TPTP Problem File: SLH0695^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : ResiduatedTransitionSystem/0001_LambdaCalculus/prob_05438_235047__14672892_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1125 ( 496 unt; 139 typ; 0 def)
% Number of atoms : 4437 (2207 equ; 0 cnn)
% Maximal formula atoms : 170 ( 4 avg)
% Number of connectives : 15572 ( 992 ~; 42 |; 418 &;11674 @)
% ( 0 <=>;2446 =>; 0 <=; 0 <~>)
% Maximal formula depth : 76 ( 7 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 495 ( 495 >; 0 *; 0 +; 0 <<)
% Number of symbols : 134 ( 131 usr; 7 con; 0-5 aty)
% Number of variables : 3926 ( 44 ^;3497 !; 385 ?;3926 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:49:00.497
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__LambdaCalculus__Olambda____calculus__Olambda_Mt__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
produc2872725455364511436lambda: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
produc674367846827345344lambda: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__LambdaCalculus__Olambda____calculus__Olambda_Mt__LambdaCalculus__Olambda____calculus__Olambda_J,type,
produc4921446539628979879lambda: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__LambdaCalculus__Olambda____calculus__Olambda_J,type,
produc3839482995950079247lambda: $tType ).
thf(ty_n_t__Set__Oset_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
set_lambda_lambda: $tType ).
thf(ty_n_t__LambdaCalculus__Olambda____calculus__Olambda,type,
lambda_lambda: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (131)
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OArr,type,
lambda_Arr: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OArr__rel,type,
lambda_Arr_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OFV,type,
lambda_FV: lambda_lambda > set_nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OIde,type,
lambda_Ide: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OIde__rel,type,
lambda_Ide_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OJoin,type,
lambda_Join: lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_ONF,type,
lambda_NF: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_ORaise,type,
lambda_Raise: nat > nat > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_ORaise__rel,type,
lambda_Raise_rel: produc674367846827345344lambda > produc674367846827345344lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OSrc,type,
lambda_Src: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OSrc__rel,type,
lambda_Src_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OSubst,type,
lambda_Subst: nat > lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OSubst__rel,type,
lambda_Subst_rel: produc2872725455364511436lambda > produc2872725455364511436lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OTrg,type,
lambda_Trg: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OTrg__rel,type,
lambda_Trg_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Obottom__up__redex,type,
lambda5654762366400029632_redex: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Obottom__up__redex__rel,type,
lambda8685541990070617401ex_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ocnv,type,
lambda_cnv: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ocontains__head__reduction,type,
lambda8135948770199036598uction: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ocontains__head__reduction__rel,type,
lambda6381018229452634563on_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oelementary__reduction,type,
lambda2670285437103676873uction: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oelementary__reduction__rel,type,
lambda2747218380695844464on_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ohead__redex,type,
lambda_head_redex: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ohead__redex__rel,type,
lambda2982300901483219379ex_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ohead__strategy,type,
lambda_head_strategy: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ohead__strategy__rel,type,
lambda8648082035647910664gy_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ohgt,type,
lambda_hgt: lambda_lambda > nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ohgt__rel,type,
lambda_hgt_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ois__head__reduction,type,
lambda5343566265203084875uction: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ois__head__reduction__rel,type,
lambda5859589972953655406on_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ois__internal__reduction,type,
lambda2591989677915507790uction: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ois__internal__reduction__rel,type,
lambda867594046455500331on_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ois__leftmost__reduction,type,
lambda5425420467215166849uction: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OApp,type,
lambda_App: lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OBeta,type,
lambda_Beta: lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OLam,type,
lambda_Lam: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_ONil,type,
lambda_Nil: lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OVar,type,
lambda_Var: nat > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_Ois__Lam,type,
lambda_is_Lam: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_Osize__lambda,type,
lambda_size_lambda: lambda_lambda > nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_Oun__Lam,type,
lambda_un_Lam: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oleftmost__strategy,type,
lambda4011002726600638587rategy: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oleftmost__strategy__rel,type,
lambda2991515046683734590gy_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Omtp,type,
lambda_mtp: nat > lambda_lambda > nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Omtp__rel,type,
lambda_mtp_rel: produc3839482995950079247lambda > produc3839482995950079247lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Onormalizable,type,
lambda_normalizable: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Onormalizing__strategy,type,
lambda5785200480123228735rategy: ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oparallel__strategy,type,
lambda8522482075158074392rategy: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oparallel__strategy__rel,type,
lambda9101494976250544865gy_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Ored,type,
lambda_red: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oreduce,type,
lambda_reduce: ( lambda_lambda > lambda_lambda ) > lambda_lambda > nat > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oreduction__strategy,type,
lambda8358424291402068104rategy: ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oresid,type,
lambda_resid: lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Osize,type,
lambda_size: lambda_lambda > nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Osize__rel,type,
lambda_size_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Osubs,type,
lambda_subs: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Osubterm,type,
lambda_subterm: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Oreduction__paths_Obottom__up__development__rel,type,
reduct1618657854384587308nt_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
inf_in1766959559283430712lambda: set_lambda_lambda > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
size_s1768714712973771222lambda: lambda_lambda > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
bot_bo1271416176687173302lambda: set_lambda_lambda ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Product__Type_OPair_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
produc7853741606629031319lambda: lambda_lambda > lambda_lambda > produc4921446539628979879lambda ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
produc8777017121224680385lambda: nat > lambda_lambda > produc3839482995950079247lambda ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__LambdaCalculus__Olambda____calculus__Olambda_Mt__LambdaCalculus__Olambda____calculus__Olambda_J,type,
produc1836381109249119870lambda: nat > produc4921446539628979879lambda > produc2872725455364511436lambda ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__LambdaCalculus__Olambda____calculus__Olambda_J,type,
produc5347854423718732792lambda: nat > produc3839482995950079247lambda > produc674367846827345344lambda ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
cohere1690309856004283859lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > set_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts_001t__Nat__Onat,type,
cohere320596682411188903ts_nat: ( nat > nat > nat ) > set_nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
conflu4927472611300549594lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts_001t__Nat__Onat,type,
confluent_rts_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
conflu5019861018119259255lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oconfluent__rts__axioms_001t__Nat__Onat,type,
conflu6972329929330909443ms_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
normal3156941126761417472lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > set_lambda_lambda > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_001t__Nat__Onat,type,
normal_sub_Cong_nat: ( nat > nat > nat ) > set_nat > nat > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
normal5162919049924913650lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > set_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts__axioms_001t__Nat__Onat,type,
normal3241730178054940488ms_nat: ( nat > nat > nat ) > set_nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opartial__magma_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
partia3924075026459072775lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
arr_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__Nat__Onat,type,
arr_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
con_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__Nat__Onat,type,
con_nat: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
ide_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oide_001t__Nat__Onat,type,
ide_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
trg_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__Nat__Onat,type,
trg_nat: ( nat > nat > nat ) > nat > nat ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
coinit1392077014771647542lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoinitial_001t__Nat__Onat,type,
coinitial_nat: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
compos7127019122090555255lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__Nat__Onat,type,
composable_nat: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
compos9183582412961930448lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001t__Nat__Onat,type,
composite_of_nat: ( nat > nat > nat ) > nat > nat > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocoterminal_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
coterm4142657152830665724lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
join_o1386658102433174083lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
joinab2519788767067900224lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
seq_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__Nat__Onat,type,
seq_nat: ( nat > nat > nat ) > nat > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
source5627170679743844886lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > set_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__Nat__Onat,type,
sources_nat: ( nat > nat > nat ) > nat > set_nat ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
target5490931411331770124lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > set_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__Nat__Onat,type,
targets_nat: ( nat > nat > nat ) > nat > set_nat ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_ax7298520571313827089lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__Nat__Onat,type,
rts_axioms_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_wi6632542615990831231lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites__axioms_001t__Nat__Onat,type,
rts_wi5665336072173770491ms_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__joins_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_wi2791606683991334555lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__joins__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_wi6344169777088970040lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
transf2236888680007955973lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation_001t__Nat__Onat_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
transf3951925986184407251lambda: ( nat > nat > nat ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( nat > lambda_lambda ) > ( nat > lambda_lambda ) > ( nat > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
weakly4554751870655413501lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001t__Nat__Onat,type,
weakly1695633097759462397rc_nat: ( nat > nat > nat ) > nat > nat ).
thf(sy_c_Set_OCollect_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
collec621444948832755541lambda: ( lambda_lambda > $o ) > set_lambda_lambda ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Oinsert_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
insert_lambda_lambda: lambda_lambda > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Wellfounded_Oaccp_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
accp_lambda_lambda: ( lambda_lambda > lambda_lambda > $o ) > lambda_lambda > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__LambdaCalculus__Olambda____calculus__Olambda_J,type,
accp_P8713979107359754310lambda: ( produc3839482995950079247lambda > produc3839482995950079247lambda > $o ) > produc3839482995950079247lambda > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__LambdaCalculus__Olambda____calculus__Olambda_Mt__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
accp_P3578032143099507075lambda: ( produc2872725455364511436lambda > produc2872725455364511436lambda > $o ) > produc2872725455364511436lambda > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
accp_P7631132853315799497lambda: ( produc674367846827345344lambda > produc674367846827345344lambda > $o ) > produc674367846827345344lambda > $o ).
thf(sy_c_member_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
member_lambda_lambda: lambda_lambda > set_lambda_lambda > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_t,type,
t: lambda_lambda ).
% Relevant facts (981)
thf(fact_0_leftmost__strategy__is__reduction__strategy,axiom,
lambda8358424291402068104rategy @ lambda4011002726600638587rategy ).
% leftmost_strategy_is_reduction_strategy
thf(fact_1_cnv__refl,axiom,
! [A: lambda_lambda] :
( ( lambda_Ide @ A )
=> ( lambda_cnv @ A @ A ) ) ).
% cnv_refl
thf(fact_2_is__leftmost__reduction__def,axiom,
( lambda5425420467215166849uction
= ( ^ [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
& ( ( lambda4011002726600638587rategy @ ( lambda_Src @ T ) )
= T ) ) ) ) ).
% is_leftmost_reduction_def
thf(fact_3_Ide__implies__Arr,axiom,
! [T2: lambda_lambda] :
( ( lambda_Ide @ T2 )
=> ( lambda_Arr @ T2 ) ) ).
% Ide_implies_Arr
thf(fact_4_Ide__Subst,axiom,
! [T2: lambda_lambda,V: lambda_lambda,N: nat] :
( ( lambda_Ide @ T2 )
=> ( ( lambda_Ide @ V )
=> ( lambda_Ide @ ( lambda_Subst @ N @ V @ T2 ) ) ) ) ).
% Ide_Subst
thf(fact_5_elementary__reduction__Raise__iff,axiom,
! [D: nat,N: nat,T2: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Raise @ D @ N @ T2 ) )
= ( lambda2670285437103676873uction @ T2 ) ) ).
% elementary_reduction_Raise_iff
thf(fact_6_cnv__sym,axiom,
! [A: lambda_lambda,B: lambda_lambda] :
( ( lambda_cnv @ A @ B )
=> ( lambda_cnv @ B @ A ) ) ).
% cnv_sym
thf(fact_7_Ide__Raise,axiom,
! [D: nat,N: nat] :
( lambda_Ide
= ( ^ [T: lambda_lambda] : ( lambda_Ide @ ( lambda_Raise @ D @ N @ T ) ) ) ) ).
% Ide_Raise
thf(fact_8_lambda__calculus_Oleftmost__strategy_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_App @ V @ Va ) ) )
=> ( ( lambda4011002726600638587rategy @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_App @ V @ Va ) ) @ U ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_App @ V @ Va ) ) )
=> ( ( lambda4011002726600638587rategy @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_App @ V @ Va ) @ ( lambda4011002726600638587rategy @ U ) ) ) ) ) ).
% lambda_calculus.leftmost_strategy.simps(6)
thf(fact_9_lambda__calculus_Oelementary__reduction_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_App @ T2 @ U ) )
= ( ( ( lambda2670285437103676873uction @ T2 )
& ( lambda_Ide @ U ) )
| ( ( lambda_Ide @ T2 )
& ( lambda2670285437103676873uction @ U ) ) ) ) ).
% lambda_calculus.elementary_reduction.simps(4)
thf(fact_10_lambda__calculus_Oelementary__reduction_Osimps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Beta @ T2 @ U ) )
= ( ( lambda_Ide @ T2 )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.elementary_reduction.simps(5)
thf(fact_11_weak__diamond,axiom,
! [A: lambda_lambda,B: lambda_lambda,B2: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( ( lambda_red @ A @ B2 )
=> ~ ! [C: lambda_lambda] :
( ( lambda_red @ B @ C )
=> ~ ( lambda_red @ B2 @ C ) ) ) ) ).
% weak_diamond
thf(fact_12_Raise__inj,axiom,
! [D: nat,N: nat,T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Raise @ D @ N @ T2 )
= ( lambda_Raise @ D @ N @ U ) )
=> ( T2 = U ) ) ).
% Raise_inj
thf(fact_13_Arr__Src,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_Arr @ ( lambda_Src @ T2 ) ) ) ).
% Arr_Src
thf(fact_14_Arr__Raise,axiom,
! [D: nat,N: nat] :
( lambda_Arr
= ( ^ [T: lambda_lambda] : ( lambda_Arr @ ( lambda_Raise @ D @ N @ T ) ) ) ) ).
% Arr_Raise
thf(fact_15_Src__Raise,axiom,
! [D: nat,N: nat,T2: lambda_lambda] :
( ( lambda_Src @ ( lambda_Raise @ D @ N @ T2 ) )
= ( lambda_Raise @ D @ N @ ( lambda_Src @ T2 ) ) ) ).
% Src_Raise
thf(fact_16_Arr__Subst,axiom,
! [V: lambda_lambda,T2: lambda_lambda,N: nat] :
( ( lambda_Arr @ V )
=> ( ( lambda_Arr @ T2 )
=> ( lambda_Arr @ ( lambda_Subst @ N @ V @ T2 ) ) ) ) ).
% Arr_Subst
thf(fact_17_church__rosser,axiom,
! [A: lambda_lambda,B: lambda_lambda] :
( ( lambda_cnv @ A @ B )
=> ? [C: lambda_lambda] :
( ( lambda_red @ A @ C )
& ( lambda_red @ B @ C ) ) ) ).
% church_rosser
thf(fact_18_red__imp__cnv,axiom,
! [A: lambda_lambda,B: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( lambda_cnv @ A @ B ) ) ).
% red_imp_cnv
thf(fact_19_Ide__iff__Src__self,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Ide @ T2 )
= ( ( lambda_Src @ T2 )
= T2 ) ) ) ).
% Ide_iff_Src_self
thf(fact_20_Ide__Src,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_Ide @ ( lambda_Src @ T2 ) ) ) ).
% Ide_Src
thf(fact_21_Coinitial__Raise__Raise,axiom,
! [T2: lambda_lambda,U: lambda_lambda,D: nat,N: nat] :
( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) )
=> ( ( lambda_Arr @ ( lambda_Raise @ D @ N @ T2 ) )
& ( lambda_Arr @ ( lambda_Raise @ D @ N @ U ) )
& ( ( lambda_Src @ ( lambda_Raise @ D @ N @ T2 ) )
= ( lambda_Src @ ( lambda_Raise @ D @ N @ U ) ) ) ) ) ).
% Coinitial_Raise_Raise
thf(fact_22_reduction__strategy__def,axiom,
( lambda8358424291402068104rategy
= ( ^ [F: lambda_lambda > lambda_lambda] :
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( ( lambda_Arr @ ( F @ T ) )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ ( F @ T ) )
= ( lambda_Src @ T ) ) ) ) ) ) ).
% reduction_strategy_def
thf(fact_23_lambda__calculus_Olambda_Oinject_I3_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda,Y41: lambda_lambda,Y42: lambda_lambda] :
( ( ( lambda_App @ X41 @ X42 )
= ( lambda_App @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% lambda_calculus.lambda.inject(3)
thf(fact_24_lambda__calculus_Olambda_Oinject_I4_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda,Y51: lambda_lambda,Y52: lambda_lambda] :
( ( ( lambda_Beta @ X51 @ X52 )
= ( lambda_Beta @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% lambda_calculus.lambda.inject(4)
thf(fact_25_Src__Src,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Src @ ( lambda_Src @ T2 ) )
= ( lambda_Src @ T2 ) ) ) ).
% Src_Src
thf(fact_26_Src__Subst,axiom,
! [T2: lambda_lambda,U: lambda_lambda,D: nat] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Arr @ U )
=> ( ( lambda_Src @ ( lambda_Subst @ D @ T2 @ U ) )
= ( lambda_Subst @ D @ ( lambda_Src @ T2 ) @ ( lambda_Src @ U ) ) ) ) ) ).
% Src_Subst
thf(fact_27_lambda__calculus_OArr_Osimps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Beta @ T2 @ U ) )
= ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(5)
thf(fact_28_lambda__calculus_OArr_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_App @ T2 @ U ) )
= ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(4)
thf(fact_29_lambda__calculus_OSrc_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Src @ ( lambda_App @ T2 @ U ) )
= ( lambda_App @ ( lambda_Src @ T2 ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.Src.simps(4)
thf(fact_30_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T3 @ U2 ) ) )
= ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ T3 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U2 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(3)
thf(fact_31_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T2 @ U ) )
= ( lambda_Src @ ( lambda_App @ T3 @ U2 ) ) )
= ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ T3 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U2 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_32_lambda__calculus_Ocnv_Ointros_I3_J,axiom,
! [A: lambda_lambda,B: lambda_lambda,C2: lambda_lambda] :
( ( lambda_cnv @ A @ B )
=> ( ( lambda_cnv @ B @ C2 )
=> ( lambda_cnv @ A @ C2 ) ) ) ).
% lambda_calculus.cnv.intros(3)
thf(fact_33_lambda__calculus_Ored_Ointros_I2_J,axiom,
! [A: lambda_lambda,B: lambda_lambda,C2: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( ( lambda_red @ B @ C2 )
=> ( lambda_red @ A @ C2 ) ) ) ).
% lambda_calculus.red.intros(2)
thf(fact_34_lambda__calculus_ORaise_Osimps_I4_J,axiom,
! [D: nat,N: nat,T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_App @ T2 @ U ) )
= ( lambda_App @ ( lambda_Raise @ D @ N @ T2 ) @ ( lambda_Raise @ D @ N @ U ) ) ) ).
% lambda_calculus.Raise.simps(4)
thf(fact_35_lambda__calculus_OSubst_Osimps_I4_J,axiom,
! [N: nat,V: lambda_lambda,T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_App @ T2 @ U ) )
= ( lambda_App @ ( lambda_Subst @ N @ V @ T2 ) @ ( lambda_Subst @ N @ V @ U ) ) ) ).
% lambda_calculus.Subst.simps(4)
thf(fact_36_lambda__calculus_Olambda_Odistinct_I19_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_App @ X41 @ X42 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(19)
thf(fact_37_lambda__calculus_Oleftmost__strategy_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V @ Va ) ) )
=> ( ( lambda4011002726600638587rategy @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V @ Va ) ) @ U ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V @ Va ) ) )
=> ( ( lambda4011002726600638587rategy @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_Beta @ V @ Va ) @ ( lambda4011002726600638587rategy @ U ) ) ) ) ) ).
% lambda_calculus.leftmost_strategy.simps(7)
thf(fact_38_lambda__calculus_OIde_Osimps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
~ ( lambda_Ide @ ( lambda_Beta @ T2 @ U ) ) ).
% lambda_calculus.Ide.simps(5)
thf(fact_39_lambda__calculus_OIde_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Ide @ ( lambda_App @ T2 @ U ) )
= ( ( lambda_Ide @ T2 )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.Ide.simps(4)
thf(fact_40_lambda__calculus_Oleftmost__strategy_Osimps_I8_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda4011002726600638587rategy @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_Beta @ T2 @ U ) ) ).
% lambda_calculus.leftmost_strategy.simps(8)
thf(fact_41_red__reduce,axiom,
! [F2: lambda_lambda > lambda_lambda,A: lambda_lambda,N: nat] :
( ( lambda8358424291402068104rategy @ F2 )
=> ( ( lambda_Ide @ A )
=> ( lambda_red @ A @ ( lambda_reduce @ F2 @ A @ N ) ) ) ) ).
% red_reduce
thf(fact_42_NF__def,axiom,
( lambda_NF
= ( ^ [A2: lambda_lambda] :
( ( lambda_Ide @ A2 )
& ! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( ( lambda_Src @ T )
= A2 ) )
=> ( lambda_Ide @ T ) ) ) ) ) ).
% NF_def
thf(fact_43_NF__reduct__is__trivial,axiom,
! [A: lambda_lambda,B: lambda_lambda] :
( ( lambda_NF @ A )
=> ( ( lambda_red @ A @ B )
=> ( A = B ) ) ) ).
% NF_reduct_is_trivial
thf(fact_44_NF__unique,axiom,
! [T2: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( lambda_red @ T2 @ U )
=> ( ( lambda_red @ T2 @ U2 )
=> ( ( lambda_NF @ U )
=> ( ( lambda_NF @ U2 )
=> ( U = U2 ) ) ) ) ) ).
% NF_unique
thf(fact_45_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ A @ ( collec621444948832755541lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X: lambda_lambda] : ( member_lambda_lambda @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_50_Collect__cong,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ! [X2: lambda_lambda] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec621444948832755541lambda @ P )
= ( collec621444948832755541lambda @ Q ) ) ) ).
% Collect_cong
thf(fact_51_elementary__reduction__bottom__up__redex,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ~ ( lambda_Ide @ T2 )
=> ( lambda2670285437103676873uction @ ( lambda5654762366400029632_redex @ T2 ) ) ) ) ).
% elementary_reduction_bottom_up_redex
thf(fact_52_elementary__reduction__head__redex,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ~ ( lambda_Ide @ ( lambda_head_redex @ T2 ) )
=> ( lambda2670285437103676873uction @ ( lambda_head_redex @ T2 ) ) ) ) ).
% elementary_reduction_head_redex
thf(fact_53_head__strategy__is__elementary,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ~ ( lambda_Ide @ ( lambda_head_strategy @ T2 ) )
=> ( lambda2670285437103676873uction @ ( lambda_head_strategy @ T2 ) ) ) ) ).
% head_strategy_is_elementary
thf(fact_54_Ide__Subst__iff,axiom,
! [N: nat,V: lambda_lambda,T2: lambda_lambda] :
( ( lambda_Ide @ ( lambda_Subst @ N @ V @ T2 ) )
= ( ( lambda_Ide @ T2 )
& ( ( member_nat @ N @ ( lambda_FV @ T2 ) )
=> ( lambda_Ide @ V ) ) ) ) ).
% Ide_Subst_iff
thf(fact_55_resid__Ide__Arr,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( ( lambda_Arr @ A )
& ( lambda_Arr @ T2 )
& ( ( lambda_Src @ A )
= ( lambda_Src @ T2 ) ) )
=> ( lambda_Ide @ ( lambda_resid @ A @ T2 ) ) ) ) ).
% resid_Ide_Arr
thf(fact_56_resid__Arr__Ide,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ A )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ A ) ) )
=> ( ( lambda_resid @ T2 @ A )
= T2 ) ) ) ).
% resid_Arr_Ide
thf(fact_57_normalizable__def,axiom,
( lambda_normalizable
= ( ^ [A2: lambda_lambda] :
( ( lambda_Ide @ A2 )
& ? [B3: lambda_lambda] :
( ( lambda_red @ A2 @ B3 )
& ( lambda_NF @ B3 ) ) ) ) ) ).
% normalizable_def
thf(fact_58_Src__head__redex,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Src @ ( lambda_head_redex @ T2 ) )
= ( lambda_Src @ T2 ) ) ) ).
% Src_head_redex
thf(fact_59_cube,axiom,
! [V: lambda_lambda,T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_resid @ ( lambda_resid @ V @ T2 ) @ ( lambda_resid @ U @ T2 ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T2 @ U ) ) ) ).
% cube
thf(fact_60_ex__un__null,axiom,
? [X2: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( lambda_resid @ X2 @ T4 )
= X2 )
& ( ( lambda_resid @ T4 @ X2 )
= X2 ) )
& ! [Y: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( lambda_resid @ Y @ T5 )
= Y )
& ( ( lambda_resid @ T5 @ Y )
= Y ) )
=> ( Y = X2 ) ) ) ).
% ex_un_null
thf(fact_61_Arr__head__strategy,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_Arr @ ( lambda_head_strategy @ T2 ) ) ) ).
% Arr_head_strategy
thf(fact_62_Arr__head__redex,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_Arr @ ( lambda_head_redex @ T2 ) ) ) ).
% Arr_head_redex
thf(fact_63_head__strategy__is__reduction__strategy,axiom,
lambda8358424291402068104rategy @ lambda_head_strategy ).
% head_strategy_is_reduction_strategy
thf(fact_64_head__strategy__Src,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_head_strategy @ ( lambda_Src @ T2 ) )
= ( lambda_head_strategy @ T2 ) ) ) ).
% head_strategy_Src
thf(fact_65_Src__head__strategy,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Src @ ( lambda_head_strategy @ T2 ) )
= ( lambda_Src @ T2 ) ) ) ).
% Src_head_strategy
thf(fact_66_normalizing__strategy__def,axiom,
( lambda5785200480123228735rategy
= ( ^ [F: lambda_lambda > lambda_lambda] :
! [A2: lambda_lambda] :
( ( lambda_normalizable @ A2 )
=> ? [N2: nat] : ( lambda_NF @ ( lambda_reduce @ F @ A2 @ N2 ) ) ) ) ) ).
% normalizing_strategy_def
thf(fact_67_resid__Arr__Src,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_resid @ T2 @ ( lambda_Src @ T2 ) )
= T2 ) ) ).
% resid_Arr_Src
thf(fact_68_rts__with__joins__axioms,axiom,
rts_wi2791606683991334555lambda @ lambda_resid ).
% rts_with_joins_axioms
thf(fact_69_is__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% is_confluent_rts
thf(fact_70_lambda__calculus_Ohead__strategy_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_App @ V @ Va ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(6)
thf(fact_71_lambda__calculus_Ohead__strategy_Osimps_I8_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_Beta @ ( lambda_Src @ T2 ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(8)
thf(fact_72_lambda__calculus_Ohead__redex_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_App @ V @ Va ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(7)
thf(fact_73_lambda__calculus_Ohead__redex_Osimps_I9_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_Beta @ ( lambda_Src @ T2 ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(9)
thf(fact_74_lambda__calculus_Ohead__strategy_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Beta @ V @ Va ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(7)
thf(fact_75_lambda__calculus_Ohead__redex_Osimps_I8_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Beta @ V @ Va ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(8)
thf(fact_76_lambda__calculus_Obottom__up__redex_Osimps_I4_J,axiom,
! [M: lambda_lambda,N3: lambda_lambda] :
( ( ~ ( lambda_Ide @ M )
=> ( ( lambda5654762366400029632_redex @ ( lambda_App @ M @ N3 ) )
= ( lambda_App @ ( lambda5654762366400029632_redex @ M ) @ ( lambda_Src @ N3 ) ) ) )
& ( ( lambda_Ide @ M )
=> ( ( lambda5654762366400029632_redex @ ( lambda_App @ M @ N3 ) )
= ( lambda_App @ M @ ( lambda5654762366400029632_redex @ N3 ) ) ) ) ) ).
% lambda_calculus.bottom_up_redex.simps(4)
thf(fact_77_partial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% partial_magma_axioms
thf(fact_78_vacuous__Subst,axiom,
! [V: lambda_lambda,I: nat,T2: lambda_lambda] :
( ( lambda_Arr @ V )
=> ( ~ ( member_nat @ I @ ( lambda_FV @ T2 ) )
=> ( ( lambda_Raise @ I @ one_one_nat @ ( lambda_Subst @ I @ V @ T2 ) )
= T2 ) ) ) ).
% vacuous_Subst
thf(fact_79_is__head__reductionI,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda2670285437103676873uction @ T2 )
=> ( ( ( lambda_head_strategy @ ( lambda_Src @ T2 ) )
= T2 )
=> ( lambda5343566265203084875uction @ T2 ) ) ) ) ).
% is_head_reductionI
thf(fact_80_NF__App__iff,axiom,
! [T1: lambda_lambda,T22: lambda_lambda] :
( ( lambda_NF @ ( lambda_App @ T1 @ T22 ) )
= ( ~ ( lambda_is_Lam @ T1 )
& ( lambda_NF @ T1 )
& ( lambda_NF @ T22 ) ) ) ).
% NF_App_iff
thf(fact_81_not__NF__elim,axiom,
! [T2: lambda_lambda] :
( ~ ( lambda_NF @ T2 )
=> ( ( lambda_Ide @ T2 )
=> ~ ! [U3: lambda_lambda] :
~ ( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U3 )
& ~ ( lambda_Ide @ U3 ) ) ) ) ).
% not_NF_elim
thf(fact_82_Con__Arr__head__redex,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_resid @ T2 @ ( lambda_head_redex @ T2 ) )
!= lambda_Nil ) ) ).
% Con_Arr_head_redex
thf(fact_83_Con__head__strategy,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_resid @ T2 @ ( lambda_head_strategy @ T2 ) )
!= lambda_Nil ) ) ).
% Con_head_strategy
thf(fact_84_NF__iff__has__no__redex,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_NF @ T2 )
= ( ~ ( ( lambda_Arr @ T2 )
& ~ ( lambda_Ide @ ( lambda8522482075158074392rategy @ T2 ) ) ) ) ) ) ).
% NF_iff_has_no_redex
thf(fact_85_Coinitial__resid__resid,axiom,
! [T2: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T2 @ V )
!= lambda_Nil )
=> ( ( ( lambda_resid @ U @ V )
!= lambda_Nil )
=> ( ( lambda_Arr @ ( lambda_resid @ T2 @ V ) )
& ( lambda_Arr @ ( lambda_resid @ U @ V ) )
& ( ( lambda_Src @ ( lambda_resid @ T2 @ V ) )
= ( lambda_Src @ ( lambda_resid @ U @ V ) ) ) ) ) ) ).
% Coinitial_resid_resid
thf(fact_86_Coinitial__iff__Con,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) )
= ( ( lambda_resid @ T2 @ U )
!= lambda_Nil ) ) ).
% Coinitial_iff_Con
thf(fact_87_contains__head__reduction__iff,axiom,
( lambda8135948770199036598uction
= ( ^ [T: lambda_lambda] :
( ( lambda_Arr @ T )
& ~ ( lambda_Ide @ ( lambda_head_redex @ T ) ) ) ) ) ).
% contains_head_reduction_iff
thf(fact_88_Con__sym,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
= ( ( lambda_resid @ U @ T2 )
!= lambda_Nil ) ) ).
% Con_sym
thf(fact_89_Cube,axiom,
! [V: lambda_lambda,T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ V @ T2 ) @ ( lambda_resid @ U @ T2 ) )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_resid @ V @ T2 ) @ ( lambda_resid @ U @ T2 ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T2 @ U ) ) ) ) ).
% Cube
thf(fact_90_Arr__not__Nil,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( T2 != lambda_Nil ) ) ).
% Arr_not_Nil
thf(fact_91_Raise__not__Nil,axiom,
! [T2: lambda_lambda,D: nat,N: nat] :
( ( T2 != lambda_Nil )
=> ( ( lambda_Raise @ D @ N @ T2 )
!= lambda_Nil ) ) ).
% Raise_not_Nil
thf(fact_92_Subst__Nil,axiom,
! [N: nat,V: lambda_lambda] :
( ( lambda_Subst @ N @ V @ lambda_Nil )
= lambda_Nil ) ).
% Subst_Nil
thf(fact_93_Subst__not__Nil,axiom,
! [V: lambda_lambda,T2: lambda_lambda,N: nat] :
( ( V != lambda_Nil )
=> ( ( T2 != lambda_Nil )
=> ( ( T2 != lambda_Nil )
=> ( ( lambda_Subst @ N @ V @ T2 )
!= lambda_Nil ) ) ) ) ).
% Subst_not_Nil
thf(fact_94_is__head__reduction__imp__contains__head__reduction,axiom,
! [T2: lambda_lambda] :
( ( lambda5343566265203084875uction @ T2 )
=> ( lambda8135948770199036598uction @ T2 ) ) ).
% is_head_reduction_imp_contains_head_reduction
thf(fact_95_parallel__strategy__is__reduction__strategy,axiom,
lambda8358424291402068104rategy @ lambda8522482075158074392rategy ).
% parallel_strategy_is_reduction_strategy
thf(fact_96_Arr__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ T2 @ U ) ) ) ).
% Arr_resid
thf(fact_97_Con__implies__Arr1,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( lambda_Arr @ T2 ) ) ).
% Con_implies_Arr1
thf(fact_98_Con__implies__Arr2,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( lambda_Arr @ U ) ) ).
