TPTP Problem File: SLH0087^1.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_01585_061238__14374866_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1423 ( 538 unt; 140 typ; 0 def)
% Number of atoms : 4149 (1401 equ; 0 cnn)
% Maximal formula atoms : 23 ( 3 avg)
% Number of connectives : 14749 ( 451 ~; 34 |; 431 &;11951 @)
% ( 0 <=>;1882 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 1236 (1236 >; 0 *; 0 +; 0 <<)
% Number of symbols : 139 ( 136 usr; 10 con; 0-5 aty)
% Number of variables : 3759 ( 97 ^;3553 !; 109 ?;3759 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:41.369
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
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 (136)
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
minus_5489901412966541873lambda: set_lambda_lambda > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_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_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_ORaise,type,
lambda_Raise: nat > nat > lambda_lambda > lambda_lambda ).
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_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_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_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_Osubterm,type,
lambda_subterm: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__LambdaCalculus__Olambda____calculus__Olambda_M_Eo_J,type,
inf_in2068736301489225125mbda_o: ( lambda_lambda > $o ) > ( lambda_lambda > $o ) > lambda_lambda > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
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_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
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_001_062_It__LambdaCalculus__Olambda____calculus__Olambda_M_Eo_J,type,
bot_bo3555140721347798439mbda_o: lambda_lambda > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: 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_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
ord_le4648583965110040310lambda: set_lambda_lambda > set_lambda_lambda > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_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_Oless__eq_001t__Set__Oset_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
ord_le1926335974945931754lambda: set_lambda_lambda > set_lambda_lambda > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
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_Ocoherent__normal__sub__rts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
cohere8844529169994681456lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > set_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Ocoherent__normal__sub__rts__axioms_001t__Nat__Onat,type,
cohere3340052296369829450ms_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_Oextensional__rts__with__joins_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
extens4217917907808584361lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
normal1556980139574411093lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > set_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_001t__Nat__Onat,type,
normal_sub_rts_nat: ( nat > nat > nat ) > set_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_OCong_H_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
normal7263485545378587287lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > set_lambda_lambda > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong_H_001t__Nat__Onat,type,
normal_sub_Cong_nat2: ( nat > nat > nat ) > set_nat > nat > nat > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
normal348130670494188410lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > set_lambda_lambda > lambda_lambda > set_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class_001t__Nat__Onat,type,
normal4883526030722344896ss_nat: ( nat > nat > nat ) > set_nat > nat > set_nat ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class__rep_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
normal6971565589081524635lambda: set_lambda_lambda > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_OCong__class__rep_001t__Nat__Onat,type,
normal3802779619712508127ep_nat: set_nat > nat ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
normal7899599941841162628lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > set_lambda_lambda > set_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Onormal__sub__rts_Ois__Cong__class_001t__Nat__Onat,type,
normal8642108778142466614ss_nat: ( nat > nat > nat ) > set_nat > set_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_Opartial__magma_Onull_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
partia1461432607296028304lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
residu6197316955523523210lambda: ( 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_Oresiduation__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
residu6479163625180095271lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_001t__Nat__Onat,type,
rts_nat: ( nat > nat > nat ) > $o ).
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_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_Ocoterminal_001t__Nat__Onat,type,
coterminal_nat: ( nat > nat > nat ) > nat > nat > $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__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_Osimulation_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula5235733974575307523lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula826895837143331878lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__to__weakly__extensional__rts_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula5637266663281507978lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__to__weakly__extensional__rts_001t__Nat__Onat_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula2844598643962246286lambda: ( nat > nat > nat ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( nat > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osub__rts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
sub_rt3575465509250962105lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > $o ) > $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__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
transf356316610583137064lambda: ( 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_Oweakly__extensional__rts_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
weakly7886464935474173793lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_001t__Nat__Onat,type,
weakly1361546452046171801ts_nat: ( nat > nat > nat ) > $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_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Ois__empty_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
is_emp4698403878761571132lambda: set_lambda_lambda > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
is_sin2795064709772026974lambda: set_lambda_lambda > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Opairwise_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
pairwi8946569470594120789lambda: ( lambda_lambda > lambda_lambda > $o ) > set_lambda_lambda > $o ).
thf(sy_c_Set_Opairwise_001t__Nat__Onat,type,
pairwise_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Set_Oremove_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
remove_lambda_lambda: lambda_lambda > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
the_el5431214018478934239lambda: set_lambda_lambda > lambda_lambda ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
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_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 ).
thf(sy_v_u,type,
u: lambda_lambda ).
% Relevant facts (1279)
thf(fact_0_cube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ).
% cube
thf(fact_1_ex__un__null,axiom,
? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( lambda_resid @ X @ T2 )
= X )
& ( ( lambda_resid @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( lambda_resid @ Y @ T3 )
= Y )
& ( ( lambda_resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% ex_un_null
thf(fact_2_Join__sym,axiom,
( lambda_Join
= ( ^ [T4: lambda_lambda,U2: lambda_lambda] : ( lambda_Join @ U2 @ T4 ) ) ) ).
% Join_sym
thf(fact_3_cong__symmetric,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ) ).
% cong_symmetric
thf(fact_4_cong__transitive,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( ( ( 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 @ T @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) ) ) ) ) ).
% cong_transitive
thf(fact_5_ide__backward__stable,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ A ) )
=> ( ide_lambda_lambda @ lambda_resid @ T ) ) ) ).
% ide_backward_stable
thf(fact_6_prfx__transitive,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) ) ) ) ).
% prfx_transitive
thf(fact_7_weak__extensionality,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( T = U ) ) ) ) ).
% weak_extensionality
thf(fact_8__092_060open_062t_A_092_060lesssim_062_At_A_092_060squnion_062_Au_092_060close_062,axiom,
ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ t @ ( lambda_Join @ t @ u ) ) ).
% \<open>t \<lesssim> t \<squnion> u\<close>
thf(fact_9__092_060open_062u_A_092_060lesssim_062_At_A_092_060squnion_062_Au_092_060close_062,axiom,
ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ u @ ( lambda_Join @ t @ u ) ) ).
% \<open>u \<lesssim> t \<squnion> u\<close>
thf(fact_10__092_060open_062u_A_092_At_A_092_060lesssim_062_A_It_A_092_060squnion_062_Au_J_A_092_At_092_060close_062,axiom,
ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ u @ t ) @ ( lambda_resid @ ( lambda_Join @ t @ u ) @ t ) ) ).
% \<open>u \ t \<lesssim> (t \<squnion> u) \ t\<close>
thf(fact_11__092_060open_062_It_A_092_060squnion_062_Au_J_A_092_At_A_092_060lesssim_062_Au_A_092_At_092_060close_062,axiom,
ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ ( lambda_Join @ t @ u ) @ t ) @ ( lambda_resid @ u @ t ) ) ).
% \<open>(t \<squnion> u) \ t \<lesssim> u \ t\<close>
thf(fact_12_is__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% is_confluent_rts
thf(fact_13_partial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% partial_magma_axioms
thf(fact_14_identities__form__coherent__normal__sub__rts,axiom,
cohere1690309856004283859lambda @ lambda_resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ).
% identities_form_coherent_normal_sub_rts
thf(fact_15_cong__implies__coterminal,axiom,
! [U: lambda_lambda,U3: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ U @ U3 ) ) ).
% cong_implies_coterminal
thf(fact_16_resid__ide_I1_J,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ T )
=> ( ( lambda_resid @ T @ A )
= T ) ) ) ).
% resid_ide(1)
thf(fact_17_coinitial__ide__are__cong,axiom,
! [A: lambda_lambda,A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_18_coinitial__ide__are__eq,axiom,
! [A: lambda_lambda,A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A2 )
=> ( A = A2 ) ) ) ) ).
% coinitial_ide_are_eq
thf(fact_19_cong__implies__coinitial,axiom,
! [U: lambda_lambda,U3: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ U @ U3 ) ) ).
% cong_implies_coinitial
thf(fact_20_cong__respects__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T5 @ U3 ) ) ) ) ).
% cong_respects_seq
thf(fact_21_cong__Ide__are__eq,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( ( lambda_Ide @ T )
=> ( ( lambda_Ide @ U )
=> ( T = U ) ) ) ) ).
% cong_Ide_are_eq
thf(fact_22_eq__Ide__are__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( T = U )
=> ( ( lambda_Ide @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ) ).
% eq_Ide_are_cong
thf(fact_23_Cong__iff__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( normal3156941126761417472lambda @ lambda_resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) @ T @ U )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).
% Cong_iff_cong
thf(fact_24_join__of__un__upto__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V2: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ 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_25_join__of__symmetric,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ lambda_resid @ U @ T @ V ) ) ).
