1.57 = ( extended_enat2 @ Nat2 ) )
% 2.85/1.57 = ( Nat = Nat2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_4_sdrop_Osimps_I1_J,axiom,
% 2.85/1.57 ! [A: $tType,S2: stream @ A] :
% 2.85/1.57 ( ( sdrop @ A @ ( zero_zero @ nat ) @ S2 )
% 2.85/1.57 = S2 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_5_chain__incr,axiom,
% 2.85/1.57 ! [A: $tType,Y: A > extended_enat,K: nat] :
% 2.85/1.57 ( ! [I: A] :
% 2.85/1.57 ? [J: A] : ( ord_less @ extended_enat @ ( Y @ I ) @ ( Y @ J ) )
% 2.85/1.57 => ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y @ J2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_6_enat__iless,axiom,
% 2.85/1.57 ! [N: extended_enat,M: nat] :
% 2.85/1.57 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 2.85/1.57 => ? [K2: nat] :
% 2.85/1.57 ( N
% 2.85/1.57 = ( extended_enat2 @ K2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_7_less__numeral__extra_I3_J,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linordered_semidom @ A @ ( type @ A ) )
% 2.85/1.57 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_8_gr__zeroI,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 2.85/1.57 => ! [N: A] :
% 2.85/1.57 ( ( N
% 2.85/1.57 != ( zero_zero @ A ) )
% 2.85/1.57 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_9_not__less__zero,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 2.85/1.57 => ! [N: A] :
% 2.85/1.57 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_10_gr__implies__not__zero,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 2.85/1.57 => ! [M: A,N: A] :
% 2.85/1.57 ( ( ord_less @ A @ M @ N )
% 2.85/1.57 => ( N
% 2.85/1.57 != ( zero_zero @ A ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_11_zero__less__iff__neq__zero,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( canoni770627133id_add @ A @ ( type @ A ) )
% 2.85/1.57 => ! [N: A] :
% 2.85/1.57 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.85/1.57 = ( N
% 2.85/1.57 != ( zero_zero @ A ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_12_fun__cong__unused__0,axiom,
% 2.85/1.57 ! [A: $tType,B: $tType,C: $tType] :
% 2.85/1.57 ( ( zero @ B @ ( type @ B ) )
% 2.85/1.57 => ! [F: ( A > B ) > C,G: C] :
% 2.85/1.57 ( ( F
% 2.85/1.57 = ( ^ [X: A > B] : G ) )
% 2.85/1.57 => ( ( F
% 2.85/1.57 @ ^ [X: A] : ( zero_zero @ B ) )
% 2.85/1.57 = G ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_13_i0__less,axiom,
% 2.85/1.57 ! [N: extended_enat] :
% 2.85/1.57 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 2.85/1.57 = ( N
% 2.85/1.57 != ( zero_zero @ extended_enat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_14_sfirst__0,axiom,
% 2.85/1.57 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.57 ( ( P @ Omega )
% 2.85/1.57 => ( ( coindu76177514sfirst @ A @ P @ Omega )
% 2.85/1.57 = ( zero_zero @ extended_enat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_15_enat__ord__simps_I2_J,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 2.85/1.57 = ( ord_less @ nat @ M @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_16_not__iless0,axiom,
% 2.85/1.57 ! [N: extended_enat] :
% 2.85/1.57 ~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_17_zero__enat__def,axiom,
% 2.85/1.57 ( ( zero_zero @ extended_enat )
% 2.85/1.57 = ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_18_enat__0__iff_I1_J,axiom,
% 2.85/1.57 ! [X2: nat] :
% 2.85/1.57 ( ( ( extended_enat2 @ X2 )
% 2.85/1.57 = ( zero_zero @ extended_enat ) )
% 2.85/1.57 = ( X2
% 2.85/1.57 = ( zero_zero @ nat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_19_enat__0__iff_I2_J,axiom,
% 2.85/1.57 ! [X2: nat] :
% 2.85/1.57 ( ( ( zero_zero @ extended_enat )
% 2.85/1.57 = ( extended_enat2 @ X2 ) )
% 2.85/1.57 = ( X2
% 2.85/1.57 = ( zero_zero @ nat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_20_less__enatE,axiom,
% 2.85/1.57 ! [N: extended_enat,M: nat] :
% 2.85/1.57 ( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
% 2.85/1.57 => ~ ! [K2: nat] :
% 2.85/1.57 ( ( N
% 2.85/1.57 = ( extended_enat2 @ K2 ) )
% 2.85/1.57 => ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_21_sfirst__eq__0,axiom,
% 2.85/1.57 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.57 ( ( ( coindu76177514sfirst @ A @ P @ Omega )
% 2.85/1.57 = ( zero_zero @ extended_enat ) )
% 2.85/1.57 = ( P @ Omega ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_22_zero__reorient,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( zero @ A @ ( type @ A ) )
% 2.85/1.57 => ! [X2: A] :
% 2.85/1.57 ( ( ( zero_zero @ A )
% 2.85/1.57 = X2 )
% 2.85/1.57 = ( X2
% 2.85/1.57 = ( zero_zero @ A ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_23_enat__less__induct,axiom,
% 2.85/1.57 ! [P: extended_enat > $o,N: extended_enat] :
% 2.85/1.57 ( ! [N2: extended_enat] :
% 2.85/1.57 ( ! [M2: extended_enat] :
% 2.85/1.57 ( ( ord_less @ extended_enat @ M2 @ N2 )
% 2.85/1.57 => ( P @ M2 ) )
% 2.85/1.57 => ( P @ N2 ) )
% 2.85/1.57 => ( P @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_24_enat_Osize_I1_J,axiom,
% 2.85/1.57 ! [Nat: nat] :
% 2.85/1.57 ( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
% 2.85/1.57 = ( zero_zero @ nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_25_until__false,axiom,
% 2.85/1.57 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1341015445_UNTIL @ A @ Phi
% 2.85/1.57 @ ^ [Xs: stream @ A] : $false )
% 2.85/1.57 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_26_alw__alw,axiom,
% 2.85/1.57 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ Phi ) )
% 2.85/1.57 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_27_neq0__conv,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ( ( N
% 2.85/1.57 != ( zero_zero @ nat ) )
% 2.85/1.57 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_28_alw__iff__sdrop,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A )
% 2.85/1.57 = ( ^ [P2: ( stream @ A ) > $o,Omega2: stream @ A] :
% 2.85/1.57 ! [M3: nat] : ( P2 @ ( sdrop @ A @ M3 @ Omega2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_29_alw__sdrop,axiom,
% 2.85/1.57 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,N: nat] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 2.85/1.57 => ( linear1386806755on_alw @ A @ Phi @ ( sdrop @ A @ N @ Xs2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_30_enat_Osize_I3_J,axiom,
% 2.85/1.57 ! [Nat: nat] :
% 2.85/1.57 ( ( size_size @ extended_enat @ ( extended_enat2 @ Nat ) )
% 2.85/1.57 = ( zero_zero @ nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_31_enat_Osimps_I6_J,axiom,
% 2.85/1.57 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 2.85/1.57 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 2.85/1.57 = ( F1 @ Nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_32_alw__False,axiom,
% 2.85/1.57 ! [A: $tType,Omega: stream @ A] :
% 2.85/1.57 ~ ( linear1386806755on_alw @ A
% 2.85/1.57 @ ^ [X: stream @ A] : $false
% 2.85/1.57 @ Omega ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_33_alw__aand,axiom,
% 2.85/1.57 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A
% 2.85/1.57 @ ^ [Xs: stream @ A] :
% 2.85/1.57 ( ( Phi @ Xs )
% 2.85/1.57 & ( Psi @ Xs ) ) )
% 2.85/1.57 = ( ^ [Xs: stream @ A] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 2.85/1.57 & ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_34_alw__mp,axiom,
% 2.85/1.57 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 2.85/1.57 => ( ( linear1386806755on_alw @ A
% 2.85/1.57 @ ^ [Xs: stream @ A] :
% 2.85/1.57 ( ( Phi @ Xs )
% 2.85/1.57 => ( Psi @ Xs ) )
% 2.85/1.57 @ Xs2 )
% 2.85/1.57 => ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_35_nat__neq__iff,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( M != N )
% 2.85/1.57 = ( ( ord_less @ nat @ M @ N )
% 2.85/1.57 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_36_less__not__refl,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ~ ( ord_less @ nat @ N @ N ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_37_less__not__refl2,axiom,
% 2.85/1.57 ! [N: nat,M: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ N @ M )
% 2.85/1.57 => ( M != N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_38_less__not__refl3,axiom,
% 2.85/1.57 ! [S2: nat,T2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ S2 @ T2 )
% 2.85/1.57 => ( S2 != T2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_39_measure__induct,axiom,
% 2.85/1.57 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ! [Y2: A] :
% 2.85/1.57 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X3 ) )
% 2.85/1.57 => ( P @ Y2 ) )
% 2.85/1.57 => ( P @ X3 ) )
% 2.85/1.57 => ( P @ A2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_40_less__irrefl__nat,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ~ ( ord_less @ nat @ N @ N ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_41_nat__less__induct,axiom,
% 2.85/1.57 ! [P: nat > $o,N: nat] :
% 2.85/1.57 ( ! [N2: nat] :
% 2.85/1.57 ( ! [M2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M2 @ N2 )
% 2.85/1.57 => ( P @ M2 ) )
% 2.85/1.57 => ( P @ N2 ) )
% 2.85/1.57 => ( P @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_42_infinite__descent,axiom,
% 2.85/1.57 ! [P: nat > $o,N: nat] :
% 2.85/1.57 ( ! [N2: nat] :
% 2.85/1.57 ( ~ ( P @ N2 )
% 2.85/1.57 => ? [M2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M2 @ N2 )
% 2.85/1.57 & ~ ( P @ M2 ) ) )
% 2.85/1.57 => ( P @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_43_linorder__neqE__nat,axiom,
% 2.85/1.57 ! [X2: nat,Y3: nat] :
% 2.85/1.57 ( ( X2 != Y3 )
% 2.85/1.57 => ( ~ ( ord_less @ nat @ X2 @ Y3 )
% 2.85/1.57 => ( ord_less @ nat @ Y3 @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_44_mem__Collect__eq,axiom,
% 2.85/1.57 ! [A: $tType,A2: A,P: A > $o] :
% 2.85/1.57 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.85/1.57 = ( P @ A2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_45_Collect__mem__eq,axiom,
% 2.85/1.57 ! [A: $tType,A3: set @ A] :
% 2.85/1.57 ( ( collect @ A
% 2.85/1.57 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 2.85/1.57 = A3 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_46_Collect__cong,axiom,
% 2.85/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( P @ X3 )
% 2.85/1.57 = ( Q @ X3 ) )
% 2.85/1.57 => ( ( collect @ A @ P )
% 2.85/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_47_ext,axiom,
% 2.85/1.57 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( F @ X3 )
% 2.85/1.57 = ( G @ X3 ) )
% 2.85/1.57 => ( F = G ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_48_measure__induct__rule,axiom,
% 2.85/1.57 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ! [Y2: A] :
% 2.85/1.57 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X3 ) )
% 2.85/1.57 => ( P @ Y2 ) )
% 2.85/1.57 => ( P @ X3 ) )
% 2.85/1.57 => ( P @ A2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_49_infinite__descent__measure,axiom,
% 2.85/1.57 ! [A: $tType,P: A > $o,V: A > nat,X2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ~ ( P @ X3 )
% 2.85/1.57 => ? [Y2: A] :
% 2.85/1.57 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
% 2.85/1.57 & ~ ( P @ Y2 ) ) )
% 2.85/1.57 => ( P @ X2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_50_size__enat__overloaded__def,axiom,
% 2.85/1.57 ( ( size_size @ extended_enat )
% 2.85/1.57 = ( extended_rec_enat @ nat
% 2.85/1.57 @ ^ [X: nat] : ( zero_zero @ nat )
% 2.85/1.57 @ ( zero_zero @ nat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_51_size__enat__def,axiom,
% 2.85/1.57 ( extended_size_enat
% 2.85/1.57 = ( extended_rec_enat @ nat
% 2.85/1.57 @ ^ [X: nat] : ( zero_zero @ nat )
% 2.85/1.57 @ ( zero_zero @ nat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_52_gr0I,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ( ( N
% 2.85/1.57 != ( zero_zero @ nat ) )
% 2.85/1.57 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_53_not__gr0,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.85/1.57 = ( N
% 2.85/1.57 = ( zero_zero @ nat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_54_not__less0,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_55_less__zeroE,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_56_gr__implies__not0,axiom,
% 2.85/1.57 ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M @ N )
% 2.85/1.57 => ( N
% 2.85/1.57 != ( zero_zero @ nat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_57_infinite__descent0,axiom,
% 2.85/1.57 ! [P: nat > $o,N: nat] :
% 2.85/1.57 ( ( P @ ( zero_zero @ nat ) )
% 2.85/1.57 => ( ! [N2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.85/1.57 => ( ~ ( P @ N2 )
% 2.85/1.57 => ? [M2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M2 @ N2 )
% 2.85/1.57 & ~ ( P @ M2 ) ) ) )
% 2.85/1.57 => ( P @ N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_58_less__nat__zero__code,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_59_infinite__descent0__measure,axiom,
% 2.85/1.57 ! [A: $tType,V: A > nat,P: A > $o,X2: A] :
% 2.85/1.57 ( ! [X3: A] :
% 2.85/1.57 ( ( ( V @ X3 )
% 2.85/1.57 = ( zero_zero @ nat ) )
% 2.85/1.57 => ( P @ X3 ) )
% 2.85/1.57 => ( ! [X3: A] :
% 2.85/1.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
% 2.85/1.57 => ( ~ ( P @ X3 )
% 2.85/1.57 => ? [Y2: A] :
% 2.85/1.57 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
% 2.85/1.57 & ~ ( P @ Y2 ) ) ) )
% 2.85/1.57 => ( P @ X2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_60_all__imp__alw,axiom,
% 2.85/1.57 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.85/1.57 ( ! [X1: stream @ A] : ( Phi @ X1 )
% 2.85/1.57 => ( linear1386806755on_alw @ A @ Phi @ Xs2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_61_alw__mono,axiom,
% 2.85/1.57 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
% 2.85/1.57 => ( ! [Xs3: stream @ A] :
% 2.85/1.57 ( ( Phi @ Xs3 )
% 2.85/1.57 => ( Psi @ Xs3 ) )
% 2.85/1.57 => ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_62_alw__cong,axiom,
% 2.85/1.57 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 2.85/1.57 => ( ! [Omega3: stream @ A] :
% 2.85/1.57 ( ( P @ Omega3 )
% 2.85/1.57 => ( ( Q1 @ Omega3 )
% 2.85/1.57 = ( Q2 @ Omega3 ) ) )
% 2.85/1.57 => ( ( linear1386806755on_alw @ A @ Q1 @ Omega )
% 2.85/1.57 = ( linear1386806755on_alw @ A @ Q2 @ Omega ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_63_alw__alwD,axiom,