% Con_implies_Arr2
thf(fact_99_Con__Raise,axiom,
! [T2: lambda_lambda,U: lambda_lambda,D: nat,N: nat] :
( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Raise @ D @ N @ T2 ) @ ( lambda_Raise @ D @ N @ U ) )
!= lambda_Nil ) ) ).
% Con_Raise
thf(fact_100_Raise__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda,K: nat,N: nat] :
( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( ( lambda_Raise @ K @ N @ ( lambda_resid @ T2 @ U ) )
= ( lambda_resid @ ( lambda_Raise @ K @ N @ T2 ) @ ( lambda_Raise @ K @ N @ U ) ) ) ) ).
% Raise_resid
thf(fact_101_resid__Subst,axiom,
! [T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda,U2: lambda_lambda,N: nat] :
( ( ( lambda_resid @ T2 @ T3 )
!= lambda_Nil )
=> ( ( ( lambda_resid @ U @ U2 )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Subst @ N @ T2 @ U ) @ ( lambda_Subst @ N @ T3 @ U2 ) )
= ( lambda_Subst @ N @ ( lambda_resid @ T2 @ T3 ) @ ( lambda_resid @ U @ U2 ) ) ) ) ) ).
% resid_Subst
thf(fact_102_Con__implies__is__Lam__iff__is__Lam,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( ( lambda_is_Lam @ T2 )
= ( lambda_is_Lam @ U ) ) ) ).
% Con_implies_is_Lam_iff_is_Lam
thf(fact_103_parallel__strategy__Src__eq,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda8522482075158074392rategy @ ( lambda_Src @ T2 ) )
= ( lambda8522482075158074392rategy @ T2 ) ) ) ).
% parallel_strategy_Src_eq
thf(fact_104_parallel__strategy__is__normalizing,axiom,
lambda5785200480123228735rategy @ lambda8522482075158074392rategy ).
% parallel_strategy_is_normalizing
thf(fact_105_head__reduction__unique,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda5343566265203084875uction @ T2 )
=> ( ( lambda5343566265203084875uction @ U )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U )
=> ( T2 = U ) ) ) ) ).
% head_reduction_unique
thf(fact_106_is__head__reduction__if,axiom,
! [U: lambda_lambda] :
( ( lambda8135948770199036598uction @ U )
=> ( ( lambda2670285437103676873uction @ U )
=> ( lambda5343566265203084875uction @ U ) ) ) ).
% is_head_reduction_if
thf(fact_107_is__head__reduction__char,axiom,
( lambda5343566265203084875uction
= ( ^ [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
& ( ( lambda_head_strategy @ ( lambda_Src @ T ) )
= T ) ) ) ) ).
% is_head_reduction_char
thf(fact_108_head__redex__is__head__reduction,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda8135948770199036598uction @ T2 )
=> ( lambda5343566265203084875uction @ ( lambda_head_redex @ T2 ) ) ) ) ).
% head_redex_is_head_reduction
thf(fact_109_ConI,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) )
=> ( ( lambda_resid @ T2 @ U )
!= lambda_Nil ) ) ) ) ).
% ConI
thf(fact_110_coinitial__char,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U )
= ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) ) ) ).
% coinitial_char
thf(fact_111_lambda__calculus_Olambda_Odisc_I6_J,axiom,
~ ( lambda_is_Lam @ lambda_Nil ) ).
% lambda_calculus.lambda.disc(6)
thf(fact_112_lambda__calculus_Olambda_Odistinct__disc_I4_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ( Lambda != lambda_Nil ) ) ).
% lambda_calculus.lambda.distinct_disc(4)
thf(fact_113_lambda__calculus_Olambda_Odistinct__disc_I3_J,axiom,
! [Lambda: lambda_lambda] :
( ( Lambda = lambda_Nil )
=> ~ ( lambda_is_Lam @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(3)
thf(fact_114_lambda__calculus_Ois__head__reduction_Osimps_I9_J,axiom,
~ ( lambda5343566265203084875uction @ lambda_Nil ) ).
% lambda_calculus.is_head_reduction.simps(9)
thf(fact_115_lambda__calculus_Oparallel__strategy_Osimps_I9_J,axiom,
( ( lambda8522482075158074392rategy @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.parallel_strategy.simps(9)
thf(fact_116_lambda__calculus_Ocontains__head__reduction_Osimps_I9_J,axiom,
~ ( lambda8135948770199036598uction @ lambda_Nil ) ).
% lambda_calculus.contains_head_reduction.simps(9)
thf(fact_117_lambda__calculus_Oparallel__strategy_Osimps_I4_J,axiom,
! [U: lambda_lambda] :
( ( lambda8522482075158074392rategy @ ( lambda_App @ lambda_Nil @ U ) )
= ( lambda_App @ ( lambda8522482075158074392rategy @ lambda_Nil ) @ ( lambda8522482075158074392rategy @ U ) ) ) ).
% lambda_calculus.parallel_strategy.simps(4)
thf(fact_118_lambda__calculus_Ois__head__reduction_Osimps_I4_J,axiom,
! [U: lambda_lambda] :
( ( lambda5343566265203084875uction @ ( lambda_App @ lambda_Nil @ U ) )
= ( ( lambda5343566265203084875uction @ lambda_Nil )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.is_head_reduction.simps(4)
thf(fact_119_lambda__calculus_Ocontains__head__reduction_Osimps_I4_J,axiom,
! [U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ lambda_Nil @ U ) )
= ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(4)
thf(fact_120_lambda__calculus_Oparallel__strategy_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda8522482075158074392rategy @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_App @ V @ Va ) ) @ ( lambda8522482075158074392rategy @ U ) ) ) ).
% lambda_calculus.parallel_strategy.simps(6)
thf(fact_121_lambda__calculus_Oparallel__strategy_Osimps_I8_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda8522482075158074392rategy @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_Beta @ ( lambda8522482075158074392rategy @ T2 ) @ ( lambda8522482075158074392rategy @ U ) ) ) ).
% lambda_calculus.parallel_strategy.simps(8)
thf(fact_122_lambda__calculus_Olambda_Odistinct_I5_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( lambda_Nil
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(5)
thf(fact_123_lambda__calculus_Oresid_Osimps_I26_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_resid @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(26)
thf(fact_124_lambda__calculus_Oresid_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(7)
thf(fact_125_lambda__calculus_Olambda_Odistinct_I7_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( lambda_Nil
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(7)
thf(fact_126_lambda__calculus_OArr_Osimps_I1_J,axiom,
~ ( lambda_Arr @ lambda_Nil ) ).
% lambda_calculus.Arr.simps(1)
thf(fact_127_lambda__calculus_OSrc_Osimps_I1_J,axiom,
( ( lambda_Src @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Src.simps(1)
thf(fact_128_lambda__calculus_OIde_Osimps_I1_J,axiom,
~ ( lambda_Ide @ lambda_Nil ) ).
% lambda_calculus.Ide.simps(1)
thf(fact_129_lambda__calculus_ORaise_Osimps_I1_J,axiom,
! [Uu: nat,Uv: nat] :
( ( lambda_Raise @ Uu @ Uv @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Raise.simps(1)
thf(fact_130_lambda__calculus_OSubst_Osimps_I1_J,axiom,
! [Uu: nat,Uv: lambda_lambda] :
( ( lambda_Subst @ Uu @ Uv @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Subst.simps(1)
thf(fact_131_lambda__calculus_Olambda_Odisc_I9_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
~ ( lambda_is_Lam @ ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.disc(9)
thf(fact_132_lambda__calculus_Oelementary__reduction_Osimps_I1_J,axiom,
~ ( lambda2670285437103676873uction @ lambda_Nil ) ).
% lambda_calculus.elementary_reduction.simps(1)
thf(fact_133_lambda__calculus_Oleftmost__strategy_Osimps_I9_J,axiom,
( ( lambda4011002726600638587rategy @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.leftmost_strategy.simps(9)
thf(fact_134_lambda__calculus_Olambda_Odisc_I10_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
~ ( lambda_is_Lam @ ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.disc(10)
thf(fact_135_lambda__calculus_Ohead__strategy_Osimps_I9_J,axiom,
( ( lambda_head_strategy @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.head_strategy.simps(9)
thf(fact_136_lambda__calculus_Ohead__redex_Osimps_I1_J,axiom,
( ( lambda_head_redex @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.head_redex.simps(1)
thf(fact_137_lambda__calculus_Obottom__up__redex_Osimps_I1_J,axiom,
( ( lambda5654762366400029632_redex @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.bottom_up_redex.simps(1)
thf(fact_138_lambda__calculus_Oparallel__strategy_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda8522482075158074392rategy @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_Beta @ V @ Va ) ) @ ( lambda8522482075158074392rategy @ U ) ) ) ).
% lambda_calculus.parallel_strategy.simps(7)
thf(fact_139_lambda__calculus_Ois__head__reduction_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda5343566265203084875uction @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( ( lambda5343566265203084875uction @ ( lambda_App @ V @ Va ) )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.is_head_reduction.simps(6)
thf(fact_140_lambda__calculus_Ois__head__reduction_Osimps_I8_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda5343566265203084875uction @ ( lambda_Beta @ T2 @ U ) )
= ( ( lambda_Ide @ T2 )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.is_head_reduction.simps(8)
thf(fact_141_lambda__calculus_OConD_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T2 @ U ) @ ( lambda_App @ T3 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T2 @ T3 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(1)
thf(fact_142_lambda__calculus_Oresid_Osimps_I31_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(31)
thf(fact_143_lambda__calculus_Oresid_Osimps_I12_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(12)
thf(fact_144_lambda__calculus_Oresid_Osimps_I3_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T2 @ U ) @ ( lambda_App @ T3 @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T2 @ U ) @ ( lambda_App @ T3 @ U2 ) )
= ( lambda_App @ ( lambda_resid @ T2 @ T3 ) @ ( lambda_resid @ U @ U2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(3)
thf(fact_145_lambda__calculus_Ocontains__head__reduction_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( ( lambda8135948770199036598uction @ ( lambda_App @ V @ Va ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(6)
thf(fact_146_lambda__calculus_Oresid_Osimps_I38_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(38)
thf(fact_147_lambda__calculus_Oresid_Osimps_I19_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(19)
thf(fact_148_lambda__calculus_Ocontains__head__reduction_Osimps_I8_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_Beta @ T2 @ U ) )
= ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(8)
thf(fact_149_lambda__calculus_Ois__head__reduction_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda5343566265203084875uction @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( ( lambda5343566265203084875uction @ ( lambda_Beta @ V @ Va ) )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.is_head_reduction.simps(7)
thf(fact_150_lambda__calculus_Ocontains__head__reduction_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V @ Va ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(7)
thf(fact_151_lambda__calculus_Oresid_Osimps_I44_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(44)
thf(fact_152_lambda__calculus_Oresid_Osimps_I43_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(43)
thf(fact_153_lambda__calculus_Oresid_Osimps_I41_J,axiom,
! [Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(41)
thf(fact_154_lambda__calculus_Oresid_Osimps_I37_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(37)
thf(fact_155_lambda__calculus_Oresid_Osimps_I36_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(36)
thf(fact_156_lambda__calculus_Oresid_Osimps_I34_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ lambda_Nil @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(34)
thf(fact_157_lambda__calculus_Oresid_Osimps_I25_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(25)
thf(fact_158_lambda__calculus_Oresid_Osimps_I24_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(24)
thf(fact_159_lambda__calculus_Oresid_Osimps_I22_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ lambda_Nil @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(22)
thf(fact_160_lambda__calculus_Oresid_Osimps_I18_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(18)
thf(fact_161_lambda__calculus_Oresid_Osimps_I17_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(17)
thf(fact_162_lambda__calculus_Oresid_Osimps_I15_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(15)
thf(fact_163_lambda__calculus_OCon__Arr__Src_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_resid @ ( lambda_Src @ T2 ) @ T2 )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_Src(2)
thf(fact_164_lambda__calculus_OCon__Arr__Src_I1_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_resid @ T2 @ ( lambda_Src @ T2 ) )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_Src(1)
thf(fact_165_lambda__calculus_OCon__implies__Coinitial3_I3_J,axiom,
! [T2: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T2 @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ U )
& ( lambda_Arr @ T2 )
& ( ( lambda_Src @ U )
= ( lambda_Src @ T2 ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(3)
thf(fact_166_lambda__calculus_OCon__implies__Coinitial3_I2_J,axiom,
! [T2: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T2 @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ V )
& ( lambda_Arr @ T2 )
& ( ( lambda_Src @ V )
= ( lambda_Src @ T2 ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(2)
thf(fact_167_lambda__calculus_OCon__implies__Coinitial3_I1_J,axiom,
! [T2: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T2 @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ V )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ V )
= ( lambda_Src @ U ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(1)
thf(fact_168_lambda__calculus_Oleftmost__strategy_Osimps_I4_J,axiom,
! [U: lambda_lambda] :
( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ lambda_Nil ) )
=> ( ( lambda4011002726600638587rategy @ ( lambda_App @ lambda_Nil @ U ) )
= ( lambda_App @ ( lambda4011002726600638587rategy @ lambda_Nil ) @ U ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ lambda_Nil ) )
=> ( ( lambda4011002726600638587rategy @ ( lambda_App @ lambda_Nil @ U ) )
= ( lambda_App @ lambda_Nil @ ( lambda4011002726600638587rategy @ U ) ) ) ) ) ).
% lambda_calculus.leftmost_strategy.simps(4)
thf(fact_169_lambda__calculus_Ohead__strategy_Osimps_I4_J,axiom,
! [U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ lambda_Nil @ U ) )
= ( lambda_App @ ( lambda_head_strategy @ lambda_Nil ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(4)
thf(fact_170_lambda__calculus_Ohead__redex_Osimps_I5_J,axiom,
! [U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ lambda_Nil @ U ) )
= ( lambda_App @ ( lambda_head_redex @ lambda_Nil ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(5)
thf(fact_171_elementary__reduction__Lam__iff,axiom,
! [T2: lambda_lambda] :
( ( lambda_is_Lam @ T2 )
=> ( ( lambda2670285437103676873uction @ T2 )
= ( lambda2670285437103676873uction @ ( lambda_un_Lam @ T2 ) ) ) ) ).
% elementary_reduction_Lam_iff
thf(fact_172_joinable__implies__coinitial,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T2 @ U )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U ) ) ).
% joinable_implies_coinitial
thf(fact_173_is__head__reduction__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda5343566265203084875uction @ T2 )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
| ( lambda5343566265203084875uction @ ( lambda_resid @ T2 @ U ) ) ) ) ) ) ).
% is_head_reduction_resid
thf(fact_174_is__head__reduction__resid_H,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda5343566265203084875uction @ T2 )
=> ( ( lambda2591989677915507790uction @ U )
=> ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) )
=> ( lambda5343566265203084875uction @ ( lambda_resid @ T2 @ U ) ) ) ) ) ).
% is_head_reduction_resid'
thf(fact_175_Ide__resid__Join,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) )
=> ( lambda_Ide @ ( lambda_resid @ U @ ( lambda_Join @ T2 @ U ) ) ) ) ).
% Ide_resid_Join
thf(fact_176_subs__parallel__strategy__Src,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_subs @ T2 @ ( lambda8522482075158074392rategy @ ( lambda_Src @ T2 ) ) ) ) ).
% subs_parallel_strategy_Src
thf(fact_177_resid__Join,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) )
=> ( ( lambda_resid @ ( lambda_Join @ T2 @ U ) @ U )
= ( lambda_resid @ T2 @ U ) ) ) ).
% resid_Join
thf(fact_178_Src__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( ( lambda_Src @ ( lambda_resid @ T2 @ U ) )
= ( lambda_Trg @ U ) ) ) ).
% Src_resid
thf(fact_179_is__internal__reduction__iff,axiom,
( lambda2591989677915507790uction
= ( ^ [T: lambda_lambda] :
( ( lambda_Arr @ T )
& ~ ( lambda8135948770199036598uction @ T ) ) ) ) ).
% is_internal_reduction_iff
thf(fact_180_Join__sym,axiom,
( lambda_Join
= ( ^ [T: lambda_lambda,U4: lambda_lambda] : ( lambda_Join @ U4 @ T ) ) ) ).
% Join_sym
thf(fact_181_weak__extensionality,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( T2 = U ) ) ) ) ).
% weak_extensionality
thf(fact_182_prfx__transitive,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ V ) ) ) ) ).
% prfx_transitive
thf(fact_183_ide__backward__stable,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ A ) )
=> ( ide_lambda_lambda @ lambda_resid @ T2 ) ) ) ).
% ide_backward_stable
thf(fact_184_cong__transitive,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T2 ) ) ) ) ) ).
% cong_transitive
thf(fact_185_cong__symmetric,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) ) ) ) ).
% cong_symmetric
thf(fact_186_Arr__Trg,axiom,
! [X3: lambda_lambda] :
( ( lambda_Arr @ X3 )
=> ( lambda_Arr @ ( lambda_Trg @ X3 ) ) ) ).
% Arr_Trg
thf(fact_187_cong__Ide__are__eq,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) )
=> ( ( lambda_Ide @ T2 )
=> ( ( lambda_Ide @ U )
=> ( T2 = U ) ) ) ) ).
% cong_Ide_are_eq
thf(fact_188_eq__Ide__are__cong,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( T2 = U )
=> ( ( lambda_Ide @ T2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) ) ) ) ).
% eq_Ide_are_cong
thf(fact_189_resid__Arr__self,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_resid @ T2 @ T2 )
= ( lambda_Trg @ T2 ) ) ) ).
% resid_Arr_self
thf(fact_190_subs__implies__prfx,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_subs @ T2 @ U )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) ) ) ).
% subs_implies_prfx
thf(fact_191_is__internal__reduction__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda2591989677915507790uction @ T2 )
=> ( ( lambda2591989677915507790uction @ U )
=> ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) )
=> ( lambda2591989677915507790uction @ ( lambda_resid @ T2 @ U ) ) ) ) ) ).
% is_internal_reduction_resid
thf(fact_192_Src__Join,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) )
=> ( ( lambda_Src @ ( lambda_Join @ T2 @ U ) )
= ( lambda_Src @ T2 ) ) ) ).
% Src_Join
thf(fact_193_cong__elementary__reductions__are__equal,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ T2 )
=> ( ( lambda2670285437103676873uction @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) )
=> ( T2 = U ) ) ) ) ).
% cong_elementary_reductions_are_equal
thf(fact_194_elementary__reduction__not__ide,axiom,
! [T2: lambda_lambda] :
( ( lambda2670285437103676873uction @ T2 )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T2 ) ) ).
% elementary_reduction_not_ide
thf(fact_195_Ide__Trg,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_Ide @ ( lambda_Trg @ T2 ) ) ) ).
% Ide_Trg
thf(fact_196_Ide__iff__Trg__self,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Ide @ T2 )
= ( ( lambda_Trg @ T2 )
= T2 ) ) ) ).
% Ide_iff_Trg_self
thf(fact_197_subs__head__redex,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_subs @ ( lambda_head_redex @ T2 ) @ T2 ) ) ).
% subs_head_redex
thf(fact_198_subs__bottom__up__redex,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_subs @ ( lambda5654762366400029632_redex @ T2 ) @ T2 ) ) ).
% subs_bottom_up_redex
thf(fact_199_cong__implies__coinitial,axiom,
! [U: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ U @ U2 ) ) ).
% cong_implies_coinitial
thf(fact_200_coinitial__ide__are__eq,axiom,
! [A: lambda_lambda,A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A4 )
=> ( A = A4 ) ) ) ) ).
% coinitial_ide_are_eq
thf(fact_201_coinitial__ide__are__cong,axiom,
! [A: lambda_lambda,A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A4 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A4 @ A ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_202_resid__ide_I1_J,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ T2 )
=> ( ( lambda_resid @ T2 @ A )
= T2 ) ) ) ).
% resid_ide(1)
thf(fact_203_subs__Ide,axiom,
! [U: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) )
=> ( lambda_subs @ U @ T2 ) ) ) ).
% subs_Ide
thf(fact_204_prfx__Join,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Join @ T2 @ U ) ) ) ) ).
% prfx_Join
thf(fact_205_resid__head__strategy__internal,axiom,
! [U: lambda_lambda] :
( ( lambda2591989677915507790uction @ U )
=> ( ( lambda_resid @ ( lambda_head_strategy @ ( lambda_Src @ U ) ) @ U )
= ( lambda_head_strategy @ ( lambda_Trg @ U ) ) ) ) ).
% resid_head_strategy_internal
thf(fact_206_internal__reduction__preserves__no__head__redex,axiom,
! [U: lambda_lambda] :
( ( lambda2591989677915507790uction @ U )
=> ( ( lambda_Ide @ ( lambda_head_strategy @ ( lambda_Src @ U ) ) )
=> ( lambda_Ide @ ( lambda_head_strategy @ ( lambda_Trg @ U ) ) ) ) ) ).
% internal_reduction_preserves_no_head_redex
thf(fact_207_Trg__Trg,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Trg @ ( lambda_Trg @ T2 ) )
= ( lambda_Trg @ T2 ) ) ) ).
% Trg_Trg
thf(fact_208_ide__char,axiom,
! [T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T2 )
= ( lambda_Ide @ T2 ) ) ).
% ide_char
thf(fact_209_prfx__char,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( lambda_Ide @ ( lambda_resid @ T2 @ U ) ) ) ).
% prfx_char
thf(fact_210_Trg__Src,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Trg @ ( lambda_Src @ T2 ) )
= ( lambda_Src @ T2 ) ) ) ).
% Trg_Src
thf(fact_211_Src__Trg,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Src @ ( lambda_Trg @ T2 ) )
= ( lambda_Trg @ T2 ) ) ) ).
% Src_Trg
thf(fact_212_Trg__Subst,axiom,
! [T2: lambda_lambda,U: lambda_lambda,D: nat] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Arr @ U )
=> ( ( lambda_Trg @ ( lambda_Subst @ D @ T2 @ U ) )
= ( lambda_Subst @ D @ ( lambda_Trg @ T2 ) @ ( lambda_Trg @ U ) ) ) ) ) ).
% Trg_Subst
thf(fact_213_resid__Src__Arr,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_resid @ ( lambda_Src @ T2 ) @ T2 )
= ( lambda_Trg @ T2 ) ) ) ).
% resid_Src_Arr
thf(fact_214_lambda__calculus_OTrg_Osimps_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Trg @ ( lambda_App @ T2 @ U ) )
= ( lambda_App @ ( lambda_Trg @ T2 ) @ ( lambda_Trg @ U ) ) ) ).
% lambda_calculus.Trg.simps(3)
thf(fact_215_lambda__calculus_OJoin_Osimps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ T2 @ U ) @ ( lambda_App @ T3 @ U2 ) )
= ( lambda_App @ ( lambda_Join @ T2 @ T3 ) @ ( lambda_Join @ U @ U2 ) ) ) ).
% lambda_calculus.Join.simps(5)
thf(fact_216_lambda__calculus_Osubs_Osimps_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( lambda_subs @ ( lambda_App @ T2 @ U ) @ ( lambda_App @ T3 @ U2 ) )
= ( ( lambda_subs @ T2 @ T3 )
& ( lambda_subs @ U @ U2 ) ) ) ).
% lambda_calculus.subs.simps(3)
thf(fact_217_lambda__calculus_OTrg_Osimps_I5_J,axiom,
( ( lambda_Trg @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Trg.simps(5)
thf(fact_218_lambda__calculus_OJoin_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(7)
thf(fact_219_lambda__calculus_OJoin_Osimps_I38_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_Join @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(38)
thf(fact_220_lambda__calculus_OJoin_Osimps_I6_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ T2 @ U ) @ ( lambda_Beta @ T3 @ U2 ) )
= ( lambda_Beta @ ( lambda_Join @ T2 @ T3 ) @ ( lambda_Join @ U @ U2 ) ) ) ).
% lambda_calculus.Join.simps(6)
thf(fact_221_lambda__calculus_Osubs_Osimps_I6_J,axiom,
! [Uv: lambda_lambda] :
~ ( lambda_subs @ lambda_Nil @ Uv ) ).
% lambda_calculus.subs.simps(6)
thf(fact_222_lambda__calculus_Osubs_Osimps_I22_J,axiom,
! [Uu: lambda_lambda] :
~ ( lambda_subs @ Uu @ lambda_Nil ) ).
% lambda_calculus.subs.simps(22)
thf(fact_223_lambda__calculus_Osubs_Osimps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( lambda_subs @ ( lambda_Beta @ T2 @ U ) @ ( lambda_Beta @ T3 @ U2 ) )
= ( ( lambda_subs @ T2 @ T3 )
& ( lambda_subs @ U @ U2 ) ) ) ).
% lambda_calculus.subs.simps(5)
thf(fact_224_lambda__calculus_Ois__internal__reduction_Osimps_I9_J,axiom,
~ ( lambda2591989677915507790uction @ lambda_Nil ) ).
% lambda_calculus.is_internal_reduction.simps(9)
thf(fact_225_lambda__calculus_Ois__internal__reduction_Osimps_I8_J,axiom,
! [Uv: lambda_lambda,Uw: lambda_lambda] :
~ ( lambda2591989677915507790uction @ ( lambda_Beta @ Uv @ Uw ) ) ).
% lambda_calculus.is_internal_reduction.simps(8)
thf(fact_226_lambda__calculus_OJoin_Osimps_I12_J,axiom,
! [Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ lambda_Nil @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(12)
thf(fact_227_lambda__calculus_OJoin_Osimps_I20_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(20)
thf(fact_228_lambda__calculus_OJoin_Osimps_I28_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(28)
thf(fact_229_lambda__calculus_OJoin_Osimps_I43_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(43)
thf(fact_230_lambda__calculus_OJoin_Osimps_I46_J,axiom,
! [Va: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ ( lambda_App @ lambda_Nil @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(46)
thf(fact_231_lambda__calculus_OJoin_Osimps_I54_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(54)
thf(fact_232_lambda__calculus_Osubs_Osimps_I11_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ V @ Va ) @ lambda_Nil ) ).
% lambda_calculus.subs.simps(11)
thf(fact_233_lambda__calculus_Osubs_Osimps_I27_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ lambda_Nil @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.subs.simps(27)
thf(fact_234_lambda__calculus_Osubs_Osimps_I16_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) ) ).
% lambda_calculus.subs.simps(16)
thf(fact_235_lambda__calculus_Osubs_Osimps_I17_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) ) ).
% lambda_calculus.subs.simps(17)
thf(fact_236_lambda__calculus_Osubs_Osimps_I21_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ Vb @ Vc ) ) ).
% lambda_calculus.subs.simps(21)
thf(fact_237_lambda__calculus_Osubs_Osimps_I30_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.subs.simps(30)
thf(fact_238_lambda__calculus_Osubs_Osimps_I36_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.subs.simps(36)
thf(fact_239_lambda__calculus_Osubs_Osimps_I37_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.subs.simps(37)
thf(fact_240_lambda__calculus_OJoin_Osimps_I31_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(31)
thf(fact_241_lambda__calculus_OJoin_Osimps_I62_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(62)
thf(fact_242_lambda__calculus_Osubs_Osimps_I18_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ V @ Va ) @ lambda_Nil ) ).
% lambda_calculus.subs.simps(18)
thf(fact_243_lambda__calculus_Osubs_Osimps_I31_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ lambda_Nil @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.subs.simps(31)
thf(fact_244_lambda__calculus_Ois__internal__reduction_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda2591989677915507790uction @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( ( lambda2591989677915507790uction @ ( lambda_App @ V @ Va ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.is_internal_reduction.simps(6)
thf(fact_245_lambda__calculus_OJoin_Osimps_I15_J,axiom,
! [Va: lambda_lambda,V: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ lambda_Nil @ Va ) @ ( lambda_Beta @ V @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(15)
thf(fact_246_lambda__calculus_OJoin_Osimps_I23_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda,V: lambda_lambda,Vd: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ Va ) @ ( lambda_Beta @ V @ Vd ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(23)
thf(fact_247_lambda__calculus_OJoin_Osimps_I24_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(24)
thf(fact_248_lambda__calculus_OJoin_Osimps_I27_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda,V: lambda_lambda,Vd: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ Va ) @ ( lambda_Beta @ V @ Vd ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(27)
thf(fact_249_lambda__calculus_OJoin_Osimps_I34_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ lambda_Nil @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(34)
thf(fact_250_lambda__calculus_OJoin_Osimps_I36_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(36)
thf(fact_251_lambda__calculus_OJoin_Osimps_I37_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(37)
thf(fact_252_lambda__calculus_OJoin_Osimps_I49_J,axiom,
! [V: lambda_lambda,Vb: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Vb ) @ ( lambda_App @ lambda_Nil @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(49)
thf(fact_253_lambda__calculus_OJoin_Osimps_I57_J,axiom,
! [V: lambda_lambda,Vd: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Vd ) @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(57)
thf(fact_254_lambda__calculus_OJoin_Osimps_I58_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(58)
thf(fact_255_lambda__calculus_OJoin_Osimps_I61_J,axiom,
! [V: lambda_lambda,Vd: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Vd ) @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(61)
thf(fact_256_lambda__calculus_OJoin_Osimps_I65_J,axiom,
! [Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ lambda_Nil @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(65)
thf(fact_257_lambda__calculus_OJoin_Osimps_I67_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(67)
thf(fact_258_lambda__calculus_OJoin_Osimps_I68_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(68)
thf(fact_259_lambda__calculus_Osubs_Osimps_I14_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ lambda_Nil @ Va ) @ ( lambda_Beta @ Vb @ Vc ) ) ).
% lambda_calculus.subs.simps(14)
thf(fact_260_lambda__calculus_Osubs_Osimps_I34_J,axiom,
! [Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ lambda_Nil @ Vc ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.subs.simps(34)
thf(fact_261_lambda__calculus_Ored_Osimps,axiom,
( lambda_red
= ( ^ [A1: lambda_lambda,A22: lambda_lambda] :
( ? [T: lambda_lambda] :
( ( A1
= ( lambda_Src @ T ) )
& ( A22
= ( lambda_Trg @ T ) )
& ( lambda_Arr @ T ) )
| ? [A2: lambda_lambda,B3: lambda_lambda,C3: lambda_lambda] :
( ( A1 = A2 )
& ( A22 = C3 )
& ( lambda_red @ A2 @ B3 )
& ( lambda_red @ B3 @ C3 ) ) ) ) ) ).
% lambda_calculus.red.simps
thf(fact_262_lambda__calculus_Ored_Ocases,axiom,
! [A12: lambda_lambda,A23: lambda_lambda] :
( ( lambda_red @ A12 @ A23 )
=> ( ! [T5: lambda_lambda] :
( ( A12
= ( lambda_Src @ T5 ) )
=> ( ( A23
= ( lambda_Trg @ T5 ) )
=> ~ ( lambda_Arr @ T5 ) ) )
=> ~ ! [B4: lambda_lambda] :
( ( lambda_red @ A12 @ B4 )
=> ~ ( lambda_red @ B4 @ A23 ) ) ) ) ).
% lambda_calculus.red.cases
thf(fact_263_lambda__calculus_Ored_Ointros_I1_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_red @ ( lambda_Src @ T2 ) @ ( lambda_Trg @ T2 ) ) ) ).
% lambda_calculus.red.intros(1)
thf(fact_264_lambda__calculus_Ocnv_Osimps,axiom,
( lambda_cnv
= ( ^ [A1: lambda_lambda,A22: lambda_lambda] :
( ? [T: lambda_lambda] :
( ( A1
= ( lambda_Src @ T ) )
& ( A22
= ( lambda_Trg @ T ) )
& ( lambda_Arr @ T ) )
| ? [T: lambda_lambda] :
( ( A1
= ( lambda_Trg @ T ) )
& ( A22
= ( lambda_Src @ T ) )
& ( lambda_Arr @ T ) )
| ? [A2: lambda_lambda,B3: lambda_lambda,C3: lambda_lambda] :
( ( A1 = A2 )
& ( A22 = C3 )
& ( lambda_cnv @ A2 @ B3 )
& ( lambda_cnv @ B3 @ C3 ) ) ) ) ) ).
% lambda_calculus.cnv.simps
thf(fact_265_lambda__calculus_Ocnv_Ocases,axiom,
! [A12: lambda_lambda,A23: lambda_lambda] :
( ( lambda_cnv @ A12 @ A23 )
=> ( ! [T5: lambda_lambda] :
( ( A12
= ( lambda_Src @ T5 ) )
=> ( ( A23
= ( lambda_Trg @ T5 ) )
=> ~ ( lambda_Arr @ T5 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( A12
= ( lambda_Trg @ T5 ) )
=> ( ( A23
= ( lambda_Src @ T5 ) )
=> ~ ( lambda_Arr @ T5 ) ) )
=> ~ ! [B4: lambda_lambda] :
( ( lambda_cnv @ A12 @ B4 )
=> ~ ( lambda_cnv @ B4 @ A23 ) ) ) ) ) ).
% lambda_calculus.cnv.cases
thf(fact_266_lambda__calculus_Ocnv_Ointros_I1_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_cnv @ ( lambda_Src @ T2 ) @ ( lambda_Trg @ T2 ) ) ) ).