% join_of_symmetric
thf(fact_26_prfx__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( lambda_Ide @ ( lambda_resid @ T @ U ) ) ) ).
% prfx_char
thf(fact_27_ide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
= ( lambda_Ide @ T ) ) ).
% ide_char
thf(fact_28_joinable__implies__coinitial,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ).
% joinable_implies_coinitial
thf(fact_29_joinable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
= ( ? [X2: lambda_lambda] : ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ X2 ) ) ) ).
% joinable_def
thf(fact_30_composable__imp__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% composable_imp_seq
thf(fact_31_resid__ide_I2_J,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ T )
=> ( ( lambda_resid @ A @ T )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ) ).
% resid_ide(2)
thf(fact_32_confluence,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% confluence
thf(fact_33_con__imp__coinitial,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ).
% con_imp_coinitial
thf(fact_34_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_35_join__of__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ V @ W )
=> ( join_o1386658102433174083lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) @ ( lambda_resid @ W @ V ) ) ) ) ).
% join_of_resid
thf(fact_36_con__with__join__of__iff_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W )
=> ( ( ( con_lambda_lambda @ lambda_resid @ U @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) )
=> ( con_lambda_lambda @ lambda_resid @ W @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_37_con__with__join__of__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_38_join__of__arr__self,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ T @ T ) ) ).
% join_of_arr_self
thf(fact_39_con__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ U @ T ) ) ).
% con_sym
thf(fact_40_resid__reflects__con,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ U @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% resid_reflects_con
thf(fact_41_apex__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ).
% apex_sym
thf(fact_42_trg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_resid @ T @ T ) ) ).
% trg_def
thf(fact_43_cong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T5 @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T5 @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_44_cong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T5 @ U ) ) ) ).
% cong_subst_left(1)
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
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_50_Collect__cong,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ! [X: lambda_lambda] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec621444948832755541lambda @ P )
= ( collec621444948832755541lambda @ Q ) ) ) ).
% Collect_cong
thf(fact_51_cong__subst__right_I2_J,axiom,
! [U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U3 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U3 ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_52_cong__subst__right_I1_J,axiom,
! [U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U3 ) ) ) ).
% cong_subst_right(1)
thf(fact_53_con__ide__are__eq,axiom,
! [A: lambda_lambda,A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A2 )
=> ( A = A2 ) ) ) ) ).
% con_ide_are_eq
thf(fact_54_con__imp__coinitial__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ? [A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A4 )
& ( con_lambda_lambda @ lambda_resid @ A4 @ T )
& ( con_lambda_lambda @ lambda_resid @ A4 @ U ) ) ) ).
% con_imp_coinitial_ax
thf(fact_55_con__target,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
=> ( ( con_lambda_lambda @ lambda_resid @ U @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ V @ U ) ) ) ) ).
% con_target
thf(fact_56_con__transitive__on__ide,axiom,
! [A: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( ide_lambda_lambda @ lambda_resid @ C )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ B )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ C )
=> ( con_lambda_lambda @ lambda_resid @ A @ C ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_57_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_58_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_59_ide__imp__con__iff__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ) ) ).
% ide_imp_con_iff_cong
thf(fact_60_prfx__implies__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% prfx_implies_con
thf(fact_61_resid__arr__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ A )
=> ( ( lambda_resid @ T @ A )
= T ) ) ) ).
% resid_arr_ide
thf(fact_62_resid__ide__arr,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ T ) ) ) ) ).
% resid_ide_arr
thf(fact_63_cong__reflexive,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) ) ) ) ).
% cong_reflexive
thf(fact_64_ide__implies__arr,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( arr_lambda_lambda @ lambda_resid @ A ) ) ).
% ide_implies_arr
thf(fact_65_prfx__reflexive,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) ) ) ).
% prfx_reflexive
thf(fact_66_con__implies__arr_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ U ) ) ).
% con_implies_arr(2)
thf(fact_67_con__implies__arr_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% con_implies_arr(1)
thf(fact_68_arrE,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( con_lambda_lambda @ lambda_resid @ T @ T ) ) ).
% arrE
thf(fact_69_arr__def,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( con_lambda_lambda @ lambda_resid @ T @ T ) ) ).
% arr_def
thf(fact_70_arr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ).
% arr_resid
thf(fact_71_arr__resid__iff__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% arr_resid_iff_con
thf(fact_72_trg__resid__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).
% trg_resid_sym
thf(fact_73_trg__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
= ( trg_lambda_lambda @ lambda_resid @ V ) ) ) ).
% trg_join_of(2)
thf(fact_74_trg__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ V ) ) ) ).
% trg_join_of(1)
thf(fact_75_composableD_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ U ) ) ).
% composableD(2)
thf(fact_76_composableD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% composableD(1)
thf(fact_77_joinable__implies__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% joinable_implies_con
thf(fact_78_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_79_ide__trg,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ).
% ide_trg
thf(fact_80_coterminal__iff__con__trg,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
= ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ).
% coterminal_iff_con_trg
thf(fact_81_coterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ) ).
% coterminal_iff\<^sub>W\<^sub>E
thf(fact_82_coterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
!= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ) ) ).
% coterminalE\<^sub>W\<^sub>E
thf(fact_83_lambda__calculus_Olambda_Oinject_I1_J,axiom,
! [X22: nat,Y2: nat] :
( ( ( lambda_Var @ X22 )
= ( lambda_Var @ Y2 ) )
= ( X22 = Y2 ) ) ).
% lambda_calculus.lambda.inject(1)
thf(fact_84_trg__trg,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ).
% trg_trg
thf(fact_85_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_86_arrI,axiom,
! [T: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ T )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% arrI
thf(fact_87_trg__ide,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( trg_lambda_lambda @ lambda_resid @ A )
= A ) ) ).
% trg_ide
thf(fact_88_arr__trg__iff__arr,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
= ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% arr_trg_iff_arr
thf(fact_89_coterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( trg_lambda_lambda @ lambda_resid @ U ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ T @ U ) ) ) ).
% coterminalI\<^sub>W\<^sub>E
thf(fact_90_lambda__calculus_OIde_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Ide @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Ide.simps(2)
thf(fact_91_confluent__rts_Oconfluence,axiom,
! [Resid: nat > nat > nat,T: nat,U: nat] :
( ( confluent_rts_nat @ Resid )
=> ( ( coinitial_nat @ Resid @ T @ U )
=> ( con_nat @ Resid @ T @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_92_confluent__rts_Oconfluence,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( conflu4927472611300549594lambda @ Resid )
=> ( ( coinit1392077014771647542lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_93_seqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
!= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ) ).
% seqE\<^sub>W\<^sub>E
thf(fact_94_in__targetsE,axiom,
! [B: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ B )
=> ~ ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B ) ) ) ).
% in_targetsE
thf(fact_95_coterminalE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
!= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ) ) ).
% coterminalE
thf(fact_96_coterminal__iff,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T5 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T5 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T5 ) ) ) ) ).
% coterminal_iff
thf(fact_97_coinitial__iff,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T5 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T5 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T5 ) ) ) ) ).
% coinitial_iff
thf(fact_98_coinitialE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
!= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ) ).
% coinitialE
thf(fact_99_join__of__arr__src_I1_J,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ A @ T @ T ) ) ) ).
% join_of_arr_src(1)
thf(fact_100_join__of__arr__src_I2_J,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ A @ T ) ) ) ).
% join_of_arr_src(2)
thf(fact_101_trg__in__targets,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( member_lambda_lambda @ ( trg_lambda_lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).
% trg_in_targets
thf(fact_102_in__sourcesE,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ A )
=> ~ ( con_lambda_lambda @ lambda_resid @ T @ A ) ) ) ).
% in_sourcesE
thf(fact_103_sources__cong__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A ) ) )
=> ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% sources_cong_closed
thf(fact_104_sources__are__cong,axiom,
! [A: lambda_lambda,T: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A ) ) ) ) ) ).
% sources_are_cong
thf(fact_105_source__is__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ).
% source_is_ide
thf(fact_106_targets__cong__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ( 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 @ T ) ) ) ) ).
% targets_cong_closed
thf(fact_107_targets__are__cong,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_108_target__is__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ).
% target_is_ide
thf(fact_109_sources__are__con,axiom,
! [A: lambda_lambda,T: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ A @ A2 ) ) ) ).
% sources_are_con
thf(fact_110_arr__has__un__source,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( Y = X ) ) ) ) ).
% arr_has_un_source
thf(fact_111_targets__resid__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).
% targets_resid_sym
thf(fact_112_targets__are__con,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ B @ B2 ) ) ) ).
% targets_are_con
thf(fact_113_arr__has__un__target,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( Y = X ) ) ) ) ).
% arr_has_un_target
thf(fact_114_resid__source__in__targets,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( member_lambda_lambda @ ( lambda_resid @ A @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).