% 2.85/1.57 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 2.85/1.57 => ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ P ) @ Omega ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_64_alwD,axiom,
% 2.85/1.57 ! [A: $tType,P: ( stream @ A ) > $o,X2: stream @ A] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A @ P @ X2 )
% 2.85/1.57 => ( P @ X2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_65_until__monoR,axiom,
% 2.85/1.57 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi_1: ( stream @ A ) > $o,Xs2: stream @ A,Psi_2: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1341015445_UNTIL @ A @ Phi @ Psi_1 @ Xs2 )
% 2.85/1.57 => ( ! [Xs3: stream @ A] :
% 2.85/1.57 ( ( Psi_1 @ Xs3 )
% 2.85/1.57 => ( Psi_2 @ Xs3 ) )
% 2.85/1.57 => ( linear1341015445_UNTIL @ A @ Phi @ Psi_2 @ Xs2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_66_until__monoL,axiom,
% 2.85/1.57 ! [A: $tType,Phi_1: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs2: stream @ A,Phi_2: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1341015445_UNTIL @ A @ Phi_1 @ Psi @ Xs2 )
% 2.85/1.57 => ( ! [Xs3: stream @ A] :
% 2.85/1.57 ( ( Phi_1 @ Xs3 )
% 2.85/1.57 => ( Phi_2 @ Xs3 ) )
% 2.85/1.57 => ( linear1341015445_UNTIL @ A @ Phi_2 @ Psi @ Xs2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_67_until__mono,axiom,
% 2.85/1.57 ! [A: $tType,Phi_1: ( stream @ A ) > $o,Psi_1: ( stream @ A ) > $o,Xs2: stream @ A,Phi_2: ( stream @ A ) > $o,Psi_2: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( linear1341015445_UNTIL @ A @ Phi_1 @ Psi_1 @ Xs2 )
% 2.85/1.57 => ( ! [Xs3: stream @ A] :
% 2.85/1.57 ( ( Phi_1 @ Xs3 )
% 2.85/1.57 => ( Phi_2 @ Xs3 ) )
% 2.85/1.57 => ( ! [Xs3: stream @ A] :
% 2.85/1.57 ( ( Psi_1 @ Xs3 )
% 2.85/1.57 => ( Psi_2 @ Xs3 ) )
% 2.85/1.57 => ( linear1341015445_UNTIL @ A @ Phi_2 @ Psi_2 @ Xs2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_68_UNTIL_Obase,axiom,
% 2.85/1.57 ! [A: $tType,Psi: ( stream @ A ) > $o,Xs2: stream @ A,Phi: ( stream @ A ) > $o] :
% 2.85/1.57 ( ( Psi @ Xs2 )
% 2.85/1.57 => ( linear1341015445_UNTIL @ A @ Phi @ Psi @ Xs2 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_69_of__nat__0__less__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linordered_semidom @ A @ ( type @ A ) )
% 2.85/1.57 => ! [N: nat] :
% 2.85/1.57 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.85/1.57 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_70_less__enat__def,axiom,
% 2.85/1.57 ( ( ord_less @ extended_enat )
% 2.85/1.57 = ( ^ [M3: extended_enat,N3: extended_enat] :
% 2.85/1.57 ( extended_case_enat @ $o
% 2.85/1.57 @ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N3 )
% 2.85/1.57 @ $false
% 2.85/1.57 @ M3 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_71_case__enat__def,axiom,
% 2.85/1.57 ! [T: $tType] :
% 2.85/1.57 ( ( extended_case_enat @ T )
% 2.85/1.57 = ( extended_rec_enat @ T ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_72_enat_Osimps_I4_J,axiom,
% 2.85/1.57 ! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
% 2.85/1.57 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
% 2.85/1.57 = ( F1 @ Nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_73_scount__eq__0__iff,axiom,
% 2.85/1.57 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.57 ( ( ( coindu1365464361scount @ A @ P @ Omega )
% 2.85/1.57 = ( zero_zero @ extended_enat ) )
% 2.85/1.57 = ( linear1386806755on_alw @ A
% 2.85/1.57 @ ^ [Xs: stream @ A] :
% 2.85/1.57 ~ ( P @ Xs )
% 2.85/1.57 @ Omega ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_74_scount__eq__0I,axiom,
% 2.85/1.57 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.57 ( ( linear1386806755on_alw @ A
% 2.85/1.57 @ ^ [Xs: stream @ A] :
% 2.85/1.57 ~ ( P @ Xs )
% 2.85/1.57 @ Omega )
% 2.85/1.57 => ( ( coindu1365464361scount @ A @ P @ Omega )
% 2.85/1.57 = ( zero_zero @ extended_enat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_75_scount__eq__0D,axiom,
% 2.85/1.57 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.57 ( ( ( coindu1365464361scount @ A @ P @ Omega )
% 2.85/1.57 = ( zero_zero @ extended_enat ) )
% 2.85/1.57 => ( linear1386806755on_alw @ A
% 2.85/1.57 @ ^ [Xs: stream @ A] :
% 2.85/1.57 ~ ( P @ Xs )
% 2.85/1.57 @ Omega ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_76_enat_Osize_I4_J,axiom,
% 2.85/1.57 ( ( size_size @ extended_enat @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 = ( zero_zero @ nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_77_of__nat__eq__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 2.85/1.57 => ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ( semiring_1_of_nat @ A @ M )
% 2.85/1.57 = ( semiring_1_of_nat @ A @ N ) )
% 2.85/1.57 = ( M = N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_78_not__infinity__eq,axiom,
% 2.85/1.57 ! [X2: extended_enat] :
% 2.85/1.57 ( ( X2
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 = ( ? [I2: nat] :
% 2.85/1.57 ( X2
% 2.85/1.57 = ( extended_enat2 @ I2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_79_not__enat__eq,axiom,
% 2.85/1.57 ! [X2: extended_enat] :
% 2.85/1.57 ( ( ! [Y4: nat] :
% 2.85/1.57 ( X2
% 2.85/1.57 != ( extended_enat2 @ Y4 ) ) )
% 2.85/1.57 = ( X2
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_80_enat__ord__simps_I4_J,axiom,
% 2.85/1.57 ! [Q3: extended_enat] :
% 2.85/1.57 ( ( ord_less @ extended_enat @ Q3 @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 = ( Q3
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_81_enat__ord__simps_I6_J,axiom,
% 2.85/1.57 ! [Q3: extended_enat] :
% 2.85/1.57 ~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q3 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_82_enat_Osimps_I5_J,axiom,
% 2.85/1.57 ! [T: $tType,F1: nat > T,F2: T] :
% 2.85/1.57 ( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 = F2 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_83_enat_Osimps_I7_J,axiom,
% 2.85/1.57 ! [T: $tType,F1: nat > T,F2: T] :
% 2.85/1.57 ( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 = F2 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_84_of__nat__eq__0__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 2.85/1.57 => ! [M: nat] :
% 2.85/1.57 ( ( ( semiring_1_of_nat @ A @ M )
% 2.85/1.57 = ( zero_zero @ A ) )
% 2.85/1.57 = ( M
% 2.85/1.57 = ( zero_zero @ nat ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_85_of__nat__0__eq__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( semiring_char_0 @ A @ ( type @ A ) )
% 2.85/1.57 => ! [N: nat] :
% 2.85/1.57 ( ( ( zero_zero @ A )
% 2.85/1.57 = ( semiring_1_of_nat @ A @ N ) )
% 2.85/1.57 = ( ( zero_zero @ nat )
% 2.85/1.57 = N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_86_of__nat__0,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( semiring_1 @ A @ ( type @ A ) )
% 2.85/1.57 => ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
% 2.85/1.57 = ( zero_zero @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_87_of__nat__less__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linordered_semidom @ A @ ( type @ A ) )
% 2.85/1.57 => ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.85/1.57 = ( ord_less @ nat @ M @ N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_88_case__enat__0,axiom,
% 2.85/1.57 ! [A: $tType,F: nat > A,I3: A] :
% 2.85/1.57 ( ( extended_case_enat @ A @ F @ I3 @ ( zero_zero @ extended_enat ) )
% 2.85/1.57 = ( F @ ( zero_zero @ nat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_89_enat_Osize_I2_J,axiom,
% 2.85/1.57 ( ( extended_size_enat @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 = ( zero_zero @ nat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_90_of__nat__eq__enat,axiom,
% 2.85/1.57 ( ( semiring_1_of_nat @ extended_enat )
% 2.85/1.57 = extended_enat2 ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_91_enat__ex__split,axiom,
% 2.85/1.57 ( ( ^ [P3: extended_enat > $o] :
% 2.85/1.57 ? [X4: extended_enat] : ( P3 @ X4 ) )
% 2.85/1.57 = ( ^ [P2: extended_enat > $o] :
% 2.85/1.57 ( ( P2 @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 | ? [X: nat] : ( P2 @ ( extended_enat2 @ X ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_92_enat_Oinducts,axiom,
% 2.85/1.57 ! [P: extended_enat > $o,Enat: extended_enat] :
% 2.85/1.57 ( ! [Nat3: nat] : ( P @ ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ( ( P @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ( P @ Enat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_93_enat_Oexhaust,axiom,
% 2.85/1.57 ! [Y3: extended_enat] :
% 2.85/1.57 ( ! [Nat3: nat] :
% 2.85/1.57 ( Y3
% 2.85/1.57 != ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ( Y3
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_94_enat3__cases,axiom,
% 2.85/1.57 ! [Y3: extended_enat,Ya: extended_enat,Yb: extended_enat] :
% 2.85/1.57 ( ( ? [Nat3: nat] :
% 2.85/1.57 ( Y3
% 2.85/1.57 = ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ( ? [Nata: nat] :
% 2.85/1.57 ( Ya
% 2.85/1.57 = ( extended_enat2 @ Nata ) )
% 2.85/1.57 => ! [Natb: nat] :
% 2.85/1.57 ( Yb
% 2.85/1.57 != ( extended_enat2 @ Natb ) ) ) )
% 2.85/1.57 => ( ( ? [Nat3: nat] :
% 2.85/1.57 ( Y3
% 2.85/1.57 = ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ( ? [Nata: nat] :
% 2.85/1.57 ( Ya
% 2.85/1.57 = ( extended_enat2 @ Nata ) )
% 2.85/1.57 => ( Yb
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) ) ) )
% 2.85/1.57 => ( ( ? [Nat3: nat] :
% 2.85/1.57 ( Y3
% 2.85/1.57 = ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ( ( Ya
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ! [Nata: nat] :
% 2.85/1.57 ( Yb
% 2.85/1.57 != ( extended_enat2 @ Nata ) ) ) )
% 2.85/1.57 => ( ( ? [Nat3: nat] :
% 2.85/1.57 ( Y3
% 2.85/1.57 = ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ( ( Ya
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ( Yb
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) ) ) )
% 2.85/1.57 => ( ( ( Y3
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ( ? [Nat3: nat] :
% 2.85/1.57 ( Ya
% 2.85/1.57 = ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ! [Nata: nat] :
% 2.85/1.57 ( Yb
% 2.85/1.57 != ( extended_enat2 @ Nata ) ) ) )
% 2.85/1.57 => ( ( ( Y3
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ( ? [Nat3: nat] :
% 2.85/1.57 ( Ya
% 2.85/1.57 = ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ( Yb
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) ) ) )
% 2.85/1.57 => ( ( ( Y3
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ( ( Ya
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ! [Nat3: nat] :
% 2.85/1.57 ( Yb
% 2.85/1.57 != ( extended_enat2 @ Nat3 ) ) ) )
% 2.85/1.57 => ~ ( ( Y3
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ( ( Ya
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ( Yb
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_95_enat2__cases,axiom,
% 2.85/1.57 ! [Y3: extended_enat,Ya: extended_enat] :
% 2.85/1.57 ( ( ? [Nat3: nat] :
% 2.85/1.57 ( Y3
% 2.85/1.57 = ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ! [Nata: nat] :
% 2.85/1.57 ( Ya
% 2.85/1.57 != ( extended_enat2 @ Nata ) ) )
% 2.85/1.57 => ( ( ? [Nat3: nat] :
% 2.85/1.57 ( Y3
% 2.85/1.57 = ( extended_enat2 @ Nat3 ) )
% 2.85/1.57 => ( Ya
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) ) )
% 2.85/1.57 => ( ( ( Y3
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ! [Nat3: nat] :
% 2.85/1.57 ( Ya
% 2.85/1.57 != ( extended_enat2 @ Nat3 ) ) )
% 2.85/1.57 => ~ ( ( Y3
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ( Ya
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_96_enat_Odistinct_I1_J,axiom,
% 2.85/1.57 ! [Nat: nat] :
% 2.85/1.57 ( ( extended_enat2 @ Nat )
% 2.85/1.57 != ( extend1396239628finity @ extended_enat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_97_enat_Odistinct_I2_J,axiom,
% 2.85/1.57 ! [Nat4: nat] :
% 2.85/1.57 ( ( extend1396239628finity @ extended_enat )
% 2.85/1.57 != ( extended_enat2 @ Nat4 ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_98_infinity__ne__i0,axiom,
% 2.85/1.57 ( ( extend1396239628finity @ extended_enat )
% 2.85/1.57 != ( zero_zero @ extended_enat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_99_of__nat__less__0__iff,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linordered_semidom @ A @ ( type @ A ) )
% 2.85/1.57 => ! [M: nat] :
% 2.85/1.57 ~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_100_of__nat__less__imp__less,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linordered_semidom @ A @ ( type @ A ) )
% 2.85/1.57 => ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
% 2.85/1.57 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_101_less__imp__of__nat__less,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( linordered_semidom @ A @ ( type @ A ) )
% 2.85/1.57 => ! [M: nat,N: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ M @ N )
% 2.85/1.57 => ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_102_infinity__ilessE,axiom,
% 2.85/1.57 ! [M: nat] :
% 2.85/1.57 ~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_103_less__infinityE,axiom,
% 2.85/1.57 ! [N: extended_enat] :
% 2.85/1.57 ( ( ord_less @ extended_enat @ N @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.57 => ~ ! [K2: nat] :
% 2.85/1.57 ( N
% 2.85/1.57 != ( extended_enat2 @ K2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_104_enat__ord__code_I4_J,axiom,
% 2.85/1.57 ! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_105_zero__less__imp__eq__int,axiom,
% 2.85/1.57 ! [K: int] :
% 2.85/1.57 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.85/1.57 => ? [N2: nat] :
% 2.85/1.57 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.85/1.57 & ( K
% 2.85/1.57 = ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_106_reals__Archimedean2,axiom,
% 2.85/1.57 ! [A: $tType] :
% 2.85/1.57 ( ( archim1804426504_field @ A @ ( type @ A ) )
% 2.85/1.57 => ! [X2: A] :
% 2.85/1.57 ? [N2: nat] : ( ord_less @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_107_times__enat__simps_I3_J,axiom,
% 2.85/1.57 ! [N: nat] :
% 2.85/1.57 ( ( ( N
% 2.85/1.57 = ( zero_zero @ nat ) )
% 2.85/1.57 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
% 2.85/1.57 = ( zero_zero @ extended_enat ) ) )
% 2.85/1.57 & ( ( N
% 2.85/1.57 != ( zero_zero @ nat ) )
% 2.85/1.57 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
% 2.85/1.57 = ( extend1396239628finity @ extended_enat ) ) ) ) ).