% lambda_calculus.cnv.intros(1)
thf(fact_267_lambda__calculus_Ocnv_Ointros_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( lambda_cnv @ ( lambda_Trg @ T2 ) @ ( lambda_Src @ T2 ) ) ) ).
% lambda_calculus.cnv.intros(2)
thf(fact_268_lambda__calculus_Ois__internal__reduction_Osimps_I4_J,axiom,
! [U: lambda_lambda] :
( ( lambda2591989677915507790uction @ ( lambda_App @ lambda_Nil @ U ) )
= ( ( lambda2591989677915507790uction @ lambda_Nil )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.is_internal_reduction.simps(4)
thf(fact_269_lambda__calculus_Ois__internal__reduction_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda2591989677915507790uction @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( ( lambda2591989677915507790uction @ ( lambda_Beta @ V @ Va ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.is_internal_reduction.simps(7)
thf(fact_270_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_271_identities__form__coherent__normal__sub__rts,axiom,
cohere1690309856004283859lambda @ lambda_resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ).
% identities_form_coherent_normal_sub_rts
thf(fact_272_cong__implies__coterminal,axiom,
! [U: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ U @ U2 ) ) ).
% cong_implies_coterminal
thf(fact_273_seq__char,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T2 @ U )
= ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Trg @ T2 )
= ( lambda_Src @ U ) ) ) ) ).
% seq_char
thf(fact_274_seqE_092_060_094sub_062_092_060Lambda_062,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T2 @ U )
=> ~ ( ( lambda_Arr @ T2 )
=> ( ( lambda_Arr @ U )
=> ( ( lambda_Trg @ T2 )
!= ( lambda_Src @ U ) ) ) ) ) ).
% seqE\<^sub>\<Lambda>
thf(fact_275_prfx__Lam__iff,axiom,
! [U: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Lam @ T2 ) ) )
= ( ( lambda_is_Lam @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_Lam @ U ) @ T2 ) ) ) ) ).
% prfx_Lam_iff
thf(fact_276_join__of__Join,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ ( lambda_Join @ T2 @ U ) ) ) ).
% join_of_Join
thf(fact_277_joinable__def,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T2 @ U )
= ( ? [X5: lambda_lambda] : ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ X5 ) ) ) ).
% joinable_def
thf(fact_278_resid__ide_I2_J,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ T2 )
=> ( ( lambda_resid @ A @ T2 )
= ( trg_lambda_lambda @ lambda_resid @ T2 ) ) ) ) ).
% resid_ide(2)
thf(fact_279_apex__sym,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) ) ).
% apex_sym
thf(fact_280_trg__def,axiom,
! [T2: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ T2 )
= ( lambda_resid @ T2 @ T2 ) ) ).
% trg_def
thf(fact_281_join__of__symmetric,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( join_o1386658102433174083lambda @ lambda_resid @ U @ T2 @ V ) ) ).
% join_of_symmetric
thf(fact_282_NF__Lam__iff,axiom,
! [T2: lambda_lambda] :
( ( lambda_NF @ ( lambda_Lam @ T2 ) )
= ( lambda_NF @ T2 ) ) ).
% NF_Lam_iff
thf(fact_283_join__of__un__upto__cong,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V2: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ V2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V2 @ V ) ) ) ) ) ).
% join_of_un_upto_cong
thf(fact_284_cong__respects__seq,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T3 @ T2 ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T3 @ U2 ) ) ) ) ).
% cong_respects_seq
thf(fact_285_trg__join__of_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ V ) ) ) ).
% trg_join_of(1)
thf(fact_286_trg__join__of_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) )
= ( trg_lambda_lambda @ lambda_resid @ V ) ) ) ).
% trg_join_of(2)
thf(fact_287_lambda__calculus_Olambda_Oinject_I2_J,axiom,
! [X32: lambda_lambda,Y3: lambda_lambda] :
( ( ( lambda_Lam @ X32 )
= ( lambda_Lam @ Y3 ) )
= ( X32 = Y3 ) ) ).
% lambda_calculus.lambda.inject(2)
thf(fact_288_trg__trg,axiom,
! [T2: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T2 ) )
= ( trg_lambda_lambda @ lambda_resid @ T2 ) ) ).
% trg_trg
thf(fact_289_trg__ide,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( trg_lambda_lambda @ lambda_resid @ A )
= A ) ) ).
% trg_ide
thf(fact_290_lambda__calculus_Olambda_Ocollapse_I2_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ( ( lambda_Lam @ ( lambda_un_Lam @ Lambda ) )
= Lambda ) ) ).
% lambda_calculus.lambda.collapse(2)
thf(fact_291_trg__char,axiom,
! [T2: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
=> ( ( trg_lambda_lambda @ lambda_resid @ T2 )
= ( lambda_Trg @ T2 ) ) )
& ( ~ ( lambda_Arr @ T2 )
=> ( ( trg_lambda_lambda @ lambda_resid @ T2 )
= lambda_Nil ) ) ) ).
% trg_char
thf(fact_292_seqI_092_060_094sub_062_092_060Lambda_062,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Trg @ T2 )
= ( lambda_Src @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T2 @ U ) ) ) ) ).
% seqI\<^sub>\<Lambda>
thf(fact_293_lambda__calculus_Olambda_Odistinct_I15_J,axiom,
! [X32: lambda_lambda,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Lam @ X32 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(15)
thf(fact_294_lambda__calculus_Olambda_Odistinct_I3_J,axiom,
! [X32: lambda_lambda] :
( lambda_Nil
!= ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.distinct(3)
thf(fact_295_lambda__calculus_Olambda_Odistinct_I17_J,axiom,
! [X32: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Lam @ X32 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(17)
thf(fact_296_lambda__calculus_OArr_Osimps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Lam @ T2 ) )
= ( lambda_Arr @ T2 ) ) ).
% lambda_calculus.Arr.simps(3)
thf(fact_297_lambda__calculus_OSrc_Osimps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Src @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda_Src @ T2 ) ) ) ).
% lambda_calculus.Src.simps(3)
thf(fact_298_lambda__calculus_OIde_Osimps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Ide @ ( lambda_Lam @ T2 ) )
= ( lambda_Ide @ T2 ) ) ).
% lambda_calculus.Ide.simps(3)
thf(fact_299_lambda__calculus_Osubs_Osimps_I2_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( lambda_subs @ ( lambda_Lam @ T2 ) @ ( lambda_Lam @ T3 ) )
= ( lambda_subs @ T2 @ T3 ) ) ).
% lambda_calculus.subs.simps(2)
thf(fact_300_lambda__calculus_OJoin_Osimps_I2_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ T2 ) @ ( lambda_Lam @ T3 ) )
= ( lambda_Lam @ ( lambda_Join @ T2 @ T3 ) ) ) ).
% lambda_calculus.Join.simps(2)
thf(fact_301_lambda__calculus_OTrg_Osimps_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_Trg @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda_Trg @ T2 ) ) ) ).
% lambda_calculus.Trg.simps(2)
thf(fact_302_lambda__calculus_Ois__head__reduction_Osimps_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda5343566265203084875uction @ ( lambda_Lam @ T2 ) )
= ( lambda5343566265203084875uction @ T2 ) ) ).
% lambda_calculus.is_head_reduction.simps(2)
thf(fact_303_lambda__calculus_Oelementary__reduction_Osimps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Lam @ T2 ) )
= ( lambda2670285437103676873uction @ T2 ) ) ).
% lambda_calculus.elementary_reduction.simps(3)
thf(fact_304_lambda__calculus_Oleftmost__strategy_Osimps_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda4011002726600638587rategy @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda4011002726600638587rategy @ T2 ) ) ) ).
% lambda_calculus.leftmost_strategy.simps(2)
thf(fact_305_lambda__calculus_Ois__internal__reduction_Osimps_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda2591989677915507790uction @ ( lambda_Lam @ T2 ) )
= ( lambda2591989677915507790uction @ T2 ) ) ).
% lambda_calculus.is_internal_reduction.simps(2)
thf(fact_306_lambda__calculus_Oparallel__strategy_Osimps_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda8522482075158074392rategy @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda8522482075158074392rategy @ T2 ) ) ) ).
% lambda_calculus.parallel_strategy.simps(2)
thf(fact_307_lambda__calculus_Ocontains__head__reduction_Osimps_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_Lam @ T2 ) )
= ( lambda8135948770199036598uction @ T2 ) ) ).
% lambda_calculus.contains_head_reduction.simps(2)
thf(fact_308_lambda__calculus_Ohead__strategy_Osimps_I2_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda_head_strategy @ T2 ) ) ) ).
% lambda_calculus.head_strategy.simps(2)
thf(fact_309_lambda__calculus_Ohead__redex_Osimps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda_head_redex @ T2 ) ) ) ).
% lambda_calculus.head_redex.simps(3)
thf(fact_310_lambda__calculus_Olambda_Odisc_I8_J,axiom,
! [X32: lambda_lambda] : ( lambda_is_Lam @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(8)
thf(fact_311_lambda__calculus_Olambda_OdiscI_I2_J,axiom,
! [Lambda: lambda_lambda,X32: lambda_lambda] :
( ( Lambda
= ( lambda_Lam @ X32 ) )
=> ( lambda_is_Lam @ Lambda ) ) ).
% lambda_calculus.lambda.discI(2)
thf(fact_312_lambda__calculus_Ois__Lam__def,axiom,
( lambda_is_Lam
= ( ^ [Lambda2: lambda_lambda] :
? [X33: lambda_lambda] :
( Lambda2
= ( lambda_Lam @ X33 ) ) ) ) ).
% lambda_calculus.is_Lam_def
thf(fact_313_lambda__calculus_Obottom__up__redex_Osimps_I3_J,axiom,
! [M: lambda_lambda] :
( ( lambda5654762366400029632_redex @ ( lambda_Lam @ M ) )
= ( lambda_Lam @ ( lambda5654762366400029632_redex @ M ) ) ) ).
% lambda_calculus.bottom_up_redex.simps(3)
thf(fact_314_lambda__calculus_Olambda_Osel_I2_J,axiom,
! [X32: lambda_lambda] :
( ( lambda_un_Lam @ ( lambda_Lam @ X32 ) )
= X32 ) ).
% lambda_calculus.lambda.sel(2)
thf(fact_315_lambda__calculus_Oresid_Osimps_I27_J,axiom,
! [V: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(27)
thf(fact_316_lambda__calculus_Oresid_Osimps_I8_J,axiom,
! [V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(8)
thf(fact_317_lambda__calculus_Oresid_Osimps_I2_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T2 ) @ ( lambda_Lam @ T3 ) )
= lambda_Nil ) )
& ( ( ( lambda_resid @ T2 @ T3 )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T2 ) @ ( lambda_Lam @ T3 ) )
= ( lambda_Lam @ ( lambda_resid @ T2 @ T3 ) ) ) ) ) ).
% lambda_calculus.resid.simps(2)
thf(fact_318_lambda__calculus_Osubs_Osimps_I9_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ V ) @ ( lambda_App @ Va @ Vb ) ) ).
% lambda_calculus.subs.simps(9)
thf(fact_319_lambda__calculus_Osubs_Osimps_I13_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ V @ Va ) @ ( lambda_Lam @ Vb ) ) ).
% lambda_calculus.subs.simps(13)
thf(fact_320_lambda__calculus_Osubs_Osimps_I25_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ Va @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.subs.simps(25)
thf(fact_321_lambda__calculus_Osubs_Osimps_I29_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.subs.simps(29)
thf(fact_322_lambda__calculus_Osubs_Osimps_I7_J,axiom,
! [V: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ V ) @ lambda_Nil ) ).
% lambda_calculus.subs.simps(7)
thf(fact_323_lambda__calculus_Osubs_Osimps_I23_J,axiom,
! [V: lambda_lambda] :
~ ( lambda_subs @ lambda_Nil @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.subs.simps(23)
thf(fact_324_lambda__calculus_Osubs_Osimps_I10_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ V ) @ ( lambda_Beta @ Va @ Vb ) ) ).
% lambda_calculus.subs.simps(10)
thf(fact_325_lambda__calculus_Osubs_Osimps_I20_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ V @ Va ) @ ( lambda_Lam @ Vb ) ) ).
% lambda_calculus.subs.simps(20)
thf(fact_326_lambda__calculus_Osubs_Osimps_I26_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ Va @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.subs.simps(26)
thf(fact_327_lambda__calculus_Osubs_Osimps_I33_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.subs.simps(33)
thf(fact_328_lambda__calculus_OJoin_Osimps_I8_J,axiom,
! [V: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ V ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(8)
thf(fact_329_lambda__calculus_OJoin_Osimps_I39_J,axiom,
! [V: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(39)
thf(fact_330_lambda__calculus_Ois__head__reduction_Osimps_I3_J,axiom,
! [Uv: lambda_lambda,Uw: lambda_lambda] :
~ ( lambda5343566265203084875uction @ ( lambda_App @ ( lambda_Lam @ Uv ) @ Uw ) ) ).
% lambda_calculus.is_head_reduction.simps(3)
thf(fact_331_lambda__calculus_Ocontains__head__reduction_Osimps_I3_J,axiom,
! [Uv: lambda_lambda,Uw: lambda_lambda] :
~ ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Lam @ Uv ) @ Uw ) ) ).
% lambda_calculus.contains_head_reduction.simps(3)
thf(fact_332_lambda__calculus_Oresid_Osimps_I10_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_App @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(10)
thf(fact_333_lambda__calculus_Oresid_Osimps_I14_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(14)
thf(fact_334_lambda__calculus_Oresid_Osimps_I29_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(29)
thf(fact_335_lambda__calculus_Oresid_Osimps_I33_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(33)
thf(fact_336_lambda__calculus_Oresid_Osimps_I11_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Beta @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(11)
thf(fact_337_lambda__calculus_Oresid_Osimps_I21_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(21)
thf(fact_338_lambda__calculus_Oresid_Osimps_I30_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(30)
thf(fact_339_lambda__calculus_Oresid_Osimps_I40_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(40)
thf(fact_340_lambda__calculus_OConD_I2_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,V2: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_Beta @ V2 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Lam @ V2 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(2)
thf(fact_341_lambda__calculus_OSrc_Osimps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Src @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T2 ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.Src.simps(5)
thf(fact_342_lambda__calculus_OSrc__eq__iff_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T3 @ U2 ) ) )
= ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ T3 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U2 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(4)
thf(fact_343_lambda__calculus_Osubs_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( lambda_subs @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) @ ( lambda_Beta @ T3 @ U2 ) )
= ( ( lambda_subs @ T2 @ T3 )
& ( lambda_subs @ U @ U2 ) ) ) ).
% lambda_calculus.subs.simps(4)
thf(fact_344_lambda__calculus_OJoin_Osimps_I10_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ V ) @ ( lambda_App @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(10)
thf(fact_345_lambda__calculus_OJoin_Osimps_I14_J,axiom,
! [Va: lambda_lambda,V: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ lambda_Nil @ Va ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(14)
thf(fact_346_lambda__calculus_OJoin_Osimps_I22_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda,V: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ Va ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(22)
thf(fact_347_lambda__calculus_OJoin_Osimps_I30_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(30)
thf(fact_348_lambda__calculus_OJoin_Osimps_I41_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(41)
thf(fact_349_lambda__calculus_OJoin_Osimps_I45_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(45)
thf(fact_350_lambda__calculus_OJoin_Osimps_I48_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ V ) @ ( lambda_App @ lambda_Nil @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(48)
thf(fact_351_lambda__calculus_OJoin_Osimps_I56_J,axiom,
! [V: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ V ) @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(56)
thf(fact_352_lambda__calculus_OJoin_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ T2 @ U ) @ ( lambda_App @ ( lambda_Lam @ T3 ) @ U2 ) )
= ( lambda_Beta @ ( lambda_Join @ T2 @ T3 ) @ ( lambda_Join @ U @ U2 ) ) ) ).
% lambda_calculus.Join.simps(4)
thf(fact_353_lambda__calculus_OJoin_Osimps_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) @ ( lambda_Beta @ T3 @ U2 ) )
= ( lambda_Beta @ ( lambda_Join @ T2 @ T3 ) @ ( lambda_Join @ U @ U2 ) ) ) ).
% lambda_calculus.Join.simps(3)
thf(fact_354_lambda__calculus_OJoin_Osimps_I11_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ V ) @ ( lambda_Beta @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(11)
thf(fact_355_lambda__calculus_OJoin_Osimps_I33_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(33)
thf(fact_356_lambda__calculus_OJoin_Osimps_I42_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(42)
thf(fact_357_lambda__calculus_OJoin_Osimps_I64_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(64)
thf(fact_358_lambda__calculus_Oleftmost__strategy_Osimps_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda4011002726600638587rategy @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) )
= ( lambda_Beta @ T2 @ U ) ) ).
% lambda_calculus.leftmost_strategy.simps(3)
thf(fact_359_lambda__calculus_Oparallel__strategy_Osimps_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda8522482075158074392rategy @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) )
= ( lambda_Beta @ ( lambda8522482075158074392rategy @ T2 ) @ ( lambda8522482075158074392rategy @ U ) ) ) ).
% lambda_calculus.parallel_strategy.simps(3)
thf(fact_360_lambda__calculus_Ois__internal__reduction_Osimps_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda2591989677915507790uction @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) )
= ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.is_internal_reduction.simps(3)
thf(fact_361_lambda__calculus_Ohead__redex_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T2 ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(4)
thf(fact_362_lambda__calculus_OConD_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V2: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T2 @ U ) @ ( lambda_Beta @ V2 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T2 @ ( lambda_Lam @ V2 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(4)
thf(fact_363_lambda__calculus_OConD_I3_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_App @ T3 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ T3 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(3)
thf(fact_364_lambda__calculus_Oresid_Osimps_I6_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T2 @ U ) @ ( lambda_App @ ( lambda_Lam @ T3 ) @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T2 @ U ) @ ( lambda_App @ ( lambda_Lam @ T3 ) @ U2 ) )
= ( lambda_Beta @ ( lambda_resid @ T2 @ T3 ) @ ( lambda_resid @ U @ U2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(6)
thf(fact_365_lambda__calculus_OArr_Oelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda_Arr @ X4 )
=> ( ( X4 != lambda_Nil )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( lambda_Arr @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(3)
thf(fact_366_lambda__calculus_OIde_Oelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda_Ide @ X4 )
=> ( ( X4 != lambda_Nil )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( lambda_Ide @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(3)
thf(fact_367_lambda__calculus_OJoin_Osimps_I60_J,axiom,
! [V: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ V ) @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(60)
thf(fact_368_lambda__calculus_OJoin_Osimps_I26_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda,V: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ Va ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(26)
thf(fact_369_lambda__calculus_Oelementary__reduction_Oelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda2670285437103676873uction @ X4 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ~ ( lambda2670285437103676873uction @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ~ ( ( ( lambda2670285437103676873uction @ T5 )
& ( lambda_Ide @ U3 ) )
| ( ( lambda_Ide @ T5 )
& ( lambda2670285437103676873uction @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.elims(2)
thf(fact_370_lambda__calculus_Ohead__strategy_Osimps_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) )
= ( lambda_Beta @ ( lambda_Src @ T2 ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(3)
thf(fact_371_lambda__calculus_Obottom__up__redex_Osimps_I5_J,axiom,
! [M: lambda_lambda,N3: lambda_lambda] :
( ( ~ ( lambda_Ide @ M )
=> ( ( lambda5654762366400029632_redex @ ( lambda_Beta @ M @ N3 ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda5654762366400029632_redex @ M ) ) @ ( lambda_Src @ N3 ) ) ) )
& ( ( lambda_Ide @ M )
=> ( ( ~ ( lambda_Ide @ N3 )
=> ( ( lambda5654762366400029632_redex @ ( lambda_Beta @ M @ N3 ) )
= ( lambda_App @ ( lambda_Lam @ M ) @ ( lambda5654762366400029632_redex @ N3 ) ) ) )
& ( ( lambda_Ide @ N3 )
=> ( ( lambda5654762366400029632_redex @ ( lambda_Beta @ M @ N3 ) )
= ( lambda_Beta @ M @ N3 ) ) ) ) ) ) ).
% lambda_calculus.bottom_up_redex.simps(5)
thf(fact_372_lambda__calculus_Oresid__head__strategy__Src_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda2591989677915507790uction @ T2 )
=> ( ( lambda5343566265203084875uction @ U )
=> ( ( seq_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( lambda_resid @ ( lambda_head_strategy @ ( lambda_Src @ T2 ) ) @ T2 )
= U ) ) ) ) ).
% lambda_calculus.resid_head_strategy_Src(1)
thf(fact_373_Coinitial__cases,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Arr @ T3 )
=> ( ( ( lambda_Src @ T2 )
= ( lambda_Src @ T3 ) )
=> ( ( ( T2 = lambda_Nil )
& ( T3 = lambda_Nil ) )
| ? [X2: nat] :
( ( T2
= ( lambda_Var @ X2 ) )
& ( T3
= ( lambda_Var @ X2 ) ) )
| ? [U3: lambda_lambda,U5: lambda_lambda] :
( ( T2
= ( lambda_Lam @ U3 ) )
& ( T3
= ( lambda_Lam @ U5 ) ) )
| ? [U3: lambda_lambda,V3: lambda_lambda,U5: lambda_lambda,V4: lambda_lambda] :
( ( T2
= ( lambda_App @ U3 @ V3 ) )
& ( T3
= ( lambda_App @ U5 @ V4 ) ) )
| ? [U3: lambda_lambda,V3: lambda_lambda,U5: lambda_lambda,V4: lambda_lambda] :
( ( T2
= ( lambda_Beta @ U3 @ V3 ) )
& ( T3
= ( lambda_Beta @ U5 @ V4 ) ) )
| ? [U3: lambda_lambda,V3: lambda_lambda,U5: lambda_lambda,V4: lambda_lambda] :
( ( T2
= ( lambda_App @ ( lambda_Lam @ U3 ) @ V3 ) )
& ( T3
= ( lambda_Beta @ U5 @ V4 ) ) )
| ? [U3: lambda_lambda,V3: lambda_lambda,U5: lambda_lambda,V4: lambda_lambda] :
( ( T2
= ( lambda_Beta @ U3 @ V3 ) )
& ( T3
= ( lambda_App @ ( lambda_Lam @ U5 ) @ V4 ) ) ) ) ) ) ) ).
% Coinitial_cases
thf(fact_374_composable__imp__seq,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T2 @ U )
=> ( seq_lambda_lambda @ lambda_resid @ T2 @ U ) ) ).
% composable_imp_seq
thf(fact_375_ArrE,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ! [I2: nat] :
( T2
!= ( lambda_Var @ I2 ) )
=> ( ! [U3: lambda_lambda] :
( T2
!= ( lambda_Lam @ U3 ) )
=> ( ! [U3: lambda_lambda,V3: lambda_lambda] :
( T2
!= ( lambda_App @ U3 @ V3 ) )
=> ~ ! [U3: lambda_lambda,V3: lambda_lambda] :
( T2
!= ( lambda_Beta @ U3 @ V3 ) ) ) ) ) ) ).
% ArrE
thf(fact_376_coterminal__iff__con__trg,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T2 @ U )
= ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T2 ) @ ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ).
% coterminal_iff_con_trg
thf(fact_377_coterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T2 @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T2 )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( trg_lambda_lambda @ lambda_resid @ T2 )
= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ) ).
% coterminal_iff\<^sub>W\<^sub>E
thf(fact_378_coterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T2 @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( trg_lambda_lambda @ lambda_resid @ T2 )
!= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ) ) ).
% coterminalE\<^sub>W\<^sub>E
thf(fact_379_lambda__calculus_Obottom__up__redex_Oelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda5654762366400029632_redex @ X4 )
= Y2 )
=> ( ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ! [X2: nat] :
( ( X4
= ( lambda_Var @ X2 ) )
=> ( Y2
!= ( lambda_Var @ X2 ) ) )
=> ( ! [M2: lambda_lambda] :
( ( X4
= ( lambda_Lam @ M2 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda5654762366400029632_redex @ M2 ) ) ) )
=> ( ! [M2: lambda_lambda,N4: lambda_lambda] :
( ( X4
= ( lambda_App @ M2 @ N4 ) )
=> ~ ( ( ~ ( lambda_Ide @ M2 )
=> ( Y2
= ( lambda_App @ ( lambda5654762366400029632_redex @ M2 ) @ ( lambda_Src @ N4 ) ) ) )
& ( ( lambda_Ide @ M2 )
=> ( Y2
= ( lambda_App @ M2 @ ( lambda5654762366400029632_redex @ N4 ) ) ) ) ) )
=> ~ ! [M2: lambda_lambda,N4: lambda_lambda] :
( ( X4
= ( lambda_Beta @ M2 @ N4 ) )
=> ~ ( ( ~ ( lambda_Ide @ M2 )
=> ( Y2
= ( lambda_App @ ( lambda_Lam @ ( lambda5654762366400029632_redex @ M2 ) ) @ ( lambda_Src @ N4 ) ) ) )
& ( ( lambda_Ide @ M2 )
=> ( ( ~ ( lambda_Ide @ N4 )
=> ( Y2
= ( lambda_App @ ( lambda_Lam @ M2 ) @ ( lambda5654762366400029632_redex @ N4 ) ) ) )
& ( ( lambda_Ide @ N4 )
=> ( Y2
= ( lambda_Beta @ M2 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.bottom_up_redex.elims
thf(fact_380_resid__reflects__con,axiom,
! [T2: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ U @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ V ) @ ( lambda_resid @ U @ V ) )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ U ) ) ) ) ).
% resid_reflects_con
thf(fact_381_con__sym,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( con_lambda_lambda @ lambda_resid @ U @ T2 ) ) ).
% con_sym
thf(fact_382_prfx__Var__iff,axiom,
! [U: lambda_lambda,I: nat] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Var @ I ) ) )
= ( U
= ( lambda_Var @ I ) ) ) ).
% prfx_Var_iff
thf(fact_383_App__Var__contains__no__head__reduction,axiom,
! [X4: nat,U: lambda_lambda] :
~ ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Var @ X4 ) @ U ) ) ).
% App_Var_contains_no_head_reduction
thf(fact_384_prfx__reflexive,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T2 ) ) ) ).
% prfx_reflexive
thf(fact_385_ide__implies__arr,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( arr_lambda_lambda @ lambda_resid @ A ) ) ).
% ide_implies_arr
thf(fact_386_cong__reflexive,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T2 ) ) ) ) ).
% cong_reflexive
thf(fact_387_resid__ide__arr,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ T2 )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ T2 ) ) ) ) ).
% resid_ide_arr
thf(fact_388_resid__arr__ide,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ T2 @ A )
=> ( ( lambda_resid @ T2 @ A )
= T2 ) ) ) ).
% resid_arr_ide
thf(fact_389_prfx__implies__con,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ U ) ) ).
% prfx_implies_con
thf(fact_390_ide__imp__con__iff__cong,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) ) ) ) ) ).
% ide_imp_con_iff_cong
thf(fact_391_ide__def,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
= ( ( con_lambda_lambda @ lambda_resid @ A @ A )
& ( ( lambda_resid @ A @ A )
= A ) ) ) ).
% ide_def
thf(fact_392_ideE,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ~ ( ( con_lambda_lambda @ lambda_resid @ A @ A )
=> ( ( lambda_resid @ A @ A )
!= A ) ) ) ).
% ideE
thf(fact_393_con__transitive__on__ide,axiom,
! [A: lambda_lambda,B: lambda_lambda,C2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( ide_lambda_lambda @ lambda_resid @ C2 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ B )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ C2 )
=> ( con_lambda_lambda @ lambda_resid @ A @ C2 ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_394_con__target,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
=> ( ( con_lambda_lambda @ lambda_resid @ U @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) @ ( lambda_resid @ V @ U ) ) ) ) ).
% con_target
thf(fact_395_con__imp__coinitial__ax,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ? [A5: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A5 )
& ( con_lambda_lambda @ lambda_resid @ A5 @ T2 )
& ( con_lambda_lambda @ lambda_resid @ A5 @ U ) ) ) ).
% con_imp_coinitial_ax
thf(fact_396_con__ide__are__eq,axiom,
! [A: lambda_lambda,A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A4 )
=> ( A = A4 ) ) ) ) ).
% con_ide_are_eq
thf(fact_397_cong__subst__right_I1_J,axiom,
! [U: lambda_lambda,U2: lambda_lambda,T2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ U2 ) ) ) ).
% cong_subst_right(1)
thf(fact_398_cong__subst__right_I2_J,axiom,
! [U: lambda_lambda,U2: lambda_lambda,T2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T2 @ U ) @ ( lambda_resid @ T2 @ U2 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T2 @ U2 ) @ ( lambda_resid @ T2 @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_399_cong__subst__left_I1_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T3 @ T2 ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( con_lambda_lambda @ lambda_resid @ T3 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_400_cong__subst__left_I2_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T3 @ T2 ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T2 @ U ) @ ( lambda_resid @ T3 @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T3 @ U ) @ ( lambda_resid @ T2 @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_401_con__implies__arr_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( arr_lambda_lambda @ lambda_resid @ U ) ) ).
% con_implies_arr(2)
thf(fact_402_con__implies__arr_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( arr_lambda_lambda @ lambda_resid @ T2 ) ) ).
% con_implies_arr(1)
thf(fact_403_arrE,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ T2 ) ) ).
% arrE
thf(fact_404_arr__def,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
= ( con_lambda_lambda @ lambda_resid @ T2 @ T2 ) ) ).
% arr_def
thf(fact_405_arr__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) ) ) ).
% arr_resid
thf(fact_406_arr__resid__iff__con,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( con_lambda_lambda @ lambda_resid @ T2 @ U ) ) ).
% arr_resid_iff_con
thf(fact_407_elementary__reduction__is__arr,axiom,
! [T2: lambda_lambda] :
( ( lambda2670285437103676873uction @ T2 )
=> ( arr_lambda_lambda @ lambda_resid @ T2 ) ) ).
% elementary_reduction_is_arr
thf(fact_408_trg__resid__sym,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) ) ) ).
% trg_resid_sym
thf(fact_409_join__of__arr__self,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ T2 @ T2 ) ) ).
% join_of_arr_self
thf(fact_410_join__of__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ V @ W )
=> ( join_o1386658102433174083lambda @ lambda_resid @ ( lambda_resid @ T2 @ V ) @ ( lambda_resid @ U @ V ) @ ( lambda_resid @ W @ V ) ) ) ) ).
% join_of_resid
thf(fact_411_con__with__join__of__iff_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ W )
=> ( ( ( con_lambda_lambda @ lambda_resid @ U @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T2 @ U ) ) )
=> ( con_lambda_lambda @ lambda_resid @ W @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_412_con__with__join__of__iff_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ T2 @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T2 ) @ ( lambda_resid @ U @ T2 ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_413_confluence,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ U ) ) ).
% confluence
thf(fact_414_con__imp__coinitial,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U ) ) ).
% con_imp_coinitial
thf(fact_415_composableD_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T2 @ U )
=> ( arr_lambda_lambda @ lambda_resid @ U ) ) ).
% composableD(2)
thf(fact_416_composableD_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T2 @ U )
=> ( arr_lambda_lambda @ lambda_resid @ T2 ) ) ).
% composableD(1)
thf(fact_417_joinable__implies__con,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T2 @ U )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ U ) ) ).
% joinable_implies_con
thf(fact_418_has__joins,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( joinab2519788767067900224lambda @ lambda_resid @ T2 @ U ) ) ).
% has_joins
thf(fact_419_ide__iff__trg__self,axiom,
! [A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A )
= ( ( trg_lambda_lambda @ lambda_resid @ A )
= A ) ) ) ).
% ide_iff_trg_self
thf(fact_420_ide__trg,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ide_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T2 ) ) ) ).
% ide_trg
thf(fact_421_lambda__calculus_Olambda_Oinject_I1_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( lambda_Var @ X22 )
= ( lambda_Var @ Y22 ) )
= ( X22 = Y22 ) ) ).
% lambda_calculus.lambda.inject(1)
thf(fact_422_con__char,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
= ( ( lambda_resid @ T2 @ U )
!= lambda_Nil ) ) ).
% con_char
thf(fact_423_arr__char,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
= ( lambda_Arr @ T2 ) ) ).
% arr_char
thf(fact_424_ideI,axiom,
! [A: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ A @ A )
=> ( ( ( lambda_resid @ A @ A )
= A )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ) ).
% ideI
thf(fact_425_arrI,axiom,
! [T2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ T2 )
=> ( arr_lambda_lambda @ lambda_resid @ T2 ) ) ).
% arrI
thf(fact_426_arr__trg__iff__arr,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T2 ) )
= ( arr_lambda_lambda @ lambda_resid @ T2 ) ) ).
% arr_trg_iff_arr
thf(fact_427_coterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T2 )
= ( trg_lambda_lambda @ lambda_resid @ U ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ T2 @ U ) ) ) ).