% resid_source_in_targets
thf(fact_115_con__imp__eq__src,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ).
% con_imp_eq_src
thf(fact_116_sources__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ V ) ) ) ).
% sources_join_of(1)
thf(fact_117_sources__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ U )
= ( source5627170679743844886lambda @ lambda_resid @ V ) ) ) ).
% sources_join_of(2)
thf(fact_118_targets__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ V ) ) ) ).
% targets_join_of(1)
thf(fact_119_targets__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
= ( target5490931411331770124lambda @ lambda_resid @ V ) ) ) ).
% targets_join_of(2)
thf(fact_120_src__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ V ) ) ) ).
% src_join_of(1)
thf(fact_121_src__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ U )
= ( weakly4554751870655413501lambda @ lambda_resid @ V ) ) ) ).
% src_join_of(2)
thf(fact_122_sources__con__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A2 )
=> ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% sources_con_closed
thf(fact_123_targets__con__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B2 )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ B2 )
=> ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% targets_con_closed
thf(fact_124_src__eqI,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= A ) ) ) ).
% src_eqI
thf(fact_125_ide__src,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ) ).
% ide_src
thf(fact_126_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_127_src__in__sources,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( member_lambda_lambda @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ).
% src_in_sources
thf(fact_128_coinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
!= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ) ).
% coinitialE\<^sub>W\<^sub>E
thf(fact_129_coinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ).
% coinitial_iff\<^sub>W\<^sub>E
thf(fact_130_composableD_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ).
% composableD(3)
thf(fact_131_seqE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
!= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ) ).
% seqE
thf(fact_132_seq__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ).
% seq_def
thf(fact_133_src__src,axiom,
! [T: lambda_lambda] :
( ( weakly4554751870655413501lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ).
% src_src
thf(fact_134_src__ide,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ A )
= A ) ) ).
% src_ide
thf(fact_135_arr__src__iff__arr,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% arr_src_iff_arr
thf(fact_136_trg__src,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ).
% trg_src
thf(fact_137_src__trg,axiom,
! [T: lambda_lambda] :
( ( weakly4554751870655413501lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ).
% src_trg
thf(fact_138_in__sourcesI,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ A )
=> ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% in_sourcesI
thf(fact_139_sources__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ).
% sources_resid
thf(fact_140_src__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ).
% src_resid
thf(fact_141_coinitialI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ) ).
% coinitialI
thf(fact_142_coinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ) ).
% coinitialI\<^sub>W\<^sub>E
thf(fact_143_coterminalI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ U ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ T @ U ) ) ) ).
% coterminalI
thf(fact_144_in__targetsI,axiom,
! [B: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B )
=> ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% in_targetsI
thf(fact_145_seqI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% seqI
thf(fact_146_seqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [U: lambda_lambda,T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% seqI\<^sub>W\<^sub>E
thf(fact_147_rts_Osources_Ocong,axiom,
sources_nat = sources_nat ).
% rts.sources.cong
thf(fact_148_rts_Osources_Ocong,axiom,
source5627170679743844886lambda = source5627170679743844886lambda ).
% rts.sources.cong
thf(fact_149_rts_Otargets_Ocong,axiom,
targets_nat = targets_nat ).
% rts.targets.cong
thf(fact_150_rts_Otargets_Ocong,axiom,
target5490931411331770124lambda = target5490931411331770124lambda ).
% rts.targets.cong
thf(fact_151_weakly__extensional__rts_Osrc_Ocong,axiom,
weakly1695633097759462397rc_nat = weakly1695633097759462397rc_nat ).
% weakly_extensional_rts.src.cong
thf(fact_152_weakly__extensional__rts_Osrc_Ocong,axiom,
weakly4554751870655413501lambda = weakly4554751870655413501lambda ).
% weakly_extensional_rts.src.cong
thf(fact_153_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: nat > nat > nat,NN: set_nat,V: nat,V2: nat,W: nat,W2: nat,T: nat,T5: 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 @ T @ V ) @ ( Resid @ T5 @ V2 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ V2 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_nat @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T5 @ W2 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_154_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,T: lambda_lambda,T5: 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 @ T @ V ) @ ( Resid @ T5 @ V2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ V2 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T5 @ W2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_155_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,U3: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( arr_nat @ Resid @ T )
=> ( ( member_nat @ U @ NN )
=> ( ( member_nat @ U3 @ NN )
=> ( ( ( sources_nat @ Resid @ U )
= ( sources_nat @ Resid @ U3 ) )
=> ( ( ( targets_nat @ Resid @ U )
= ( targets_nat @ Resid @ U3 ) )
=> ( ( ( sources_nat @ Resid @ T )
= ( sources_nat @ Resid @ U ) )
=> ( ( member_nat @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U3 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_156_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U3 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U )
= ( source5627170679743844886lambda @ Resid @ U3 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U )
= ( target5490931411331770124lambda @ Resid @ U3 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_157_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,U: nat,U3: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ U @ U3 )
=> ( ( con_nat @ Resid @ T @ U )
=> ( ( ( sources_nat @ Resid @ T5 )
= ( sources_nat @ Resid @ U3 ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U3 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_158_coherent__normal__sub__rts_OCong__subst_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ U @ U3 )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( ( source5627170679743844886lambda @ Resid @ T5 )
= ( source5627170679743844886lambda @ Resid @ U3 ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U3 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(2)
thf(fact_159_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,U: nat,U3: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ U @ U3 )
=> ( ( con_nat @ Resid @ T @ U )
=> ( ( ( sources_nat @ Resid @ T5 )
= ( sources_nat @ Resid @ U3 ) )
=> ( con_nat @ Resid @ T5 @ U3 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_160_coherent__normal__sub__rts_OCong__subst_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ U @ U3 )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( ( source5627170679743844886lambda @ Resid @ T5 )
= ( source5627170679743844886lambda @ Resid @ U3 ) )
=> ( con_lambda_lambda @ Resid @ T5 @ U3 ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst(1)
thf(fact_161_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,T5: nat,U3: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( ( sources_nat @ Resid @ T )
= ( sources_nat @ Resid @ U ) )
=> ( ( ( sources_nat @ Resid @ T5 )
= ( sources_nat @ Resid @ U3 ) )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ U @ U3 )
=> ( ( con_nat @ Resid @ T @ U )
= ( con_nat @ Resid @ T5 @ U3 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_162_coherent__normal__sub__rts_OCong__subst__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T5 )
= ( source5627170679743844886lambda @ Resid @ U3 ) )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ U @ U3 )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
= ( con_lambda_lambda @ Resid @ T5 @ U3 ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong_subst_con
thf(fact_163_residuation_Oide_Ocong,axiom,
ide_nat = ide_nat ).
% residuation.ide.cong
thf(fact_164_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_165_residuation_Ocon_Ocong,axiom,
con_nat = con_nat ).
% residuation.con.cong
thf(fact_166_residuation_Ocon_Ocong,axiom,
con_lambda_lambda = con_lambda_lambda ).
% residuation.con.cong
thf(fact_167_residuation_Oarr_Ocong,axiom,
arr_nat = arr_nat ).
% residuation.arr.cong
thf(fact_168_residuation_Oarr_Ocong,axiom,
arr_lambda_lambda = arr_lambda_lambda ).
% residuation.arr.cong
thf(fact_169_residuation_Otrg_Ocong,axiom,
trg_nat = trg_nat ).
% residuation.trg.cong
thf(fact_170_residuation_Otrg_Ocong,axiom,
trg_lambda_lambda = trg_lambda_lambda ).
% residuation.trg.cong
thf(fact_171_rts_Oseq_Ocong,axiom,
seq_nat = seq_nat ).
% rts.seq.cong
thf(fact_172_rts_Oseq_Ocong,axiom,
seq_lambda_lambda = seq_lambda_lambda ).
% rts.seq.cong
thf(fact_173_normal__sub__rts_OCong_Ocong,axiom,
normal_sub_Cong_nat = normal_sub_Cong_nat ).
% normal_sub_rts.Cong.cong
thf(fact_174_normal__sub__rts_OCong_Ocong,axiom,
normal3156941126761417472lambda = normal3156941126761417472lambda ).
% normal_sub_rts.Cong.cong
thf(fact_175_rts_Ojoin__of_Ocong,axiom,
join_o1386658102433174083lambda = join_o1386658102433174083lambda ).
% rts.join_of.cong
thf(fact_176_rts_Ocoinitial_Ocong,axiom,
coinitial_nat = coinitial_nat ).
% rts.coinitial.cong
thf(fact_177_rts_Ocoinitial_Ocong,axiom,
coinit1392077014771647542lambda = coinit1392077014771647542lambda ).
% rts.coinitial.cong
thf(fact_178_rts_Ocomposable_Ocong,axiom,
compos7127019122090555255lambda = compos7127019122090555255lambda ).