% 2.85/1.57
% 2.85/1.57 thf(fact_108_times__enat__simps_I4_J,axiom,
% 2.85/1.57 ! [M: nat] :
% 2.85/1.57 ( ( ( M
% 2.85/1.57 = ( zero_zero @ nat ) )
% 2.85/1.57 => ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 = ( zero_zero @ extended_enat ) ) )
% 2.85/1.58 & ( ( M
% 2.85/1.58 != ( zero_zero @ nat ) )
% 2.85/1.58 => ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_109_the__enat__0,axiom,
% 2.85/1.58 ( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
% 2.85/1.58 = ( zero_zero @ nat ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_110_scount__finite,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A
% 2.85/1.58 @ ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( P @ Xs ) )
% 2.85/1.58 @ Omega )
% 2.85/1.58 => ( ord_less @ extended_enat @ ( coindu1365464361scount @ A @ P @ Omega ) @ ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_111_ev__ev,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ ( linear505997466_on_ev @ A @ Phi ) )
% 2.85/1.58 = ( linear505997466_on_ev @ A @ Phi ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_112_of__nat__mult,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [M: nat,N: nat] :
% 2.85/1.58 ( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
% 2.85/1.58 = ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_113_times__enat__simps_I2_J,axiom,
% 2.85/1.58 ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_114_not__alw__not,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xt: stream @ A] :
% 2.85/1.58 ~ ( Phi @ Xt )
% 2.85/1.58 @ Xs ) )
% 2.85/1.58 = ( linear505997466_on_ev @ A @ Phi ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_115_not__ev__not,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( linear505997466_on_ev @ A
% 2.85/1.58 @ ^ [Xt: stream @ A] :
% 2.85/1.58 ~ ( Phi @ Xt )
% 2.85/1.58 @ Xs ) )
% 2.85/1.58 = ( linear1386806755on_alw @ A @ Phi ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_116_int__int__eq,axiom,
% 2.85/1.58 ! [M: nat,N: nat] :
% 2.85/1.58 ( ( ( semiring_1_of_nat @ int @ M )
% 2.85/1.58 = ( semiring_1_of_nat @ int @ N ) )
% 2.85/1.58 = ( M = N ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_117_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( ab_semigroup_mult @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.85/1.58 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 2.85/1.58 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_118_mult_Oassoc,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semigroup_mult @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.85/1.58 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 2.85/1.58 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_119_mult_Ocommute,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( ab_semigroup_mult @ A @ ( type @ A ) )
% 2.85/1.58 => ( ( times_times @ A )
% 2.85/1.58 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_120_mult_Oleft__commute,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( ab_semigroup_mult @ A @ ( type @ A ) )
% 2.85/1.58 => ! [B2: A,A2: A,C2: A] :
% 2.85/1.58 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
% 2.85/1.58 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_121_ev__or,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( Phi @ Xs )
% 2.85/1.58 | ( Psi @ Xs ) ) )
% 2.85/1.58 = ( ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ Phi @ Xs )
% 2.85/1.58 | ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_122_ev_Obase,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.85/1.58 ( ( Phi @ Xs2 )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Phi @ Xs2 ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_123_ev__mono,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.85/1.58 => ( ! [Xs3: stream @ A] :
% 2.85/1.58 ( ( Phi @ Xs3 )
% 2.85/1.58 => ( Psi @ Xs3 ) )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_124_ev__False,axiom,
% 2.85/1.58 ! [A: $tType,Omega: stream @ A] :
% 2.85/1.58 ~ ( linear505997466_on_ev @ A
% 2.85/1.58 @ ^ [X: stream @ A] : $false
% 2.85/1.58 @ Omega ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_125_less__int__code_I1_J,axiom,
% 2.85/1.58 ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_126_ev__alw__imp__alw__ev,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 2.85/1.58 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_127_ev__cong,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear1386806755on_alw @ A @ P @ Omega )
% 2.85/1.58 => ( ! [Omega3: stream @ A] :
% 2.85/1.58 ( ( P @ Omega3 )
% 2.85/1.58 => ( ( Q1 @ Omega3 )
% 2.85/1.58 = ( Q2 @ Omega3 ) ) )
% 2.85/1.58 => ( ( linear505997466_on_ev @ A @ Q1 @ Omega )
% 2.85/1.58 = ( linear505997466_on_ev @ A @ Q2 @ Omega ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_128_ev__iff__sdrop,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A )
% 2.85/1.58 = ( ^ [P2: ( stream @ A ) > $o,Omega2: stream @ A] :
% 2.85/1.58 ? [M3: nat] : ( P2 @ ( sdrop @ A @ M3 @ Omega2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_129_mult__of__nat__commute,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [X2: nat,Y3: A] :
% 2.85/1.58 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ X2 ) @ Y3 )
% 2.85/1.58 = ( times_times @ A @ Y3 @ ( semiring_1_of_nat @ A @ X2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_130_imult__is__0,axiom,
% 2.85/1.58 ! [M: extended_enat,N: extended_enat] :
% 2.85/1.58 ( ( ( times_times @ extended_enat @ M @ N )
% 2.85/1.58 = ( zero_zero @ extended_enat ) )
% 2.85/1.58 = ( ( M
% 2.85/1.58 = ( zero_zero @ extended_enat ) )
% 2.85/1.58 | ( N
% 2.85/1.58 = ( zero_zero @ extended_enat ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_131_alw__ev__imp__ev__alw,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.58 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega )
% 2.85/1.58 => ( linear505997466_on_ev @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( P @ Xs )
% 2.85/1.58 & ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Xs ) )
% 2.85/1.58 @ Omega ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_132_ev__alw__alw__impl,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 2.85/1.58 => ( ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 2.85/1.58 @ Xs2 )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_133_alw__impl__ev__alw,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.85/1.58 ( ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( Phi @ Xs )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 2.85/1.58 @ Xs2 )
% 2.85/1.58 => ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ Phi @ Xs )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 2.85/1.58 @ Xs2 ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_134_alw__alw__impl__ev,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( linear1386806755on_alw @ A @ Phi @ Xs )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs ) ) )
% 2.85/1.58 = ( ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs )
% 2.85/1.58 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Psi ) @ Xs ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_135_ev__alw__impl__ev,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.85/1.58 => ( ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( Phi @ Xs )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs ) )
% 2.85/1.58 @ Xs2 )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_136_not__alw__iff,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.58 ( ( ~ ( linear1386806755on_alw @ A @ P @ Omega ) )
% 2.85/1.58 = ( linear505997466_on_ev @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( P @ Xs )
% 2.85/1.58 @ Omega ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_137_ev__alw__impl,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.85/1.58 => ( ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( Phi @ Xs )
% 2.85/1.58 => ( Psi @ Xs ) )
% 2.85/1.58 @ Xs2 )
% 2.85/1.58 => ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_138_ev__alw__aand,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A,Psi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
% 2.85/1.58 => ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Psi ) @ Xs2 )
% 2.85/1.58 => ( linear505997466_on_ev @ A
% 2.85/1.58 @ ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ( ( Phi @ Xs )
% 2.85/1.58 & ( Psi @ Xs ) ) )
% 2.85/1.58 @ Xs2 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_139_not__ev__iff,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.58 ( ( ~ ( linear505997466_on_ev @ A @ P @ Omega ) )
% 2.85/1.58 = ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( P @ Xs )
% 2.85/1.58 @ Omega ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_140_not__alw,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( linear1386806755on_alw @ A @ Phi @ Xs ) )
% 2.85/1.58 = ( linear505997466_on_ev @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( Phi @ Xs ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_141_not__ev,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o] :
% 2.85/1.58 ( ( ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( linear505997466_on_ev @ A @ Phi @ Xs ) )
% 2.85/1.58 = ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( Phi @ Xs ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_142_the__enat_Osimps,axiom,
% 2.85/1.58 ! [N: nat] :
% 2.85/1.58 ( ( extended_the_enat @ ( extended_enat2 @ N ) )
% 2.85/1.58 = N ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_143_alw__ev__sdrop,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,M: nat,Xs2: stream @ A] :
% 2.85/1.58 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ ( sdrop @ A @ M @ Xs2 ) )
% 2.85/1.58 => ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs2 ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_144_ex__less__of__nat__mult,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( archim1804426504_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [X2: A,Y3: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
% 2.85/1.58 => ? [N2: nat] : ( ord_less @ A @ Y3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_145_imult__is__infinity,axiom,
% 2.85/1.58 ! [A2: extended_enat,B2: extended_enat] :
% 2.85/1.58 ( ( ( times_times @ extended_enat @ A2 @ B2 )
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 = ( ( ( A2
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 & ( B2
% 2.85/1.58 != ( zero_zero @ extended_enat ) ) )
% 2.85/1.58 | ( ( B2
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 & ( A2
% 2.85/1.58 != ( zero_zero @ extended_enat ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_146_enat__0__less__mult__iff,axiom,
% 2.85/1.58 ! [M: extended_enat,N: extended_enat] :
% 2.85/1.58 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N ) )
% 2.85/1.58 = ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M )
% 2.85/1.58 & ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_147_scount__infinite__iff,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.58 ( ( ( coindu1365464361scount @ A @ P @ Omega )
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 = ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_148_scount__infinite,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.58 ( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega )
% 2.85/1.58 => ( ( coindu1365464361scount @ A @ P @ Omega )
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_149_sfirst__finite,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.58 ( ( ord_less @ extended_enat @ ( coindu76177514sfirst @ A @ P @ Omega ) @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 = ( linear505997466_on_ev @ A @ P @ Omega ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_150_imult__infinity__right,axiom,
% 2.85/1.58 ! [N: extended_enat] :
% 2.85/1.58 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 2.85/1.58 => ( ( times_times @ extended_enat @ N @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_151_imult__infinity,axiom,
% 2.85/1.58 ! [N: extended_enat] :
% 2.85/1.58 ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
% 2.85/1.58 => ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ N )
% 2.85/1.58 = ( extend1396239628finity @ extended_enat ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_152_enat__the__enat,axiom,
% 2.85/1.58 ! [N: extended_enat] :
% 2.85/1.58 ( ( N
% 2.85/1.58 != ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 => ( ( extended_enat2 @ ( extended_the_enat @ N ) )
% 2.85/1.58 = N ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_153_mult__cancel__right,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiri1923998003cancel @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,C2: A,B2: A] :
% 2.85/1.58 ( ( ( times_times @ A @ A2 @ C2 )
% 2.85/1.58 = ( times_times @ A @ B2 @ C2 ) )
% 2.85/1.58 = ( ( C2
% 2.85/1.58 = ( zero_zero @ A ) )
% 2.85/1.58 | ( A2 = B2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_154_mult__cancel__left,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiri1923998003cancel @ A @ ( type @ A ) )
% 2.85/1.58 => ! [C2: A,A2: A,B2: A] :
% 2.85/1.58 ( ( ( times_times @ A @ C2 @ A2 )
% 2.85/1.58 = ( times_times @ A @ C2 @ B2 ) )
% 2.85/1.58 = ( ( C2
% 2.85/1.58 = ( zero_zero @ A ) )
% 2.85/1.58 | ( A2 = B2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_155_mult__eq__0__iff,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiri1193490041visors @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ( times_times @ A @ A2 @ B2 )
% 2.85/1.58 = ( zero_zero @ A ) )
% 2.85/1.58 = ( ( A2
% 2.85/1.58 = ( zero_zero @ A ) )
% 2.85/1.58 | ( B2
% 2.85/1.58 = ( zero_zero @ A ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_156_mult__zero__right,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( mult_zero @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 = ( zero_zero @ A ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_157_mult__zero__left,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( mult_zero @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 = ( zero_zero @ A ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_158_mult__cancel2,axiom,
% 2.85/1.58 ! [M: nat,K: nat,N: nat] :
% 2.85/1.58 ( ( ( times_times @ nat @ M @ K )
% 2.85/1.58 = ( times_times @ nat @ N @ K ) )
% 2.85/1.58 = ( ( M = N )