% coterminalI\<^sub>W\<^sub>E
thf(fact_428_lambda__calculus_Olambda_Odistinct_I11_J,axiom,
! [X22: nat,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Var @ X22 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(11)
thf(fact_429_lambda__calculus_Olambda_Odistinct_I1_J,axiom,
! [X22: nat] :
( lambda_Nil
!= ( lambda_Var @ X22 ) ) ).
% lambda_calculus.lambda.distinct(1)
thf(fact_430_lambda__calculus_Olambda_Odistinct_I13_J,axiom,
! [X22: nat,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Var @ X22 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(13)
thf(fact_431_lambda__calculus_Olambda_Odistinct_I9_J,axiom,
! [X22: nat,X32: lambda_lambda] :
( ( lambda_Var @ X22 )
!= ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.distinct(9)
thf(fact_432_lambda__calculus_OArr_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Arr @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Arr.simps(2)
thf(fact_433_lambda__calculus_OSrc__eq__iff_I1_J,axiom,
! [I: nat,I3: nat] :
( ( ( lambda_Src @ ( lambda_Var @ I ) )
= ( lambda_Src @ ( lambda_Var @ I3 ) ) )
= ( I = I3 ) ) ).
% lambda_calculus.Src_eq_iff(1)
thf(fact_434_lambda__calculus_OSrc_Osimps_I2_J,axiom,
! [I: nat] :
( ( lambda_Src @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Src.simps(2)
thf(fact_435_lambda__calculus_OIde_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Ide @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Ide.simps(2)
thf(fact_436_lambda__calculus_Osubs_Osimps_I1_J,axiom,
! [I: nat,I3: nat] :
( ( lambda_subs @ ( lambda_Var @ I ) @ ( lambda_Var @ I3 ) )
= ( I = I3 ) ) ).
% lambda_calculus.subs.simps(1)
thf(fact_437_lambda__calculus_OTrg_Osimps_I1_J,axiom,
! [I: nat] :
( ( lambda_Trg @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Trg.simps(1)
thf(fact_438_lambda__calculus_Ois__head__reduction_Osimps_I1_J,axiom,
! [Uu: nat] :
~ ( lambda5343566265203084875uction @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.is_head_reduction.simps(1)
thf(fact_439_lambda__calculus_Oelementary__reduction_Osimps_I2_J,axiom,
! [Uu: nat] :
~ ( lambda2670285437103676873uction @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.elementary_reduction.simps(2)
thf(fact_440_lambda__calculus_Oleftmost__strategy_Osimps_I1_J,axiom,
! [X4: nat] :
( ( lambda4011002726600638587rategy @ ( lambda_Var @ X4 ) )
= ( lambda_Var @ X4 ) ) ).
% lambda_calculus.leftmost_strategy.simps(1)
thf(fact_441_lambda__calculus_Ois__internal__reduction_Osimps_I1_J,axiom,
! [Uu: nat] : ( lambda2591989677915507790uction @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.is_internal_reduction.simps(1)
thf(fact_442_lambda__calculus_Oparallel__strategy_Osimps_I1_J,axiom,
! [I: nat] :
( ( lambda8522482075158074392rategy @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.parallel_strategy.simps(1)
thf(fact_443_lambda__calculus_Ocontains__head__reduction_Osimps_I1_J,axiom,
! [Uu: nat] :
~ ( lambda8135948770199036598uction @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.contains_head_reduction.simps(1)
thf(fact_444_lambda__calculus_Ohead__strategy_Osimps_I1_J,axiom,
! [I: nat] :
( ( lambda_head_strategy @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.head_strategy.simps(1)
thf(fact_445_lambda__calculus_Ohead__redex_Osimps_I2_J,axiom,
! [X4: nat] :
( ( lambda_head_redex @ ( lambda_Var @ X4 ) )
= ( lambda_Var @ X4 ) ) ).
% lambda_calculus.head_redex.simps(2)
thf(fact_446_lambda__calculus_Obottom__up__redex_Osimps_I2_J,axiom,
! [X4: nat] :
( ( lambda5654762366400029632_redex @ ( lambda_Var @ X4 ) )
= ( lambda_Var @ X4 ) ) ).
% lambda_calculus.bottom_up_redex.simps(2)
thf(fact_447_lambda__calculus_Olambda_Odisc_I7_J,axiom,
! [X22: nat] :
~ ( lambda_is_Lam @ ( lambda_Var @ X22 ) ) ).
% lambda_calculus.lambda.disc(7)
thf(fact_448_lambda__calculus_Oresid_Osimps_I1_J,axiom,
! [I: nat,I3: nat] :
( ( ( I = I3 )
=> ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I3 ) )
= ( lambda_Var @ I ) ) )
& ( ( I != I3 )
=> ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I3 ) )
= lambda_Nil ) ) ) ).
% lambda_calculus.resid.simps(1)
thf(fact_449_lambda__calculus_Osubs_Osimps_I28_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Var @ Vb ) @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.subs.simps(28)
thf(fact_450_lambda__calculus_Osubs_Osimps_I12_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
~ ( lambda_subs @ ( lambda_App @ V @ Va ) @ ( lambda_Var @ Vb ) ) ).
% lambda_calculus.subs.simps(12)
thf(fact_451_lambda__calculus_Osubs_Osimps_I32_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.subs.simps(32)
thf(fact_452_lambda__calculus_Osubs_Osimps_I19_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
~ ( lambda_subs @ ( lambda_Beta @ V @ Va ) @ ( lambda_Var @ Vb ) ) ).
% lambda_calculus.subs.simps(19)
thf(fact_453_lambda__calculus_OJoin_Osimps_I1_J,axiom,
! [X4: nat,X6: nat] :
( ( ( X4 = X6 )
=> ( ( lambda_Join @ ( lambda_Var @ X4 ) @ ( lambda_Var @ X6 ) )
= ( lambda_Var @ X4 ) ) )
& ( ( X4 != X6 )
=> ( ( lambda_Join @ ( lambda_Var @ X4 ) @ ( lambda_Var @ X6 ) )
= lambda_Nil ) ) ) ).
% lambda_calculus.Join.simps(1)
thf(fact_454_lambda__calculus_Osubs_Osimps_I24_J,axiom,
! [Va: nat,V: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Var @ Va ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.subs.simps(24)
thf(fact_455_lambda__calculus_Osubs_Osimps_I8_J,axiom,
! [V: lambda_lambda,Va: nat] :
~ ( lambda_subs @ ( lambda_Lam @ V ) @ ( lambda_Var @ Va ) ) ).
% lambda_calculus.subs.simps(8)
thf(fact_456_lambda__calculus_Oparallel__strategy_Osimps_I5_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda8522482075158074392rategy @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_Var @ V ) ) @ ( lambda8522482075158074392rategy @ U ) ) ) ).
% lambda_calculus.parallel_strategy.simps(5)
thf(fact_457_lambda__calculus_Oresid_Osimps_I32_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(32)
thf(fact_458_lambda__calculus_Oresid_Osimps_I13_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(13)
thf(fact_459_lambda__calculus_Oresid_Osimps_I39_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(39)
thf(fact_460_lambda__calculus_Oresid_Osimps_I20_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(20)
thf(fact_461_lambda__calculus_Oresid_Osimps_I28_J,axiom,
! [Va: nat,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Va ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(28)
thf(fact_462_lambda__calculus_Oresid_Osimps_I9_J,axiom,
! [V: lambda_lambda,Va: nat] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Var @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(9)
thf(fact_463_lambda__calculus_OIde_Oelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda_Ide @ X4 )
=> ( ! [Uu2: nat] :
( X4
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ~ ( lambda_Ide @ T5 ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(2)
thf(fact_464_lambda__calculus_Osubs_Osimps_I15_J,axiom,
! [Vd: nat,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_Var @ Vd ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) ) ).
% lambda_calculus.subs.simps(15)
thf(fact_465_lambda__calculus_Osubs_Osimps_I35_J,axiom,
! [Vd: nat,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.subs.simps(35)
thf(fact_466_lambda__calculus_OJoin_Osimps_I55_J,axiom,
! [V: nat,Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Var @ V ) @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(55)
thf(fact_467_lambda__calculus_OJoin_Osimps_I51_J,axiom,
! [V: nat,Vb: nat,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Var @ V ) @ ( lambda_App @ ( lambda_Var @ Vb ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(51)
thf(fact_468_lambda__calculus_OJoin_Osimps_I50_J,axiom,
! [Vb: nat,Va: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ ( lambda_App @ ( lambda_Var @ Vb ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(50)
thf(fact_469_lambda__calculus_OJoin_Osimps_I47_J,axiom,
! [V: nat,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Var @ V ) @ ( lambda_App @ lambda_Nil @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(47)
thf(fact_470_lambda__calculus_OJoin_Osimps_I44_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Var @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(44)
thf(fact_471_lambda__calculus_OJoin_Osimps_I29_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
( ( lambda_Join @ ( lambda_App @ V @ Va ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(29)
thf(fact_472_lambda__calculus_OJoin_Osimps_I21_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda,V: nat] :
( ( lambda_Join @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ Va ) @ ( lambda_Var @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(21)
thf(fact_473_lambda__calculus_OJoin_Osimps_I17_J,axiom,
! [Vb: nat,Va: lambda_lambda,V: nat] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Var @ Vb ) @ Va ) @ ( lambda_Var @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(17)
thf(fact_474_lambda__calculus_OJoin_Osimps_I16_J,axiom,
! [Vb: nat,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Var @ Vb ) @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(16)
thf(fact_475_lambda__calculus_OJoin_Osimps_I13_J,axiom,
! [Va: lambda_lambda,V: nat] :
( ( lambda_Join @ ( lambda_App @ lambda_Nil @ Va ) @ ( lambda_Var @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(13)
thf(fact_476_lambda__calculus_OJoin_Osimps_I63_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(63)
thf(fact_477_lambda__calculus_OJoin_Osimps_I32_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
( ( lambda_Join @ ( lambda_Beta @ V @ Va ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(32)
thf(fact_478_lambda__calculus_OJoin_Osimps_I40_J,axiom,
! [Va: nat,V: lambda_lambda] :
( ( lambda_Join @ ( lambda_Var @ Va ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(40)
thf(fact_479_lambda__calculus_OJoin_Osimps_I9_J,axiom,
! [V: lambda_lambda,Va: nat] :
( ( lambda_Join @ ( lambda_Lam @ V ) @ ( lambda_Var @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(9)
thf(fact_480_lambda__calculus_Ois__internal__reduction_Osimps_I5_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda2591989677915507790uction @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( ( lambda2591989677915507790uction @ ( lambda_Var @ V ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.is_internal_reduction.simps(5)
thf(fact_481_lambda__calculus_Ocontains__head__reduction_Osimps_I5_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( ( lambda8135948770199036598uction @ ( lambda_Var @ V ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(5)
thf(fact_482_lambda__calculus_Ois__head__reduction_Osimps_I5_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda5343566265203084875uction @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( ( lambda5343566265203084875uction @ ( lambda_Var @ V ) )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.is_head_reduction.simps(5)
thf(fact_483_lambda__calculus_Oleftmost__strategy_Osimps_I5_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V ) ) )
=> ( ( lambda4011002726600638587rategy @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V ) ) @ U ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V ) ) )
=> ( ( lambda4011002726600638587rategy @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( lambda_App @ ( lambda_Var @ V ) @ ( lambda4011002726600638587rategy @ U ) ) ) ) ) ).
% lambda_calculus.leftmost_strategy.simps(5)
thf(fact_484_lambda__calculus_Ohead__strategy_Osimps_I5_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Var @ V ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(5)
thf(fact_485_lambda__calculus_Ohead__redex_Osimps_I6_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Var @ V ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(6)
thf(fact_486_lambda__calculus_Oresid_Osimps_I16_J,axiom,
! [Vd: nat,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vd ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(16)
thf(fact_487_lambda__calculus_Oresid_Osimps_I23_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: nat,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(23)
thf(fact_488_lambda__calculus_Oresid_Osimps_I35_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: nat,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_Var @ Vd ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(35)
thf(fact_489_lambda__calculus_Oresid_Osimps_I42_J,axiom,
! [Vd: nat,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(42)
thf(fact_490_lambda__calculus_Ocontains__head__reduction_Ocases,axiom,
! [X4: lambda_lambda] :
( ! [Uu2: nat] :
( X4
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T5: lambda_lambda] :
( X4
!= ( lambda_Lam @ T5 ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ( ! [U3: lambda_lambda] :
( X4
!= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_Beta @ T5 @ U3 ) )
=> ( X4 = lambda_Nil ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.cases
thf(fact_491_lambda__calculus_Oelementary__reduction_Ocases,axiom,
! [X4: lambda_lambda] :
( ( X4 != lambda_Nil )
=> ( ! [Uu2: nat] :
( X4
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T5: lambda_lambda] :
( X4
!= ( lambda_Lam @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_App @ T5 @ U3 ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.cases
thf(fact_492_lambda__calculus_Ohead__redex_Ocases,axiom,
! [X4: lambda_lambda] :
( ( X4 != lambda_Nil )
=> ( ! [X2: nat] :
( X4
!= ( lambda_Var @ X2 ) )
=> ( ! [T5: lambda_lambda] :
( X4
!= ( lambda_Lam @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ! [U3: lambda_lambda] :
( X4
!= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.head_redex.cases
thf(fact_493_lambda__calculus_Olambda_Oexhaust,axiom,
! [Y2: lambda_lambda] :
( ( Y2 != lambda_Nil )
=> ( ! [X23: nat] :
( Y2
!= ( lambda_Var @ X23 ) )
=> ( ! [X34: lambda_lambda] :
( Y2
!= ( lambda_Lam @ X34 ) )
=> ( ! [X412: lambda_lambda,X422: lambda_lambda] :
( Y2
!= ( lambda_App @ X412 @ X422 ) )
=> ~ ! [X512: lambda_lambda,X522: lambda_lambda] :
( Y2
!= ( lambda_Beta @ X512 @ X522 ) ) ) ) ) ) ).
% lambda_calculus.lambda.exhaust
thf(fact_494_lambda__calculus_OTrg_Ocases,axiom,
! [X4: lambda_lambda] :
( ! [I2: nat] :
( X4
!= ( lambda_Var @ I2 ) )
=> ( ! [T5: lambda_lambda] :
( X4
!= ( lambda_Lam @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_App @ T5 @ U3 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( lambda_Beta @ T5 @ U3 ) )
=> ( X4 = lambda_Nil ) ) ) ) ) ).
% lambda_calculus.Trg.cases
thf(fact_495_lambda__calculus_OArr_Oelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda_Arr @ X4 )
=> ( ! [Uu2: nat] :
( X4
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ~ ( lambda_Arr @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(2)
thf(fact_496_lambda__calculus_OJoin_Osimps_I19_J,axiom,
! [Vb: nat,Va: lambda_lambda,V: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Var @ Vb ) @ Va ) @ ( lambda_Beta @ V @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(19)
thf(fact_497_lambda__calculus_OJoin_Osimps_I25_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda,V: nat] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ Va ) @ ( lambda_Var @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(25)
thf(fact_498_lambda__calculus_OJoin_Osimps_I35_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: nat,Vc: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(35)
thf(fact_499_lambda__calculus_OJoin_Osimps_I53_J,axiom,
! [V: lambda_lambda,Vc: lambda_lambda,Vb: nat,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ V @ Vc ) @ ( lambda_App @ ( lambda_Var @ Vb ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(53)
thf(fact_500_lambda__calculus_OJoin_Osimps_I59_J,axiom,
! [V: nat,Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Var @ V ) @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(59)
thf(fact_501_lambda__calculus_OJoin_Osimps_I66_J,axiom,
! [Vd: nat,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(66)
thf(fact_502_lambda__calculus_Osubs_Oelims_I2_J,axiom,
! [X4: lambda_lambda,Xa: lambda_lambda] :
( ( lambda_subs @ X4 @ Xa )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ! [I4: nat] :
( ( Xa
= ( lambda_Var @ I4 ) )
=> ( I2 != I4 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ! [T6: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T6 ) )
=> ~ ( lambda_subs @ T5 @ T6 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_App @ T6 @ U5 ) )
=> ~ ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ~ ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ~ ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.subs.elims(2)
thf(fact_503_lambda__calculus_OJoin_Osimps_I52_J,axiom,
! [V: lambda_lambda,Vb: nat,Va: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ V ) @ ( lambda_App @ ( lambda_Var @ Vb ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(52)
thf(fact_504_lambda__calculus_OJoin_Osimps_I18_J,axiom,
! [Vb: nat,Va: lambda_lambda,V: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Var @ Vb ) @ Va ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(18)
thf(fact_505_lambda__calculus_OArr_Oelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda_Arr @ X4 )
= Y2 )
=> ( ( ( X4 = lambda_Nil )
=> Y2 )
=> ( ( ? [Uu2: nat] :
( X4
= ( lambda_Var @ Uu2 ) )
=> ~ Y2 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
= ( ~ ( lambda_Arr @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(1)
thf(fact_506_lambda__calculus_OSrc_Oelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Src @ X4 )
= Y2 )
=> ( ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ( Y2
!= ( lambda_Var @ I2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda_Src @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_Src @ T5 ) @ ( lambda_Src @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T5 ) ) @ ( lambda_Src @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Src.elims
thf(fact_507_lambda__calculus_OIde_Oelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda_Ide @ X4 )
= Y2 )
=> ( ( ( X4 = lambda_Nil )
=> Y2 )
=> ( ( ? [Uu2: nat] :
( X4
= ( lambda_Var @ Uu2 ) )
=> ~ Y2 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
= ( ~ ( lambda_Ide @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) )
=> ~ ( ? [T5: lambda_lambda,U3: lambda_lambda] :
( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> Y2 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(1)
thf(fact_508_lambda__calculus_Osubs_Oelims_I3_J,axiom,
! [X4: lambda_lambda,Xa: lambda_lambda] :
( ~ ( lambda_subs @ X4 @ Xa )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ! [I4: nat] :
( ( Xa
= ( lambda_Var @ I4 ) )
=> ( I2 = I4 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ! [T6: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T6 ) )
=> ( lambda_subs @ T5 @ T6 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_App @ T6 @ U5 ) )
=> ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) )
=> ( ( X4 != lambda_Nil )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( Xa != lambda_Nil ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ! [Va2: nat] :
( Xa
!= ( lambda_Var @ Va2 ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ! [Va2: lambda_lambda,Vb2: lambda_lambda] :
( Xa
!= ( lambda_App @ Va2 @ Vb2 ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ! [Va2: lambda_lambda,Vb2: lambda_lambda] :
( Xa
!= ( lambda_Beta @ Va2 @ Vb2 ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( Xa != lambda_Nil ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ! [Vb2: nat] :
( Xa
!= ( lambda_Var @ Vb2 ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ! [Vb2: lambda_lambda] :
( Xa
!= ( lambda_Lam @ Vb2 ) ) )
=> ( ( ? [Va2: lambda_lambda] :
( X4
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ! [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
!= ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ( ? [Vd2: nat,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Var @ Vd2 ) @ Va2 ) )
=> ! [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
!= ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Va2 ) )
=> ! [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
!= ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Va2 ) )
=> ! [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
!= ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( Xa != lambda_Nil ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ! [Vb2: nat] :
( Xa
!= ( lambda_Var @ Vb2 ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ! [Vb2: lambda_lambda] :
( Xa
!= ( lambda_Lam @ Vb2 ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ! [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
!= ( lambda_App @ Vb2 @ Vc2 ) ) )
=> ( ( Xa != lambda_Nil )
=> ( ( ( X4 = lambda_Nil )
=> ! [V3: lambda_lambda] :
( Xa
!= ( lambda_Lam @ V3 ) ) )
=> ( ( ? [Va2: nat] :
( X4
= ( lambda_Var @ Va2 ) )
=> ! [V3: lambda_lambda] :
( Xa
!= ( lambda_Lam @ V3 ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ! [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
!= ( lambda_App @ V3 @ Va2 ) ) )
=> ( ( ? [Vb2: nat] :
( X4
= ( lambda_Var @ Vb2 ) )
=> ! [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
!= ( lambda_App @ V3 @ Va2 ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ! [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
!= ( lambda_Beta @ V3 @ Va2 ) ) )
=> ~ ( ? [Vb2: nat] :
( X4
= ( lambda_Var @ Vb2 ) )
=> ! [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
!= ( lambda_Beta @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.subs.elims(3)
thf(fact_509_lambda__calculus_Osubs_Oelims_I1_J,axiom,
! [X4: lambda_lambda,Xa: lambda_lambda,Y2: $o] :
( ( ( lambda_subs @ X4 @ Xa )
= Y2 )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ! [I4: nat] :
( ( Xa
= ( lambda_Var @ I4 ) )
=> ( Y2
= ( I2 != I4 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ! [T6: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T6 ) )
=> ( Y2
= ( ~ ( lambda_subs @ T5 @ T6 ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_App @ T6 @ U5 ) )
=> ( Y2
= ( ~ ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ( Y2
= ( ~ ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ( Y2
= ( ~ ( ( lambda_subs @ T5 @ T6 )
& ( lambda_subs @ U3 @ U5 ) ) ) ) ) )
=> ( ( ( X4 = lambda_Nil )
=> Y2 )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ( Xa = lambda_Nil )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: nat] :
( Xa
= ( lambda_Var @ Va2 ) )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: lambda_lambda,Vb2: lambda_lambda] :
( Xa
= ( lambda_App @ Va2 @ Vb2 ) )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: lambda_lambda,Vb2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Va2 @ Vb2 ) )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ? [Vb2: nat] :
( Xa
= ( lambda_Var @ Vb2 ) )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ? [Vb2: lambda_lambda] :
( Xa
= ( lambda_Lam @ Vb2 ) )
=> Y2 ) )
=> ( ( ? [Va2: lambda_lambda] :
( X4
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Vb2 @ Vc2 ) )
=> Y2 ) )
=> ( ( ? [Vd2: nat,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Var @ Vd2 ) @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Vb2 @ Vc2 ) )
=> Y2 ) )
=> ( ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Vb2 @ Vc2 ) )
=> Y2 ) )
=> ( ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Vb2 @ Vc2 ) )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vb2: nat] :
( Xa
= ( lambda_Var @ Vb2 ) )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vb2: lambda_lambda] :
( Xa
= ( lambda_Lam @ Vb2 ) )
=> Y2 ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ Vb2 @ Vc2 ) )
=> Y2 ) )
=> ( ( ( Xa = lambda_Nil )
=> Y2 )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> Y2 ) )
=> ( ( ? [Va2: nat] :
( X4
= ( lambda_Var @ Va2 ) )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> Y2 ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ V3 @ Va2 ) )
=> Y2 ) )
=> ( ( ? [Vb2: nat] :
( X4
= ( lambda_Var @ Vb2 ) )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ V3 @ Va2 ) )
=> Y2 ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Va2 ) )
=> Y2 ) )
=> ~ ( ? [Vb2: nat] :
( X4
= ( lambda_Var @ Vb2 ) )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Va2 ) )
=> Y2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.subs.elims(1)
thf(fact_510_lambda__calculus_OJoin_Oelims,axiom,
! [X4: lambda_lambda,Xa: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Join @ X4 @ Xa )
= Y2 )
=> ( ! [X2: nat] :
( ( X4
= ( lambda_Var @ X2 ) )
=> ! [X7: nat] :
( ( Xa
= ( lambda_Var @ X7 ) )
=> ~ ( ( ( X2 = X7 )
=> ( Y2
= ( lambda_Var @ X2 ) ) )
& ( ( X2 != X7 )
=> ( Y2 = lambda_Nil ) ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ! [T6: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T6 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda_Join @ T5 @ T6 ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda_Join @ T5 @ T6 ) @ ( lambda_Join @ U3 @ U5 ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_App @ ( lambda_Lam @ T6 ) @ U5 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda_Join @ T5 @ T6 ) @ ( lambda_Join @ U3 @ U5 ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_App @ T6 @ U5 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_Join @ T5 @ T6 ) @ ( lambda_Join @ U3 @ U5 ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda_Join @ T5 @ T6 ) @ ( lambda_Join @ U3 @ U5 ) ) ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: nat] :
( Xa
= ( lambda_Var @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: lambda_lambda,Vb2: lambda_lambda] :
( Xa
= ( lambda_App @ Va2 @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: lambda_lambda,Vb2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Va2 @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Va2: lambda_lambda] :
( X4
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Va2: lambda_lambda] :
( X4
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( ? [V3: nat] :
( Xa
= ( lambda_Var @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Va2: lambda_lambda] :
( X4
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Va2: lambda_lambda] :
( X4
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( ? [V3: lambda_lambda,Vb2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: nat,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: nat,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) )
=> ( ? [V3: nat] :
( Xa
= ( lambda_Var @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: nat,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: nat,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) )
=> ( ? [V3: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( ? [V3: nat] :
( Xa
= ( lambda_Var @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( ? [V3: lambda_lambda,Vd2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Vd2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( ? [V3: nat] :
( Xa
= ( lambda_Var @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( ? [V3: lambda_lambda,Vd2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Vd2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ? [Vb2: nat] :
( Xa
= ( lambda_Var @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ? [Vb2: lambda_lambda] :
( Xa
= ( lambda_Lam @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vb2: nat] :
( Xa
= ( lambda_Var @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vb2: lambda_lambda] :
( Xa
= ( lambda_Lam @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ lambda_Nil @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vd2: nat,Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Va2: nat] :
( X4
= ( lambda_Var @ Va2 ) )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ V3 @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: nat] :
( X4
= ( lambda_Var @ Vb2 ) )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ V3 @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [Va2: lambda_lambda] :
( Xa
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: nat] :
( X4
= ( lambda_Var @ V3 ) )
=> ( ? [Va2: lambda_lambda] :
( Xa
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: lambda_lambda] :
( Xa
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [Vb2: nat,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: nat] :
( X4
= ( lambda_Var @ V3 ) )
=> ( ? [Vb2: nat,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Vb2: nat,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: nat] :
( X4
= ( lambda_Var @ V3 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: nat] :
( X4
= ( lambda_Var @ V3 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ~ ( ? [Vb2: nat] :
( X4
= ( lambda_Var @ Vb2 ) )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Join.elims
thf(fact_511_lambda__calculus_Oparallel__strategy_Oelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda8522482075158074392rategy @ X4 )
= Y2 )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ( Y2
!= ( lambda_Var @ I2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda8522482075158074392rategy @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda8522482075158074392rategy @ T5 ) @ ( lambda8522482075158074392rategy @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda8522482075158074392rategy @ lambda_Nil ) @ ( lambda8522482075158074392rategy @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_Var @ V3 ) ) @ ( lambda8522482075158074392rategy @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_App @ V3 @ Va2 ) ) @ ( lambda8522482075158074392rategy @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_Beta @ V3 @ Va2 ) ) @ ( lambda8522482075158074392rategy @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda8522482075158074392rategy @ T5 ) @ ( lambda8522482075158074392rategy @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.parallel_strategy.elims
thf(fact_512_lambda__calculus_Ois__internal__reduction_Oelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda2591989677915507790uction @ X4 )
= Y2 )
=> ( ( ? [Uu2: nat] :
( X4
= ( lambda_Var @ Uu2 ) )
=> ~ Y2 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
= ( ~ ( lambda2591989677915507790uction @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda2591989677915507790uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda2591989677915507790uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda2591989677915507790uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda2591989677915507790uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ( ? [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X4
= ( lambda_Beta @ Uv2 @ Uw2 ) )
=> Y2 )
=> ~ ( ( X4 = lambda_Nil )
=> Y2 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_internal_reduction.elims(1)
thf(fact_513_lambda__calculus_Ois__internal__reduction_Oelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda2591989677915507790uction @ X4 )
=> ( ! [Uu2: nat] :
( X4
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ~ ( lambda2591989677915507790uction @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ~ ( ( lambda2591989677915507790uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ~ ( ( lambda2591989677915507790uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda2591989677915507790uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ~ ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda2591989677915507790uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_internal_reduction.elims(2)
thf(fact_514_lambda__calculus_Ois__internal__reduction_Oelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda2591989677915507790uction @ X4 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( lambda2591989677915507790uction @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( lambda2591989677915507790uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( lambda2591989677915507790uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda2591989677915507790uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda2591989677915507790uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X4
!= ( lambda_Beta @ Uv2 @ Uw2 ) )
=> ( X4 = lambda_Nil ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_internal_reduction.elims(3)
thf(fact_515_lambda__calculus_Ocontains__head__reduction_Oelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda8135948770199036598uction @ X4 )
= Y2 )
=> ( ( ? [Uu2: nat] :
( X4
= ( lambda_Var @ Uu2 ) )
=> Y2 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
= ( ~ ( lambda8135948770199036598uction @ T5 ) ) ) )
=> ( ( ? [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> Y2 )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> Y2 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.elims(1)
thf(fact_516_lambda__calculus_Ocontains__head__reduction_Oelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda8135948770199036598uction @ X4 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ~ ( lambda8135948770199036598uction @ T5 ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ~ ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.elims(2)
thf(fact_517_lambda__calculus_Ocontains__head__reduction_Oelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda8135948770199036598uction @ X4 )
=> ( ! [Uu2: nat] :
( X4
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( lambda8135948770199036598uction @ T5 ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ( X4 = lambda_Nil ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.elims(3)
thf(fact_518_lambda__calculus_Ois__head__reduction_Oelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda5343566265203084875uction @ X4 )
= Y2 )
=> ( ( ? [Uu2: nat] :
( X4
= ( lambda_Var @ Uu2 ) )
=> Y2 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
= ( ~ ( lambda5343566265203084875uction @ T5 ) ) ) )
=> ( ( ? [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> Y2 )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda5343566265203084875uction @ lambda_Nil )
& ( lambda_Ide @ U3 ) ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda5343566265203084875uction @ ( lambda_Var @ V3 ) )
& ( lambda_Ide @ U3 ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda5343566265203084875uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda5343566265203084875uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> Y2 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_head_reduction.elims(1)
thf(fact_519_lambda__calculus_Ois__head__reduction_Oelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda5343566265203084875uction @ X4 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ~ ( lambda5343566265203084875uction @ T5 ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ~ ( ( lambda5343566265203084875uction @ lambda_Nil )
& ( lambda_Ide @ U3 ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ~ ( ( lambda5343566265203084875uction @ ( lambda_Var @ V3 ) )
& ( lambda_Ide @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda5343566265203084875uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda5343566265203084875uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_head_reduction.elims(2)
thf(fact_520_lambda__calculus_Ois__head__reduction_Oelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda5343566265203084875uction @ X4 )
=> ( ! [Uu2: nat] :
( X4
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( lambda5343566265203084875uction @ T5 ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X4
!= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( lambda5343566265203084875uction @ lambda_Nil )
& ( lambda_Ide @ U3 ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( lambda5343566265203084875uction @ ( lambda_Var @ V3 ) )
& ( lambda_Ide @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda5343566265203084875uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda5343566265203084875uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) )
=> ( X4 = lambda_Nil ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_head_reduction.elims(3)
thf(fact_521_lambda__calculus_Oelementary__reduction_Oelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda2670285437103676873uction @ X4 )
= Y2 )
=> ( ( ( X4 = lambda_Nil )
=> Y2 )
=> ( ( ? [Uu2: nat] :
( X4
= ( lambda_Var @ Uu2 ) )
=> Y2 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
= ( ~ ( lambda2670285437103676873uction @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
= ( ~ ( ( ( lambda2670285437103676873uction @ T5 )
& ( lambda_Ide @ U3 ) )
| ( ( lambda_Ide @ T5 )
& ( lambda2670285437103676873uction @ U3 ) ) ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.elims(1)
thf(fact_522_lambda__calculus_Oelementary__reduction_Oelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda2670285437103676873uction @ X4 )
=> ( ( X4 != lambda_Nil )
=> ( ! [Uu2: nat] :
( X4
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( lambda2670285437103676873uction @ T5 ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( ( lambda2670285437103676873uction @ T5 )
& ( lambda_Ide @ U3 ) )
| ( ( lambda_Ide @ T5 )
& ( lambda2670285437103676873uction @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.elims(3)
thf(fact_523_lambda__calculus_Oleftmost__strategy_Oelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda4011002726600638587rategy @ X4 )
= Y2 )
=> ( ! [X2: nat] :
( ( X4
= ( lambda_Var @ X2 ) )
=> ( Y2
!= ( lambda_Var @ X2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda4011002726600638587rategy @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ T5 @ U3 ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ~ ( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ lambda_Nil ) )
=> ( Y2
= ( lambda_App @ ( lambda4011002726600638587rategy @ lambda_Nil ) @ U3 ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ lambda_Nil ) )
=> ( Y2
= ( lambda_App @ lambda_Nil @ ( lambda4011002726600638587rategy @ U3 ) ) ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ~ ( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V3 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V3 ) ) @ U3 ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V3 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda_Var @ V3 ) @ ( lambda4011002726600638587rategy @ U3 ) ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_App @ V3 @ Va2 ) ) @ U3 ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ ( lambda4011002726600638587rategy @ U3 ) ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V3 @ Va2 ) ) @ U3 ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda4011002726600638587rategy @ U3 ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ T5 @ U3 ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.leftmost_strategy.elims
thf(fact_524_lambda__calculus_Ohead__strategy_Oelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_head_strategy @ X4 )
= Y2 )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ( Y2
!= ( lambda_Var @ I2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda_head_strategy @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda_Src @ T5 ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_head_strategy @ lambda_Nil ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Var @ V3 ) ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_head_strategy @ ( lambda_App @ V3 @ Va2 ) ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Beta @ V3 @ Va2 ) ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda_Src @ T5 ) @ ( lambda_Src @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.head_strategy.elims
thf(fact_525_lambda__calculus_Ohead__redex_Oelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_head_redex @ X4 )
= Y2 )
=> ( ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ! [X2: nat] :
( ( X4
= ( lambda_Var @ X2 ) )
=> ( Y2
!= ( lambda_Var @ X2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda_head_redex @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T5 ) ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_head_redex @ lambda_Nil ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_head_redex @ ( lambda_Var @ V3 ) ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_head_redex @ ( lambda_App @ V3 @ Va2 ) ) @ ( lambda_Src @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_head_redex @ ( lambda_Beta @ V3 @ Va2 ) ) @ ( lambda_Src @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda_Src @ T5 ) @ ( lambda_Src @ U3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.head_redex.elims
thf(fact_526_ConE,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( ( lambda_resid @ T2 @ T3 )
!= lambda_Nil )
=> ( ! [I2: nat] :
( ( T2
= ( lambda_Var @ I2 ) )
=> ( ( T3
= ( lambda_Var @ I2 ) )
=> ( ( lambda_resid @ T2 @ T3 )
!= ( lambda_Var @ I2 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( T2
= ( lambda_Lam @ U3 ) )
=> ! [U5: lambda_lambda] :
( ( T3
= ( lambda_Lam @ U5 ) )
=> ( ( ( lambda_resid @ U3 @ U5 )
!= lambda_Nil )
=> ( ( lambda_resid @ T2 @ T3 )
!= ( lambda_Lam @ ( lambda_resid @ U3 @ U5 ) ) ) ) ) )
=> ( ! [U3: lambda_lambda,V3: lambda_lambda] :
( ( T2
= ( lambda_App @ U3 @ V3 ) )
=> ! [U5: lambda_lambda,V4: lambda_lambda] :
( ( T3
= ( lambda_App @ U5 @ V4 ) )
=> ( ( ( lambda_resid @ U3 @ U5 )
!= lambda_Nil )
=> ( ( ( lambda_resid @ V3 @ V4 )
!= lambda_Nil )
=> ( ( lambda_resid @ T2 @ T3 )
!= ( lambda_App @ ( lambda_resid @ U3 @ U5 ) @ ( lambda_resid @ V3 @ V4 ) ) ) ) ) ) )
=> ( ! [U3: lambda_lambda,V3: lambda_lambda] :
( ( T2
= ( lambda_Beta @ U3 @ V3 ) )
=> ! [U5: lambda_lambda,V4: lambda_lambda] :
( ( T3
= ( lambda_Beta @ U5 @ V4 ) )
=> ( ( ( lambda_resid @ U3 @ U5 )
!= lambda_Nil )
=> ( ( ( lambda_resid @ V3 @ V4 )
!= lambda_Nil )
=> ( ( lambda_resid @ T2 @ T3 )
!= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ V3 @ V4 ) @ ( lambda_resid @ U3 @ U5 ) ) ) ) ) ) )
=> ( ! [U3: lambda_lambda,V3: lambda_lambda] :
( ( T2
= ( lambda_App @ ( lambda_Lam @ U3 ) @ V3 ) )
=> ! [U5: lambda_lambda,V4: lambda_lambda] :
( ( T3
= ( lambda_Beta @ U5 @ V4 ) )
=> ( ( ( lambda_resid @ U3 @ U5 )
!= lambda_Nil )
=> ( ( ( lambda_resid @ V3 @ V4 )
!= lambda_Nil )
=> ( ( lambda_resid @ T2 @ T3 )
!= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ V3 @ V4 ) @ ( lambda_resid @ U3 @ U5 ) ) ) ) ) ) )
=> ~ ! [U3: lambda_lambda,V3: lambda_lambda] :
( ( T2
= ( lambda_Beta @ U3 @ V3 ) )
=> ! [U5: lambda_lambda,V4: lambda_lambda] :
( ( T3
= ( lambda_App @ ( lambda_Lam @ U5 ) @ V4 ) )
=> ( ( ( lambda_resid @ U3 @ U5 )
!= lambda_Nil )
=> ( ( ( lambda_resid @ V3 @ V4 )
!= lambda_Nil )
=> ( ( lambda_resid @ T2 @ T3 )
!= ( lambda_Beta @ ( lambda_resid @ U3 @ U5 ) @ ( lambda_resid @ V3 @ V4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ConE
thf(fact_527_rts__with__joins_Ohas__joins,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T2: lambda_lambda,U: lambda_lambda] :
( ( rts_wi2791606683991334555lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T2 @ U )
=> ( joinab2519788767067900224lambda @ Resid @ T2 @ U ) ) ) ).