% rts.composable.cong
thf(fact_179_rts_Ojoinable_Ocong,axiom,
joinab2519788767067900224lambda = joinab2519788767067900224lambda ).
% rts.joinable.cong
thf(fact_180_rts_Ocoterminal_Ocong,axiom,
coterminal_nat = coterminal_nat ).
% rts.coterminal.cong
thf(fact_181_rts_Ocoterminal_Ocong,axiom,
coterm4142657152830665724lambda = coterm4142657152830665724lambda ).
% rts.coterminal.cong
thf(fact_182_partial__magma_Ointro,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ? [X4: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP @ X4 @ T3 )
= X4 )
& ( ( OP @ T3 @ X4 )
= X4 ) )
& ! [Y3: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ Y3 @ T2 )
= Y3 )
& ( ( OP @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X4 ) ) )
=> ( partia3924075026459072775lambda @ OP ) ) ).
% partial_magma.intro
thf(fact_183_partial__magma_Oex__un__null,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ X @ T2 )
= X )
& ( ( OP @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP @ Y @ T3 )
= Y )
& ( ( OP @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_184_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X3: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( OP2 @ X3 @ T4 )
= X3 )
& ( ( OP2 @ T4 @ X3 )
= X3 ) )
& ! [Y4: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( OP2 @ Y4 @ T4 )
= Y4 )
& ( ( OP2 @ T4 @ Y4 )
= Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_185_arr__iff__has__target,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( ( target5490931411331770124lambda @ lambda_resid @ T )
!= bot_bo1271416176687173302lambda ) ) ).
% arr_iff_has_target
thf(fact_186_arr__iff__has__source,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( ( source5627170679743844886lambda @ lambda_resid @ T )
!= bot_bo1271416176687173302lambda ) ) ).
% arr_iff_has_source
thf(fact_187_rts__axioms_Ointro,axiom,
! [Resid: nat > nat > nat] :
( ! [T3: nat] :
( ( arr_nat @ Resid @ T3 )
=> ( ide_nat @ Resid @ ( trg_nat @ Resid @ T3 ) ) )
=> ( ! [A4: nat,T3: nat] :
( ( ide_nat @ Resid @ A4 )
=> ( ( con_nat @ Resid @ T3 @ A4 )
=> ( ( Resid @ T3 @ A4 )
= T3 ) ) )
=> ( ! [A4: nat,T3: nat] :
( ( ide_nat @ Resid @ A4 )
=> ( ( con_nat @ Resid @ A4 @ T3 )
=> ( ide_nat @ Resid @ ( Resid @ A4 @ T3 ) ) ) )
=> ( ! [T3: nat,U4: nat] :
( ( con_nat @ Resid @ T3 @ U4 )
=> ? [A5: nat] :
( ( ide_nat @ Resid @ A5 )
& ( con_nat @ Resid @ A5 @ T3 )
& ( con_nat @ Resid @ A5 @ U4 ) ) )
=> ( ! [T3: nat,U4: nat,V3: nat] :
( ( ide_nat @ Resid @ ( Resid @ T3 @ U4 ) )
=> ( ( con_nat @ Resid @ U4 @ V3 )
=> ( con_nat @ Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ V3 @ U4 ) ) ) )
=> ( rts_axioms_nat @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_188_rts__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( arr_lambda_lambda @ Resid @ T3 )
=> ( ide_lambda_lambda @ Resid @ ( trg_lambda_lambda @ Resid @ T3 ) ) )
=> ( ! [A4: lambda_lambda,T3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A4 )
=> ( ( con_lambda_lambda @ Resid @ T3 @ A4 )
=> ( ( Resid @ T3 @ A4 )
= T3 ) ) )
=> ( ! [A4: lambda_lambda,T3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A4 )
=> ( ( con_lambda_lambda @ Resid @ A4 @ T3 )
=> ( ide_lambda_lambda @ Resid @ ( Resid @ A4 @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ Resid @ T3 @ U4 )
=> ? [A5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A5 )
& ( con_lambda_lambda @ Resid @ A5 @ T3 )
& ( con_lambda_lambda @ Resid @ A5 @ U4 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda,V3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ ( Resid @ T3 @ U4 ) )
=> ( ( con_lambda_lambda @ Resid @ U4 @ V3 )
=> ( con_lambda_lambda @ Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ V3 @ U4 ) ) ) )
=> ( rts_ax7298520571313827089lambda @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_189_rts__axioms__def,axiom,
( rts_axioms_nat
= ( ^ [Resid2: nat > nat > nat] :
( ! [T4: nat] :
( ( arr_nat @ Resid2 @ T4 )
=> ( ide_nat @ Resid2 @ ( trg_nat @ Resid2 @ T4 ) ) )
& ! [A6: nat,T4: nat] :
( ( ide_nat @ Resid2 @ A6 )
=> ( ( con_nat @ Resid2 @ T4 @ A6 )
=> ( ( Resid2 @ T4 @ A6 )
= T4 ) ) )
& ! [A6: nat,T4: nat] :
( ( ide_nat @ Resid2 @ A6 )
=> ( ( con_nat @ Resid2 @ A6 @ T4 )
=> ( ide_nat @ Resid2 @ ( Resid2 @ A6 @ T4 ) ) ) )
& ! [T4: nat,U2: nat] :
( ( con_nat @ Resid2 @ T4 @ U2 )
=> ? [A6: nat] :
( ( ide_nat @ Resid2 @ A6 )
& ( con_nat @ Resid2 @ A6 @ T4 )
& ( con_nat @ Resid2 @ A6 @ U2 ) ) )
& ! [T4: nat,U2: nat,V4: nat] :
( ( ide_nat @ Resid2 @ ( Resid2 @ T4 @ U2 ) )
=> ( ( con_nat @ Resid2 @ U2 @ V4 )
=> ( con_nat @ Resid2 @ ( Resid2 @ T4 @ U2 ) @ ( Resid2 @ V4 @ U2 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_190_rts__axioms__def,axiom,
( rts_ax7298520571313827089lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( arr_lambda_lambda @ Resid2 @ T4 )
=> ( ide_lambda_lambda @ Resid2 @ ( trg_lambda_lambda @ Resid2 @ T4 ) ) )
& ! [A6: lambda_lambda,T4: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A6 )
=> ( ( con_lambda_lambda @ Resid2 @ T4 @ A6 )
=> ( ( Resid2 @ T4 @ A6 )
= T4 ) ) )
& ! [A6: lambda_lambda,T4: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A6 )
=> ( ( con_lambda_lambda @ Resid2 @ A6 @ T4 )
=> ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ A6 @ T4 ) ) ) )
& ! [T4: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ Resid2 @ T4 @ U2 )
=> ? [A6: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A6 )
& ( con_lambda_lambda @ Resid2 @ A6 @ T4 )
& ( con_lambda_lambda @ Resid2 @ A6 @ U2 ) ) )
& ! [T4: lambda_lambda,U2: lambda_lambda,V4: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ T4 @ U2 ) )
=> ( ( con_lambda_lambda @ Resid2 @ U2 @ V4 )
=> ( con_lambda_lambda @ Resid2 @ ( Resid2 @ T4 @ U2 ) @ ( Resid2 @ V4 @ U2 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_191_coherent__normal__sub__rts__axioms__def,axiom,
( cohere3340052296369829450ms_nat
= ( ^ [Resid2: nat > nat > nat,NN2: set_nat] :
! [T4: nat,U2: nat,U5: nat] :
( ( arr_nat @ Resid2 @ T4 )
=> ( ( member_nat @ U2 @ NN2 )
=> ( ( member_nat @ U5 @ NN2 )
=> ( ( ( sources_nat @ Resid2 @ U2 )
= ( sources_nat @ Resid2 @ U5 ) )
=> ( ( ( targets_nat @ Resid2 @ U2 )
= ( targets_nat @ Resid2 @ U5 ) )
=> ( ( ( sources_nat @ Resid2 @ T4 )
= ( sources_nat @ Resid2 @ U2 ) )
=> ( ( member_nat @ ( Resid2 @ ( Resid2 @ T4 @ U2 ) @ ( Resid2 @ T4 @ U5 ) ) @ NN2 )
& ( member_nat @ ( Resid2 @ ( Resid2 @ T4 @ U5 ) @ ( Resid2 @ T4 @ U2 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_192_coherent__normal__sub__rts__axioms__def,axiom,
( cohere8844529169994681456lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda,NN2: set_lambda_lambda] :
! [T4: lambda_lambda,U2: lambda_lambda,U5: lambda_lambda] :
( ( arr_lambda_lambda @ Resid2 @ T4 )
=> ( ( member_lambda_lambda @ U2 @ NN2 )
=> ( ( member_lambda_lambda @ U5 @ NN2 )
=> ( ( ( source5627170679743844886lambda @ Resid2 @ U2 )
= ( source5627170679743844886lambda @ Resid2 @ U5 ) )
=> ( ( ( target5490931411331770124lambda @ Resid2 @ U2 )
= ( target5490931411331770124lambda @ Resid2 @ U5 ) )
=> ( ( ( source5627170679743844886lambda @ Resid2 @ T4 )
= ( source5627170679743844886lambda @ Resid2 @ U2 ) )
=> ( ( member_lambda_lambda @ ( Resid2 @ ( Resid2 @ T4 @ U2 ) @ ( Resid2 @ T4 @ U5 ) ) @ NN2 )
& ( member_lambda_lambda @ ( Resid2 @ ( Resid2 @ T4 @ U5 ) @ ( Resid2 @ T4 @ U2 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_193_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: nat > nat > nat,NN: set_nat] :
( ! [T3: nat,U4: nat,U6: nat] :
( ( arr_nat @ Resid @ T3 )
=> ( ( member_nat @ U4 @ NN )
=> ( ( member_nat @ U6 @ NN )
=> ( ( ( sources_nat @ Resid @ U4 )
= ( sources_nat @ Resid @ U6 ) )
=> ( ( ( targets_nat @ Resid @ U4 )
= ( targets_nat @ Resid @ U6 ) )
=> ( ( ( sources_nat @ Resid @ T3 )
= ( sources_nat @ Resid @ U4 ) )
=> ( ( member_nat @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ T3 @ U6 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T3 @ U6 ) @ ( Resid @ T3 @ U4 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere3340052296369829450ms_nat @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_194_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda] :
( ! [T3: lambda_lambda,U4: lambda_lambda,U6: lambda_lambda] :
( ( arr_lambda_lambda @ Resid @ T3 )
=> ( ( member_lambda_lambda @ U4 @ NN )
=> ( ( member_lambda_lambda @ U6 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U4 )
= ( source5627170679743844886lambda @ Resid @ U6 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U4 )
= ( target5490931411331770124lambda @ Resid @ U6 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T3 )
= ( source5627170679743844886lambda @ Resid @ U4 ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ T3 @ U6 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T3 @ U6 ) @ ( Resid @ T3 @ U4 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere8844529169994681456lambda @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_195_composite__of__arr__target,axiom,
! [T: lambda_lambda,B: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ T @ B @ T ) ) ) ).