% 2.85/1.58 | ( K
% 2.85/1.58 = ( zero_zero @ nat ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_159_mult__cancel1,axiom,
% 2.85/1.58 ! [K: nat,M: nat,N: nat] :
% 2.85/1.58 ( ( ( times_times @ nat @ K @ M )
% 2.85/1.58 = ( times_times @ nat @ K @ N ) )
% 2.85/1.58 = ( ( M = N )
% 2.85/1.58 | ( K
% 2.85/1.58 = ( zero_zero @ nat ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_160_mult__0__right,axiom,
% 2.85/1.58 ! [M: nat] :
% 2.85/1.58 ( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
% 2.85/1.58 = ( zero_zero @ nat ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_161_mult__is__0,axiom,
% 2.85/1.58 ! [M: nat,N: nat] :
% 2.85/1.58 ( ( ( times_times @ nat @ M @ N )
% 2.85/1.58 = ( zero_zero @ nat ) )
% 2.85/1.58 = ( ( M
% 2.85/1.58 = ( zero_zero @ nat ) )
% 2.85/1.58 | ( N
% 2.85/1.58 = ( zero_zero @ nat ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_162_mult__less__cancel2,axiom,
% 2.85/1.58 ! [M: nat,K: nat,N: nat] :
% 2.85/1.58 ( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
% 2.85/1.58 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.85/1.58 & ( ord_less @ nat @ M @ N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_163_nat__0__less__mult__iff,axiom,
% 2.85/1.58 ! [M: nat,N: nat] :
% 2.85/1.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
% 2.85/1.58 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
% 2.85/1.58 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_164_times__enat__simps_I1_J,axiom,
% 2.85/1.58 ! [M: nat,N: nat] :
% 2.85/1.58 ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 2.85/1.58 = ( extended_enat2 @ ( times_times @ nat @ M @ N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_165_times__int__code_I1_J,axiom,
% 2.85/1.58 ! [K: int] :
% 2.85/1.58 ( ( times_times @ int @ K @ ( zero_zero @ int ) )
% 2.85/1.58 = ( zero_zero @ int ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_166_times__int__code_I2_J,axiom,
% 2.85/1.58 ! [L: int] :
% 2.85/1.58 ( ( times_times @ int @ ( zero_zero @ int ) @ L )
% 2.85/1.58 = ( zero_zero @ int ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_167_mult__0,axiom,
% 2.85/1.58 ! [N: nat] :
% 2.85/1.58 ( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
% 2.85/1.58 = ( zero_zero @ nat ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_168_mult__less__mono1,axiom,
% 2.85/1.58 ! [I3: nat,J3: nat,K: nat] :
% 2.85/1.58 ( ( ord_less @ nat @ I3 @ J3 )
% 2.85/1.58 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.85/1.58 => ( ord_less @ nat @ ( times_times @ nat @ I3 @ K ) @ ( times_times @ nat @ J3 @ K ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_169_mult__less__mono2,axiom,
% 2.85/1.58 ! [I3: nat,J3: nat,K: nat] :
% 2.85/1.58 ( ( ord_less @ nat @ I3 @ J3 )
% 2.85/1.58 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.85/1.58 => ( ord_less @ nat @ ( times_times @ nat @ K @ I3 ) @ ( times_times @ nat @ K @ J3 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_170_zmult__zless__mono2,axiom,
% 2.85/1.58 ! [I3: int,J3: int,K: int] :
% 2.85/1.58 ( ( ord_less @ int @ I3 @ J3 )
% 2.85/1.58 => ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 2.85/1.58 => ( ord_less @ int @ ( times_times @ int @ K @ I3 ) @ ( times_times @ int @ K @ J3 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_171_linorder__neqE__linordered__idom,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_idom @ A @ ( type @ A ) )
% 2.85/1.58 => ! [X2: A,Y3: A] :
% 2.85/1.58 ( ( X2 != Y3 )
% 2.85/1.58 => ( ~ ( ord_less @ A @ X2 @ Y3 )
% 2.85/1.58 => ( ord_less @ A @ Y3 @ X2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_172_zmult__zless__mono2__lemma,axiom,
% 2.85/1.58 ! [I3: int,J3: int,K: nat] :
% 2.85/1.58 ( ( ord_less @ int @ I3 @ J3 )
% 2.85/1.58 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.85/1.58 => ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I3 ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J3 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_173_times__enat__def,axiom,
% 2.85/1.58 ( ( times_times @ extended_enat )
% 2.85/1.58 = ( ^ [M3: extended_enat,N3: extended_enat] :
% 2.85/1.58 ( extended_case_enat @ extended_enat
% 2.85/1.58 @ ^ [O: nat] :
% 2.85/1.58 ( extended_case_enat @ extended_enat
% 2.85/1.58 @ ^ [P4: nat] : ( extended_enat2 @ ( times_times @ nat @ O @ P4 ) )
% 2.85/1.58 @ ( if @ extended_enat
% 2.85/1.58 @ ( O
% 2.85/1.58 = ( zero_zero @ nat ) )
% 2.85/1.58 @ ( zero_zero @ extended_enat )
% 2.85/1.58 @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 @ N3 )
% 2.85/1.58 @ ( if @ extended_enat
% 2.85/1.58 @ ( N3
% 2.85/1.58 = ( zero_zero @ extended_enat ) )
% 2.85/1.58 @ ( zero_zero @ extended_enat )
% 2.85/1.58 @ ( extend1396239628finity @ extended_enat ) )
% 2.85/1.58 @ M3 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_174_mult__not__zero,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( mult_zero @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ( times_times @ A @ A2 @ B2 )
% 2.85/1.58 != ( zero_zero @ A ) )
% 2.85/1.58 => ( ( A2
% 2.85/1.58 != ( zero_zero @ A ) )
% 2.85/1.58 & ( B2
% 2.85/1.58 != ( zero_zero @ A ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_175_divisors__zero,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiri1193490041visors @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ( times_times @ A @ A2 @ B2 )
% 2.85/1.58 = ( zero_zero @ A ) )
% 2.85/1.58 => ( ( A2
% 2.85/1.58 = ( zero_zero @ A ) )
% 2.85/1.58 | ( B2
% 2.85/1.58 = ( zero_zero @ A ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_176_no__zero__divisors,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiri1193490041visors @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( A2
% 2.85/1.58 != ( zero_zero @ A ) )
% 2.85/1.58 => ( ( B2
% 2.85/1.58 != ( zero_zero @ A ) )
% 2.85/1.58 => ( ( times_times @ A @ A2 @ B2 )
% 2.85/1.58 != ( zero_zero @ A ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_177_mult__left__cancel,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiri1923998003cancel @ A @ ( type @ A ) )
% 2.85/1.58 => ! [C2: A,A2: A,B2: A] :
% 2.85/1.58 ( ( C2
% 2.85/1.58 != ( zero_zero @ A ) )
% 2.85/1.58 => ( ( ( times_times @ A @ C2 @ A2 )
% 2.85/1.58 = ( times_times @ A @ C2 @ B2 ) )
% 2.85/1.58 = ( A2 = B2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_178_mult__right__cancel,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( semiri1923998003cancel @ A @ ( type @ A ) )
% 2.85/1.58 => ! [C2: A,A2: A,B2: A] :
% 2.85/1.58 ( ( C2
% 2.85/1.58 != ( zero_zero @ A ) )
% 2.85/1.58 => ( ( ( times_times @ A @ A2 @ C2 )
% 2.85/1.58 = ( times_times @ A @ B2 @ C2 ) )
% 2.85/1.58 = ( A2 = B2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_179_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord893533164strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ B2 )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 2.85/1.58 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_180_mult__less__cancel__right__disj,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,C2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 2.85/1.58 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 2.85/1.58 & ( ord_less @ A @ A2 @ B2 ) )
% 2.85/1.58 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 2.85/1.58 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_181_mult__strict__right__mono,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord20386208strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ B2 )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 2.85/1.58 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_182_mult__strict__right__mono__neg,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [B2: A,A2: A,C2: A] :
% 2.85/1.58 ( ( ord_less @ A @ B2 @ A2 )
% 2.85/1.58 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_183_mult__less__cancel__left__disj,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [C2: A,A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
% 2.85/1.58 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 2.85/1.58 & ( ord_less @ A @ A2 @ B2 ) )
% 2.85/1.58 | ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 2.85/1.58 & ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_184_mult__strict__left__mono,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord20386208strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ B2 )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 2.85/1.58 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_185_mult__strict__left__mono__neg,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [B2: A,A2: A,C2: A] :
% 2.85/1.58 ( ( ord_less @ A @ B2 @ A2 )
% 2.85/1.58 => ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_186_mult__less__cancel__left__pos,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [C2: A,A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 2.85/1.58 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
% 2.85/1.58 = ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_187_mult__less__cancel__left__neg,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [C2: A,A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
% 2.85/1.58 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_188_zero__less__mult__pos2,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord20386208strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [B2: A,A2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_189_zero__less__mult__pos,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord20386208strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 => ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_190_zero__less__mult__iff,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
% 2.85/1.58 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 2.85/1.58 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_191_mult__pos__neg2,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord20386208strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_192_mult__pos__pos,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord20386208strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.85/1.58 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_193_mult__pos__neg,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord20386208strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_194_mult__neg__pos,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord20386208strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.85/1.58 => ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_195_mult__less__0__iff,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 2.85/1.58 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 2.85/1.58 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_196_not__square__less__zero,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_ring @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_197_mult__neg__neg,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linord581940658strict @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_198_nat__mult__less__cancel__disj,axiom,
% 2.85/1.58 ! [K: nat,M: nat,N: nat] :
% 2.85/1.58 ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 2.85/1.58 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.85/1.58 & ( ord_less @ nat @ M @ N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_199_transfer__int__nat__relations_I2_J,axiom,
% 2.85/1.58 ! [X2: nat,Y3: nat] :
% 2.85/1.58 ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ X2 ) @ ( semiring_1_of_nat @ int @ Y3 ) )
% 2.85/1.58 = ( ord_less @ nat @ X2 @ Y3 ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_200_transfer__int__nat__numerals_I1_J,axiom,
% 2.85/1.58 ( ( zero_zero @ int )
% 2.85/1.58 = ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_201_transfer__int__nat__relations_I1_J,axiom,
% 2.85/1.58 ! [X2: nat,Y3: nat] :
% 2.85/1.58 ( ( ( semiring_1_of_nat @ int @ X2 )
% 2.85/1.58 = ( semiring_1_of_nat @ int @ Y3 ) )
% 2.85/1.58 = ( X2 = Y3 ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_202_int__if__cong,axiom,
% 2.85/1.58 ! [P: $o,X2: nat,Y3: nat] :
% 2.85/1.58 ( ( P
% 2.85/1.58 => ( ( semiring_1_of_nat @ int @ X2 )
% 2.85/1.58 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X2 @ Y3 ) ) ) )
% 2.85/1.58 & ( ~ P
% 2.85/1.58 => ( ( semiring_1_of_nat @ int @ Y3 )
% 2.85/1.58 = ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X2 @ Y3 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_203_nat__mult__eq__cancel__disj,axiom,
% 2.85/1.58 ! [K: nat,M: nat,N: nat] :
% 2.85/1.58 ( ( ( times_times @ nat @ K @ M )
% 2.85/1.58 = ( times_times @ nat @ K @ N ) )
% 2.85/1.58 = ( ( K
% 2.85/1.58 = ( zero_zero @ nat ) )
% 2.85/1.58 | ( M = N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_204_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
% 2.85/1.58 ! [X2: nat,Y3: nat] :
% 2.85/1.58 ( ( times_times @ int @ ( semiring_1_of_nat @ int @ X2 ) @ ( semiring_1_of_nat @ int @ Y3 ) )
% 2.85/1.58 = ( semiring_1_of_nat @ int @ ( times_times @ nat @ X2 @ Y3 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_205_nat__mult__less__cancel1,axiom,
% 2.85/1.58 ! [K: nat,M: nat,N: nat] :
% 2.85/1.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.85/1.58 => ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
% 2.85/1.58 = ( ord_less @ nat @ M @ N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_206_nat__mult__eq__cancel1,axiom,
% 2.85/1.58 ! [K: nat,M: nat,N: nat] :
% 2.85/1.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 2.85/1.58 => ( ( ( times_times @ nat @ K @ M )
% 2.85/1.58 = ( times_times @ nat @ K @ N ) )
% 2.85/1.58 = ( M = N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_207_linordered__field__class_Osign__simps_I45_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
% 2.85/1.58 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
% 2.85/1.58 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_208_linordered__field__class_Osign__simps_I44_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
% 2.85/1.58 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 & ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
% 2.85/1.58 | ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 & ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_209_linordered__field__no__ub,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [X5: A] :