% rts_with_joins.has_joins
thf(fact_528_confluent__rts_Oconfluence,axiom,
! [Resid: nat > nat > nat,T2: nat,U: nat] :
( ( confluent_rts_nat @ Resid )
=> ( ( coinitial_nat @ Resid @ T2 @ U )
=> ( con_nat @ Resid @ T2 @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_529_confluent__rts_Oconfluence,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T2: lambda_lambda,U: lambda_lambda] :
( ( conflu4927472611300549594lambda @ Resid )
=> ( ( coinit1392077014771647542lambda @ Resid @ T2 @ U )
=> ( con_lambda_lambda @ Resid @ T2 @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_530_seqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T2 @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( trg_lambda_lambda @ lambda_resid @ T2 )
!= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ) ).
% seqE\<^sub>W\<^sub>E
thf(fact_531_in__targetsE,axiom,
! [B: lambda_lambda,T2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ B )
=> ~ ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T2 ) @ B ) ) ) ).
% in_targetsE
thf(fact_532_Raise__0,axiom,
! [D: nat,T2: lambda_lambda] :
( ( lambda_Raise @ D @ zero_zero_nat @ T2 )
= T2 ) ).
% Raise_0
thf(fact_533_target__is__ide,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ).
% target_is_ide
thf(fact_534_targets__are__cong,axiom,
! [B: lambda_lambda,T2: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_535_targets__cong__closed,axiom,
! [B: lambda_lambda,T2: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B ) ) )
=> ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) ) ) ) ).
% targets_cong_closed
thf(fact_536_arr__has__un__target,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( Y = X2 ) ) ) ) ).
% arr_has_un_target
thf(fact_537_targets__resid__sym,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) ) ) ).
% targets_resid_sym
thf(fact_538_targets__are__con,axiom,
! [B: lambda_lambda,T2: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( con_lambda_lambda @ lambda_resid @ B @ B2 ) ) ) ).
% targets_are_con
thf(fact_539_con__imp__eq__src,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T2 )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ).
% con_imp_eq_src
thf(fact_540_targets__join__of_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ V ) ) ) ).
% targets_join_of(1)
thf(fact_541_targets__join__of_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) )
= ( target5490931411331770124lambda @ lambda_resid @ V ) ) ) ).
% targets_join_of(2)
thf(fact_542_src__join__of_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T2 )
= ( weakly4554751870655413501lambda @ lambda_resid @ V ) ) ) ).
% src_join_of(1)
thf(fact_543_src__join__of_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ U )
= ( weakly4554751870655413501lambda @ lambda_resid @ V ) ) ) ).
% src_join_of(2)
thf(fact_544_targets__con__closed,axiom,
! [B: lambda_lambda,T2: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B2 )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ B2 )
=> ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) ) ) ) ) ).
% targets_con_closed
thf(fact_545_ide__src,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ide_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T2 ) ) ) ).
% ide_src
thf(fact_546_ide__iff__src__self,axiom,
! [A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A )
= ( ( weakly4554751870655413501lambda @ lambda_resid @ A )
= A ) ) ) ).
% ide_iff_src_self
thf(fact_547_src__eqI,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ T2 )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T2 )
= A ) ) ) ).
% src_eqI
thf(fact_548_trg__in__targets,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( member_lambda_lambda @ ( trg_lambda_lambda @ lambda_resid @ T2 ) @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) ) ) ).
% trg_in_targets
thf(fact_549_coinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T2 )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( weakly4554751870655413501lambda @ lambda_resid @ T2 )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ).
% coinitial_iff\<^sub>W\<^sub>E
thf(fact_550_coinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T2 )
!= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ) ).
% coinitialE\<^sub>W\<^sub>E
thf(fact_551_coterminal__iff,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T2 @ T3 )
= ( ( arr_lambda_lambda @ lambda_resid @ T2 )
& ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= ( target5490931411331770124lambda @ lambda_resid @ T3 ) ) ) ) ).
% coterminal_iff
thf(fact_552_coterminalE,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T2 @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
!= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ) ) ).
% coterminalE
thf(fact_553_src__src,axiom,
! [T2: lambda_lambda] :
( ( weakly4554751870655413501lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T2 ) )
= ( weakly4554751870655413501lambda @ lambda_resid @ T2 ) ) ).
% src_src
thf(fact_554_src__ide,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ A )
= A ) ) ).
% src_ide
thf(fact_555_arr__src__iff__arr,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T2 ) )
= ( arr_lambda_lambda @ lambda_resid @ T2 ) ) ).
% arr_src_iff_arr
thf(fact_556_trg__src,axiom,
! [T2: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T2 ) )
= ( weakly4554751870655413501lambda @ lambda_resid @ T2 ) ) ).
% trg_src
thf(fact_557_src__trg,axiom,
! [T2: lambda_lambda] :
( ( weakly4554751870655413501lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T2 ) )
= ( trg_lambda_lambda @ lambda_resid @ T2 ) ) ).
% src_trg
thf(fact_558_src__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ).
% src_resid
thf(fact_559_coinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( ( weakly4554751870655413501lambda @ lambda_resid @ T2 )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U ) ) ) ).
% coinitialI\<^sub>W\<^sub>E
thf(fact_560_coterminalI,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= ( target5490931411331770124lambda @ lambda_resid @ U ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ T2 @ U ) ) ) ).
% coterminalI
thf(fact_561_src__char,axiom,
! [T2: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T2 )
= ( lambda_Src @ T2 ) ) )
& ( ~ ( lambda_Arr @ T2 )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T2 )
= lambda_Nil ) ) ) ).
% src_char
thf(fact_562_in__targetsI,axiom,
! [B: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T2 ) @ B )
=> ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) ) ) ) ).
% in_targetsI
thf(fact_563_seqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [U: lambda_lambda,T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T2 )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T2 @ U ) ) ) ) ).
% seqI\<^sub>W\<^sub>E
thf(fact_564_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_565_lambda__calculus_Oreduce_Osimps_I1_J,axiom,
! [F2: lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( lambda_reduce @ F2 @ A @ zero_zero_nat )
= A ) ).
% lambda_calculus.reduce.simps(1)
thf(fact_566_residuation_Oide_Ocong,axiom,
ide_nat = ide_nat ).
% residuation.ide.cong
thf(fact_567_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_568_lambda__calculus_OTrg_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Trg @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U ) @ ( lambda_Trg @ T2 ) ) ) ).
% lambda_calculus.Trg.simps(4)
thf(fact_569_residuation_Otrg_Ocong,axiom,
trg_lambda_lambda = trg_lambda_lambda ).
% residuation.trg.cong
thf(fact_570_rts_Oseq_Ocong,axiom,
seq_nat = seq_nat ).
% rts.seq.cong
thf(fact_571_rts_Oseq_Ocong,axiom,
seq_lambda_lambda = seq_lambda_lambda ).
% rts.seq.cong
thf(fact_572_rts_Ojoin__of_Ocong,axiom,
join_o1386658102433174083lambda = join_o1386658102433174083lambda ).
% rts.join_of.cong
thf(fact_573_rts_Ocoinitial_Ocong,axiom,
coinitial_nat = coinitial_nat ).
% rts.coinitial.cong
thf(fact_574_rts_Ocoinitial_Ocong,axiom,
coinit1392077014771647542lambda = coinit1392077014771647542lambda ).
% rts.coinitial.cong
thf(fact_575_rts_Ojoinable_Ocong,axiom,
joinab2519788767067900224lambda = joinab2519788767067900224lambda ).
% rts.joinable.cong
thf(fact_576_rts_Ocoterminal_Ocong,axiom,
coterm4142657152830665724lambda = coterm4142657152830665724lambda ).
% rts.coterminal.cong
thf(fact_577_partial__magma_Ointro,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ? [X8: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP @ X8 @ T5 )
= X8 )
& ( ( OP @ T5 @ X8 )
= X8 ) )
& ! [Y4: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( OP @ Y4 @ T4 )
= Y4 )
& ( ( OP @ T4 @ Y4 )
= Y4 ) )
=> ( Y4 = X8 ) ) )
=> ( partia3924075026459072775lambda @ OP ) ) ).
% partial_magma.intro
thf(fact_578_partial__magma_Oex__un__null,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ? [X2: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( OP @ X2 @ T4 )
= X2 )
& ( ( OP @ T4 @ X2 )
= X2 ) )
& ! [Y: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP @ Y @ T5 )
= Y )
& ( ( OP @ T5 @ Y )
= Y ) )
=> ( Y = X2 ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_579_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X: lambda_lambda] :
( ! [T: lambda_lambda] :
( ( ( OP2 @ X @ T )
= X )
& ( ( OP2 @ T @ X )
= X ) )
& ! [Y5: lambda_lambda] :
( ! [T: lambda_lambda] :
( ( ( OP2 @ Y5 @ T )
= Y5 )
& ( ( OP2 @ T @ Y5 )
= Y5 ) )
=> ( Y5 = X ) ) ) ) ) ).
% partial_magma_def
thf(fact_580_lambda__calculus_Oresid_Osimps_I4_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T2 @ U ) @ ( lambda_Beta @ T3 @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T2 @ U ) @ ( lambda_Beta @ T3 @ U2 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U @ U2 ) @ ( lambda_resid @ T2 @ T3 ) ) ) ) ) ).
% lambda_calculus.resid.simps(4)
thf(fact_581_lambda__calculus_Onon__extensional_I2_J,axiom,
( ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Beta @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) ) )
!= ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) ) ).
% lambda_calculus.non_extensional(2)
thf(fact_582_lambda__calculus_Ocong__nontrivial_I3_J,axiom,
~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_App @ ( lambda_App @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) @ ( lambda_App @ ( lambda_Beta @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) @ ( lambda_App @ ( lambda_App @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) ) ) ) ).
% lambda_calculus.cong_nontrivial(3)
thf(fact_583_lambda__calculus_Oresid_Osimps_I5_J,axiom,
! [T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) @ ( lambda_Beta @ T3 @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T2 @ T3 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) @ ( lambda_Beta @ T3 @ U2 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U @ U2 ) @ ( lambda_resid @ T2 @ T3 ) ) ) ) ) ).
% lambda_calculus.resid.simps(5)
thf(fact_584_lambda__calculus_Osubs__example_I2_J,axiom,
~ ( lambda_subs @ ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_Beta @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ).
% lambda_calculus.subs_example(2)
thf(fact_585_lambda__calculus_Ocong__nontrivial_I1_J,axiom,
coinit1392077014771647542lambda @ lambda_resid @ ( lambda_App @ ( lambda_App @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) @ ( lambda_App @ ( lambda_Beta @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) ).
% lambda_calculus.cong_nontrivial(1)
thf(fact_586_lambda__calculus_Ocong__nontrivial_I2_J,axiom,
coterm4142657152830665724lambda @ lambda_resid @ ( lambda_App @ ( lambda_App @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) @ ( lambda_App @ ( lambda_Beta @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Lam @ ( lambda_App @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) ).
% lambda_calculus.cong_nontrivial(2)
thf(fact_587_lambda__calculus_Oresid_Oelims,axiom,
! [X4: lambda_lambda,Xa: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_resid @ X4 @ Xa )
= Y2 )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ! [I4: nat] :
( ( Xa
= ( lambda_Var @ I4 ) )
=> ~ ( ( ( I2 = I4 )
=> ( Y2
= ( lambda_Var @ I2 ) ) )
& ( ( I2 != I4 )
=> ( Y2 = lambda_Nil ) ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ! [T6: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T6 ) )
=> ~ ( ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
=> ( Y2 = lambda_Nil ) )
& ( ( ( lambda_resid @ T5 @ T6 )
!= lambda_Nil )
=> ( Y2
= ( lambda_Lam @ ( lambda_resid @ T5 @ T6 ) ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_App @ T6 @ U5 ) )
=> ~ ( ( ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U3 @ U5 )
= lambda_Nil ) )
=> ( Y2 = lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U3 @ U5 )
= lambda_Nil ) )
=> ( Y2
= ( lambda_App @ ( lambda_resid @ T5 @ T6 ) @ ( lambda_resid @ U3 @ U5 ) ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ~ ( ( ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U3 @ U5 )
= lambda_Nil ) )
=> ( Y2 = lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U3 @ U5 )
= lambda_Nil ) )
=> ( Y2
= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U3 @ U5 ) @ ( lambda_resid @ T5 @ T6 ) ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T6 @ U5 ) )
=> ~ ( ( ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U3 @ U5 )
= lambda_Nil ) )
=> ( Y2 = lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U3 @ U5 )
= lambda_Nil ) )
=> ( Y2
= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U3 @ U5 ) @ ( lambda_resid @ T5 @ T6 ) ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ! [T6: lambda_lambda,U5: lambda_lambda] :
( ( Xa
= ( lambda_App @ ( lambda_Lam @ T6 ) @ U5 ) )
=> ~ ( ( ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U3 @ U5 )
= lambda_Nil ) )
=> ( Y2 = lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T5 @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U3 @ U5 )
= lambda_Nil ) )
=> ( Y2
= ( lambda_Beta @ ( lambda_resid @ T5 @ T6 ) @ ( lambda_resid @ U3 @ U5 ) ) ) ) ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: nat] :
( Xa
= ( lambda_Var @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: lambda_lambda,Vb2: lambda_lambda] :
( Xa
= ( lambda_App @ Va2 @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda] :
( X4
= ( lambda_Lam @ V3 ) )
=> ( ? [Va2: lambda_lambda,Vb2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Va2 @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ? [Vb2: nat] :
( Xa
= ( lambda_Var @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ V3 @ Va2 ) )
=> ( ? [Vb2: lambda_lambda] :
( Xa
= ( lambda_Lam @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Va2: lambda_lambda] :
( X4
= ( lambda_App @ lambda_Nil @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Vb2 @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vd2: nat,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Var @ Vd2 ) @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Vb2 @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Vb2 @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Va2 ) )
=> ( ? [Vb2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_Beta @ Vb2 @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vb2: nat] :
( Xa
= ( lambda_Var @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vb2: lambda_lambda] :
( Xa
= ( lambda_Lam @ Vb2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ lambda_Nil @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vd2: nat,Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( Xa
= ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Vc2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( Xa = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Va2: nat] :
( X4
= ( lambda_Var @ Va2 ) )
=> ( ? [V3: lambda_lambda] :
( Xa
= ( lambda_Lam @ V3 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ V3 @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ? [Vb2: nat] :
( X4
= ( lambda_Var @ Vb2 ) )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_App @ V3 @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ( ( ( X4 = lambda_Nil )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) )
=> ~ ( ? [Vb2: nat] :
( X4
= ( lambda_Var @ Vb2 ) )
=> ( ? [V3: lambda_lambda,Va2: lambda_lambda] :
( Xa
= ( lambda_Beta @ V3 @ Va2 ) )
=> ( Y2 != lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.resid.elims
thf(fact_588_lambda__calculus_Onon__extensional_I1_J,axiom,
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Beta @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) @ ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) @ ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_Beta @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Beta @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) ) ) ) ) ) ) ).
% lambda_calculus.non_extensional(1)
thf(fact_589_lambda__calculus_Osubs__example_I1_J,axiom,
ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_Beta @ ( lambda_Var @ zero_zero_nat ) @ ( lambda_Var @ zero_zero_nat ) ) ) @ ( lambda_Beta @ ( lambda_Var @ one_one_nat ) @ ( lambda_App @ ( lambda_Lam @ ( lambda_Var @ zero_zero_nat ) ) @ ( lambda_Var @ zero_zero_nat ) ) ) ) ).
% lambda_calculus.subs_example(1)
thf(fact_590_lambda__calculus_OTrg_Oelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Trg @ X4 )
= Y2 )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ( Y2
!= ( lambda_Var @ I2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda_Trg @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_Trg @ T5 ) @ ( lambda_Trg @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U3 ) @ ( lambda_Trg @ T5 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( Y2 != lambda_Nil ) ) ) ) ) ) ) ).
% lambda_calculus.Trg.elims
thf(fact_591_raise__Subst,axiom,
! [T2: lambda_lambda,V: lambda_lambda,P2: nat,N: nat] :
( ( T2 != lambda_Nil )
=> ( ( V != lambda_Nil )
=> ( ( lambda_Raise @ zero_zero_nat @ P2 @ ( lambda_Subst @ N @ V @ T2 ) )
= ( lambda_Subst @ ( plus_plus_nat @ P2 @ N ) @ V @ ( lambda_Raise @ zero_zero_nat @ P2 @ T2 ) ) ) ) ) ).
% raise_Subst
thf(fact_592_arr__iff__has__target,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
= ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
!= bot_bo1271416176687173302lambda ) ) ).
% arr_iff_has_target
thf(fact_593_rts__axioms_Ointro,axiom,
! [Resid: nat > nat > nat] :
( ! [T5: nat] :
( ( arr_nat @ Resid @ T5 )
=> ( ide_nat @ Resid @ ( trg_nat @ Resid @ T5 ) ) )
=> ( ! [A5: nat,T5: nat] :
( ( ide_nat @ Resid @ A5 )
=> ( ( con_nat @ Resid @ T5 @ A5 )
=> ( ( Resid @ T5 @ A5 )
= T5 ) ) )
=> ( ! [A5: nat,T5: nat] :
( ( ide_nat @ Resid @ A5 )
=> ( ( con_nat @ Resid @ A5 @ T5 )
=> ( ide_nat @ Resid @ ( Resid @ A5 @ T5 ) ) ) )
=> ( ! [T5: nat,U3: nat] :
( ( con_nat @ Resid @ T5 @ U3 )
=> ? [A6: nat] :
( ( ide_nat @ Resid @ A6 )
& ( con_nat @ Resid @ A6 @ T5 )
& ( con_nat @ Resid @ A6 @ U3 ) ) )
=> ( ! [T5: nat,U3: nat,V3: nat] :
( ( ide_nat @ Resid @ ( Resid @ T5 @ U3 ) )
=> ( ( con_nat @ Resid @ U3 @ V3 )
=> ( con_nat @ Resid @ ( Resid @ T5 @ U3 ) @ ( Resid @ V3 @ U3 ) ) ) )
=> ( rts_axioms_nat @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_594_rts__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( arr_lambda_lambda @ Resid @ T5 )
=> ( ide_lambda_lambda @ Resid @ ( trg_lambda_lambda @ Resid @ T5 ) ) )
=> ( ! [A5: lambda_lambda,T5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A5 )
=> ( ( con_lambda_lambda @ Resid @ T5 @ A5 )
=> ( ( Resid @ T5 @ A5 )
= T5 ) ) )
=> ( ! [A5: lambda_lambda,T5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A5 )
=> ( ( con_lambda_lambda @ Resid @ A5 @ T5 )
=> ( ide_lambda_lambda @ Resid @ ( Resid @ A5 @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( con_lambda_lambda @ Resid @ T5 @ U3 )
=> ? [A6: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A6 )
& ( con_lambda_lambda @ Resid @ A6 @ T5 )
& ( con_lambda_lambda @ Resid @ A6 @ U3 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,V3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ ( Resid @ T5 @ U3 ) )
=> ( ( con_lambda_lambda @ Resid @ U3 @ V3 )
=> ( con_lambda_lambda @ Resid @ ( Resid @ T5 @ U3 ) @ ( Resid @ V3 @ U3 ) ) ) )
=> ( rts_ax7298520571313827089lambda @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_595_rts__axioms__def,axiom,
( rts_axioms_nat
= ( ^ [Resid2: nat > nat > nat] :
( ! [T: nat] :
( ( arr_nat @ Resid2 @ T )
=> ( ide_nat @ Resid2 @ ( trg_nat @ Resid2 @ T ) ) )
& ! [A2: nat,T: nat] :
( ( ide_nat @ Resid2 @ A2 )
=> ( ( con_nat @ Resid2 @ T @ A2 )
=> ( ( Resid2 @ T @ A2 )
= T ) ) )
& ! [A2: nat,T: nat] :
( ( ide_nat @ Resid2 @ A2 )
=> ( ( con_nat @ Resid2 @ A2 @ T )
=> ( ide_nat @ Resid2 @ ( Resid2 @ A2 @ T ) ) ) )
& ! [T: nat,U4: nat] :
( ( con_nat @ Resid2 @ T @ U4 )
=> ? [A2: nat] :
( ( ide_nat @ Resid2 @ A2 )
& ( con_nat @ Resid2 @ A2 @ T )
& ( con_nat @ Resid2 @ A2 @ U4 ) ) )
& ! [T: nat,U4: nat,V5: nat] :
( ( ide_nat @ Resid2 @ ( Resid2 @ T @ U4 ) )
=> ( ( con_nat @ Resid2 @ U4 @ V5 )
=> ( con_nat @ Resid2 @ ( Resid2 @ T @ U4 ) @ ( Resid2 @ V5 @ U4 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_596_rts__axioms__def,axiom,
( rts_ax7298520571313827089lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T: lambda_lambda] :
( ( arr_lambda_lambda @ Resid2 @ T )
=> ( ide_lambda_lambda @ Resid2 @ ( trg_lambda_lambda @ Resid2 @ T ) ) )
& ! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A2 )
=> ( ( con_lambda_lambda @ Resid2 @ T @ A2 )
=> ( ( Resid2 @ T @ A2 )
= T ) ) )
& ! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A2 )
=> ( ( con_lambda_lambda @ Resid2 @ A2 @ T )
=> ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ A2 @ T ) ) ) )
& ! [T: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ Resid2 @ T @ U4 )
=> ? [A2: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A2 )
& ( con_lambda_lambda @ Resid2 @ A2 @ T )
& ( con_lambda_lambda @ Resid2 @ A2 @ U4 ) ) )
& ! [T: lambda_lambda,U4: lambda_lambda,V5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ T @ U4 ) )
=> ( ( con_lambda_lambda @ Resid2 @ U4 @ V5 )
=> ( con_lambda_lambda @ Resid2 @ ( Resid2 @ T @ U4 ) @ ( Resid2 @ V5 @ U4 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_597_Raise__plus,axiom,
! [D: nat,M3: nat,N: nat,T2: lambda_lambda] :
( ( lambda_Raise @ D @ ( plus_plus_nat @ M3 @ N ) @ T2 )
= ( lambda_Raise @ ( plus_plus_nat @ D @ M3 ) @ N @ ( lambda_Raise @ D @ M3 @ T2 ) ) ) ).
% Raise_plus
thf(fact_598_raise__Raise,axiom,
! [P2: nat,N: nat,K: nat,T2: lambda_lambda] :
( ( lambda_Raise @ zero_zero_nat @ P2 @ ( lambda_Raise @ N @ K @ T2 ) )
= ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Raise @ zero_zero_nat @ P2 @ T2 ) ) ) ).
% raise_Raise
thf(fact_599_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_600_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_601_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_602_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y2 ) )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_603_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y2: nat] :
( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_604_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_605_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_606_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_607_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_608_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_609_lambda__calculus_Otargets__simps_I1_J,axiom,
( ( target5490931411331770124lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.targets_simps(1)
thf(fact_610_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_611_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_612_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_613_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_614_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_615_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_616_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_617_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B3: nat] : ( plus_plus_nat @ B3 @ A2 ) ) ) ).
% add.commute
thf(fact_618_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_619_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_620_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_621_targets__Raise,axiom,
! [T2: lambda_lambda,D: nat,N: nat] :
( ( lambda_Arr @ T2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Raise @ D @ N @ T2 ) )
= ( insert_lambda_lambda @ ( lambda_Raise @ D @ N @ ( lambda_Trg @ T2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% targets_Raise
thf(fact_622_targets__char_092_060_094sub_062_092_060Lambda_062,axiom,
! [T2: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= ( insert_lambda_lambda @ ( lambda_Trg @ T2 ) @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( lambda_Arr @ T2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= bot_bo1271416176687173302lambda ) ) ) ).
% targets_char\<^sub>\<Lambda>
thf(fact_623_Raise__Subst_H,axiom,
! [T2: lambda_lambda,V: lambda_lambda,K: nat,N: nat,P2: nat] :
( ( T2 != lambda_Nil )
=> ( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( lambda_Raise @ K @ P2 @ ( lambda_Subst @ N @ V @ T2 ) )
= ( lambda_Subst @ ( plus_plus_nat @ P2 @ N ) @ V @ ( lambda_Raise @ K @ P2 @ T2 ) ) ) ) ) ) ).
% Raise_Subst'
thf(fact_624_Raise__plus_H,axiom,
! [D2: nat,D: nat,N: nat,M3: nat,T2: lambda_lambda] :
( ( ord_less_eq_nat @ D2 @ ( plus_plus_nat @ D @ N ) )
=> ( ( ord_less_eq_nat @ D @ D2 )
=> ( ( lambda_Raise @ D @ ( plus_plus_nat @ M3 @ N ) @ T2 )
= ( lambda_Raise @ D2 @ M3 @ ( lambda_Raise @ D @ N @ T2 ) ) ) ) ) ).
% Raise_plus'
thf(fact_625_Raise__Raise,axiom,
! [I: nat,N: nat,P2: nat,K: nat,T2: lambda_lambda] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( lambda_Raise @ I @ P2 @ ( lambda_Raise @ N @ K @ T2 ) )
= ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Raise @ I @ P2 @ T2 ) ) ) ) ).
% Raise_Raise
thf(fact_626_raise__plus,axiom,
! [D: nat,N: nat,M3: nat,T2: lambda_lambda] :
( ( ord_less_eq_nat @ D @ N )
=> ( ( lambda_Raise @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) @ T2 )
= ( lambda_Raise @ D @ M3 @ ( lambda_Raise @ zero_zero_nat @ N @ T2 ) ) ) ) ).
% raise_plus
thf(fact_627_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_628_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_629_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_630_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_631_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_632_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_633_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_634_lambda__calculus_Otargets__simps_I2_J,axiom,
! [X4: nat] :
( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Var @ X4 ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X4 ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.targets_simps(2)
thf(fact_635_targets__simp,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= ( insert_lambda_lambda @ ( lambda_Trg @ T2 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% targets_simp
thf(fact_636_lambda__calculus_Otargets__simps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_App @ T2 @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ T2 ) @ ( lambda_Trg @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.targets_simps(4)
thf(fact_637_lambda__calculus_Otargets__simps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Lam @ T2 ) )
= ( insert_lambda_lambda @ ( lambda_Lam @ ( lambda_Trg @ T2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.targets_simps(3)
thf(fact_638_targets__subst,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Arr @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Subst @ zero_zero_nat @ T2 @ U ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ T2 ) @ ( lambda_Trg @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% targets_subst
thf(fact_639_lambda__calculus_Otargets__simps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Beta @ T2 @ U ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U ) @ ( lambda_Trg @ T2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.targets_simps(5)
thf(fact_640_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_641_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_642_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_643_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_644_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_645_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_646_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ) ).
% less_eqE
thf(fact_647_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_648_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_649_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_650_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_651_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_652_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_653_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_654_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_655_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_656_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_657_add__nonneg__eq__0__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_658_add__nonpos__eq__0__iff,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X4 @ Y2 )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_659_lambda__calculus_ORaise_Osimps_I2_J,axiom,
! [D: nat,I: nat,N: nat] :
( ( ( ord_less_eq_nat @ D @ I )
=> ( ( lambda_Raise @ D @ N @ ( lambda_Var @ I ) )
= ( lambda_Var @ ( plus_plus_nat @ I @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ D @ I )
=> ( ( lambda_Raise @ D @ N @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ) ) ).
% lambda_calculus.Raise.simps(2)
thf(fact_660_Con__implies__Coinitial__ind,axiom,
! [T2: lambda_lambda,U: lambda_lambda,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( lambda_size @ T2 ) @ ( lambda_size @ U ) ) @ N )
=> ( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) ) ) ) ).
% Con_implies_Coinitial_ind
thf(fact_661_Coinitial__implies__Con__ind,axiom,
! [T2: lambda_lambda,N: nat,U: lambda_lambda] :
( ( ord_less_eq_nat @ ( lambda_size @ ( lambda_Src @ T2 ) ) @ N )
=> ( ( ( lambda_Arr @ T2 )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T2 )
= ( lambda_Src @ U ) ) )
=> ( ( lambda_resid @ T2 @ U )
!= lambda_Nil ) ) ) ).