% composite_of_arr_target
thf(fact_196_composite__of__source__arr,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ A @ T @ T ) ) ) ).
% composite_of_source_arr
thf(fact_197_composite__of__ide__arr,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ A @ T @ T )
= ( con_lambda_lambda @ lambda_resid @ T @ A ) ) ) ).
% composite_of_ide_arr
thf(fact_198_composite__of__arr__ide,axiom,
! [B: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T @ B @ T )
= ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) @ B ) ) ) ).
% composite_of_arr_ide
thf(fact_199_composite__ofE,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) ) ) ) ).
% composite_ofE
thf(fact_200_composite__of__cancel__left,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,U3: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T @ U3 @ V )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) ) ) ) ).
% composite_of_cancel_left
thf(fact_201_composite__of__def,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) ) ) ).
% composite_of_def
thf(fact_202_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_203_composite__of__unq__upto__cong,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda,V2: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ 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_204_con__prfx__composite__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( con_lambda_lambda @ lambda_resid @ T @ V ) ) ) ).
% con_prfx_composite_of(2)
thf(fact_205_con__prfx__composite__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W )
=> ( con_lambda_lambda @ lambda_resid @ T @ W ) ) ).
% con_prfx_composite_of(1)
thf(fact_206_resid__composite__of_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( compos9183582412961930448lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ ( lambda_resid @ V @ T ) ) @ ( lambda_resid @ W @ V ) ) ) ) ).
% resid_composite_of(4)
thf(fact_207_resid__composite__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ U ) ) ) ).
% resid_composite_of(2)
thf(fact_208_resid__composite__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ W @ T ) ) ) ) ).
% resid_composite_of(1)
thf(fact_209_bounded__imp__con,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T5 @ U3 @ V )
=> ( con_lambda_lambda @ lambda_resid @ T @ T5 ) ) ) ).
% bounded_imp_con
thf(fact_210_con__composite__of__iff,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,W: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
= ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ W @ T ) @ U ) ) ) ).
% con_composite_of_iff
thf(fact_211_arr__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( arr_lambda_lambda @ lambda_resid @ V ) ) ).
% arr_composite_of
thf(fact_212_sources__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ V )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ).
% sources_composite_of
thf(fact_213_targets__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ V )
= ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).
% targets_composite_of
thf(fact_214_trg__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ V )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ).
% trg_composite_of
thf(fact_215_src__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ V )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ).
% src_composite_of
thf(fact_216_join__ofE,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ~ ( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
=> ~ ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V ) ) ) ).
% join_ofE
thf(fact_217_join__of__def,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
= ( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
& ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V ) ) ) ).
% join_of_def
thf(fact_218_composable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
= ( ? [X2: lambda_lambda] : ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ X2 ) ) ) ).
% composable_def
thf(fact_219_resid__composite__of_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ 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 @ T ) @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ T ) @ U ) @ ( lambda_resid @ V @ W ) ) ) ) ) ) ).
% resid_composite_of(3)
thf(fact_220_composite__ofI,axiom,
! [U: lambda_lambda,V: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V ) ) ) ).
% composite_ofI
thf(fact_221_join__ofI,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V ) ) ) ).
% join_ofI
thf(fact_222_rts_Ocomposite__of_Ocong,axiom,
composite_of_nat = composite_of_nat ).
% rts.composite_of.cong
thf(fact_223_rts_Ocomposite__of_Ocong,axiom,
compos9183582412961930448lambda = compos9183582412961930448lambda ).
% rts.composite_of.cong
thf(fact_224_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,T5: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( composite_of_nat @ Resid @ T @ U @ T5 )
=> ( ( member_nat @ U @ NN )
=> ( ( member_nat @ ( Resid @ T5 @ T ) @ NN )
& ( member_nat @ ( Resid @ T @ T5 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_225_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,T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ T5 )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_226_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T: nat,T5: nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( ( composite_of_nat @ Resid @ U @ T @ T5 )
=> ( ( member_nat @ U @ NN )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T5 @ T ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_227_coherent__normal__sub__rts_OCong__composite__of__normal__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ T5 )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T5 @ T ) ) ) ) ).