% 2.85/1.58 ? [X1: A] : ( ord_less @ A @ X5 @ X1 ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_210_linordered__field__no__lb,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [X5: A] :
% 2.85/1.58 ? [Y5: A] : ( ord_less @ A @ Y5 @ X5 ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_211_linordered__field__class_Osign__simps_I25_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [B2: A,A2: A,C2: A] :
% 2.85/1.58 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
% 2.85/1.58 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_212_linordered__field__class_Osign__simps_I24_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ( ( times_times @ A )
% 2.85/1.58 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_213_linordered__field__class_Osign__simps_I23_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A,C2: A] :
% 2.85/1.58 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
% 2.85/1.58 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_214_scount__eq,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( coindu1365464361scount @ A )
% 2.85/1.58 = ( ^ [P2: ( stream @ A ) > $o,Omega2: stream @ A] :
% 2.85/1.58 ( if @ extended_enat @ ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P2 ) @ Omega2 ) @ ( extend1396239628finity @ extended_enat )
% 2.85/1.58 @ ( extended_enat2
% 2.85/1.58 @ ( finite_card @ nat
% 2.85/1.58 @ ( collect @ nat
% 2.85/1.58 @ ^ [I2: nat] : ( P2 @ ( sdrop @ A @ I2 @ Omega2 ) ) ) ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_215_semiring__normalization__rules_I10_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 = ( zero_zero @ A ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_216_semiring__normalization__rules_I7_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ( ( times_times @ A )
% 2.85/1.58 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_217_semiring__normalization__rules_I13_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 2.85/1.58 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 2.85/1.58 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ ( times_times @ A @ Ly @ Ry ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_218_semiring__normalization__rules_I14_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 2.85/1.58 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 2.85/1.58 = ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ ( times_times @ A @ Rx @ Ry ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_219_semiring__normalization__rules_I15_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 2.85/1.58 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 2.85/1.58 = ( times_times @ A @ Rx @ ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Ry ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_220_semiring__normalization__rules_I16_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [Lx: A,Ly: A,Rx: A] :
% 2.85/1.58 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
% 2.85/1.58 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ly ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_221_semiring__normalization__rules_I17_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [Lx: A,Ly: A,Rx: A] :
% 2.85/1.58 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
% 2.85/1.58 = ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ Rx ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_222_semiring__normalization__rules_I18_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [Lx: A,Rx: A,Ry: A] :
% 2.85/1.58 ( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
% 2.85/1.58 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ry ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_223_semiring__normalization__rules_I19_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [Lx: A,Rx: A,Ry: A] :
% 2.85/1.58 ( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
% 2.85/1.58 = ( times_times @ A @ Rx @ ( times_times @ A @ Lx @ Ry ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_224_scount__eq__card,axiom,
% 2.85/1.58 ! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A
% 2.85/1.58 @ ( linear1386806755on_alw @ A
% 2.85/1.58 @ ^ [Xs: stream @ A] :
% 2.85/1.58 ~ ( P @ Xs ) )
% 2.85/1.58 @ Omega )
% 2.85/1.58 => ( ( coindu1365464361scount @ A @ P @ Omega )
% 2.85/1.58 = ( extended_enat2
% 2.85/1.58 @ ( finite_card @ nat
% 2.85/1.58 @ ( collect @ nat
% 2.85/1.58 @ ^ [I2: nat] : ( P @ ( sdrop @ A @ I2 @ Omega ) ) ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_225_semiring__normalization__rules_I9_J,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( comm_semiring_1 @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 = ( zero_zero @ A ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_226_card__Collect__less__nat,axiom,
% 2.85/1.58 ! [N: nat] :
% 2.85/1.58 ( ( finite_card @ nat
% 2.85/1.58 @ ( collect @ nat
% 2.85/1.58 @ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) )
% 2.85/1.58 = N ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_227_sdrop__wait,axiom,
% 2.85/1.58 ! [A: $tType,Phi: ( stream @ A ) > $o,Xs2: stream @ A] :
% 2.85/1.58 ( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
% 2.85/1.58 => ( Phi @ ( sdrop @ A @ ( linear1335279038n_wait @ A @ Phi @ Xs2 ) @ Xs2 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_228_transfer__morphism__int__nat,axiom,
% 2.85/1.58 ( nat_tr1645093318rphism @ nat @ int @ ( semiring_1_of_nat @ int )
% 2.85/1.58 @ ^ [N3: nat] : $true ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_229_zero__less__ceiling,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( archim1727834104eiling @ A @ ( type @ A ) )
% 2.85/1.58 => ! [X2: A] :
% 2.85/1.58 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
% 2.85/1.58 = ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_230_ceiling__zero,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( archim1727834104eiling @ A @ ( type @ A ) )
% 2.85/1.58 => ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
% 2.85/1.58 = ( zero_zero @ int ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_231_ceiling__of__nat,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( archim1727834104eiling @ A @ ( type @ A ) )
% 2.85/1.58 => ! [N: nat] :
% 2.85/1.58 ( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
% 2.85/1.58 = ( semiring_1_of_nat @ int @ N ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_232_ceiling__less__cancel,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( archim1727834104eiling @ A @ ( type @ A ) )
% 2.85/1.58 => ! [X2: A,Y3: A] :
% 2.85/1.58 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y3 ) )
% 2.85/1.58 => ( ord_less @ A @ X2 @ Y3 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_233_zero__less__nat__eq,axiom,
% 2.85/1.58 ! [Z: int] :
% 2.85/1.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z ) )
% 2.85/1.58 = ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_234_ex__inverse__of__nat__less,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( archim1804426504_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [X2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
% 2.85/1.58 => ? [N2: nat] :
% 2.85/1.58 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
% 2.85/1.58 & ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_235_inverse__zero,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( division_ring @ A @ ( type @ A ) )
% 2.85/1.58 => ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
% 2.85/1.58 = ( zero_zero @ A ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_236_inverse__nonzero__iff__nonzero,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( division_ring @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( ( inverse_inverse @ A @ A2 )
% 2.85/1.58 = ( zero_zero @ A ) )
% 2.85/1.58 = ( A2
% 2.85/1.58 = ( zero_zero @ A ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_237_inverse__mult__distrib,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
% 2.85/1.58 = ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_238_nat__int,axiom,
% 2.85/1.58 ! [N: nat] :
% 2.85/1.58 ( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
% 2.85/1.58 = N ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_239_inverse__positive__iff__positive,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 2.85/1.58 = ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_240_inverse__negative__iff__negative,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.85/1.58 = ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_241_inverse__less__iff__less__neg,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.85/1.58 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_242_inverse__less__iff__less,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 2.85/1.58 => ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.85/1.58 = ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_243_nat__0,axiom,
% 2.85/1.58 ( ( nat2 @ ( zero_zero @ int ) )
% 2.85/1.58 = ( zero_zero @ nat ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_244_zless__nat__conj,axiom,
% 2.85/1.58 ! [W: int,Z: int] :
% 2.85/1.58 ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 2.85/1.58 = ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
% 2.85/1.58 & ( ord_less @ int @ W @ Z ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_245_zless__nat__eq__int__zless,axiom,
% 2.85/1.58 ! [M: nat,Z: int] :
% 2.85/1.58 ( ( ord_less @ nat @ M @ ( nat2 @ Z ) )
% 2.85/1.58 = ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_246_positive__imp__inverse__positive,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 => ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_247_negative__imp__inverse__negative,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_248_inverse__positive__imp__positive,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
% 2.85/1.58 => ( ( A2
% 2.85/1.58 != ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_249_inverse__negative__imp__negative,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ( A2
% 2.85/1.58 != ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_250_less__imp__inverse__less__neg,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ B2 )
% 2.85/1.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_251_inverse__less__imp__less__neg,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
% 2.85/1.58 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 2.85/1.58 => ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(fact_252_less__imp__inverse__less,axiom,
% 2.85/1.58 ! [A: $tType] :
% 2.85/1.58 ( ( linordered_field @ A @ ( type @ A ) )
% 2.85/1.58 => ! [A2: A,B2: A] :
% 2.85/1.58 ( ( ord_less @ A @ A2 @ B2 )
% 2.85/1.58 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
% 2.85/1.58 => ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
% 2.85/1.58 semiri1923998003cancel @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
% 2.85/1.58 linord893533164strict @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
% 2.85/1.58 linord20386208strict @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 2.85/1.58 semiri1193490041visors @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
% 2.85/1.58 linord581940658strict @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 2.85/1.58 linordered_semidom @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
% 2.85/1.58 ab_semigroup_mult @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
% 2.85/1.58 linordered_ring @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 2.85/1.58 linordered_idom @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 2.85/1.58 comm_semiring_1 @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
% 2.85/1.58 semigroup_mult @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 2.85/1.58 semiring_char_0 @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 2.85/1.58 semiring_1 @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
% 2.85/1.58 mult_zero @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 2.85/1.58 zero @ int @ ( type @ int ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_1,axiom,
% 2.85/1.58 semiri1923998003cancel @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_2,axiom,
% 2.85/1.58 linord893533164strict @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.85/1.58 canoni770627133id_add @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_3,axiom,
% 2.85/1.58 linord20386208strict @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_4,axiom,
% 2.85/1.58 semiri1193490041visors @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_5,axiom,
% 2.85/1.58 linordered_semidom @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_6,axiom,
% 2.85/1.58 ab_semigroup_mult @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_7,axiom,
% 2.85/1.58 comm_semiring_1 @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_8,axiom,
% 2.85/1.58 semigroup_mult @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_9,axiom,
% 2.85/1.58 semiring_char_0 @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Rings_Osemiring__1_10,axiom,
% 2.85/1.58 semiring_1 @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Rings_Omult__zero_11,axiom,
% 2.85/1.58 mult_zero @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Nat_Onat___Groups_Ozero_12,axiom,
% 2.85/1.58 zero @ nat @ ( type @ nat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_13,axiom,
% 2.85/1.58 canoni770627133id_add @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_14,axiom,
% 2.85/1.58 semiri1193490041visors @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_15,axiom,
% 2.85/1.58 ab_semigroup_mult @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_16,axiom,
% 2.85/1.58 comm_semiring_1 @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_17,axiom,
% 2.85/1.58 semigroup_mult @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_18,axiom,
% 2.85/1.58 semiring_char_0 @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_19,axiom,
% 2.85/1.58 semiring_1 @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_20,axiom,
% 2.85/1.58 mult_zero @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(tcon_Extended__Nat_Oenat___Groups_Ozero_21,axiom,
% 2.85/1.58 zero @ extended_enat @ ( type @ extended_enat ) ).