% Coinitial_implies_Con_ind
thf(fact_662_Con__Src,axiom,
! [T2: lambda_lambda,U: lambda_lambda,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( lambda_size @ T2 ) @ ( lambda_size @ U ) ) @ N )
=> ( ( ( lambda_resid @ T2 @ U )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Src @ T2 ) @ ( lambda_Src @ U ) )
!= lambda_Nil ) ) ) ).
% Con_Src
thf(fact_663_size__Raise,axiom,
! [D: nat,N: nat,T2: lambda_lambda] :
( ( lambda_size @ ( lambda_Raise @ D @ N @ T2 ) )
= ( lambda_size @ T2 ) ) ).
% size_Raise
thf(fact_664_lambda__calculus_Osize_Osimps_I1_J,axiom,
( ( lambda_size @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.size.simps(1)
thf(fact_665_lambda__calculus_Osize_Osimps_I2_J,axiom,
! [Uu: nat] :
( ( lambda_size @ ( lambda_Var @ Uu ) )
= one_one_nat ) ).
% lambda_calculus.size.simps(2)
thf(fact_666_lambda__calculus_Osize_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_size @ ( lambda_App @ T2 @ U ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T2 ) @ ( lambda_size @ U ) ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(4)
thf(fact_667_lambda__calculus_Osize_Osimps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_size @ ( lambda_Beta @ T2 @ U ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T2 ) @ one_one_nat ) @ ( lambda_size @ U ) ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(5)
thf(fact_668_lambda__calculus_Osize_Osimps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_size @ ( lambda_Lam @ T2 ) )
= ( plus_plus_nat @ ( lambda_size @ T2 ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(3)
thf(fact_669_lambda__calculus_Osize_Oelims,axiom,
! [X4: lambda_lambda,Y2: nat] :
( ( ( lambda_size @ X4 )
= Y2 )
=> ( ( ( X4 = lambda_Nil )
=> ( Y2 != zero_zero_nat ) )
=> ( ( ? [Uu2: nat] :
( X4
= ( lambda_Var @ Uu2 ) )
=> ( Y2 != one_one_nat ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( plus_plus_nat @ ( lambda_size @ T5 ) @ one_one_nat ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
!= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T5 ) @ ( lambda_size @ U3 ) ) @ one_one_nat ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T5 ) @ one_one_nat ) @ ( lambda_size @ U3 ) ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% lambda_calculus.size.elims
thf(fact_670_hgt__resid__ide,axiom,
! [U: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( ( lambda_subs @ U @ T2 )
=> ( ord_less_eq_nat @ ( lambda_hgt @ ( lambda_resid @ T2 @ U ) ) @ ( lambda_hgt @ T2 ) ) ) ) ).
% hgt_resid_ide
thf(fact_671_Subst__Raise,axiom,
! [V: lambda_lambda,D: nat,M3: nat,N: nat,T2: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ D @ M3 )
=> ( ( ord_less_eq_nat @ M3 @ ( plus_plus_nat @ N @ D ) )
=> ( ( lambda_Subst @ M3 @ V @ ( lambda_Raise @ D @ ( suc @ N ) @ T2 ) )
= ( lambda_Raise @ D @ N @ T2 ) ) ) ) ) ).
% Subst_Raise
thf(fact_672_Subst__raise,axiom,
! [V: lambda_lambda,M3: nat,N: nat,T2: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ M3 @ N )
=> ( ( lambda_Subst @ M3 @ V @ ( lambda_Raise @ zero_zero_nat @ ( suc @ N ) @ T2 ) )
= ( lambda_Raise @ zero_zero_nat @ N @ T2 ) ) ) ) ).
% Subst_raise
thf(fact_673_hgt__Raise,axiom,
! [L: nat,K: nat,T2: lambda_lambda] :
( ( lambda_hgt @ ( lambda_Raise @ L @ K @ T2 ) )
= ( lambda_hgt @ T2 ) ) ).
% hgt_Raise
thf(fact_674_Raise__Suc,axiom,
! [D: nat,N: nat,T2: lambda_lambda] :
( ( lambda_Raise @ D @ ( suc @ N ) @ T2 )
= ( lambda_Raise @ D @ one_one_nat @ ( lambda_Raise @ D @ N @ T2 ) ) ) ).
% Raise_Suc
thf(fact_675_substitution__lemma,axiom,
! [V: lambda_lambda,W: lambda_lambda,N: nat,T2: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( W != lambda_Nil )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Subst @ zero_zero_nat @ W @ T2 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Subst @ N @ V @ W ) @ ( lambda_Subst @ ( suc @ N ) @ V @ T2 ) ) ) ) ) ).
% substitution_lemma
thf(fact_676_Subst__Subst,axiom,
! [V: lambda_lambda,W: lambda_lambda,M3: nat,N: nat,T2: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( W != lambda_Nil )
=> ( ( lambda_Subst @ ( plus_plus_nat @ M3 @ N ) @ W @ ( lambda_Subst @ M3 @ V @ T2 ) )
= ( lambda_Subst @ M3 @ ( lambda_Subst @ N @ W @ V ) @ ( lambda_Subst @ ( suc @ ( plus_plus_nat @ M3 @ N ) ) @ W @ T2 ) ) ) ) ) ).
% Subst_Subst
thf(fact_677_Raise__subst,axiom,
! [N: nat,K: nat,V: lambda_lambda,T2: lambda_lambda] :
( ( lambda_Raise @ N @ K @ ( lambda_Subst @ zero_zero_nat @ V @ T2 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Raise @ N @ K @ V ) @ ( lambda_Raise @ ( suc @ N ) @ K @ T2 ) ) ) ).
% Raise_subst
thf(fact_678_Raise__Subst,axiom,
! [P2: nat,N: nat,K: nat,V: lambda_lambda,T2: lambda_lambda] :
( ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Subst @ P2 @ V @ T2 ) )
= ( lambda_Subst @ P2 @ ( lambda_Raise @ N @ K @ V ) @ ( lambda_Raise @ ( suc @ ( plus_plus_nat @ P2 @ N ) ) @ K @ T2 ) ) ) ).
% Raise_Subst
thf(fact_679_lambda__calculus_Ohgt_Osimps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( lambda_hgt @ ( lambda_Lam @ T2 ) )
= ( lambda_hgt @ T2 ) ) ).
% lambda_calculus.hgt.simps(3)
thf(fact_680_lambda__calculus_ORaise_Osimps_I5_J,axiom,
! [D: nat,N: nat,T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_Beta @ ( lambda_Raise @ ( suc @ D ) @ N @ T2 ) @ ( lambda_Raise @ D @ N @ U ) ) ) ).
% lambda_calculus.Raise.simps(5)
thf(fact_681_lambda__calculus_OSubst_Osimps_I5_J,axiom,
! [N: nat,V: lambda_lambda,T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_Beta @ T2 @ U ) )
= ( lambda_Beta @ ( lambda_Subst @ ( suc @ N ) @ V @ T2 ) @ ( lambda_Subst @ N @ V @ U ) ) ) ).
% lambda_calculus.Subst.simps(5)
thf(fact_682_lambda__calculus_ORaise_Osimps_I3_J,axiom,
! [D: nat,N: nat,T2: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda_Raise @ ( suc @ D ) @ N @ T2 ) ) ) ).
% lambda_calculus.Raise.simps(3)
thf(fact_683_lambda__calculus_OSubst_Osimps_I3_J,axiom,
! [N: nat,V: lambda_lambda,T2: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda_Subst @ ( suc @ N ) @ V @ T2 ) ) ) ).
% lambda_calculus.Subst.simps(3)
thf(fact_684_lambda__calculus_Ohgt_Osimps_I1_J,axiom,
( ( lambda_hgt @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.hgt.simps(1)
thf(fact_685_lambda__calculus_Ohgt_Osimps_I2_J,axiom,
! [Uu: nat] :
( ( lambda_hgt @ ( lambda_Var @ Uu ) )
= zero_zero_nat ) ).
% lambda_calculus.hgt.simps(2)
thf(fact_686_lambda__calculus_Ohgt_Osimps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_hgt @ ( lambda_App @ T2 @ U ) )
= ( plus_plus_nat @ ( lambda_hgt @ T2 ) @ ( lambda_hgt @ U ) ) ) ).
% lambda_calculus.hgt.simps(4)
thf(fact_687_lambda__calculus_Oreduce_Osimps_I2_J,axiom,
! [F2: lambda_lambda > lambda_lambda,A: lambda_lambda,N: nat] :
( ( lambda_reduce @ F2 @ A @ ( suc @ N ) )
= ( lambda_reduce @ F2 @ ( lambda_Trg @ ( F2 @ A ) ) @ N ) ) ).
% lambda_calculus.reduce.simps(2)
thf(fact_688_lambda__calculus_Oreduce_Oelims,axiom,
! [X4: lambda_lambda > lambda_lambda,Xa: lambda_lambda,Xb: nat,Y2: lambda_lambda] :
( ( ( lambda_reduce @ X4 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xb = zero_zero_nat )
=> ( Y2 != Xa ) )
=> ~ ! [N5: nat] :
( ( Xb
= ( suc @ N5 ) )
=> ( Y2
!= ( lambda_reduce @ X4 @ ( lambda_Trg @ ( X4 @ Xa ) ) @ N5 ) ) ) ) ) ).
% lambda_calculus.reduce.elims
thf(fact_689_lambda__calculus_ORaise_Oelims,axiom,
! [X4: nat,Xa: nat,Xb: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Raise @ X4 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xb = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ! [I2: nat] :
( ( Xb
= ( lambda_Var @ I2 ) )
=> ~ ( ( ( ord_less_eq_nat @ X4 @ I2 )
=> ( Y2
= ( lambda_Var @ ( plus_plus_nat @ I2 @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ X4 @ I2 )
=> ( Y2
= ( lambda_Var @ I2 ) ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( Xb
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda_Raise @ ( suc @ X4 ) @ Xa @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xb
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_Raise @ X4 @ Xa @ T5 ) @ ( lambda_Raise @ X4 @ Xa @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xb
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda_Raise @ ( suc @ X4 ) @ Xa @ T5 ) @ ( lambda_Raise @ X4 @ Xa @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Raise.elims
thf(fact_690_hgt__resid__App__head__redex,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_App @ T2 @ U ) )
=> ( ~ ( lambda_Ide @ ( lambda_head_redex @ ( lambda_App @ T2 @ U ) ) )
=> ( ord_less_nat @ ( lambda_hgt @ ( lambda_resid @ ( lambda_App @ T2 @ U ) @ ( lambda_head_redex @ ( lambda_App @ T2 @ U ) ) ) ) @ ( lambda_hgt @ ( lambda_App @ T2 @ U ) ) ) ) ) ).
% hgt_resid_App_head_redex
thf(fact_691_sources__Raise,axiom,
! [T2: lambda_lambda,D: nat,N: nat] :
( ( lambda_Arr @ T2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Raise @ D @ N @ T2 ) )
= ( insert_lambda_lambda @ ( lambda_Raise @ D @ N @ ( lambda_Src @ T2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% sources_Raise
thf(fact_692_elementary__reduction__decreases__hgt,axiom,
! [U: lambda_lambda,T2: lambda_lambda] :
( ( lambda2670285437103676873uction @ U )
=> ( ( lambda_subs @ U @ T2 )
=> ( ord_less_nat @ ( lambda_hgt @ ( lambda_resid @ T2 @ U ) ) @ ( lambda_hgt @ T2 ) ) ) ) ).
% elementary_reduction_decreases_hgt
thf(fact_693_sources__cong__closed,axiom,
! [A: lambda_lambda,T2: lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A4 @ A ) ) )
=> ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) ) ) ) ).
% sources_cong_closed
thf(fact_694_sources__are__cong,axiom,
! [A: lambda_lambda,T2: lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A4 @ A ) ) ) ) ) ).
% sources_are_cong
thf(fact_695_source__is__ide,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ).
% source_is_ide
thf(fact_696_arr__has__un__source,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( Y = X2 ) ) ) ) ).
% arr_has_un_source
thf(fact_697_sources__are__con,axiom,
! [A: lambda_lambda,T2: lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( con_lambda_lambda @ lambda_resid @ A @ A4 ) ) ) ).
% sources_are_con
thf(fact_698_resid__source__in__targets,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( member_lambda_lambda @ ( lambda_resid @ A @ T2 ) @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) ) ) ).
% resid_source_in_targets
thf(fact_699_sources__join__of_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
= ( source5627170679743844886lambda @ lambda_resid @ V ) ) ) ).
% sources_join_of(1)
thf(fact_700_sources__join__of_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ U )
= ( source5627170679743844886lambda @ lambda_resid @ V ) ) ) ).
% sources_join_of(2)
thf(fact_701_Raise__Var,axiom,
! [D: nat,N: nat,I: nat] :
( ( lambda_Raise @ D @ N @ ( lambda_Var @ I ) )
= ( lambda_Var @ ( if_nat @ ( ord_less_nat @ I @ D ) @ I @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% Raise_Var
thf(fact_702_in__sourcesE,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ A )
=> ~ ( con_lambda_lambda @ lambda_resid @ T2 @ A ) ) ) ).
% in_sourcesE
thf(fact_703_sources__con__closed,axiom,
! [A: lambda_lambda,T2: lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A4 )
=> ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) ) ) ) ) ).
% sources_con_closed
thf(fact_704_arr__iff__has__source,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
= ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
!= bot_bo1271416176687173302lambda ) ) ).
% arr_iff_has_source
thf(fact_705_src__in__sources,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( member_lambda_lambda @ ( weakly4554751870655413501lambda @ lambda_resid @ T2 ) @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) ) ) ).
% src_in_sources
thf(fact_706_join__of__arr__src_I2_J,axiom,
! [T2: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ A @ T2 ) ) ) ).
% join_of_arr_src(2)
thf(fact_707_join__of__arr__src_I1_J,axiom,
! [T2: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ A @ T2 @ T2 ) ) ) ).
% join_of_arr_src(1)
thf(fact_708_coinitialE,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
!= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ) ).
% coinitialE
thf(fact_709_coinitial__iff,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ T3 )
= ( ( arr_lambda_lambda @ lambda_resid @ T2 )
& ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) ) ) ).
% coinitial_iff
thf(fact_710_composableD_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T2 @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ).
% composableD(3)
thf(fact_711_seq__def,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T2 @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T2 )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ).
% seq_def
thf(fact_712_seqE,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T2 @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
!= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ) ).
% seqE
thf(fact_713_sources__char_092_060_094sub_062_092_060Lambda_062,axiom,
! [T2: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
= ( insert_lambda_lambda @ ( lambda_Src @ T2 ) @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( lambda_Arr @ T2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
= bot_bo1271416176687173302lambda ) ) ) ).
% sources_char\<^sub>\<Lambda>
thf(fact_714_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_715_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_716_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_717_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_718_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_719_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_720_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_721_in__sourcesI,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ T2 @ A )
=> ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) ) ) ) ).
% in_sourcesI
thf(fact_722_lambda__calculus_Osources__simps_I1_J,axiom,
( ( source5627170679743844886lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.sources_simps(1)
thf(fact_723_sources__resid,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ).
% sources_resid
thf(fact_724_coinitialI,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
= ( source5627170679743844886lambda @ lambda_resid @ U ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U ) ) ) ).
% coinitialI
thf(fact_725_lambda__calculus_Osources__simps_I2_J,axiom,
! [X4: nat] :
( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Var @ X4 ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X4 ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.sources_simps(2)
thf(fact_726_seqI,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= ( source5627170679743844886lambda @ lambda_resid @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T2 @ U ) ) ) ) ).
% seqI
thf(fact_727_sources__simp,axiom,
! [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
= ( insert_lambda_lambda @ ( lambda_Src @ T2 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% sources_simp
thf(fact_728_lambda__calculus_Osources__simps_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_App @ T2 @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T2 ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_simps(4)
thf(fact_729_lambda__calculus_Osources__simps_I3_J,axiom,
! [T2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Lam @ T2 ) )
= ( insert_lambda_lambda @ ( lambda_Lam @ ( lambda_Src @ T2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.sources_simps(3)
thf(fact_730_sources__subst,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T2 )
=> ( ( lambda_Arr @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Subst @ zero_zero_nat @ T2 @ U ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Src @ T2 ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% sources_subst
thf(fact_731_lambda__calculus_Osources__simps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Beta @ T2 @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T2 ) ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_simps(5)
thf(fact_732_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_733_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_734_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_735_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_736_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_737_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_738_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_739_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_740_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_741_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_742_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_743_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_744_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_745_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_746_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_747_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_748_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_749_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C: nat] :
( ( B
= ( plus_plus_nat @ A @ C ) )
=> ( C = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_750_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_751_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_752_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: nat > nat > nat,NN: set_nat,V: nat,V2: nat,W: nat,W2: nat,T2: nat,T3: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( member_nat @ V @ NN )
=> ( ( member_nat @ V2 @ NN )
=> ( ( member_nat @ W @ NN )
=> ( ( member_nat @ W2 @ NN )
=> ( ( ( sources_nat @ Resid @ V )
= ( sources_nat @ Resid @ W ) )
=> ( ( ( sources_nat @ Resid @ V2 )
= ( sources_nat @ Resid @ W2 ) )
=> ( ( ( targets_nat @ Resid @ W )
= ( targets_nat @ Resid @ W2 ) )
=> ( ( ( member_nat @ ( Resid @ ( Resid @ T2 @ V ) @ ( Resid @ T3 @ V2 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T3 @ V2 ) @ ( Resid @ T2 @ V ) ) @ NN ) )
=> ( ( member_nat @ ( Resid @ ( Resid @ T2 @ W ) @ ( Resid @ T3 @ W2 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T3 @ W2 ) @ ( Resid @ T2 @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_753_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,V: lambda_lambda,V2: lambda_lambda,W: lambda_lambda,W2: lambda_lambda,T2: lambda_lambda,T3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ V @ NN )
=> ( ( member_lambda_lambda @ V2 @ NN )
=> ( ( member_lambda_lambda @ W @ NN )
=> ( ( member_lambda_lambda @ W2 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ V )
= ( source5627170679743844886lambda @ Resid @ W ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ V2 )
= ( source5627170679743844886lambda @ Resid @ W2 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ W )
= ( target5490931411331770124lambda @ Resid @ W2 ) )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T2 @ V ) @ ( Resid @ T3 @ V2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T3 @ V2 ) @ ( Resid @ T2 @ V ) ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T2 @ W ) @ ( Resid @ T3 @ W2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T3 @ W2 ) @ ( Resid @ T2 @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_754_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T2: nat,U: nat,U2: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( arr_nat @ Resid @ T2 )
=> ( ( member_nat @ U @ NN )
=> ( ( member_nat @ U2 @ NN )
=> ( ( ( sources_nat @ Resid @ U )
= ( sources_nat @ Resid @ U2 ) )
=> ( ( ( targets_nat @ Resid @ U )
= ( targets_nat @ Resid @ U2 ) )
=> ( ( ( sources_nat @ Resid @ T2 )
= ( sources_nat @ Resid @ U ) )
=> ( ( member_nat @ ( Resid @ ( Resid @ T2 @ U ) @ ( Resid @ T2 @ U2 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T2 @ U2 ) @ ( Resid @ T2 @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_755_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T2: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T2 )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U2 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U )
= ( source5627170679743844886lambda @ Resid @ U2 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U )
= ( target5490931411331770124lambda @ Resid @ U2 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T2 )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T2 @ U ) @ ( Resid @ T2 @ U2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T2 @ U2 ) @ ( Resid @ T2 @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_756_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_757_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_758_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_759_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_760_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_761_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_762_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_763_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ( P @ N5 )
=> ( P @ ( suc @ N5 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_764_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_765_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_766_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_767_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_le_one
thf(fact_768_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_769_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_770_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_771_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_772_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_773_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_774_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_775_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_776_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_777_mtp__Raise__ind,axiom,
! [L: nat,D: nat,T2: lambda_lambda,S: nat,X4: nat,K: nat] :
( ( ord_less_eq_nat @ L @ D )
=> ( ( ord_less_eq_nat @ ( lambda_size @ T2 ) @ S )
=> ( ( lambda_mtp @ ( plus_plus_nat @ ( plus_plus_nat @ X4 @ D ) @ K ) @ ( lambda_Raise @ L @ K @ T2 ) )
= ( lambda_mtp @ ( plus_plus_nat @ X4 @ D ) @ T2 ) ) ) ) ).
% mtp_Raise_ind
thf(fact_778_coterminal__def,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T2 @ U )
= ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) @ ( target5490931411331770124lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% coterminal_def
thf(fact_779_mtpE__eq__Raise,axiom,
! [X4: nat,D: nat,K: nat,T2: lambda_lambda] :
( ( ord_less_nat @ X4 @ D )
=> ( ( lambda_mtp @ X4 @ ( lambda_Raise @ D @ K @ T2 ) )
= ( lambda_mtp @ X4 @ T2 ) ) ) ).
% mtpE_eq_Raise
thf(fact_780_mtp__Raise_H,axiom,
! [L: nat,K: nat,T2: lambda_lambda] :
( ( lambda_mtp @ L @ ( lambda_Raise @ L @ ( suc @ K ) @ T2 ) )
= zero_zero_nat ) ).
% mtp_Raise'
thf(fact_781_mtp_092_060_094sub_0620__Subst__cancel,axiom,
! [D: nat,U: lambda_lambda,T2: lambda_lambda] :
( ( lambda_mtp @ zero_zero_nat @ ( lambda_Subst @ ( suc @ D ) @ U @ T2 ) )
= ( lambda_mtp @ zero_zero_nat @ T2 ) ) ).
% mtp\<^sub>0_Subst_cancel
thf(fact_782_mtp__Subst__cancel,axiom,
! [K: nat,D: nat,U: lambda_lambda,T2: lambda_lambda] :
( ( lambda_mtp @ K @ ( lambda_Subst @ ( plus_plus_nat @ ( suc @ D ) @ K ) @ U @ T2 ) )
= ( lambda_mtp @ K @ T2 ) ) ).
% mtp_Subst_cancel
thf(fact_783_mtp__Raise,axiom,
! [L: nat,D: nat,X4: nat,K: nat,T2: lambda_lambda] :
( ( ord_less_eq_nat @ L @ D )
=> ( ( lambda_mtp @ ( plus_plus_nat @ ( plus_plus_nat @ X4 @ D ) @ K ) @ ( lambda_Raise @ L @ K @ T2 ) )
= ( lambda_mtp @ ( plus_plus_nat @ X4 @ D ) @ T2 ) ) ) ).
% mtp_Raise
thf(fact_784_mtp__gt__0__iff__in__FV,axiom,
! [X4: nat,T2: lambda_lambda] :
( ( ord_less_nat @ zero_zero_nat @ ( lambda_mtp @ X4 @ T2 ) )
= ( member_nat @ X4 @ ( lambda_FV @ T2 ) ) ) ).
% mtp_gt_0_iff_in_FV
thf(fact_785_sources__eqI,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) @ ( source5627170679743844886lambda @ lambda_resid @ T3 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T2 )
= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) ) ).
% sources_eqI
thf(fact_786_targets__eqI,axiom,
! [T2: lambda_lambda,T3: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T2 )
= ( target5490931411331770124lambda @ lambda_resid @ T3 ) ) ) ).
% targets_eqI
thf(fact_787_mtp__raise,axiom,
! [X4: nat,D: nat,T2: lambda_lambda] :
( ( lambda_mtp @ ( plus_plus_nat @ X4 @ ( suc @ D ) ) @ ( lambda_Raise @ zero_zero_nat @ D @ T2 ) )
= ( lambda_mtp @ ( suc @ X4 ) @ T2 ) ) ).
% mtp_raise
thf(fact_788_elementary__reduction__nonincreases__mtp,axiom,
! [U: lambda_lambda,T2: lambda_lambda,X4: nat] :
( ( lambda2670285437103676873uction @ U )
=> ( ( lambda_subs @ U @ T2 )
=> ( ord_less_eq_nat @ ( lambda_mtp @ X4 @ ( lambda_resid @ T2 @ U ) ) @ ( lambda_mtp @ X4 @ T2 ) ) ) ) ).
% elementary_reduction_nonincreases_mtp
thf(fact_789_con__imp__common__source,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) @ ( source5627170679743844886lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% con_imp_common_source
thf(fact_790_coinitial__def,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T2 @ U )
= ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) @ ( source5627170679743844886lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% coinitial_def
thf(fact_791_lambda__calculus_Omtp_Osimps_I1_J,axiom,
! [X4: nat] :
( ( lambda_mtp @ X4 @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.mtp.simps(1)
thf(fact_792_lambda__calculus_Omtp_Osimps_I4_J,axiom,
! [X4: nat,T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_mtp @ X4 @ ( lambda_App @ T2 @ U ) )
= ( plus_plus_nat @ ( lambda_mtp @ X4 @ T2 ) @ ( lambda_mtp @ X4 @ U ) ) ) ).
% lambda_calculus.mtp.simps(4)
thf(fact_793_lambda__calculus_Omtp_Osimps_I3_J,axiom,
! [X4: nat,T2: lambda_lambda] :
( ( lambda_mtp @ X4 @ ( lambda_Lam @ T2 ) )
= ( lambda_mtp @ ( suc @ X4 ) @ T2 ) ) ).
% lambda_calculus.mtp.simps(3)
thf(fact_794_lambda__calculus_Omtp_Osimps_I2_J,axiom,
! [Z: nat,X4: nat] :
( ( ( Z = X4 )
=> ( ( lambda_mtp @ X4 @ ( lambda_Var @ Z ) )
= one_one_nat ) )
& ( ( Z != X4 )
=> ( ( lambda_mtp @ X4 @ ( lambda_Var @ Z ) )
= zero_zero_nat ) ) ) ).
% lambda_calculus.mtp.simps(2)
thf(fact_795_mtp__Subst,axiom,
! [K: nat,U: lambda_lambda,T2: lambda_lambda] :
( ( lambda_mtp @ K @ ( lambda_Subst @ K @ U @ T2 ) )
= ( plus_plus_nat @ ( lambda_mtp @ ( suc @ K ) @ T2 ) @ ( times_times_nat @ ( lambda_mtp @ K @ ( lambda_Raise @ zero_zero_nat @ K @ U ) ) @ ( lambda_mtp @ K @ T2 ) ) ) ) ).
% mtp_Subst
thf(fact_796_hgt__Subst,axiom,
! [U: lambda_lambda,K: nat,T2: lambda_lambda] :
( ( lambda_Arr @ U )
=> ( ( lambda_hgt @ ( lambda_Subst @ K @ U @ T2 ) )
= ( plus_plus_nat @ ( lambda_hgt @ T2 ) @ ( times_times_nat @ ( lambda_hgt @ U ) @ ( lambda_mtp @ K @ T2 ) ) ) ) ) ).
% hgt_Subst
thf(fact_797_mtp__Subst_H,axiom,
! [X4: nat,D: nat,U: lambda_lambda,T2: lambda_lambda] :
( ( lambda_mtp @ ( plus_plus_nat @ X4 @ ( suc @ D ) ) @ ( lambda_Subst @ D @ U @ T2 ) )
= ( plus_plus_nat @ ( lambda_mtp @ ( plus_plus_nat @ X4 @ ( suc @ ( suc @ D ) ) ) @ T2 ) @ ( times_times_nat @ ( lambda_mtp @ ( suc @ X4 ) @ U ) @ ( lambda_mtp @ D @ T2 ) ) ) ) ).
% mtp_Subst'
thf(fact_798_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_799_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_800_nat__1__eq__mult__iff,axiom,
! [M3: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M3 @ N ) )
= ( ( M3 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_801_nat__mult__eq__1__iff,axiom,
! [M3: nat,N: nat] :
( ( ( times_times_nat @ M3 @ N )
= one_one_nat )
= ( ( M3 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_802_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_803_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B3: nat] : ( times_times_nat @ B3 @ A2 ) ) ) ).
% mult.commute
thf(fact_804_mult_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_805_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_806_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_807_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_808_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_809_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_810_mult__eq__self__implies__10,axiom,
! [M3: nat,N: nat] :
( ( M3
= ( times_times_nat @ M3 @ N ) )
=> ( ( N = one_one_nat )
| ( M3 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_811_less__1__mult,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M3 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M3 @ N ) ) ) ) ).
% less_1_mult
thf(fact_812_mult__left__le,axiom,
! [C2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).
% mult_left_le
thf(fact_813_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_814_mtp0__subst__le,axiom,
! [U: lambda_lambda,T2: lambda_lambda] : ( ord_less_eq_nat @ ( lambda_mtp @ zero_zero_nat @ ( lambda_Subst @ zero_zero_nat @ U @ T2 ) ) @ ( plus_plus_nat @ ( lambda_mtp @ one_one_nat @ T2 ) @ ( times_times_nat @ ( lambda_mtp @ zero_zero_nat @ U ) @ ( ord_max_nat @ one_one_nat @ ( lambda_mtp @ zero_zero_nat @ T2 ) ) ) ) ) ).
% mtp0_subst_le
thf(fact_815_max__add__distrib__left,axiom,
! [X4: nat,Y2: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X4 @ Y2 ) @ Z )
= ( ord_max_nat @ ( plus_plus_nat @ X4 @ Z ) @ ( plus_plus_nat @ Y2 @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_816_max__add__distrib__right,axiom,
! [X4: nat,Y2: nat,Z: nat] :
( ( plus_plus_nat @ X4 @ ( ord_max_nat @ Y2 @ Z ) )
= ( ord_max_nat @ ( plus_plus_nat @ X4 @ Y2 ) @ ( plus_plus_nat @ X4 @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_817_lambda__calculus_Omtp_Osimps_I5_J,axiom,
! [X4: nat,T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_mtp @ X4 @ ( lambda_Beta @ T2 @ U ) )
= ( plus_plus_nat @ ( lambda_mtp @ ( suc @ X4 ) @ T2 ) @ ( times_times_nat @ ( lambda_mtp @ X4 @ U ) @ ( ord_max_nat @ one_one_nat @ ( lambda_mtp @ zero_zero_nat @ T2 ) ) ) ) ) ).
% lambda_calculus.mtp.simps(5)
thf(fact_818_lambda__calculus_Ohgt_Osimps_I5_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_hgt @ ( lambda_Beta @ T2 @ U ) )
= ( suc @ ( plus_plus_nat @ ( lambda_hgt @ T2 ) @ ( times_times_nat @ ( lambda_hgt @ U ) @ ( ord_max_nat @ one_one_nat @ ( lambda_mtp @ zero_zero_nat @ T2 ) ) ) ) ) ) ).
% lambda_calculus.hgt.simps(5)
thf(fact_819_lambda__calculus_Omtp_Oelims,axiom,
! [X4: nat,Xa: lambda_lambda,Y2: nat] :
( ( ( lambda_mtp @ X4 @ Xa )
= Y2 )
=> ( ( ( Xa = lambda_Nil )
=> ( Y2 != zero_zero_nat ) )
=> ( ! [Z2: nat] :
( ( Xa
= ( lambda_Var @ Z2 ) )
=> ~ ( ( ( Z2 = X4 )
=> ( Y2 = one_one_nat ) )
& ( ( Z2 != X4 )
=> ( Y2 = zero_zero_nat ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_mtp @ ( suc @ X4 ) @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xa
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
!= ( plus_plus_nat @ ( lambda_mtp @ X4 @ T5 ) @ ( lambda_mtp @ X4 @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( plus_plus_nat @ ( lambda_mtp @ ( suc @ X4 ) @ T5 ) @ ( times_times_nat @ ( lambda_mtp @ X4 @ U3 ) @ ( ord_max_nat @ one_one_nat @ ( lambda_mtp @ zero_zero_nat @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.mtp.elims
thf(fact_820_lambda__calculus_Ohgt_Oelims,axiom,
! [X4: lambda_lambda,Y2: nat] :
( ( ( lambda_hgt @ X4 )
= Y2 )
=> ( ( ( X4 = lambda_Nil )
=> ( Y2 != zero_zero_nat ) )
=> ( ( ? [Uu2: nat] :
( X4
= ( lambda_Var @ Uu2 ) )
=> ( Y2 != zero_zero_nat ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_hgt @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
!= ( plus_plus_nat @ ( lambda_hgt @ T5 ) @ ( lambda_hgt @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( suc @ ( plus_plus_nat @ ( lambda_hgt @ T5 ) @ ( times_times_nat @ ( lambda_hgt @ U3 ) @ ( ord_max_nat @ one_one_nat @ ( lambda_mtp @ zero_zero_nat @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.hgt.elims
thf(fact_821_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_822_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_823_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_824_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_825_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_826_subterm__implies__smaller,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T2 @ U )
=> ( ord_less_nat @ ( lambda_size @ T2 ) @ ( lambda_size @ U ) ) ) ).