% coherent_normal_sub_rts.Cong_composite_of_normal_arr
thf(fact_228_coherent__normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat] :
( ( cohere320596682411188903ts_nat @ Resid @ NN )
=> ( cohere3340052296369829450ms_nat @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(2)
thf(fact_229_coherent__normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( cohere8844529169994681456lambda @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(2)
thf(fact_230_normal__sub__rts__axioms__def,axiom,
( normal3241730178054940488ms_nat
= ( ^ [Resid2: nat > nat > nat,NN2: set_nat] :
( ! [T4: nat] :
( ( member_nat @ T4 @ NN2 )
=> ( arr_nat @ Resid2 @ T4 ) )
& ! [A6: nat] :
( ( ide_nat @ Resid2 @ A6 )
=> ( member_nat @ A6 @ NN2 ) )
& ! [U2: nat,T4: nat] :
( ( member_nat @ U2 @ NN2 )
=> ( ( coinitial_nat @ Resid2 @ T4 @ U2 )
=> ( member_nat @ ( Resid2 @ U2 @ T4 ) @ NN2 ) ) )
& ! [U2: nat,T4: nat] :
( ( member_nat @ U2 @ NN2 )
=> ( ( member_nat @ ( Resid2 @ T4 @ U2 ) @ NN2 )
=> ( member_nat @ T4 @ NN2 ) ) )
& ! [U2: nat,T4: nat] :
( ( member_nat @ U2 @ NN2 )
=> ( ( seq_nat @ Resid2 @ U2 @ T4 )
=> ? [X2: nat] : ( composite_of_nat @ Resid2 @ U2 @ T4 @ X2 ) ) )
& ! [U2: nat,T4: nat] :
( ( member_nat @ U2 @ NN2 )
=> ( ( seq_nat @ Resid2 @ T4 @ U2 )
=> ? [X2: nat] : ( composite_of_nat @ Resid2 @ T4 @ U2 @ X2 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_231_normal__sub__rts__axioms__def,axiom,
( normal5162919049924913650lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda,NN2: set_lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( member_lambda_lambda @ T4 @ NN2 )
=> ( arr_lambda_lambda @ Resid2 @ T4 ) )
& ! [A6: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A6 )
=> ( member_lambda_lambda @ A6 @ NN2 ) )
& ! [U2: lambda_lambda,T4: lambda_lambda] :
( ( member_lambda_lambda @ U2 @ NN2 )
=> ( ( coinit1392077014771647542lambda @ Resid2 @ T4 @ U2 )
=> ( member_lambda_lambda @ ( Resid2 @ U2 @ T4 ) @ NN2 ) ) )
& ! [U2: lambda_lambda,T4: lambda_lambda] :
( ( member_lambda_lambda @ U2 @ NN2 )
=> ( ( member_lambda_lambda @ ( Resid2 @ T4 @ U2 ) @ NN2 )
=> ( member_lambda_lambda @ T4 @ NN2 ) ) )
& ! [U2: lambda_lambda,T4: lambda_lambda] :
( ( member_lambda_lambda @ U2 @ NN2 )
=> ( ( seq_lambda_lambda @ Resid2 @ U2 @ T4 )
=> ? [X2: lambda_lambda] : ( compos9183582412961930448lambda @ Resid2 @ U2 @ T4 @ X2 ) ) )
& ! [U2: lambda_lambda,T4: lambda_lambda] :
( ( member_lambda_lambda @ U2 @ NN2 )
=> ( ( seq_lambda_lambda @ Resid2 @ T4 @ U2 )
=> ? [X2: lambda_lambda] : ( compos9183582412961930448lambda @ Resid2 @ T4 @ U2 @ X2 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_232_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_nat,Resid: nat > nat > nat] :
( ! [T3: nat] :
( ( member_nat @ T3 @ NN )
=> ( arr_nat @ Resid @ T3 ) )
=> ( ! [A4: nat] :
( ( ide_nat @ Resid @ A4 )
=> ( member_nat @ A4 @ NN ) )
=> ( ! [U4: nat,T3: nat] :
( ( member_nat @ U4 @ NN )
=> ( ( coinitial_nat @ Resid @ T3 @ U4 )
=> ( member_nat @ ( Resid @ U4 @ T3 ) @ NN ) ) )
=> ( ! [U4: nat,T3: nat] :
( ( member_nat @ U4 @ NN )
=> ( ( member_nat @ ( Resid @ T3 @ U4 ) @ NN )
=> ( member_nat @ T3 @ NN ) ) )
=> ( ! [U4: nat,T3: nat] :
( ( member_nat @ U4 @ NN )
=> ( ( seq_nat @ Resid @ U4 @ T3 )
=> ? [X_1: nat] : ( composite_of_nat @ Resid @ U4 @ T3 @ X_1 ) ) )
=> ( ! [U4: nat,T3: nat] :
( ( member_nat @ U4 @ NN )
=> ( ( seq_nat @ Resid @ T3 @ U4 )
=> ? [X_1: nat] : ( composite_of_nat @ Resid @ T3 @ U4 @ X_1 ) ) )
=> ( normal3241730178054940488ms_nat @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_233_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_lambda_lambda,Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( member_lambda_lambda @ T3 @ NN )
=> ( arr_lambda_lambda @ Resid @ T3 ) )
=> ( ! [A4: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A4 )
=> ( member_lambda_lambda @ A4 @ NN ) )
=> ( ! [U4: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ( ( coinit1392077014771647542lambda @ Resid @ T3 @ U4 )
=> ( member_lambda_lambda @ ( Resid @ U4 @ T3 ) @ NN ) ) )
=> ( ! [U4: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T3 @ U4 ) @ NN )
=> ( member_lambda_lambda @ T3 @ NN ) ) )
=> ( ! [U4: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ( ( seq_lambda_lambda @ Resid @ U4 @ T3 )
=> ? [X_1: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ U4 @ T3 @ X_1 ) ) )
=> ( ! [U4: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ( ( seq_lambda_lambda @ Resid @ T3 @ U4 )
=> ? [X_1: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ T3 @ U4 @ X_1 ) ) )
=> ( normal5162919049924913650lambda @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_234_coterminal__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
= ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% coterminal_def
thf(fact_235_coinitial__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
= ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% coinitial_def
thf(fact_236_con__imp__common__source,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% con_imp_common_source
thf(fact_237_targets__eqI,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T5 ) ) ) ).
% targets_eqI
thf(fact_238_sources__eqI,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T5 ) ) ) ).
% sources_eqI
thf(fact_239_empty__Collect__eq,axiom,
! [P: lambda_lambda > $o] :
( ( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ P ) )
= ( ! [X3: lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_240_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_241_empty__iff,axiom,
! [C: lambda_lambda] :
~ ( member_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ).
% empty_iff
thf(fact_242_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_243_all__not__in__conv,axiom,
! [A3: set_lambda_lambda] :
( ( ! [X3: lambda_lambda] :
~ ( member_lambda_lambda @ X3 @ A3 ) )
= ( A3 = bot_bo1271416176687173302lambda ) ) ).
% all_not_in_conv
thf(fact_244_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_245_Collect__empty__eq,axiom,
! [P: lambda_lambda > $o] :
( ( ( collec621444948832755541lambda @ P )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_246_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_247_IntI,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_248_IntI,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ A3 )
=> ( ( member_lambda_lambda @ C @ B3 )
=> ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_249_Int__iff,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
= ( ( member_nat @ C @ A3 )
& ( member_nat @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_250_Int__iff,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) )
= ( ( member_lambda_lambda @ C @ A3 )
& ( member_lambda_lambda @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_251_disjoint__iff__not__equal,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A3 )
=> ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ B3 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_252_disjoint__iff__not__equal,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B3 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_253_Int__empty__right,axiom,
! [A3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A3 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_right
thf(fact_254_Int__empty__right,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_255_Int__empty__left,axiom,
! [B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ B3 )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_left
thf(fact_256_Int__empty__left,axiom,
! [B3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B3 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_257_disjoint__iff,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A3 )
=> ~ ( member_lambda_lambda @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_258_disjoint__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ~ ( member_nat @ X3 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_259_Int__emptyI,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A3 )
=> ~ ( member_lambda_lambda @ X @ B3 ) )
=> ( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= bot_bo1271416176687173302lambda ) ) ).
% Int_emptyI
thf(fact_260_Int__emptyI,axiom,
! [A3: set_nat,B3: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ~ ( member_nat @ X @ B3 ) )
=> ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_261_IntE,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ~ ( member_nat @ C @ B3 ) ) ) ).
% IntE
thf(fact_262_IntE,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) )
=> ~ ( ( member_lambda_lambda @ C @ A3 )
=> ~ ( member_lambda_lambda @ C @ B3 ) ) ) ).
% IntE
thf(fact_263_IntD1,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ A3 ) ) ).
% IntD1
thf(fact_264_IntD1,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) )
=> ( member_lambda_lambda @ C @ A3 ) ) ).
% IntD1
thf(fact_265_IntD2,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ B3 ) ) ).
% IntD2
thf(fact_266_IntD2,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) )
=> ( member_lambda_lambda @ C @ B3 ) ) ).
% IntD2
thf(fact_267_Int__assoc,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C2 )
= ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_268_Int__assoc,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,C2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) @ C2 )
= ( inf_in1766959559283430712lambda @ A3 @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_269_Int__absorb,axiom,
! [A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_270_Int__absorb,axiom,
! [A3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_271_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A7: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A7 ) ) ) ).
% Int_commute
thf(fact_272_Int__commute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A7: set_lambda_lambda,B4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ B4 @ A7 ) ) ) ).
% Int_commute
thf(fact_273_Int__left__absorb,axiom,
! [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
= ( inf_inf_set_nat @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_274_Int__left__absorb,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A3 @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) )
= ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_275_Int__left__commute,axiom,
! [A3: set_nat,B3: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C2 ) )
= ( inf_inf_set_nat @ B3 @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_276_Int__left__commute,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,C2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A3 @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) )
= ( inf_in1766959559283430712lambda @ B3 @ ( inf_in1766959559283430712lambda @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_277_emptyE,axiom,
! [A: lambda_lambda] :
~ ( member_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ).
% emptyE
thf(fact_278_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_279_equals0D,axiom,
! [A3: set_lambda_lambda,A: lambda_lambda] :
( ( A3 = bot_bo1271416176687173302lambda )
=> ~ ( member_lambda_lambda @ A @ A3 ) ) ).
% equals0D
thf(fact_280_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_281_equals0I,axiom,
! [A3: set_lambda_lambda] :
( ! [Y3: lambda_lambda] :
~ ( member_lambda_lambda @ Y3 @ A3 )
=> ( A3 = bot_bo1271416176687173302lambda ) ) ).
% equals0I
thf(fact_282_equals0I,axiom,
! [A3: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_283_ex__in__conv,axiom,
! [A3: set_lambda_lambda] :
( ( ? [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A3 ) )
= ( A3 != bot_bo1271416176687173302lambda ) ) ).