% 2.85/1.58
% 2.85/1.58 thf(help_If_3_1_T,axiom,
% 2.85/1.58 ! [P: $o] :
% 2.85/1.58 ( ( P = $true )
% 2.85/1.58 | ( P = $false ) ) ).
% 2.85/1.58
% 2.85/1.58 thf(help_If_2_1_T,axiom,
% 2.85/1.58 ! [A: $tType,X2: A,Y3: A] :
% 2.85/1.58 ( ( if @ A @ $false @ X2 @ Y3 )
% 2.85/1.58 = Y3 ) ).
% 2.85/1.58
% 2.85/1.58 thf(help_If_1_1_T,axiom,
% 2.85/1.58 ! [A: $tType,X2: A,Y3: A] :
% 2.85/1.58 ( ( if @ A @ $true @ X2 @ Y3 )
% 2.85/1.58 = X2 ) ).
% 2.85/1.58
% 2.85/1.58 thf(conj_0,conjecture,
% 2.85/1.58 ( ( ord_less @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ ( coindu76177514sfirst @ a @ p @ omega ) )
% 2.85/1.58 = ( ~ ( p @ ( sdrop @ a @ ( zero_zero @ nat ) @ omega ) ) ) ) ).
% 2.85/1.58 ------------------------
% 2.85/1.60 % (20647)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/2Mi)
% 2.85/1.60 % (20646)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/2Mi)
% 2.85/1.60 % (20647)Instruction limit reached!
% 2.85/1.60 % (20647)------------------------------
% 2.85/1.60 % (20647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.85/1.60 % (20647)Termination reason: Unknown
% 2.85/1.60 % (20647)Termination phase: shuffling
% 2.85/1.60
% 2.85/1.60 % (20647)Memory used [KB]: 1279
% 2.85/1.60 % (20644)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/4Mi)
% 2.85/1.60 % (20647)Time elapsed: 0.002 s
% 2.85/1.60 % (20647)Instructions burned: 2 (million)
% 2.85/1.60 % (20647)------------------------------
% 2.85/1.60 % (20647)------------------------------
% 2.94/1.60 % (20646)Instruction limit reached!
% 2.94/1.60 % (20646)------------------------------
% 2.94/1.60 % (20646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.60 % (20646)Termination reason: Unknown
% 2.94/1.60 % (20646)Termination phase: shuffling
% 2.94/1.60
% 2.94/1.60 % (20646)Memory used [KB]: 1279
% 2.94/1.60 % (20646)Time elapsed: 0.003 s
% 2.94/1.60 % (20646)Instructions burned: 4 (million)
% 2.94/1.60 % (20646)------------------------------
% 2.94/1.60 % (20646)------------------------------
% 2.94/1.60 % (20644)Instruction limit reached!
% 2.94/1.60 % (20644)------------------------------
% 2.94/1.60 % (20644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.60 % (20644)Termination reason: Unknown
% 2.94/1.60 % (20644)Termination phase: shuffling
% 2.94/1.60
% 2.94/1.60 % (20644)Memory used [KB]: 1407
% 2.94/1.60 % (20644)Time elapsed: 0.003 s
% 2.94/1.60 % (20644)Instructions burned: 6 (million)
% 2.94/1.60 % (20644)------------------------------
% 2.94/1.60 % (20644)------------------------------
% 2.94/1.60 % (20643)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/183Mi)
% 2.94/1.60 % (20645)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/27Mi)
% 2.94/1.60 % (20649)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/18Mi)
% 2.94/1.60 % (20648)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/275Mi)
% 2.94/1.61 % (20650)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/3Mi)
% 2.94/1.61 % (20651)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/37Mi)
% 2.94/1.61 % (20650)Instruction limit reached!
% 2.94/1.61 % (20650)------------------------------
% 2.94/1.61 % (20650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (20650)Termination reason: Unknown
% 2.94/1.61 % (20650)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (20650)Memory used [KB]: 1279
% 2.94/1.61 % (20650)Time elapsed: 0.002 s
% 2.94/1.61 % (20650)Instructions burned: 3 (million)
% 2.94/1.61 % (20650)------------------------------
% 2.94/1.61 % (20650)------------------------------
% 2.94/1.61 % (20652)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/15Mi)
% 2.94/1.61 % (20652)Instruction limit reached!
% 2.94/1.61 % (20652)------------------------------
% 2.94/1.61 % (20652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (20652)Termination reason: Unknown
% 2.94/1.61 % (20652)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (20652)Memory used [KB]: 1663
% 2.94/1.61 % (20652)Time elapsed: 0.008 s
% 2.94/1.61 % (20652)Instructions burned: 17 (million)
% 2.94/1.61 % (20652)------------------------------
% 2.94/1.61 % (20652)------------------------------
% 2.94/1.62 % (20649)Instruction limit reached!
% 2.94/1.62 % (20649)------------------------------
% 2.94/1.62 % (20649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (20649)Termination reason: Unknown
% 2.94/1.62 % (20649)Termination phase: Property scanning
% 2.94/1.62
% 2.94/1.62 % (20649)Memory used [KB]: 1663
% 2.94/1.62 % (20649)Time elapsed: 0.014 s
% 2.94/1.62 % (20649)Instructions burned: 18 (million)
% 2.94/1.62 % (20649)------------------------------
% 2.94/1.62 % (20649)------------------------------
% 2.94/1.62 % (20653)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/3Mi)
% 2.94/1.62 % (20653)Instruction limit reached!
% 2.94/1.62 % (20653)------------------------------
% 2.94/1.62 % (20653)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (20653)Termination reason: Unknown
% 2.94/1.62 % (20653)Termination phase: shuffling
% 2.94/1.62
% 2.94/1.62 % (20653)Memory used [KB]: 1279
% 2.94/1.62 % (20653)Time elapsed: 0.002 s
% 2.94/1.62 % (20653)Instructions burned: 3 (million)
% 2.94/1.62 % (20653)------------------------------
% 2.94/1.62 % (20653)------------------------------
% 2.94/1.62 % (20651)Instruction limit reached!
% 2.94/1.62 % (20651)------------------------------
% 2.94/1.62 % (20651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (20651)Termination reason: Unknown
% 2.94/1.62 % (20651)Termination phase: Property scanning
% 2.94/1.62
% 2.94/1.62 % (20651)Memory used [KB]: 2046
% 2.94/1.62 % (20651)Time elapsed: 0.012 s
% 2.94/1.62 % (20651)Instructions burned: 40 (million)
% 2.94/1.62 % (20651)------------------------------
% 2.94/1.62 % (20651)------------------------------
% 2.94/1.62 % (20645)Instruction limit reached!
% 2.94/1.62 % (20645)------------------------------
% 2.94/1.62 % (20645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (20645)Termination reason: Unknown
% 2.94/1.62 % (20645)Termination phase: Property scanning
% 2.94/1.62
% 2.94/1.62 % (20645)Memory used [KB]: 1663
% 2.94/1.62 % (20645)Time elapsed: 0.018 s
% 2.94/1.62 % (20645)Instructions burned: 27 (million)
% 2.94/1.62 % (20645)------------------------------
% 2.94/1.62 % (20645)------------------------------
% 2.94/1.62 % (20654)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/1041Mi)
% 2.94/1.63 % (20656)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/16Mi)
% 2.94/1.63 % (20657)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/3Mi)
% 2.94/1.63 % (20657)Instruction limit reached!
% 2.94/1.63 % (20657)------------------------------
% 2.94/1.63 % (20657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (20657)Termination reason: Unknown
% 2.94/1.63 % (20657)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (20657)Memory used [KB]: 1407
% 2.94/1.63 % (20657)Time elapsed: 0.003 s
% 2.94/1.63 % (20657)Instructions burned: 5 (million)
% 2.94/1.63 % (20657)------------------------------
% 2.94/1.63 % (20657)------------------------------
% 2.94/1.63 % (20656)Instruction limit reached!
% 2.94/1.63 % (20656)------------------------------
% 2.94/1.63 % (20656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (20656)Termination reason: Unknown
% 2.94/1.63 % (20656)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (20656)Memory used [KB]: 1663
% 2.94/1.63 % (20656)Time elapsed: 0.007 s
% 2.94/1.63 % (20656)Instructions burned: 18 (million)
% 2.94/1.63 % (20656)------------------------------
% 2.94/1.63 % (20656)------------------------------
% 2.94/1.63 % (20655)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/7Mi)
% 2.94/1.64 % (20658)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/3Mi)
% 2.94/1.64 % (20655)Instruction limit reached!
% 2.94/1.64 % (20655)------------------------------
% 2.94/1.64 % (20655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (20655)Termination reason: Unknown
% 2.94/1.64 % (20655)Termination phase: shuffling
% 2.94/1.64
% 2.94/1.64 % (20655)Memory used [KB]: 1407
% 2.94/1.64 % (20655)Time elapsed: 0.006 s
% 2.94/1.64 % (20655)Instructions burned: 7 (million)
% 2.94/1.64 % (20655)------------------------------
% 2.94/1.64 % (20655)------------------------------
% 2.94/1.64 % (20659)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/7Mi)
% 2.94/1.64 % (20658)Instruction limit reached!