% subterm_implies_smaller
thf(fact_827_lambda__calculus_OSubst_Oelims,axiom,
! [X4: nat,Xa: lambda_lambda,Xb: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Subst @ X4 @ Xa @ Xb )
= Y2 )
=> ( ( ( Xb = lambda_Nil )
=> ( Y2 != lambda_Nil ) )
=> ( ! [I2: nat] :
( ( Xb
= ( lambda_Var @ I2 ) )
=> ~ ( ( ( ord_less_nat @ X4 @ I2 )
=> ( Y2
= ( lambda_Var @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) )
& ( ~ ( ord_less_nat @ X4 @ I2 )
=> ( ( ( X4 = I2 )
=> ( Y2
= ( lambda_Raise @ zero_zero_nat @ X4 @ Xa ) ) )
& ( ( X4 != I2 )
=> ( Y2
= ( lambda_Var @ I2 ) ) ) ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( Xb
= ( lambda_Lam @ T5 ) )
=> ( Y2
!= ( lambda_Lam @ ( lambda_Subst @ ( suc @ X4 ) @ Xa @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xb
= ( lambda_App @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_App @ ( lambda_Subst @ X4 @ Xa @ T5 ) @ ( lambda_Subst @ X4 @ Xa @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xb
= ( lambda_Beta @ T5 @ U3 ) )
=> ( Y2
!= ( lambda_Beta @ ( lambda_Subst @ ( suc @ X4 ) @ Xa @ T5 ) @ ( lambda_Subst @ X4 @ Xa @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Subst.elims
thf(fact_828_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_829_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_830_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_831_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_832_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_833_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_834_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_835_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_836_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_837_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_838_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_839_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_840_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_841_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_842_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_843_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_844_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_845_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_846_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_847_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_848_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_849_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_850_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_851_lambda__calculus_Osubterm_Ointros_I3_J,axiom,
! [U: lambda_lambda,T2: lambda_lambda] : ( lambda_subterm @ U @ ( lambda_App @ T2 @ U ) ) ).
% lambda_calculus.subterm.intros(3)
thf(fact_852_lambda__calculus_Osubterm_Ointros_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] : ( lambda_subterm @ T2 @ ( lambda_App @ T2 @ U ) ) ).
% lambda_calculus.subterm.intros(2)
thf(fact_853_lambda__calculus_Osubterm__lemmas_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T2 @ ( lambda_App @ T2 @ U ) )
& ( lambda_subterm @ U @ ( lambda_App @ T2 @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(3)
thf(fact_854_lambda__calculus_Osubterm_Ointros_I5_J,axiom,
! [U: lambda_lambda,T2: lambda_lambda] : ( lambda_subterm @ U @ ( lambda_Beta @ T2 @ U ) ) ).
% lambda_calculus.subterm.intros(5)
thf(fact_855_lambda__calculus_Osubterm_Ointros_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] : ( lambda_subterm @ T2 @ ( lambda_Beta @ T2 @ U ) ) ).
% lambda_calculus.subterm.intros(4)
thf(fact_856_lambda__calculus_Osubterm__lemmas_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T2 @ ( lambda_Beta @ T2 @ U ) )
& ( lambda_subterm @ U @ ( lambda_Beta @ T2 @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(4)
thf(fact_857_lambda__calculus_Osubterm__lemmas_I1_J,axiom,
! [T2: lambda_lambda] : ( lambda_subterm @ T2 @ ( lambda_Lam @ T2 ) ) ).
% lambda_calculus.subterm_lemmas(1)
thf(fact_858_lambda__calculus_Osubterm_Ointros_I6_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( lambda_subterm @ T2 @ U )
=> ( ( lambda_subterm @ U @ V )
=> ( lambda_subterm @ T2 @ V ) ) ) ).
% lambda_calculus.subterm.intros(6)
thf(fact_859_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_860_lambda__calculus_Osubterm__lemmas_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T2 @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) )
& ( lambda_subterm @ U @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(2)
thf(fact_861_lambda__calculus_Osubterm_Osimps,axiom,
( lambda_subterm
= ( ^ [A1: lambda_lambda,A22: lambda_lambda] :
( ? [T: lambda_lambda] :
( ( A1 = T )
& ( A22
= ( lambda_Lam @ T ) ) )
| ? [T: lambda_lambda,U4: lambda_lambda] :
( ( A1 = T )
& ( A22
= ( lambda_App @ T @ U4 ) ) )
| ? [T: lambda_lambda,U4: lambda_lambda] :
( ( A1 = U4 )
& ( A22
= ( lambda_App @ T @ U4 ) ) )
| ? [T: lambda_lambda,U4: lambda_lambda] :
( ( A1 = T )
& ( A22
= ( lambda_Beta @ T @ U4 ) ) )
| ? [T: lambda_lambda,U4: lambda_lambda] :
( ( A1 = U4 )
& ( A22
= ( lambda_Beta @ T @ U4 ) ) )
| ? [T: lambda_lambda,U4: lambda_lambda,V5: lambda_lambda] :
( ( A1 = T )
& ( A22 = V5 )
& ( lambda_subterm @ T @ U4 )
& ( lambda_subterm @ U4 @ V5 ) ) ) ) ) ).
% lambda_calculus.subterm.simps
thf(fact_862_lambda__calculus_Osubterm_Ocases,axiom,
! [A12: lambda_lambda,A23: lambda_lambda] :
( ( lambda_subterm @ A12 @ A23 )
=> ( ( A23
!= ( lambda_Lam @ A12 ) )
=> ( ! [U3: lambda_lambda] :
( A23
!= ( lambda_App @ A12 @ U3 ) )
=> ( ! [T5: lambda_lambda] :
( A23
!= ( lambda_App @ T5 @ A12 ) )
=> ( ! [U3: lambda_lambda] :
( A23
!= ( lambda_Beta @ A12 @ U3 ) )
=> ( ! [T5: lambda_lambda] :
( A23
!= ( lambda_Beta @ T5 @ A12 ) )
=> ~ ! [U3: lambda_lambda] :
( ( lambda_subterm @ A12 @ U3 )
=> ~ ( lambda_subterm @ U3 @ A23 ) ) ) ) ) ) ) ) ).
% lambda_calculus.subterm.cases
thf(fact_863_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_864_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_865_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N2: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_866_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N2: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_867_lambda__calculus_OSubst_Osimps_I2_J,axiom,
! [N: nat,I: nat,V: lambda_lambda] :
( ( ( ord_less_nat @ N @ I )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Var @ I ) )
= ( lambda_Var @ ( minus_minus_nat @ I @ one_one_nat ) ) ) )
& ( ~ ( ord_less_nat @ N @ I )
=> ( ( ( N = I )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Var @ I ) )
= ( lambda_Raise @ zero_zero_nat @ N @ V ) ) )
& ( ( N != I )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ) ) ) ) ).
% lambda_calculus.Subst.simps(2)
thf(fact_868_lambda__calculus_Ohgt_Opelims,axiom,
! [X4: lambda_lambda,Y2: nat] :
( ( ( lambda_hgt @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda_hgt_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ( ( Y2 = zero_zero_nat )
=> ~ ( accp_lambda_lambda @ lambda_hgt_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ( ( Y2 = zero_zero_nat )
=> ~ ( accp_lambda_lambda @ lambda_hgt_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_hgt @ T5 ) )
=> ~ ( accp_lambda_lambda @ lambda_hgt_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( plus_plus_nat @ ( lambda_hgt @ T5 ) @ ( lambda_hgt @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_hgt_rel @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( suc @ ( plus_plus_nat @ ( lambda_hgt @ T5 ) @ ( times_times_nat @ ( lambda_hgt @ U3 ) @ ( ord_max_nat @ one_one_nat @ ( lambda_mtp @ zero_zero_nat @ T5 ) ) ) ) ) )
=> ~ ( accp_lambda_lambda @ lambda_hgt_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.hgt.pelims
thf(fact_869_lambda__calculus_Osize_Opelims,axiom,
! [X4: lambda_lambda,Y2: nat] :
( ( ( lambda_size @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda_size_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ( ( Y2 = zero_zero_nat )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ( ( Y2 = one_one_nat )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( plus_plus_nat @ ( lambda_size @ T5 ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T5 ) @ ( lambda_size @ U3 ) ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T5 ) @ one_one_nat ) @ ( lambda_size @ U3 ) ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.size.pelims
thf(fact_870_lambda__calculus_Obottom__up__redex_Opelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda5654762366400029632_redex @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda8685541990070617401ex_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda8685541990070617401ex_rel @ lambda_Nil ) ) )
=> ( ! [X2: nat] :
( ( X4
= ( lambda_Var @ X2 ) )
=> ( ( Y2
= ( lambda_Var @ X2 ) )
=> ~ ( accp_lambda_lambda @ lambda8685541990070617401ex_rel @ ( lambda_Var @ X2 ) ) ) )
=> ( ! [M2: lambda_lambda] :
( ( X4
= ( lambda_Lam @ M2 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda5654762366400029632_redex @ M2 ) ) )
=> ~ ( accp_lambda_lambda @ lambda8685541990070617401ex_rel @ ( lambda_Lam @ M2 ) ) ) )
=> ( ! [M2: lambda_lambda,N4: lambda_lambda] :
( ( X4
= ( lambda_App @ M2 @ N4 ) )
=> ( ( ( ~ ( lambda_Ide @ M2 )
=> ( Y2
= ( lambda_App @ ( lambda5654762366400029632_redex @ M2 ) @ ( lambda_Src @ N4 ) ) ) )
& ( ( lambda_Ide @ M2 )
=> ( Y2
= ( lambda_App @ M2 @ ( lambda5654762366400029632_redex @ N4 ) ) ) ) )
=> ~ ( accp_lambda_lambda @ lambda8685541990070617401ex_rel @ ( lambda_App @ M2 @ N4 ) ) ) )
=> ~ ! [M2: lambda_lambda,N4: lambda_lambda] :
( ( X4
= ( lambda_Beta @ M2 @ N4 ) )
=> ( ( ( ~ ( lambda_Ide @ M2 )
=> ( Y2
= ( lambda_App @ ( lambda_Lam @ ( lambda5654762366400029632_redex @ M2 ) ) @ ( lambda_Src @ N4 ) ) ) )
& ( ( lambda_Ide @ M2 )
=> ( ( ~ ( lambda_Ide @ N4 )
=> ( Y2
= ( lambda_App @ ( lambda_Lam @ M2 ) @ ( lambda5654762366400029632_redex @ N4 ) ) ) )
& ( ( lambda_Ide @ N4 )
=> ( Y2
= ( lambda_Beta @ M2 @ N4 ) ) ) ) ) )
=> ~ ( accp_lambda_lambda @ lambda8685541990070617401ex_rel @ ( lambda_Beta @ M2 @ N4 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.bottom_up_redex.pelims
thf(fact_871_lambda__calculus_OTrg_Opelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Trg @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda_Trg_rel @ X4 )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ( ( Y2
= ( lambda_Var @ I2 ) )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ ( lambda_Var @ I2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda_Trg @ T5 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_Trg @ T5 ) @ ( lambda_Trg @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U3 ) @ ( lambda_Trg @ T5 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Trg.pelims
thf(fact_872_lambda__calculus_OSrc_Opelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Src @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda_Src_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ lambda_Nil ) ) )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ( ( Y2
= ( lambda_Var @ I2 ) )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ ( lambda_Var @ I2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda_Src @ T5 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_Src @ T5 ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T5 ) ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Src.pelims
thf(fact_873_lambda__calculus_Ohead__redex_Opelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_head_redex @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ lambda_Nil ) ) )
=> ( ! [X2: nat] :
( ( X4
= ( lambda_Var @ X2 ) )
=> ( ( Y2
= ( lambda_Var @ X2 ) )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ ( lambda_Var @ X2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda_head_redex @ T5 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T5 ) ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_head_redex @ lambda_Nil ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ ( lambda_App @ lambda_Nil @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Var @ V3 ) ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_head_redex @ ( lambda_App @ V3 @ Va2 ) ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Beta @ V3 @ Va2 ) ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ ( lambda_Src @ T5 ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2982300901483219379ex_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.head_redex.pelims
thf(fact_874_lambda__calculus_Ohead__strategy_Opelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_head_strategy @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ X4 )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ( ( Y2
= ( lambda_Var @ I2 ) )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ ( lambda_Var @ I2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda_head_strategy @ T5 ) ) )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ ( lambda_Src @ T5 ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_head_strategy @ lambda_Nil ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ ( lambda_App @ lambda_Nil @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Var @ V3 ) ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_App @ V3 @ Va2 ) ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Beta @ V3 @ Va2 ) ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ ( lambda_Src @ T5 ) @ ( lambda_Src @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda8648082035647910664gy_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.head_strategy.pelims
thf(fact_875_lambda__calculus_Oleftmost__strategy_Opelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda4011002726600638587rategy @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ X4 )
=> ( ! [X2: nat] :
( ( X4
= ( lambda_Var @ X2 ) )
=> ( ( Y2
= ( lambda_Var @ X2 ) )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ ( lambda_Var @ X2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda4011002726600638587rategy @ T5 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ lambda_Nil ) )
=> ( Y2
= ( lambda_App @ ( lambda4011002726600638587rategy @ lambda_Nil ) @ U3 ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ lambda_Nil ) )
=> ( Y2
= ( lambda_App @ lambda_Nil @ ( lambda4011002726600638587rategy @ U3 ) ) ) ) )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ ( lambda_App @ lambda_Nil @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V3 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V3 ) ) @ U3 ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Var @ V3 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda_Var @ V3 ) @ ( lambda4011002726600638587rategy @ U3 ) ) ) ) )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_App @ V3 @ Va2 ) ) @ U3 ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ ( lambda4011002726600638587rategy @ U3 ) ) ) ) )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( ( ~ ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V3 @ Va2 ) ) @ U3 ) ) )
& ( ( lambda_Ide @ ( lambda4011002726600638587rategy @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( Y2
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda4011002726600638587rategy @ U3 ) ) ) ) )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda2991515046683734590gy_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.leftmost_strategy.pelims
thf(fact_876_reduction__paths_Obottom__up__development_Opinduct,axiom,
! [A0: lambda_lambda,P: lambda_lambda > $o] :
( ( accp_lambda_lambda @ reduct1618657854384587308nt_rel @ A0 )
=> ( ! [T5: lambda_lambda] :
( ( accp_lambda_lambda @ reduct1618657854384587308nt_rel @ T5 )
=> ( ( ~ ( ~ ( lambda_Arr @ T5 )
| ( lambda_Ide @ T5 ) )
=> ( P @ ( lambda_resid @ T5 @ ( lambda5654762366400029632_redex @ T5 ) ) ) )
=> ( P @ T5 ) ) )
=> ( P @ A0 ) ) ) ).
% reduction_paths.bottom_up_development.pinduct
thf(fact_877_lambda__calculus_Oparallel__strategy_Opelims,axiom,
! [X4: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda8522482075158074392rategy @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ X4 )
=> ( ! [I2: nat] :
( ( X4
= ( lambda_Var @ I2 ) )
=> ( ( Y2
= ( lambda_Var @ I2 ) )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ ( lambda_Var @ I2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda8522482075158074392rategy @ T5 ) ) )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ ( lambda8522482075158074392rategy @ T5 ) @ ( lambda8522482075158074392rategy @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda8522482075158074392rategy @ lambda_Nil ) @ ( lambda8522482075158074392rategy @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ ( lambda_App @ lambda_Nil @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_Var @ V3 ) ) @ ( lambda8522482075158074392rategy @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_App @ V3 @ Va2 ) ) @ ( lambda8522482075158074392rategy @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda8522482075158074392rategy @ ( lambda_Beta @ V3 @ Va2 ) ) @ ( lambda8522482075158074392rategy @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ ( lambda8522482075158074392rategy @ T5 ) @ ( lambda8522482075158074392rategy @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda9101494976250544865gy_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.parallel_strategy.pelims
thf(fact_878_lambda__calculus_Oelementary__reduction_Opelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda2670285437103676873uction @ X4 )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ lambda_Nil ) )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_Lam @ T5 ) )
=> ( lambda2670285437103676873uction @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_App @ T5 @ U3 ) )
=> ( ( ( lambda2670285437103676873uction @ T5 )
& ( lambda_Ide @ U3 ) )
| ( ( lambda_Ide @ T5 )
& ( lambda2670285437103676873uction @ U3 ) ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_Beta @ T5 @ U3 ) )
=> ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.pelims(3)
thf(fact_879_lambda__calculus_Oelementary__reduction_Opelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda2670285437103676873uction @ X4 )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ X4 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_Lam @ T5 ) )
=> ~ ( lambda2670285437103676873uction @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_App @ T5 @ U3 ) )
=> ~ ( ( ( lambda2670285437103676873uction @ T5 )
& ( lambda_Ide @ U3 ) )
| ( ( lambda_Ide @ T5 )
& ( lambda2670285437103676873uction @ U3 ) ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.pelims(2)
thf(fact_880_lambda__calculus_Oelementary__reduction_Opelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda2670285437103676873uction @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda2670285437103676873uction @ T5 ) )
=> ~ ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( ( ( lambda2670285437103676873uction @ T5 )
& ( lambda_Ide @ U3 ) )
| ( ( lambda_Ide @ T5 )
& ( lambda2670285437103676873uction @ U3 ) ) ) )
=> ~ ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda2747218380695844464on_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.pelims(1)
thf(fact_881_lambda__calculus_OIde_Opelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda_Ide @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ( Y2
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Ide @ T5 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(1)
thf(fact_882_lambda__calculus_OArr_Opelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda_Arr @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ( Y2
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Arr @ T5 ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(1)
thf(fact_883_lambda__calculus_OIde_Opelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda_Ide @ X4 )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X4 )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T5 ) )
=> ~ ( lambda_Ide @ T5 ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T5 @ U3 ) )
=> ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(2)
thf(fact_884_lambda__calculus_OArr_Opelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda_Arr @ X4 )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ lambda_Nil ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T5 ) )
=> ( lambda_Arr @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T5 @ U3 ) )
=> ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T5 @ U3 ) )
=> ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(3)
thf(fact_885_lambda__calculus_OArr_Opelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda_Arr @ X4 )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X4 )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T5 ) )
=> ~ ( lambda_Arr @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T5 @ U3 ) )
=> ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(2)
thf(fact_886_lambda__calculus_OIde_Opelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda_Ide @ X4 )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X4 )
=> ( ( ( X4 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ lambda_Nil ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T5 ) )
=> ( lambda_Ide @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T5 @ U3 ) )
=> ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(3)
thf(fact_887_lambda__calculus_Ois__head__reduction_Opelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda5343566265203084875uction @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ X4 )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda5343566265203084875uction @ T5 ) )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( Y2
= ( ( lambda5343566265203084875uction @ lambda_Nil )
& ( lambda_Ide @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ lambda_Nil @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda5343566265203084875uction @ ( lambda_Var @ V3 ) )
& ( lambda_Ide @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda5343566265203084875uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda5343566265203084875uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_head_reduction.pelims(1)
thf(fact_888_lambda__calculus_Ois__head__reduction_Opelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda5343566265203084875uction @ X4 )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ X4 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_Lam @ T5 ) )
=> ~ ( lambda5343566265203084875uction @ T5 ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ lambda_Nil @ U3 ) )
=> ~ ( ( lambda5343566265203084875uction @ lambda_Nil )
& ( lambda_Ide @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ~ ( ( lambda5343566265203084875uction @ ( lambda_Var @ V3 ) )
& ( lambda_Ide @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda5343566265203084875uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda5343566265203084875uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_head_reduction.pelims(2)
thf(fact_889_lambda__calculus_Ois__head__reduction_Opelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda5343566265203084875uction @ X4 )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ X4 )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_Lam @ T5 ) )
=> ( lambda5343566265203084875uction @ T5 ) ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( lambda5343566265203084875uction @ lambda_Nil )
& ( lambda_Ide @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( lambda5343566265203084875uction @ ( lambda_Var @ V3 ) )
& ( lambda_Ide @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda5343566265203084875uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda5343566265203084875uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Ide @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ ( lambda_Beta @ T5 @ U3 ) )
=> ( ( lambda_Ide @ T5 )
& ( lambda_Ide @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda5859589972953655406on_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_head_reduction.pelims(3)
thf(fact_890_lambda__calculus_Ocontains__head__reduction_Opelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda8135948770199036598uction @ X4 )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ X4 )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_Lam @ T5 ) )
=> ( lambda8135948770199036598uction @ T5 ) ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_Beta @ T5 @ U3 ) )
=> ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.pelims(3)
thf(fact_891_lambda__calculus_Ocontains__head__reduction_Opelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda8135948770199036598uction @ X4 )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ X4 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_Lam @ T5 ) )
=> ~ ( lambda8135948770199036598uction @ T5 ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ lambda_Nil @ U3 ) )
=> ~ ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_Beta @ T5 @ U3 ) )
=> ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.pelims(2)
thf(fact_892_lambda__calculus_Ocontains__head__reduction_Opelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda8135948770199036598uction @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ X4 )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda8135948770199036598uction @ T5 ) )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( Y2
= ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ lambda_Nil @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ ( lambda_Beta @ T5 @ U3 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda6381018229452634563on_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.pelims(1)
thf(fact_893_lambda__calculus_Ois__internal__reduction_Opelims_I1_J,axiom,
! [X4: lambda_lambda,Y2: $o] :
( ( ( lambda2591989677915507790uction @ X4 )
= Y2 )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ X4 )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ( Y2
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda2591989677915507790uction @ T5 ) )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( Y2
= ( ( lambda2591989677915507790uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ lambda_Nil @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda2591989677915507790uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda2591989677915507790uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( Y2
= ( ( lambda2591989677915507790uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) ) ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( ( X4
= ( lambda_Beta @ Uv2 @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_Beta @ Uv2 @ Uw2 ) ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ( ~ Y2
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_internal_reduction.pelims(1)
thf(fact_894_lambda__calculus_Ois__internal__reduction_Opelims_I2_J,axiom,
! [X4: lambda_lambda] :
( ( lambda2591989677915507790uction @ X4 )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ X4 )
=> ( ! [Uu2: nat] :
( ( X4
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_Lam @ T5 ) )
=> ~ ( lambda2591989677915507790uction @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ~ ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ lambda_Nil @ U3 ) )
=> ~ ( ( lambda2591989677915507790uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ~ ( ( lambda2591989677915507790uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda2591989677915507790uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ~ ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ~ ( ( lambda2591989677915507790uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_internal_reduction.pelims(2)
thf(fact_895_lambda__calculus_Ois__internal__reduction_Opelims_I3_J,axiom,
! [X4: lambda_lambda] :
( ~ ( lambda2591989677915507790uction @ X4 )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ X4 )
=> ( ! [T5: lambda_lambda] :
( ( X4
= ( lambda_Lam @ T5 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_Lam @ T5 ) )
=> ( lambda2591989677915507790uction @ T5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
=> ( ( lambda_Arr @ T5 )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [U3: lambda_lambda] :
( ( X4
= ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ lambda_Nil @ U3 ) )
=> ( ( lambda2591989677915507790uction @ lambda_Nil )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: nat,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Var @ V3 ) @ U3 ) )
=> ( ( lambda2591989677915507790uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda2591989677915507790uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U3: lambda_lambda] :
( ( X4
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U3 ) )
=> ( ( lambda2591989677915507790uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U3 ) ) ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( ( X4
= ( lambda_Beta @ Uv2 @ Uw2 ) )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ ( lambda_Beta @ Uv2 @ Uw2 ) ) )
=> ~ ( ( X4 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda867594046455500331on_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.is_internal_reduction.pelims(3)
thf(fact_896_lambda__calculus_Olambda_Osize__gen_I3_J,axiom,
! [X32: lambda_lambda] :
( ( lambda_size_lambda @ ( lambda_Lam @ X32 ) )
= ( plus_plus_nat @ ( lambda_size_lambda @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size_gen(3)
thf(fact_897_lambda__calculus_Olambda_Osize__gen_I5_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_size_lambda @ ( lambda_Beta @ X51 @ X52 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size_lambda @ X51 ) @ ( lambda_size_lambda @ X52 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size_gen(5)
thf(fact_898_lambda__calculus_Olambda_Osize__gen_I1_J,axiom,
( ( lambda_size_lambda @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size_gen(1)
thf(fact_899_lambda__calculus_Olambda_Osize__gen_I2_J,axiom,
! [X22: nat] :
( ( lambda_size_lambda @ ( lambda_Var @ X22 ) )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size_gen(2)
thf(fact_900_lambda__calculus_Olambda_Osize__gen_I4_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_size_lambda @ ( lambda_App @ X41 @ X42 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size_lambda @ X41 ) @ ( lambda_size_lambda @ X42 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size_gen(4)
thf(fact_901_lambda__calculus_Oresid__head__strategy__Src_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( lambda2591989677915507790uction @ T2 )
=> ( ( lambda5343566265203084875uction @ U )
=> ( ( seq_lambda_lambda @ lambda_resid @ T2 @ U )
=> ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ ( lambda_Join @ ( lambda_head_strategy @ ( lambda_Src @ T2 ) ) @ T2 ) ) ) ) ) ).
% lambda_calculus.resid_head_strategy_Src(2)
thf(fact_902_lambda__calculus_Olambda_Osize_I8_J,axiom,
! [X32: lambda_lambda] :
( ( size_s1768714712973771222lambda @ ( lambda_Lam @ X32 ) )
= ( plus_plus_nat @ ( size_s1768714712973771222lambda @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size(8)
thf(fact_903_composite__of__unq__upto__cong,axiom,
! [U: lambda_lambda,T2: lambda_lambda,V: lambda_lambda,V2: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ V2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V2 @ V ) ) ) ) ) ).
% composite_of_unq_upto_cong
thf(fact_904_composite__of__ide__self,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( compos9183582412961930448lambda @ lambda_resid @ A @ A @ A ) ) ).
% composite_of_ide_self
thf(fact_905_composite__of__def,axiom,
! [U: lambda_lambda,T2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ ( lambda_resid @ V @ U ) ) ) ) ) ).
% composite_of_def
thf(fact_906_composite__of__cancel__left,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda,U2: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U2 @ V )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) ) ) ) ).
% composite_of_cancel_left
thf(fact_907_composite__ofE,axiom,
! [U: lambda_lambda,T2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ ( lambda_resid @ V @ U ) ) ) ) ) ) ).
% composite_ofE
thf(fact_908_arr__composite__of,axiom,
! [U: lambda_lambda,T2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V )
=> ( arr_lambda_lambda @ lambda_resid @ V ) ) ).
% arr_composite_of
thf(fact_909_con__prfx__composite__of_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ V ) ) ) ).
% con_prfx_composite_of(2)
thf(fact_910_con__prfx__composite__of_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ W )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ W ) ) ).
% con_prfx_composite_of(1)
thf(fact_911_resid__composite__of_I4_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( compos9183582412961930448lambda @ lambda_resid @ ( lambda_resid @ T2 @ V ) @ ( lambda_resid @ U @ ( lambda_resid @ V @ T2 ) ) @ ( lambda_resid @ W @ V ) ) ) ) ).
% resid_composite_of(4)
thf(fact_912_resid__composite__of_I2_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T2 ) @ U ) ) ) ).
% resid_composite_of(2)
thf(fact_913_resid__composite__of_I1_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T2 ) @ ( lambda_resid @ W @ T2 ) ) ) ) ).
% resid_composite_of(1)
thf(fact_914_bounded__imp__con,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T3 @ U2 @ V )
=> ( con_lambda_lambda @ lambda_resid @ T2 @ T3 ) ) ) ).
% bounded_imp_con
thf(fact_915_con__composite__of__iff,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda,W: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
= ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ W @ T2 ) @ U ) ) ) ).
% con_composite_of_iff
thf(fact_916_sources__composite__of,axiom,
! [U: lambda_lambda,T2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ V )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ).
% sources_composite_of
thf(fact_917_targets__composite__of,axiom,
! [U: lambda_lambda,T2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ V )
= ( target5490931411331770124lambda @ lambda_resid @ T2 ) ) ) ).
% targets_composite_of
thf(fact_918_trg__composite__of,axiom,
! [U: lambda_lambda,T2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ V )
= ( trg_lambda_lambda @ lambda_resid @ T2 ) ) ) ).
% trg_composite_of
thf(fact_919_src__composite__of,axiom,
! [U: lambda_lambda,T2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ V )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ).
% src_composite_of
thf(fact_920_join__ofE,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
=> ~ ( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ ( lambda_resid @ U @ T2 ) @ V )
=> ~ ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T2 @ U ) @ V ) ) ) ).
% join_ofE
thf(fact_921_join__of__def,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V )
= ( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ ( lambda_resid @ U @ T2 ) @ V )
& ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T2 @ U ) @ V ) ) ) ).
% join_of_def
thf(fact_922_composable__def,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T2 @ U )
= ( ? [X5: lambda_lambda] : ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ X5 ) ) ) ).
% composable_def
thf(fact_923_resid__composite__of_I3_J,axiom,
! [T2: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ W ) @ ( lambda_resid @ ( lambda_resid @ V @ T2 ) @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ T2 ) @ U ) @ ( lambda_resid @ V @ W ) ) ) ) ) ) ).
% resid_composite_of(3)
thf(fact_924_composite__of__arr__ide,axiom,
! [B: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ B @ T2 )
= ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ T2 ) @ B ) ) ) ).
% composite_of_arr_ide
thf(fact_925_composite__of__ide__arr,axiom,
! [A: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ A @ T2 @ T2 )
= ( con_lambda_lambda @ lambda_resid @ T2 @ A ) ) ) ).
% composite_of_ide_arr
thf(fact_926_composite__of__source__arr,axiom,
! [T2: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T2 ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ A @ T2 @ T2 ) ) ) ).
% composite_of_source_arr
thf(fact_927_composite__of__arr__target,axiom,
! [T2: lambda_lambda,B: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T2 )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T2 ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ T2 @ B @ T2 ) ) ) ).
% composite_of_arr_target
thf(fact_928_composite__ofI,axiom,
! [U: lambda_lambda,V: lambda_lambda,T2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ ( lambda_resid @ V @ U ) ) ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ U @ T2 @ V ) ) ) ).
% composite_ofI
thf(fact_929_join__ofI,axiom,
! [T2: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T2 @ ( lambda_resid @ U @ T2 ) @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T2 @ U ) @ V )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T2 @ U @ V ) ) ) ).