% ex_in_conv
thf(fact_284_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_285_boolean__algebra_Oconj__zero__right,axiom,
! [X5: nat > $o] :
( ( inf_inf_nat_o @ X5 @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_286_boolean__algebra_Oconj__zero__right,axiom,
! [X5: lambda_lambda > $o] :
( ( inf_in2068736301489225125mbda_o @ X5 @ bot_bo3555140721347798439mbda_o )
= bot_bo3555140721347798439mbda_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_287_boolean__algebra_Oconj__zero__right,axiom,
! [X5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X5 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_288_boolean__algebra_Oconj__zero__right,axiom,
! [X5: set_nat] :
( ( inf_inf_set_nat @ X5 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_289_boolean__algebra_Oconj__zero__left,axiom,
! [X5: nat > $o] :
( ( inf_inf_nat_o @ bot_bot_nat_o @ X5 )
= bot_bot_nat_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_290_boolean__algebra_Oconj__zero__left,axiom,
! [X5: lambda_lambda > $o] :
( ( inf_in2068736301489225125mbda_o @ bot_bo3555140721347798439mbda_o @ X5 )
= bot_bo3555140721347798439mbda_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_291_boolean__algebra_Oconj__zero__left,axiom,
! [X5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X5 )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_292_boolean__algebra_Oconj__zero__left,axiom,
! [X5: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X5 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_293_inf__bot__right,axiom,
! [X5: nat > $o] :
( ( inf_inf_nat_o @ X5 @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
% inf_bot_right
thf(fact_294_inf__bot__right,axiom,
! [X5: lambda_lambda > $o] :
( ( inf_in2068736301489225125mbda_o @ X5 @ bot_bo3555140721347798439mbda_o )
= bot_bo3555140721347798439mbda_o ) ).
% inf_bot_right
thf(fact_295_inf__bot__right,axiom,
! [X5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X5 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_right
thf(fact_296_inf__bot__right,axiom,
! [X5: set_nat] :
( ( inf_inf_set_nat @ X5 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_297_inf__bot__left,axiom,
! [X5: nat > $o] :
( ( inf_inf_nat_o @ bot_bot_nat_o @ X5 )
= bot_bot_nat_o ) ).
% inf_bot_left
thf(fact_298_inf__bot__left,axiom,
! [X5: lambda_lambda > $o] :
( ( inf_in2068736301489225125mbda_o @ bot_bo3555140721347798439mbda_o @ X5 )
= bot_bo3555140721347798439mbda_o ) ).
% inf_bot_left
thf(fact_299_inf__bot__left,axiom,
! [X5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X5 )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_left
thf(fact_300_inf__bot__left,axiom,
! [X5: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X5 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_301_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_302_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_303_inf_Oidem,axiom,
! [A: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ A )
= A ) ).
% inf.idem
thf(fact_304_inf__idem,axiom,
! [X5: set_nat] :
( ( inf_inf_set_nat @ X5 @ X5 )
= X5 ) ).
% inf_idem
thf(fact_305_inf__idem,axiom,
! [X5: nat] :
( ( inf_inf_nat @ X5 @ X5 )
= X5 ) ).
% inf_idem
thf(fact_306_inf__idem,axiom,
! [X5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X5 @ X5 )
= X5 ) ).
% inf_idem
thf(fact_307_inf_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_308_inf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_309_inf_Oleft__idem,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ A @ B ) )
= ( inf_in1766959559283430712lambda @ A @ B ) ) ).
% inf.left_idem
thf(fact_310_inf__left__idem,axiom,
! [X5: set_nat,Y5: set_nat] :
( ( inf_inf_set_nat @ X5 @ ( inf_inf_set_nat @ X5 @ Y5 ) )
= ( inf_inf_set_nat @ X5 @ Y5 ) ) ).
% inf_left_idem
thf(fact_311_inf__left__idem,axiom,
! [X5: nat,Y5: nat] :
( ( inf_inf_nat @ X5 @ ( inf_inf_nat @ X5 @ Y5 ) )
= ( inf_inf_nat @ X5 @ Y5 ) ) ).
% inf_left_idem
thf(fact_312_inf__left__idem,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X5 @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) )
= ( inf_in1766959559283430712lambda @ X5 @ Y5 ) ) ).
% inf_left_idem
thf(fact_313_inf__right__idem,axiom,
! [X5: set_nat,Y5: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X5 @ Y5 ) @ Y5 )
= ( inf_inf_set_nat @ X5 @ Y5 ) ) ).
% inf_right_idem
thf(fact_314_inf__right__idem,axiom,
! [X5: nat,Y5: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ Y5 )
= ( inf_inf_nat @ X5 @ Y5 ) ) ).
% inf_right_idem
thf(fact_315_inf__right__idem,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) @ Y5 )
= ( inf_in1766959559283430712lambda @ X5 @ Y5 ) ) ).
% inf_right_idem
thf(fact_316_inf_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_317_inf_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_318_inf_Oright__idem,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ B )
= ( inf_in1766959559283430712lambda @ A @ B ) ) ).
% inf.right_idem
thf(fact_319_bot__set__def,axiom,
( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ bot_bo3555140721347798439mbda_o ) ) ).
% bot_set_def
thf(fact_320_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_321_inf__left__commute,axiom,
! [X5: set_nat,Y5: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X5 @ ( inf_inf_set_nat @ Y5 @ Z ) )
= ( inf_inf_set_nat @ Y5 @ ( inf_inf_set_nat @ X5 @ Z ) ) ) ).
% inf_left_commute
thf(fact_322_inf__left__commute,axiom,
! [X5: nat,Y5: nat,Z: nat] :
( ( inf_inf_nat @ X5 @ ( inf_inf_nat @ Y5 @ Z ) )
= ( inf_inf_nat @ Y5 @ ( inf_inf_nat @ X5 @ Z ) ) ) ).
% inf_left_commute
thf(fact_323_inf__left__commute,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda,Z: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X5 @ ( inf_in1766959559283430712lambda @ Y5 @ Z ) )
= ( inf_in1766959559283430712lambda @ Y5 @ ( inf_in1766959559283430712lambda @ X5 @ Z ) ) ) ).
% inf_left_commute
thf(fact_324_inf_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_325_inf_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_326_inf_Oleft__commute,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda,C: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ B @ ( inf_in1766959559283430712lambda @ A @ C ) )
= ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_327_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B3
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A @ B3 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_328_boolean__algebra__cancel_Oinf2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( inf_inf_nat @ K @ B ) )
=> ( ( inf_inf_nat @ A @ B3 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_329_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_lambda_lambda,K: set_lambda_lambda,B: set_lambda_lambda,A: set_lambda_lambda] :
( ( B3
= ( inf_in1766959559283430712lambda @ K @ B ) )
=> ( ( inf_in1766959559283430712lambda @ A @ B3 )
= ( inf_in1766959559283430712lambda @ K @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_330_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A3
= ( inf_inf_set_nat @ K @ A ) )
=> ( ( inf_inf_set_nat @ A3 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_331_boolean__algebra__cancel_Oinf1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( inf_inf_nat @ K @ A ) )
=> ( ( inf_inf_nat @ A3 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_332_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_lambda_lambda,K: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A3
= ( inf_in1766959559283430712lambda @ K @ A ) )
=> ( ( inf_in1766959559283430712lambda @ A3 @ B )
= ( inf_in1766959559283430712lambda @ K @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_333_inf__commute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [X3: set_lambda_lambda,Y4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_334_inf_Ocommute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ B5 @ A6 ) ) ) ).
% inf.commute
thf(fact_335_inf__assoc,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda,Z: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) @ Z )
= ( inf_in1766959559283430712lambda @ X5 @ ( inf_in1766959559283430712lambda @ Y5 @ Z ) ) ) ).
% inf_assoc
thf(fact_336_inf_Oassoc,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C )
= ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) ) ) ).
% inf.assoc
thf(fact_337_inf__sup__aci_I1_J,axiom,
( inf_in1766959559283430712lambda
= ( ^ [X3: set_lambda_lambda,Y4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_338_inf__sup__aci_I2_J,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda,Z: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) @ Z )
= ( inf_in1766959559283430712lambda @ X5 @ ( inf_in1766959559283430712lambda @ Y5 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_339_inf__sup__aci_I3_J,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda,Z: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X5 @ ( inf_in1766959559283430712lambda @ Y5 @ Z ) )
= ( inf_in1766959559283430712lambda @ Y5 @ ( inf_in1766959559283430712lambda @ X5 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_340_inf__sup__aci_I4_J,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X5 @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) )
= ( inf_in1766959559283430712lambda @ X5 @ Y5 ) ) ).