% 2.94/1.64 % (20658)------------------------------
% 2.94/1.64 % (20658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (20658)Termination reason: Unknown
% 2.94/1.64 % (20658)Termination phase: shuffling
% 2.94/1.64
% 2.94/1.64 % (20658)Memory used [KB]: 1279
% 2.94/1.64 % (20658)Time elapsed: 0.004 s
% 2.94/1.64 % (20658)Instructions burned: 3 (million)
% 2.94/1.64 % (20658)------------------------------
% 2.94/1.64 % (20658)------------------------------
% 2.94/1.64 % (20659)Instruction limit reached!
% 2.94/1.64 % (20659)------------------------------
% 2.94/1.64 % (20659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (20659)Termination reason: Unknown
% 2.94/1.64 % (20659)Termination phase: shuffling
% 2.94/1.64
% 2.94/1.64 % (20659)Memory used [KB]: 1535
% 2.94/1.64 % (20659)Time elapsed: 0.004 s
% 2.94/1.64 % (20659)Instructions burned: 9 (million)
% 2.94/1.64 % (20659)------------------------------
% 2.94/1.64 % (20659)------------------------------
% 2.94/1.64 % (20660)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_20370 for (2999ds/3Mi)
% 2.94/1.64 % (20648)First to succeed.
% 2.94/1.64 % (20660)Instruction limit reached!
% 2.94/1.64 % (20660)------------------------------
% 2.94/1.64 % (20660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (20660)Termination reason: Unknown
% 2.94/1.64 % (20660)Termination phase: shuffling
% 2.94/1.64
% 2.94/1.64 % (20660)Memory used [KB]: 1407
% 2.94/1.64 % (20660)Time elapsed: 0.002 s
% 2.94/1.64 % (20660)Instructions burned: 3 (million)
% 2.94/1.64 % (20660)------------------------------
% 2.94/1.64 % (20660)------------------------------
% 2.94/1.65 % (20648)Refutation found. Thanks to Tanya!
% 2.94/1.65 % SZS status Theorem for DTF2THF_20370
% 2.94/1.65 % SZS output start Proof for DTF2THF_20370
% 2.94/1.65 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.94/1.65 thf(type_def_7, type, int: $tType).
% 2.94/1.65 thf(type_def_8, type, stream: ($tType) > $tType).
% 2.94/1.65 thf(type_def_9, type, extended_enat: $tType).
% 2.94/1.65 thf(type_def_10, type, nat: $tType).
% 2.94/1.65 thf(type_def_11, type, set: ($tType) > $tType).
% 2.94/1.65 thf(type_def_12, type, a: $tType).
% 2.94/1.65 thf(func_def_0, type, extended_enat: $tType).
% 2.94/1.65 thf(func_def_1, type, stream: ($tType) > $tType).
% 2.94/1.65 thf(func_def_2, type, set: ($tType) > $tType).
% 2.94/1.65 thf(func_def_3, type, nat: $tType).
% 2.94/1.65 thf(func_def_4, type, int: $tType).
% 2.94/1.65 thf(func_def_5, type, itself: ($tType) > $tType).
% 2.94/1.65 thf(func_def_6, type, a: $tType).
% 2.94/1.65 thf(func_def_7, type, zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_8, type, field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_9, type, mult_zero: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_10, type, semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_11, type, semiring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_12, type, division_ring: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_13, type, semigroup_mult: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_14, type, comm_semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_15, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_16, type, linordered_ring: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_17, type, linordered_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_18, type, ab_semigroup_mult: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_19, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_20, type, linord581940658strict: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_21, type, archim1727834104eiling: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_22, type, semiri1193490041visors: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_23, type, linord20386208strict: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_24, type, archim1804426504_field: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_25, type, canoni770627133id_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_26, type, linord893533164strict: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_27, type, semiri1923998003cancel: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.94/1.65 thf(func_def_28, type, archimedean_ceiling: !>[X0: $tType]:(X0 > int)).
% 2.94/1.65 thf(func_def_29, type, coindu1365464361scount: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > extended_enat)).
% 2.94/1.65 thf(func_def_30, type, coindu76177514sfirst: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > extended_enat)).
% 2.94/1.65 thf(func_def_31, type, extended_enat2: nat > extended_enat).
% 2.94/1.65 thf(func_def_32, type, extended_case_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 2.94/1.65 thf(func_def_33, type, extended_rec_enat: !>[X0: $tType]:((nat > X0) > X0 > extended_enat > X0)).
% 2.94/1.65 thf(func_def_34, type, extended_size_enat: extended_enat > nat).
% 2.94/1.65 thf(func_def_35, type, extend1396239628finity: !>[X0: $tType]:(X0)).
% 2.94/1.65 thf(func_def_36, type, extended_the_enat: extended_enat > nat).
% 2.94/1.65 thf(func_def_37, type, inverse_inverse: !>[X0: $tType]:(X0 > X0)).
% 2.94/1.65 thf(func_def_38, type, finite_card: !>[X0: $tType]:(set @ X0 > nat)).
% 2.94/1.65 thf(func_def_39, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.94/1.65 thf(func_def_40, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.94/1.65 thf(func_def_41, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 2.94/1.65 thf(func_def_42, type, nat2: int > nat).
% 2.94/1.65 thf(func_def_43, type, linear1341015445_UNTIL: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0 > $o)).
% 2.94/1.65 thf(func_def_44, type, linear1386806755on_alw: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > $o)).
% 2.94/1.65 thf(func_def_45, type, linear505997466_on_ev: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > $o)).
% 2.94/1.65 thf(func_def_46, type, linear1335279038n_wait: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0 > nat)).
% 2.94/1.65 thf(func_def_47, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 2.94/1.65 thf(func_def_48, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.94/1.65 thf(func_def_49, type, nat_tr1645093318rphism: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > $o) > $o)).
% 2.94/1.65 thf(func_def_50, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.94/1.65 thf(func_def_51, type, type: !>[X0: $tType]:(itself @ X0)).
% 2.94/1.65 thf(func_def_52, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.94/1.65 thf(func_def_53, type, sdrop: !>[X0: $tType]:(nat > stream @ X0 > stream @ X0)).
% 2.94/1.65 thf(func_def_54, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.94/1.65 thf(func_def_55, type, p: stream @ a > $o).
% 2.94/1.65 thf(func_def_56, type, omega: stream @ a).
% 2.94/1.65 thf(func_def_71, type, sP0: extended_enat > extended_enat > extended_enat > $o).
% 2.94/1.65 thf(func_def_72, type, sP1: extended_enat > extended_enat > extended_enat > $o).
% 2.94/1.65 thf(func_def_73, type, sP2: extended_enat > extended_enat > extended_enat > $o).
% 2.94/1.65 thf(func_def_74, type, sP3: extended_enat > extended_enat > extended_enat > $o).
% 2.94/1.65 thf(func_def_75, type, sP4: extended_enat > extended_enat > extended_enat > $o).
% 2.94/1.65 thf(func_def_76, type, sP5: extended_enat > extended_enat > extended_enat > $o).
% 2.94/1.65 thf(func_def_77, type, sP6: extended_enat > extended_enat > extended_enat > $o).
% 2.94/1.65 thf(func_def_78, type, sP7: extended_enat > extended_enat > $o).
% 2.94/1.65 thf(func_def_79, type, sK8: extended_enat > nat).
% 2.94/1.65 thf(func_def_80, type, sK9: extended_enat > nat).
% 2.94/1.65 thf(func_def_81, type, sK10: extended_enat > nat).
% 2.94/1.65 thf(func_def_82, type, sK11: extended_enat > nat).
% 2.94/1.65 thf(func_def_83, type, sK12: extended_enat > nat).
% 2.94/1.65 thf(func_def_84, type, sK13: extended_enat > nat).
% 2.94/1.65 thf(func_def_85, type, sK14: extended_enat > nat).
% 2.94/1.65 thf(func_def_86, type, sK15: extended_enat > nat).
% 2.94/1.65 thf(func_def_87, type, sK16: extended_enat > nat).
% 2.94/1.65 thf(func_def_88, type, sK17: extended_enat > nat).
% 2.94/1.65 thf(func_def_89, type, sK18: extended_enat > nat).
% 2.94/1.65 thf(func_def_90, type, sK19: extended_enat > nat).
% 2.94/1.65 thf(func_def_91, type, sK20: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_92, type, sK21: extended_enat > nat).
% 2.94/1.65 thf(func_def_93, type, sK22: extended_enat > nat).
% 2.94/1.65 thf(func_def_94, type, sK23: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_95, type, sK24: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_96, type, sK25: !>[X0: $tType]:((stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_97, type, sK26: extended_enat > nat).
% 2.94/1.65 thf(func_def_98, type, sK27: (nat > $o) > nat).
% 2.94/1.65 thf(func_def_99, type, sK28: !>[X0: $tType]:((X0 > extended_enat) > X0)).
% 2.94/1.65 thf(func_def_100, type, sK29: !>[X0: $tType]:(nat > (X0 > extended_enat) > X0)).
% 2.94/1.65 thf(func_def_101, type, sK30: (extended_enat > $o) > extended_enat).
% 2.94/1.65 thf(func_def_102, type, sK31: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_103, type, sK32: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_104, type, sK33: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.94/1.65 thf(func_def_105, type, sK34: (nat > $o) > nat).
% 2.94/1.65 thf(func_def_106, type, sK35: nat > extended_enat > nat).
% 2.94/1.65 thf(func_def_107, type, sK36: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_108, type, sK37: extended_enat > nat).
% 2.94/1.65 thf(func_def_109, type, sK38: (nat > $o) > nat).
% 2.94/1.65 thf(func_def_110, type, sK39: !>[X0: $tType]:((X0 > nat) > (X0 > $o) > X0)).
% 2.94/1.65 thf(func_def_111, type, sK40: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.94/1.65 thf(func_def_112, type, sK41: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.94/1.65 thf(func_def_113, type, sK42: !>[X0: $tType]:((X0 > $o) > (X0 > nat) > X0)).
% 2.94/1.65 thf(func_def_114, type, sK43: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.94/1.65 thf(func_def_115, type, sK44: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_116, type, sK45: !>[X0: $tType]:((stream @ X0 > $o) > (stream @ X0 > $o) > stream @ X0)).
% 2.94/1.65 thf(func_def_117, type, sK46: extended_enat > nat).
% 2.94/1.65 thf(func_def_118, type, sK47: extended_enat > nat).
% 2.94/1.65 thf(func_def_119, type, sK48: extended_enat > nat).
% 2.94/1.65 thf(func_def_120, type, sK49: extended_enat > nat).
% 2.94/1.65 thf(func_def_121, type, sK50: (extended_enat > $o) > nat).
% 2.94/1.65 thf(func_def_122, type, sK51: extended_enat > nat).
% 2.94/1.65 thf(func_def_124, type, ph53: !>[X0: $tType]:(X0)).
% 2.94/1.65 thf(f1289,plain,(
% 2.94/1.65 $false),
% 2.94/1.65 inference(subsumption_resolution,[status(thm)],[f1288,f1277])).
% 2.94/1.65 thf(f1277,plain,(
% 2.94/1.65 ((p @ omega) != $true)),
% 2.94/1.65 inference(forward_demodulation,[status(thm)],[f1276,f1157])).
% 2.94/1.65 thf(f1157,plain,(
% 2.94/1.65 ( ! [X0 : $tType,X1 : stream @ X0] : (((sdrop @ X0 @ zero_zero @ nat @ X1) = X1)) )),
% 2.94/1.65 inference(cnf_transformation,[status(thm)],[f815])).
% 2.94/1.65 thf(f815,plain,(
% 2.94/1.65 ! [X0 : $tType,X1 : stream @ X0] : ((sdrop @ X0 @ zero_zero @ nat @ X1) = X1)),
% 2.94/1.65 inference(rectify,[status(thm)],[f5])).
% 2.94/1.65 thf(f5,axiom,(
% 2.94/1.65 ! [X0 : $tType,X10 : stream @ X0] : ((sdrop @ X0 @ zero_zero @ nat @ X10) = X10)),
% 2.94/1.65 file('/export/starexec/sandbox2/tmp/tmp.plLMjpaOMh/DTF2THF_20370.p',fact_4_sdrop_Osimps_I1_J)).
% 2.94/1.65 thf(f1276,plain,(
% 2.94/1.65 ((p @ (sdrop @ a @ zero_zero @ nat @ omega)) != $true)),
% 2.94/1.65 inference(subsumption_resolution,[status(thm)],[f1179,f1132])).