% join_ofI
thf(fact_930_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T2: nat,U: nat,T3: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( composite_of_nat @ Resid @ T2 @ U @ T3 )
=> ( ( member_nat @ U @ NN )
=> ( ( member_nat @ ( Resid @ T3 @ T2 ) @ NN )
& ( member_nat @ ( Resid @ T2 @ T3 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_931_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T2 @ U @ T3 )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T3 @ T2 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T2 @ T3 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_932_lambda__calculus_Olambda_Osize_I6_J,axiom,
( ( size_s1768714712973771222lambda @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size(6)
thf(fact_933_lambda__calculus_Olambda_Osize_I7_J,axiom,
! [X22: nat] :
( ( size_s1768714712973771222lambda @ ( lambda_Var @ X22 ) )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size(7)
thf(fact_934_lambda__calculus_Olambda_Osize_I9_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( ( size_s1768714712973771222lambda @ ( lambda_App @ X41 @ X42 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1768714712973771222lambda @ X41 ) @ ( size_s1768714712973771222lambda @ X42 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size(9)
thf(fact_935_lambda__calculus_Olambda_Osize_I10_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( ( size_s1768714712973771222lambda @ ( lambda_Beta @ X51 @ X52 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1768714712973771222lambda @ X51 ) @ ( size_s1768714712973771222lambda @ X52 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size(10)
thf(fact_936_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_nat,Resid: nat > nat > nat] :
( ! [T5: nat] :
( ( member_nat @ T5 @ NN )
=> ( arr_nat @ Resid @ T5 ) )
=> ( ! [A5: nat] :
( ( ide_nat @ Resid @ A5 )
=> ( member_nat @ A5 @ NN ) )
=> ( ! [U3: nat,T5: nat] :
( ( member_nat @ U3 @ NN )
=> ( ( coinitial_nat @ Resid @ T5 @ U3 )
=> ( member_nat @ ( Resid @ U3 @ T5 ) @ NN ) ) )
=> ( ! [U3: nat,T5: nat] :
( ( member_nat @ U3 @ NN )
=> ( ( member_nat @ ( Resid @ T5 @ U3 ) @ NN )
=> ( member_nat @ T5 @ NN ) ) )
=> ( ! [U3: nat,T5: nat] :
( ( member_nat @ U3 @ NN )
=> ( ( seq_nat @ Resid @ U3 @ T5 )
=> ? [X_1: nat] : ( composite_of_nat @ Resid @ U3 @ T5 @ X_1 ) ) )
=> ( ! [U3: nat,T5: nat] :
( ( member_nat @ U3 @ NN )
=> ( ( seq_nat @ Resid @ T5 @ U3 )
=> ? [X_1: nat] : ( composite_of_nat @ Resid @ T5 @ U3 @ X_1 ) ) )
=> ( normal3241730178054940488ms_nat @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_937_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_lambda_lambda,Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( member_lambda_lambda @ T5 @ NN )
=> ( arr_lambda_lambda @ Resid @ T5 ) )
=> ( ! [A5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A5 )
=> ( member_lambda_lambda @ A5 @ NN ) )
=> ( ! [U3: lambda_lambda,T5: lambda_lambda] :
( ( member_lambda_lambda @ U3 @ NN )
=> ( ( coinit1392077014771647542lambda @ Resid @ T5 @ U3 )
=> ( member_lambda_lambda @ ( Resid @ U3 @ T5 ) @ NN ) ) )
=> ( ! [U3: lambda_lambda,T5: lambda_lambda] :
( ( member_lambda_lambda @ U3 @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T5 @ U3 ) @ NN )
=> ( member_lambda_lambda @ T5 @ NN ) ) )
=> ( ! [U3: lambda_lambda,T5: lambda_lambda] :
( ( member_lambda_lambda @ U3 @ NN )
=> ( ( seq_lambda_lambda @ Resid @ U3 @ T5 )
=> ? [X_1: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ U3 @ T5 @ X_1 ) ) )
=> ( ! [U3: lambda_lambda,T5: lambda_lambda] :
( ( member_lambda_lambda @ U3 @ NN )
=> ( ( seq_lambda_lambda @ Resid @ T5 @ U3 )
=> ? [X_1: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ T5 @ U3 @ X_1 ) ) )
=> ( normal5162919049924913650lambda @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_938_normal__sub__rts__axioms__def,axiom,
( normal3241730178054940488ms_nat
= ( ^ [Resid2: nat > nat > nat,NN2: set_nat] :
( ! [T: nat] :
( ( member_nat @ T @ NN2 )
=> ( arr_nat @ Resid2 @ T ) )
& ! [A2: nat] :
( ( ide_nat @ Resid2 @ A2 )
=> ( member_nat @ A2 @ NN2 ) )
& ! [U4: nat,T: nat] :
( ( member_nat @ U4 @ NN2 )
=> ( ( coinitial_nat @ Resid2 @ T @ U4 )
=> ( member_nat @ ( Resid2 @ U4 @ T ) @ NN2 ) ) )
& ! [U4: nat,T: nat] :
( ( member_nat @ U4 @ NN2 )
=> ( ( member_nat @ ( Resid2 @ T @ U4 ) @ NN2 )
=> ( member_nat @ T @ NN2 ) ) )
& ! [U4: nat,T: nat] :
( ( member_nat @ U4 @ NN2 )
=> ( ( seq_nat @ Resid2 @ U4 @ T )
=> ? [X5: nat] : ( composite_of_nat @ Resid2 @ U4 @ T @ X5 ) ) )
& ! [U4: nat,T: nat] :
( ( member_nat @ U4 @ NN2 )
=> ( ( seq_nat @ Resid2 @ T @ U4 )
=> ? [X5: nat] : ( composite_of_nat @ Resid2 @ T @ U4 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_939_normal__sub__rts__axioms__def,axiom,
( normal5162919049924913650lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda,NN2: set_lambda_lambda] :
( ! [T: lambda_lambda] :
( ( member_lambda_lambda @ T @ NN2 )
=> ( arr_lambda_lambda @ Resid2 @ T ) )
& ! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A2 )
=> ( member_lambda_lambda @ A2 @ NN2 ) )
& ! [U4: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN2 )
=> ( ( coinit1392077014771647542lambda @ Resid2 @ T @ U4 )
=> ( member_lambda_lambda @ ( Resid2 @ U4 @ T ) @ NN2 ) ) )
& ! [U4: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN2 )
=> ( ( member_lambda_lambda @ ( Resid2 @ T @ U4 ) @ NN2 )
=> ( member_lambda_lambda @ T @ NN2 ) ) )
& ! [U4: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN2 )
=> ( ( seq_lambda_lambda @ Resid2 @ U4 @ T )
=> ? [X5: lambda_lambda] : ( compos9183582412961930448lambda @ Resid2 @ U4 @ T @ X5 ) ) )
& ! [U4: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN2 )
=> ( ( seq_lambda_lambda @ Resid2 @ T @ U4 )
=> ? [X5: lambda_lambda] : ( compos9183582412961930448lambda @ Resid2 @ T @ U4 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_940_Cong__iff__cong,axiom,
! [T2: lambda_lambda,U: lambda_lambda] :
( ( normal3156941126761417472lambda @ lambda_resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) @ T2 @ U )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T2 @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T2 ) ) ) ) ).
% Cong_iff_cong
thf(fact_941_rts__with__composites__axioms_Ointro,axiom,
! [Resid: nat > nat > nat] :
( ! [T5: nat,U3: nat] :
( ( seq_nat @ Resid @ T5 @ U3 )
=> ( composable_nat @ Resid @ T5 @ U3 ) )
=> ( rts_wi5665336072173770491ms_nat @ Resid ) ) ).
% rts_with_composites_axioms.intro
thf(fact_942_rts__with__composites__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( seq_lambda_lambda @ Resid @ T5 @ U3 )
=> ( compos7127019122090555255lambda @ Resid @ T5 @ U3 ) )
=> ( rts_wi6632542615990831231lambda @ Resid ) ) ).
% rts_with_composites_axioms.intro
thf(fact_943_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T2: nat,T3: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( composite_of_nat @ Resid @ U @ T2 @ T3 )
=> ( ( member_nat @ U @ NN )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T3 @ T2 ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_944_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T2: lambda_lambda,T3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T2 @ T3 )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T3 @ T2 ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_945_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T2: nat,U: nat,T3: nat,U2: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( ( sources_nat @ Resid @ T2 )
= ( sources_nat @ Resid @ U ) )
=> ( ( ( sources_nat @ Resid @ T3 )
= ( sources_nat @ Resid @ U2 ) )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T2 @ T3 )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ U @ U2 )
=> ( ( con_nat @ Resid @ T2 @ U )
= ( con_nat @ Resid @ T3 @ U2 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_946_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T2: lambda_lambda,U: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ T2 )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T3 )
= ( source5627170679743844886lambda @ Resid @ U2 ) )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T2 @ T3 )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ U @ U2 )
=> ( ( con_lambda_lambda @ Resid @ T2 @ U )
= ( con_lambda_lambda @ Resid @ T3 @ U2 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_947_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T2: nat,T3: nat,U: nat,U2: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T2 @ T3 )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ U @ U2 )
=> ( ( con_nat @ Resid @ T2 @ U )
=> ( ( ( sources_nat @ Resid @ T3 )
= ( sources_nat @ Resid @ U2 ) )
=> ( con_nat @ Resid @ T3 @ U2 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_948_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T2 @ T3 )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ U @ U2 )
=> ( ( con_lambda_lambda @ Resid @ T2 @ U )
=> ( ( ( source5627170679743844886lambda @ Resid @ T3 )
= ( source5627170679743844886lambda @ Resid @ U2 ) )
=> ( con_lambda_lambda @ Resid @ T3 @ U2 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_949_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T2: nat,T3: nat,U: nat,U2: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T2 @ T3 )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ U @ U2 )
=> ( ( con_nat @ Resid @ T2 @ U )
=> ( ( ( sources_nat @ Resid @ T3 )
= ( sources_nat @ Resid @ U2 ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ ( Resid @ T2 @ U ) @ ( Resid @ T3 @ U2 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_950_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T2: lambda_lambda,T3: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T2 @ T3 )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ U @ U2 )
=> ( ( con_lambda_lambda @ Resid @ T2 @ U )
=> ( ( ( source5627170679743844886lambda @ Resid @ T3 )
= ( source5627170679743844886lambda @ Resid @ U2 ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ ( Resid @ T2 @ U ) @ ( Resid @ T3 @ U2 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_951_rts__with__composites__axioms__def,axiom,
( rts_wi5665336072173770491ms_nat
= ( ^ [Resid2: nat > nat > nat] :
! [T: nat,U4: nat] :
( ( seq_nat @ Resid2 @ T @ U4 )
=> ( composable_nat @ Resid2 @ T @ U4 ) ) ) ) ).
% rts_with_composites_axioms_def
thf(fact_952_rts__with__composites__axioms__def,axiom,
( rts_wi6632542615990831231lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
! [T: lambda_lambda,U4: lambda_lambda] :
( ( seq_lambda_lambda @ Resid2 @ T @ U4 )
=> ( compos7127019122090555255lambda @ Resid2 @ T @ U4 ) ) ) ) ).
% rts_with_composites_axioms_def
thf(fact_953_rts__with__joins__axioms__def,axiom,
( rts_wi6344169777088970040lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
! [T: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ Resid2 @ T @ U4 )
=> ( joinab2519788767067900224lambda @ Resid2 @ T @ U4 ) ) ) ) ).
% rts_with_joins_axioms_def
thf(fact_954_rts__with__joins__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( con_lambda_lambda @ Resid @ T5 @ U3 )
=> ( joinab2519788767067900224lambda @ Resid @ T5 @ U3 ) )
=> ( rts_wi6344169777088970040lambda @ Resid ) ) ).
% rts_with_joins_axioms.intro
thf(fact_955_rts__with__joins_Oaxioms_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( rts_wi2791606683991334555lambda @ Resid )
=> ( rts_wi6344169777088970040lambda @ Resid ) ) ).
% rts_with_joins.axioms(2)
thf(fact_956_confluent__rts__axioms_Ointro,axiom,
! [Resid: nat > nat > nat] :
( ! [T5: nat,U3: nat] :
( ( coinitial_nat @ Resid @ T5 @ U3 )
=> ( con_nat @ Resid @ T5 @ U3 ) )
=> ( conflu6972329929330909443ms_nat @ Resid ) ) ).
% confluent_rts_axioms.intro
thf(fact_957_confluent__rts__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid @ T5 @ U3 )
=> ( con_lambda_lambda @ Resid @ T5 @ U3 ) )
=> ( conflu5019861018119259255lambda @ Resid ) ) ).
% confluent_rts_axioms.intro
thf(fact_958_confluent__rts__axioms__def,axiom,
( conflu6972329929330909443ms_nat
= ( ^ [Resid2: nat > nat > nat] :
! [T: nat,U4: nat] :
( ( coinitial_nat @ Resid2 @ T @ U4 )
=> ( con_nat @ Resid2 @ T @ U4 ) ) ) ) ).
% confluent_rts_axioms_def
thf(fact_959_confluent__rts__axioms__def,axiom,
( conflu5019861018119259255lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
! [T: lambda_lambda,U4: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid2 @ T @ U4 )
=> ( con_lambda_lambda @ Resid2 @ T @ U4 ) ) ) ) ).
% confluent_rts_axioms_def
thf(fact_960_confluent__rts_Oaxioms_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( conflu4927472611300549594lambda @ Resid )
=> ( conflu5019861018119259255lambda @ Resid ) ) ).
% confluent_rts.axioms(2)
thf(fact_961_transformation_Onaturality3,axiom,
! [A3: nat > nat > nat,B5: lambda_lambda > lambda_lambda > lambda_lambda,F3: nat > lambda_lambda,G: nat > lambda_lambda,Tau: nat > lambda_lambda,F2: nat] :
( ( transf3951925986184407251lambda @ A3 @ B5 @ F3 @ G @ Tau )
=> ( ( arr_nat @ A3 @ F2 )
=> ( join_o1386658102433174083lambda @ B5 @ ( Tau @ ( weakly1695633097759462397rc_nat @ A3 @ F2 ) ) @ ( F3 @ F2 ) @ ( Tau @ F2 ) ) ) ) ).
% transformation.naturality3
thf(fact_962_transformation_Onaturality3,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F3: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,F2: lambda_lambda] :
( ( transf2236888680007955973lambda @ A3 @ B5 @ F3 @ G @ Tau )
=> ( ( arr_lambda_lambda @ A3 @ F2 )
=> ( join_o1386658102433174083lambda @ B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A3 @ F2 ) ) @ ( F3 @ F2 ) @ ( Tau @ F2 ) ) ) ) ).
% transformation.naturality3
thf(fact_963_lambda__calculus_Oresid_Ocases,axiom,
! [X4: produc4921446539628979879lambda] :
( ! [I2: nat,I4: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ I2 ) @ ( lambda_Var @ I4 ) ) )
=> ( ! [T5: lambda_lambda,T6: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ T5 ) @ ( lambda_Lam @ T6 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ T5 @ U3 ) @ ( lambda_App @ T6 @ U5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ T5 @ U3 ) @ ( lambda_Beta @ T6 @ U5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) @ ( lambda_Beta @ T6 @ U5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ T5 @ U3 ) @ ( lambda_App @ ( lambda_Lam @ T6 ) @ U5 ) ) )
=> ( ! [Uv2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ Uv2 ) )
=> ( ! [V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_Var @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_App @ Va2 @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_Beta @ Va2 @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ ( lambda_Var @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ ( lambda_Lam @ Vb2 ) ) )
=> ( ! [Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ! [Vd2: nat,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vd2 ) @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_Var @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_Lam @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ lambda_Nil @ Vc2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vd2: nat,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Vc2 ) ) )
=> ( ! [Uu2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ Uu2 @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: nat,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Va2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: lambda_lambda,Vb2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ Va2 @ Vb2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: lambda_lambda,Vb2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ Va2 @ Vb2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Vb2: nat,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Vb2 ) @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ Vb2 ) @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ Vb2 @ Vc2 ) @ ( lambda_App @ lambda_Nil @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Vd2: nat,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ Vb2 @ Vc2 ) @ ( lambda_App @ ( lambda_Var @ Vd2 ) @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ Vb2 @ Vc2 ) @ ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ Vb2 @ Vc2 ) @ ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vb2: nat,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Vb2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ Vb2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vd2: nat,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ~ ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.resid.cases
thf(fact_964_transformation_Opreserves__trg,axiom,
! [A3: nat > nat > nat,B5: lambda_lambda > lambda_lambda > lambda_lambda,F3: nat > lambda_lambda,G: nat > lambda_lambda,Tau: nat > lambda_lambda,F2: nat] :
( ( transf3951925986184407251lambda @ A3 @ B5 @ F3 @ G @ Tau )
=> ( ( ide_nat @ A3 @ F2 )
=> ( ( trg_lambda_lambda @ B5 @ ( Tau @ F2 ) )
= ( G @ ( trg_nat @ A3 @ F2 ) ) ) ) ) ).
% transformation.preserves_trg
thf(fact_965_transformation_Opreserves__trg,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F3: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,F2: lambda_lambda] :
( ( transf2236888680007955973lambda @ A3 @ B5 @ F3 @ G @ Tau )
=> ( ( ide_lambda_lambda @ A3 @ F2 )
=> ( ( trg_lambda_lambda @ B5 @ ( Tau @ F2 ) )
= ( G @ ( trg_lambda_lambda @ A3 @ F2 ) ) ) ) ) ).
% transformation.preserves_trg
thf(fact_966_transformation_Opreserves__src,axiom,
! [A3: nat > nat > nat,B5: lambda_lambda > lambda_lambda > lambda_lambda,F3: nat > lambda_lambda,G: nat > lambda_lambda,Tau: nat > lambda_lambda,F2: nat] :
( ( transf3951925986184407251lambda @ A3 @ B5 @ F3 @ G @ Tau )
=> ( ( ide_nat @ A3 @ F2 )
=> ( ( weakly4554751870655413501lambda @ B5 @ ( Tau @ F2 ) )
= ( F3 @ ( weakly1695633097759462397rc_nat @ A3 @ F2 ) ) ) ) ) ).
% transformation.preserves_src
thf(fact_967_transformation_Opreserves__src,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F3: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,F2: lambda_lambda] :
( ( transf2236888680007955973lambda @ A3 @ B5 @ F3 @ G @ Tau )
=> ( ( ide_lambda_lambda @ A3 @ F2 )
=> ( ( weakly4554751870655413501lambda @ B5 @ ( Tau @ F2 ) )
= ( F3 @ ( weakly4554751870655413501lambda @ A3 @ F2 ) ) ) ) ) ).
% transformation.preserves_src
thf(fact_968_transformation_Onaturality1,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F3: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,F2: lambda_lambda] :
( ( transf2236888680007955973lambda @ A3 @ B5 @ F3 @ G @ Tau )
=> ( ( arr_lambda_lambda @ A3 @ F2 )
=> ( ( B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A3 @ F2 ) ) @ ( F3 @ F2 ) )
= ( Tau @ ( trg_lambda_lambda @ A3 @ F2 ) ) ) ) ) ).
% transformation.naturality1
thf(fact_969_lambda__calculus_OJoin_Ocases,axiom,
! [X4: produc4921446539628979879lambda] :
( ! [X2: nat,X7: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ X2 ) @ ( lambda_Var @ X7 ) ) )
=> ( ! [T5: lambda_lambda,T6: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ T5 ) @ ( lambda_Lam @ T6 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) @ ( lambda_Beta @ T6 @ U5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ T5 @ U3 ) @ ( lambda_App @ ( lambda_Lam @ T6 ) @ U5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ T5 @ U3 ) @ ( lambda_App @ T6 @ U5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ T5 @ U3 ) @ ( lambda_Beta @ T6 @ U5 ) ) )
=> ( ! [Uv2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ Uv2 ) )
=> ( ! [V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_Var @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_App @ Va2 @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_Beta @ Va2 @ Vb2 ) ) )
=> ( ! [Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Va2 ) @ lambda_Nil ) )
=> ( ! [Va2: lambda_lambda,V3: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Va2 ) @ ( lambda_Var @ V3 ) ) )
=> ( ! [Va2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Va2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: lambda_lambda,V3: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Va2 ) @ ( lambda_Beta @ V3 @ Vb2 ) ) )
=> ( ! [Vb2: nat,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) @ lambda_Nil ) )
=> ( ! [Vb2: nat,Va2: lambda_lambda,V3: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) @ ( lambda_Var @ V3 ) ) )
=> ( ! [Vb2: nat,Va2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Vb2: nat,Va2: lambda_lambda,V3: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) @ ( lambda_Beta @ V3 @ Vc2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) @ lambda_Nil ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda,V3: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) @ ( lambda_Var @ V3 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda,V3: lambda_lambda,Vd2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) @ ( lambda_Beta @ V3 @ Vd2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) @ lambda_Nil ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda,V3: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) @ ( lambda_Var @ V3 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda,V3: lambda_lambda,Vd2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) @ ( lambda_Beta @ V3 @ Vd2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ ( lambda_Var @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ ( lambda_Lam @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_Var @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_Lam @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ lambda_Nil @ Vc2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vd2: nat,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Vc2 ) ) )
=> ( ! [Uu2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ Uu2 @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: nat,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Va2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: lambda_lambda,Vb2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ Va2 @ Vb2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: lambda_lambda,Vb2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ Va2 @ Vb2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Vb2: nat,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Vb2 ) @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ Vb2 ) @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_App @ lambda_Nil @ Va2 ) ) )
=> ( ! [V3: nat,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ V3 ) @ ( lambda_App @ lambda_Nil @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_App @ lambda_Nil @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Vb2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Vb2 ) @ ( lambda_App @ lambda_Nil @ Va2 ) ) )
=> ( ! [Vb2: nat,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) ) )
=> ( ! [V3: nat,Vb2: nat,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ V3 ) @ ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Vb2: nat,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Vc2: lambda_lambda,Vb2: nat,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Vc2 ) @ ( lambda_App @ ( lambda_Var @ Vb2 ) @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) ) )
=> ( ! [V3: nat,Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ V3 ) @ ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Vd2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Vd2 ) @ ( lambda_App @ ( lambda_App @ Vb2 @ Vc2 ) @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) ) )
=> ( ! [V3: nat,Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ V3 ) @ ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Vd2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Vd2 ) @ ( lambda_App @ ( lambda_Beta @ Vb2 @ Vc2 ) @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vb2: nat,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Vb2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ Vb2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vd2: nat,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ~ ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Join.cases
thf(fact_970_lambda__calculus_Osubs_Ocases,axiom,
! [X4: produc4921446539628979879lambda] :
( ! [I2: nat,I4: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ I2 ) @ ( lambda_Var @ I4 ) ) )
=> ( ! [T5: lambda_lambda,T6: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ T5 ) @ ( lambda_Lam @ T6 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ T5 @ U3 ) @ ( lambda_App @ T6 @ U5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) @ ( lambda_Beta @ T6 @ U5 ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda,T6: lambda_lambda,U5: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ T5 @ U3 ) @ ( lambda_Beta @ T6 @ U5 ) ) )
=> ( ! [Uv2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ Uv2 ) )
=> ( ! [V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_Var @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_App @ Va2 @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ V3 ) @ ( lambda_Beta @ Va2 @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ ( lambda_Var @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ V3 @ Va2 ) @ ( lambda_Lam @ Vb2 ) ) )
=> ( ! [Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ! [Vd2: nat,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vd2 ) @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: nat] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_Var @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_Lam @ Vb2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ V3 @ Va2 ) @ ( lambda_App @ Vb2 @ Vc2 ) ) )
=> ( ! [Uu2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ Uu2 @ lambda_Nil ) )
=> ( ! [V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: nat,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Va2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: lambda_lambda,Vb2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ Va2 @ Vb2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [Va2: lambda_lambda,Vb2: lambda_lambda,V3: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ Va2 @ Vb2 ) @ ( lambda_Lam @ V3 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Vb2: nat,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Vb2 ) @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ Vb2 ) @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Beta @ Vb2 @ Vc2 ) @ ( lambda_App @ V3 @ Va2 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ lambda_Nil @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vb2: nat,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Var @ Vb2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vb2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_Lam @ Vb2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ lambda_Nil @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vd2: nat,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ( ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) )
=> ~ ! [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda,V3: lambda_lambda,Va2: lambda_lambda] :
( X4
!= ( produc7853741606629031319lambda @ ( lambda_App @ ( lambda_Beta @ Vd2 @ Ve2 ) @ Vc2 ) @ ( lambda_Beta @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.subs.cases
thf(fact_971_lambda__calculus_OSubst_Ocases,axiom,
! [X4: produc2872725455364511436lambda] :
( ! [Uu2: nat,Uv2: lambda_lambda] :
( X4
!= ( produc1836381109249119870lambda @ Uu2 @ ( produc7853741606629031319lambda @ Uv2 @ lambda_Nil ) ) )
=> ( ! [N5: nat,V3: lambda_lambda,I2: nat] :
( X4
!= ( produc1836381109249119870lambda @ N5 @ ( produc7853741606629031319lambda @ V3 @ ( lambda_Var @ I2 ) ) ) )
=> ( ! [N5: nat,V3: lambda_lambda,T5: lambda_lambda] :
( X4
!= ( produc1836381109249119870lambda @ N5 @ ( produc7853741606629031319lambda @ V3 @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [N5: nat,V3: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( produc1836381109249119870lambda @ N5 @ ( produc7853741606629031319lambda @ V3 @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ~ ! [N5: nat,V3: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( produc1836381109249119870lambda @ N5 @ ( produc7853741606629031319lambda @ V3 @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ).
% lambda_calculus.Subst.cases
thf(fact_972_lambda__calculus_OSubst_Opelims,axiom,
! [X4: nat,Xa: lambda_lambda,Xb: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Subst @ X4 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P3578032143099507075lambda @ lambda_Subst_rel @ ( produc1836381109249119870lambda @ X4 @ ( produc7853741606629031319lambda @ Xa @ Xb ) ) )
=> ( ( ( Xb = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_P3578032143099507075lambda @ lambda_Subst_rel @ ( produc1836381109249119870lambda @ X4 @ ( produc7853741606629031319lambda @ Xa @ lambda_Nil ) ) ) ) )
=> ( ! [I2: nat] :
( ( Xb
= ( lambda_Var @ I2 ) )
=> ( ( ( ( ord_less_nat @ X4 @ I2 )
=> ( Y2
= ( lambda_Var @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) )
& ( ~ ( ord_less_nat @ X4 @ I2 )
=> ( ( ( X4 = I2 )
=> ( Y2
= ( lambda_Raise @ zero_zero_nat @ X4 @ Xa ) ) )
& ( ( X4 != I2 )
=> ( Y2
= ( lambda_Var @ I2 ) ) ) ) ) )
=> ~ ( accp_P3578032143099507075lambda @ lambda_Subst_rel @ ( produc1836381109249119870lambda @ X4 @ ( produc7853741606629031319lambda @ Xa @ ( lambda_Var @ I2 ) ) ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( Xb
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda_Subst @ ( suc @ X4 ) @ Xa @ T5 ) ) )
=> ~ ( accp_P3578032143099507075lambda @ lambda_Subst_rel @ ( produc1836381109249119870lambda @ X4 @ ( produc7853741606629031319lambda @ Xa @ ( lambda_Lam @ T5 ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xb
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_Subst @ X4 @ Xa @ T5 ) @ ( lambda_Subst @ X4 @ Xa @ U3 ) ) )
=> ~ ( accp_P3578032143099507075lambda @ lambda_Subst_rel @ ( produc1836381109249119870lambda @ X4 @ ( produc7853741606629031319lambda @ Xa @ ( lambda_App @ T5 @ U3 ) ) ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xb
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ ( lambda_Subst @ ( suc @ X4 ) @ Xa @ T5 ) @ ( lambda_Subst @ X4 @ Xa @ U3 ) ) )
=> ~ ( accp_P3578032143099507075lambda @ lambda_Subst_rel @ ( produc1836381109249119870lambda @ X4 @ ( produc7853741606629031319lambda @ Xa @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Subst.pelims
thf(fact_973_lambda__calculus_Omtp_Ocases,axiom,
! [X4: produc3839482995950079247lambda] :
( ! [X2: nat] :
( X4
!= ( produc8777017121224680385lambda @ X2 @ lambda_Nil ) )
=> ( ! [X2: nat,Z2: nat] :
( X4
!= ( produc8777017121224680385lambda @ X2 @ ( lambda_Var @ Z2 ) ) )
=> ( ! [X2: nat,T5: lambda_lambda] :
( X4
!= ( produc8777017121224680385lambda @ X2 @ ( lambda_Lam @ T5 ) ) )
=> ( ! [X2: nat,T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( produc8777017121224680385lambda @ X2 @ ( lambda_App @ T5 @ U3 ) ) )
=> ~ ! [X2: nat,T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( produc8777017121224680385lambda @ X2 @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ).
% lambda_calculus.mtp.cases
thf(fact_974_lambda__calculus_Omtp_Opelims,axiom,
! [X4: nat,Xa: lambda_lambda,Y2: nat] :
( ( ( lambda_mtp @ X4 @ Xa )
= Y2 )
=> ( ( accp_P8713979107359754310lambda @ lambda_mtp_rel @ ( produc8777017121224680385lambda @ X4 @ Xa ) )
=> ( ( ( Xa = lambda_Nil )
=> ( ( Y2 = zero_zero_nat )
=> ~ ( accp_P8713979107359754310lambda @ lambda_mtp_rel @ ( produc8777017121224680385lambda @ X4 @ lambda_Nil ) ) ) )
=> ( ! [Z2: nat] :
( ( Xa
= ( lambda_Var @ Z2 ) )
=> ( ( ( ( Z2 = X4 )
=> ( Y2 = one_one_nat ) )
& ( ( Z2 != X4 )
=> ( Y2 = zero_zero_nat ) ) )
=> ~ ( accp_P8713979107359754310lambda @ lambda_mtp_rel @ ( produc8777017121224680385lambda @ X4 @ ( lambda_Var @ Z2 ) ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_mtp @ ( suc @ X4 ) @ T5 ) )
=> ~ ( accp_P8713979107359754310lambda @ lambda_mtp_rel @ ( produc8777017121224680385lambda @ X4 @ ( lambda_Lam @ T5 ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xa
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( plus_plus_nat @ ( lambda_mtp @ X4 @ T5 ) @ ( lambda_mtp @ X4 @ U3 ) ) )
=> ~ ( accp_P8713979107359754310lambda @ lambda_mtp_rel @ ( produc8777017121224680385lambda @ X4 @ ( lambda_App @ T5 @ U3 ) ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( plus_plus_nat @ ( lambda_mtp @ ( suc @ X4 ) @ T5 ) @ ( times_times_nat @ ( lambda_mtp @ X4 @ U3 ) @ ( ord_max_nat @ one_one_nat @ ( lambda_mtp @ zero_zero_nat @ T5 ) ) ) ) )
=> ~ ( accp_P8713979107359754310lambda @ lambda_mtp_rel @ ( produc8777017121224680385lambda @ X4 @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.mtp.pelims
thf(fact_975_lambda__calculus_ORaise_Ocases,axiom,
! [X4: produc674367846827345344lambda] :
( ! [Uu2: nat,Uv2: nat] :
( X4
!= ( produc5347854423718732792lambda @ Uu2 @ ( produc8777017121224680385lambda @ Uv2 @ lambda_Nil ) ) )
=> ( ! [D3: nat,N5: nat,I2: nat] :
( X4
!= ( produc5347854423718732792lambda @ D3 @ ( produc8777017121224680385lambda @ N5 @ ( lambda_Var @ I2 ) ) ) )
=> ( ! [D3: nat,N5: nat,T5: lambda_lambda] :
( X4
!= ( produc5347854423718732792lambda @ D3 @ ( produc8777017121224680385lambda @ N5 @ ( lambda_Lam @ T5 ) ) ) )
=> ( ! [D3: nat,N5: nat,T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( produc5347854423718732792lambda @ D3 @ ( produc8777017121224680385lambda @ N5 @ ( lambda_App @ T5 @ U3 ) ) ) )
=> ~ ! [D3: nat,N5: nat,T5: lambda_lambda,U3: lambda_lambda] :
( X4
!= ( produc5347854423718732792lambda @ D3 @ ( produc8777017121224680385lambda @ N5 @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ).
% lambda_calculus.Raise.cases
thf(fact_976_gbinomial__1,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ one_one_nat )
= A ) ).
% gbinomial_1
thf(fact_977_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_978_lambda__calculus_ORaise_Opelims,axiom,
! [X4: nat,Xa: nat,Xb: lambda_lambda,Y2: lambda_lambda] :
( ( ( lambda_Raise @ X4 @ Xa @ Xb )
= Y2 )
=> ( ( accp_P7631132853315799497lambda @ lambda_Raise_rel @ ( produc5347854423718732792lambda @ X4 @ ( produc8777017121224680385lambda @ Xa @ Xb ) ) )
=> ( ( ( Xb = lambda_Nil )
=> ( ( Y2 = lambda_Nil )
=> ~ ( accp_P7631132853315799497lambda @ lambda_Raise_rel @ ( produc5347854423718732792lambda @ X4 @ ( produc8777017121224680385lambda @ Xa @ lambda_Nil ) ) ) ) )
=> ( ! [I2: nat] :
( ( Xb
= ( lambda_Var @ I2 ) )
=> ( ( ( ( ord_less_eq_nat @ X4 @ I2 )
=> ( Y2
= ( lambda_Var @ ( plus_plus_nat @ I2 @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ X4 @ I2 )
=> ( Y2
= ( lambda_Var @ I2 ) ) ) )
=> ~ ( accp_P7631132853315799497lambda @ lambda_Raise_rel @ ( produc5347854423718732792lambda @ X4 @ ( produc8777017121224680385lambda @ Xa @ ( lambda_Var @ I2 ) ) ) ) ) )
=> ( ! [T5: lambda_lambda] :
( ( Xb
= ( lambda_Lam @ T5 ) )
=> ( ( Y2
= ( lambda_Lam @ ( lambda_Raise @ ( suc @ X4 ) @ Xa @ T5 ) ) )
=> ~ ( accp_P7631132853315799497lambda @ lambda_Raise_rel @ ( produc5347854423718732792lambda @ X4 @ ( produc8777017121224680385lambda @ Xa @ ( lambda_Lam @ T5 ) ) ) ) ) )
=> ( ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xb
= ( lambda_App @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_App @ ( lambda_Raise @ X4 @ Xa @ T5 ) @ ( lambda_Raise @ X4 @ Xa @ U3 ) ) )
=> ~ ( accp_P7631132853315799497lambda @ lambda_Raise_rel @ ( produc5347854423718732792lambda @ X4 @ ( produc8777017121224680385lambda @ Xa @ ( lambda_App @ T5 @ U3 ) ) ) ) ) )
=> ~ ! [T5: lambda_lambda,U3: lambda_lambda] :
( ( Xb
= ( lambda_Beta @ T5 @ U3 ) )
=> ( ( Y2
= ( lambda_Beta @ ( lambda_Raise @ ( suc @ X4 ) @ Xa @ T5 ) @ ( lambda_Raise @ X4 @ Xa @ U3 ) ) )
=> ~ ( accp_P7631132853315799497lambda @ lambda_Raise_rel @ ( produc5347854423718732792lambda @ X4 @ ( produc8777017121224680385lambda @ Xa @ ( lambda_Beta @ T5 @ U3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Raise.pelims
thf(fact_979_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_980_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y2: nat] :
( ( if_nat @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y2: nat] :
( ( if_nat @ $true @ X4 @ Y2 )
= X4 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
lambda_Ide @ t ).
thf(conj_1,conjecture,
( ( lambda2670285437103676873uction @ ( lambda4011002726600638587rategy @ t ) )
| ( lambda_Ide @ ( lambda4011002726600638587rategy @ t ) ) ) ).
%------------------------------------------------------------------------------