% 2.94/1.65 thf(f1132,plain,(
% 2.94/1.65 ( ! [X0 : $tType,X2 : stream @ X0 > $o,X3 : stream @ X0,X1 : nat] : (($true != (ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu76177514sfirst @ X0 @ X2 @ X3))) | ((X2 @ (sdrop @ X0 @ X1 @ X3)) != $true)) )),
% 2.94/1.65 inference(cnf_transformation,[status(thm)],[f881])).
% 2.94/1.65 thf(f881,plain,(
% 2.94/1.65 ! [X0 : $tType,X1 : nat,X2 : stream @ X0 > $o,X3 : stream @ X0] : (((X2 @ (sdrop @ X0 @ X1 @ X3)) != $true) | ($true != (ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu76177514sfirst @ X0 @ X2 @ X3))))),
% 2.94/1.65 inference(ennf_transformation,[status(thm)],[f800])).
% 2.94/1.65 thf(f800,plain,(
% 2.94/1.65 ! [X0 : $tType,X1 : nat,X2 : stream @ X0 > $o,X3 : stream @ X0] : (($true = (ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu76177514sfirst @ X0 @ X2 @ X3))) => ((X2 @ (sdrop @ X0 @ X1 @ X3)) != $true))),
% 2.94/1.65 inference(flattening,[status(thm)],[f336])).
% 2.94/1.65 thf(f336,plain,(
% 2.94/1.65 ! [X0 : $tType,X1 : nat,X2 : stream @ X0 > $o,X3 : stream @ X0] : (($true = (ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu76177514sfirst @ X0 @ X2 @ X3))) => ~((X2 @ (sdrop @ X0 @ X1 @ X3)) = $true))),
% 2.94/1.65 inference(fool_elimination,[status(thm)],[f335])).
% 2.94/1.65 thf(f335,plain,(
% 2.94/1.65 ! [X0 : $tType,X1 : nat,X2 : stream @ X0 > $o,X3 : stream @ X0] : ((ord_less @ extended_enat @ (extended_enat2 @ X1) @ (coindu76177514sfirst @ X0 @ X2 @ X3)) => ~(X2 @ (sdrop @ X0 @ X1 @ X3)))),
% 2.94/1.65 inference(rectify,[status(thm)],[f2])).
% 2.94/1.65 thf(f2,axiom,(
% 2.94/1.65 ! [X0 : $tType,X5 : nat,X6 : stream @ X0 > $o,X7 : stream @ X0] : ((ord_less @ extended_enat @ (extended_enat2 @ X5) @ (coindu76177514sfirst @ X0 @ X6 @ X7)) => ~(X6 @ (sdrop @ X0 @ X5 @ X7)))),
% 2.94/1.65 file('/export/starexec/sandbox2/tmp/tmp.plLMjpaOMh/DTF2THF_20370.p',fact_1_less__sfirstD)).
% 2.94/1.65 thf(f1179,plain,(
% 2.94/1.65 ((ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)) = $true) | ((p @ (sdrop @ a @ zero_zero @ nat @ omega)) != $true)),
% 2.94/1.65 inference(cnf_transformation,[status(thm)],[f1007])).
% 2.94/1.65 thf(f1007,plain,(
% 2.94/1.65 (((ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)) != $true) | ((p @ (sdrop @ a @ zero_zero @ nat @ omega)) = $true)) & (((ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)) = $true) | ((p @ (sdrop @ a @ zero_zero @ nat @ omega)) != $true))),
% 2.94/1.65 inference(nnf_transformation,[status(thm)],[f856])).
% 2.94/1.65 thf(f856,plain,(
% 2.94/1.65 ((p @ (sdrop @ a @ zero_zero @ nat @ omega)) != $true) <~> ((ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)) = $true)),
% 2.94/1.65 inference(ennf_transformation,[status(thm)],[f797])).
% 2.94/1.65 thf(f797,plain,(
% 2.94/1.65 ~(((p @ (sdrop @ a @ zero_zero @ nat @ omega)) != $true) <=> ((ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)) = $true))),
% 2.94/1.65 inference(flattening,[status(thm)],[f464])).
% 2.94/1.65 thf(f464,plain,(
% 2.94/1.65 ~(~((p @ (sdrop @ a @ zero_zero @ nat @ omega)) = $true) <=> ((ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)) = $true))),
% 2.94/1.65 inference(fool_elimination,[status(thm)],[f463])).
% 2.94/1.65 thf(f463,plain,(
% 2.94/1.65 ~(~(p @ (sdrop @ a @ zero_zero @ nat @ omega)) = (ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)))),
% 2.94/1.65 inference(rectify,[status(thm)],[f295])).
% 2.94/1.65 thf(f295,negated_conjecture,(
% 2.94/1.65 ~(~(p @ (sdrop @ a @ zero_zero @ nat @ omega)) = (ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)))),
% 2.94/1.65 inference(negated_conjecture,[status(cth)],[f294])).
% 2.94/1.65 thf(f294,conjecture,(
% 2.94/1.65 (~(p @ (sdrop @ a @ zero_zero @ nat @ omega)) = (ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)))),
% 2.94/1.65 file('/export/starexec/sandbox2/tmp/tmp.plLMjpaOMh/DTF2THF_20370.p',conj_0)).
% 2.94/1.65 thf(f1288,plain,(
% 2.94/1.65 ((p @ omega) = $true)),
% 2.94/1.65 inference(trivial_inequality_removal,[status(thm)],[f1285])).
% 2.94/1.65 thf(f1285,plain,(
% 2.94/1.65 (zero_zero @ extended_enat != zero_zero @ extended_enat) | ((p @ omega) = $true)),
% 2.94/1.65 inference(superposition,[status(thm)],[f1152,f1281])).
% 2.94/1.65 thf(f1281,plain,(
% 2.94/1.65 (zero_zero @ extended_enat = (coindu76177514sfirst @ a @ p @ omega))),
% 2.94/1.65 inference(trivial_inequality_removal,[status(thm)],[f1280])).
% 2.94/1.65 thf(f1280,plain,(
% 2.94/1.65 (zero_zero @ extended_enat = (coindu76177514sfirst @ a @ p @ omega)) | ($true != $true)),
% 2.94/1.65 inference(superposition,[status(thm)],[f1275,f1236])).
% 2.94/1.65 thf(f1236,plain,(
% 2.94/1.65 ( ! [X0 : extended_enat] : (((ord_less @ extended_enat @ zero_zero @ extended_enat @ X0) = $true) | (zero_zero @ extended_enat = X0)) )),
% 2.94/1.65 inference(cnf_transformation,[status(thm)],[f1041])).
% 2.94/1.65 thf(f1041,plain,(
% 2.94/1.65 ! [X0 : extended_enat] : ((((ord_less @ extended_enat @ zero_zero @ extended_enat @ X0) = $true) | (zero_zero @ extended_enat = X0)) & ((zero_zero @ extended_enat != X0) | ((ord_less @ extended_enat @ zero_zero @ extended_enat @ X0) != $true)))),
% 2.94/1.65 inference(nnf_transformation,[status(thm)],[f412])).
% 2.94/1.65 thf(f412,plain,(
% 2.94/1.65 ! [X0 : extended_enat] : (((ord_less @ extended_enat @ zero_zero @ extended_enat @ X0) = $true) <=> (zero_zero @ extended_enat != X0))),
% 2.94/1.65 inference(fool_elimination,[status(thm)],[f411])).
% 2.94/1.65 thf(f411,plain,(
% 2.94/1.65 ! [X0 : extended_enat] : ((zero_zero @ extended_enat != X0) = (ord_less @ extended_enat @ zero_zero @ extended_enat @ X0))),
% 2.94/1.65 inference(rectify,[status(thm)],[f14])).
% 2.94/1.65 thf(f14,axiom,(
% 2.94/1.65 ! [X5 : extended_enat] : ((zero_zero @ extended_enat != X5) = (ord_less @ extended_enat @ zero_zero @ extended_enat @ X5))),
% 2.94/1.65 file('/export/starexec/sandbox2/tmp/tmp.plLMjpaOMh/DTF2THF_20370.p',fact_13_i0__less)).
% 2.94/1.65 thf(f1275,plain,(
% 2.94/1.65 ((ord_less @ extended_enat @ zero_zero @ extended_enat @ (coindu76177514sfirst @ a @ p @ omega)) != $true)),
% 2.94/1.65 inference(forward_demodulation,[status(thm)],[f1274,f1149])).
% 2.94/1.65 thf(f1149,plain,(
% 2.94/1.65 (zero_zero @ extended_enat = (extended_enat2 @ zero_zero @ nat))),
% 2.94/1.65 inference(cnf_transformation,[status(thm)],[f18])).
% 2.94/1.65 thf(f18,axiom,(
% 2.94/1.65 (zero_zero @ extended_enat = (extended_enat2 @ zero_zero @ nat))),
% 2.94/1.65 file('/export/starexec/sandbox2/tmp/tmp.plLMjpaOMh/DTF2THF_20370.p',fact_17_zero__enat__def)).
% 2.94/1.65 thf(f1274,plain,(
% 2.94/1.65 ((ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)) != $true)),
% 2.94/1.65 inference(subsumption_resolution,[status(thm)],[f1180,f1132])).
% 2.94/1.65 thf(f1180,plain,(
% 2.94/1.65 ((p @ (sdrop @ a @ zero_zero @ nat @ omega)) = $true) | ((ord_less @ extended_enat @ (extended_enat2 @ zero_zero @ nat) @ (coindu76177514sfirst @ a @ p @ omega)) != $true)),
% 2.94/1.65 inference(cnf_transformation,[status(thm)],[f1007])).
% 2.94/1.65 thf(f1152,plain,(
% 2.94/1.65 ( ! [X0 : $tType,X2 : stream @ X0,X1 : stream @ X0 > $o] : ((zero_zero @ extended_enat != (coindu76177514sfirst @ X0 @ X1 @ X2)) | ($true = (X1 @ X2))) )),
% 2.94/1.65 inference(cnf_transformation,[status(thm)],[f992])).
% 2.94/1.65 thf(f992,plain,(
% 2.94/1.65 ! [X0 : $tType,X1 : stream @ X0 > $o,X2 : stream @ X0] : (((zero_zero @ extended_enat = (coindu76177514sfirst @ X0 @ X1 @ X2)) | ($true != (X1 @ X2))) & (($true = (X1 @ X2)) | (zero_zero @ extended_enat != (coindu76177514sfirst @ X0 @ X1 @ X2))))),
% 2.94/1.65 inference(nnf_transformation,[status(thm)],[f768])).
% 2.94/1.65 thf(f768,plain,(
% 2.94/1.65 ! [X0 : $tType,X1 : stream @ X0 > $o,X2 : stream @ X0] : ((zero_zero @ extended_enat = (coindu76177514sfirst @ X0 @ X1 @ X2)) <=> ($true = (X1 @ X2)))),
% 2.94/1.65 inference(fool_elimination,[status(thm)],[f767])).
% 2.94/1.65 thf(f767,plain,(
% 2.94/1.65 ! [X0 : $tType,X1 : stream @ X0 > $o,X2 : stream @ X0] : ((X1 @ X2) = (zero_zero @ extended_enat = (coindu76177514sfirst @ X0 @ X1 @ X2)))),
% 2.94/1.65 inference(rectify,[status(thm)],[f22])).
% 2.94/1.65 thf(f22,axiom,(
% 2.94/1.65 ! [X0 : $tType,X6 : stream @ X0 > $o,X7 : stream @ X0] : ((X6 @ X7) = ((coindu76177514sfirst @ X0 @ X6 @ X7) = zero_zero @ extended_enat))),
% 2.94/1.65 file('/export/starexec/sandbox2/tmp/tmp.plLMjpaOMh/DTF2THF_20370.p',fact_21_sfirst__eq__0)).
% 2.94/1.65 % SZS output end Proof for DTF2THF_20370
% 2.94/1.65 % (20648)------------------------------
% 2.94/1.65 % (20648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.65 % (20648)Termination reason: Refutation
% 2.94/1.65
% 2.94/1.65 % (20648)Memory used [KB]: 6652
% 2.94/1.65 % (20648)Time elapsed: 0.041 s
% 2.94/1.65 % (20648)Instructions burned: 64 (million)
% 2.94/1.65 % (20648)------------------------------
% 2.94/1.65 % (20648)------------------------------
% 2.94/1.65 % (20642)Success in time 0.06 s
%------------------------------------------------------------------------------