% inf_sup_aci(4)
thf(fact_341_rts__with__composites__axioms__def,axiom,
( rts_wi6632542615990831231lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
! [T4: lambda_lambda,U2: lambda_lambda] :
( ( seq_lambda_lambda @ Resid2 @ T4 @ U2 )
=> ( compos7127019122090555255lambda @ Resid2 @ T4 @ U2 ) ) ) ) ).
% rts_with_composites_axioms_def
thf(fact_342_rts__with__composites__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( seq_lambda_lambda @ Resid @ T3 @ U4 )
=> ( compos7127019122090555255lambda @ Resid @ T3 @ U4 ) )
=> ( rts_wi6632542615990831231lambda @ Resid ) ) ).
% rts_with_composites_axioms.intro
thf(fact_343_rts__with__joins__axioms__def,axiom,
( rts_wi6344169777088970040lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
! [T4: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ Resid2 @ T4 @ U2 )
=> ( joinab2519788767067900224lambda @ Resid2 @ T4 @ U2 ) ) ) ) ).
% rts_with_joins_axioms_def
thf(fact_344_rts__with__joins__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ Resid @ T3 @ U4 )
=> ( joinab2519788767067900224lambda @ Resid @ T3 @ U4 ) )
=> ( rts_wi6344169777088970040lambda @ Resid ) ) ).
% rts_with_joins_axioms.intro
thf(fact_345_confluent__rts__axioms__def,axiom,
( conflu5019861018119259255lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
! [T4: lambda_lambda,U2: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid2 @ T4 @ U2 )
=> ( con_lambda_lambda @ Resid2 @ T4 @ U2 ) ) ) ) ).
% confluent_rts_axioms_def
thf(fact_346_confluent__rts__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid @ T3 @ U4 )
=> ( con_lambda_lambda @ Resid @ T3 @ U4 ) )
=> ( conflu5019861018119259255lambda @ Resid ) ) ).
% confluent_rts_axioms.intro
thf(fact_347_confluent__rts_Oaxioms_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( conflu4927472611300549594lambda @ Resid )
=> ( conflu5019861018119259255lambda @ Resid ) ) ).
% confluent_rts.axioms(2)
thf(fact_348_bot__empty__eq,axiom,
( bot_bo3555140721347798439mbda_o
= ( ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ bot_bo1271416176687173302lambda ) ) ) ).
% bot_empty_eq
thf(fact_349_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_350_Collect__empty__eq__bot,axiom,
! [P: lambda_lambda > $o] :
( ( ( collec621444948832755541lambda @ P )
= bot_bo1271416176687173302lambda )
= ( P = bot_bo3555140721347798439mbda_o ) ) ).
% Collect_empty_eq_bot
thf(fact_351_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_352_lambda__calculus_Otargets__simps_I2_J,axiom,
! [X5: nat] :
( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Var @ X5 ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X5 ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.targets_simps(2)
thf(fact_353_lambda__calculus_Osources__simps_I2_J,axiom,
! [X5: nat] :
( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Var @ X5 ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X5 ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.sources_simps(2)
thf(fact_354_transformation_Onaturality3,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,F2: lambda_lambda] :
( ( transf2236888680007955973lambda @ A3 @ B3 @ F @ G @ Tau )
=> ( ( arr_lambda_lambda @ A3 @ F2 )
=> ( join_o1386658102433174083lambda @ B3 @ ( Tau @ ( weakly4554751870655413501lambda @ A3 @ F2 ) ) @ ( F @ F2 ) @ ( Tau @ F2 ) ) ) ) ).
% transformation.naturality3
thf(fact_355_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A3: nat > nat > nat,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: nat > lambda_lambda,B: nat,T: nat] :
( ( simula2844598643962246286lambda @ A3 @ B3 @ F )
=> ( ( member_nat @ B @ ( targets_nat @ A3 @ T ) )
=> ( ( trg_lambda_lambda @ B3 @ ( F @ T ) )
= ( F @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_356_simulation__to__weakly__extensional__rts_Opreserves__trg,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,B: lambda_lambda,T: lambda_lambda] :
( ( simula5637266663281507978lambda @ A3 @ B3 @ F )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ A3 @ T ) )
=> ( ( trg_lambda_lambda @ B3 @ ( F @ T ) )
= ( F @ B ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_trg
thf(fact_357_insertCI,axiom,
! [A: lambda_lambda,B3: set_lambda_lambda,B: lambda_lambda] :
( ( ~ ( member_lambda_lambda @ A @ B3 )
=> ( A = B ) )
=> ( member_lambda_lambda @ A @ ( insert_lambda_lambda @ B @ B3 ) ) ) ).
% insertCI
thf(fact_358_insertCI,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_359_insert__iff,axiom,
! [A: lambda_lambda,B: lambda_lambda,A3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ ( insert_lambda_lambda @ B @ A3 ) )
= ( ( A = B )
| ( member_lambda_lambda @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_360_insert__iff,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
= ( ( A = B )
| ( member_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_361_insert__absorb2,axiom,
! [X5: lambda_lambda,A3: set_lambda_lambda] :
( ( insert_lambda_lambda @ X5 @ ( insert_lambda_lambda @ X5 @ A3 ) )
= ( insert_lambda_lambda @ X5 @ A3 ) ) ).
% insert_absorb2
thf(fact_362_insert__absorb2,axiom,
! [X5: nat,A3: set_nat] :
( ( insert_nat @ X5 @ ( insert_nat @ X5 @ A3 ) )
= ( insert_nat @ X5 @ A3 ) ) ).
% insert_absorb2
thf(fact_363_singletonI,axiom,
! [A: lambda_lambda] : ( member_lambda_lambda @ A @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) ).
% singletonI
thf(fact_364_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_365_Int__insert__right__if1,axiom,
! [A: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_366_Int__insert__right__if1,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ A3 )
=> ( ( inf_in1766959559283430712lambda @ A3 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_367_Int__insert__right__if0,axiom,
! [A: nat,A3: set_nat,B3: set_nat] :
( ~ ( member_nat @ A @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A @ B3 ) )
= ( inf_inf_set_nat @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_368_Int__insert__right__if0,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A @ A3 )
=> ( ( inf_in1766959559283430712lambda @ A3 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_369_insert__inter__insert,axiom,
! [A: nat,A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_370_insert__inter__insert,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ A3 ) @ ( insert_lambda_lambda @ A @ B3 ) )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_371_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B3: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_372_Int__insert__left__if1,axiom,
! [A: lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ B3 ) @ C2 )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_373_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B3: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C2 )
= ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_374_Int__insert__left__if0,axiom,
! [A: lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ B3 ) @ C2 )
= ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_375_disjoint__insert_I2_J,axiom,
! [A3: set_lambda_lambda,B: lambda_lambda,B3: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A3 @ ( insert_lambda_lambda @ B @ B3 ) ) )
= ( ~ ( member_lambda_lambda @ B @ A3 )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_376_disjoint__insert_I2_J,axiom,
! [A3: set_nat,B: nat,B3: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ ( insert_nat @ B @ B3 ) ) )
= ( ~ ( member_nat @ B @ A3 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_377_disjoint__insert_I1_J,axiom,
! [B3: set_lambda_lambda,A: lambda_lambda,A3: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ B3 @ ( insert_lambda_lambda @ A @ A3 ) )
= bot_bo1271416176687173302lambda )
= ( ~ ( member_lambda_lambda @ A @ B3 )
& ( ( inf_in1766959559283430712lambda @ B3 @ A3 )
= bot_bo1271416176687173302lambda ) ) ) ).
% disjoint_insert(1)
thf(fact_378_disjoint__insert_I1_J,axiom,
! [B3: set_nat,A: nat,A3: set_nat] :
( ( ( inf_inf_set_nat @ B3 @ ( insert_nat @ A @ A3 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B3 )
& ( ( inf_inf_set_nat @ B3 @ A3 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_379_insert__disjoint_I2_J,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ A3 ) @ B3 ) )
= ( ~ ( member_lambda_lambda @ A @ B3 )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_380_insert__disjoint_I2_J,axiom,
! [A: nat,A3: set_nat,B3: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ B3 ) )
= ( ~ ( member_nat @ A @ B3 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_381_insert__disjoint_I1_J,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ A3 ) @ B3 )
= bot_bo1271416176687173302lambda )
= ( ~ ( member_lambda_lambda @ A @ B3 )
& ( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= bot_bo1271416176687173302lambda ) ) ) ).
% insert_disjoint(1)
thf(fact_382_insert__disjoint_I1_J,axiom,
! [A: nat,A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A3 ) @ B3 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B3 )
& ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_383_insertE,axiom,
! [A: lambda_lambda,B: lambda_lambda,A3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ ( insert_lambda_lambda @ B @ A3 ) )
=> ( ( A != B )
=> ( member_lambda_lambda @ A @ A3 ) ) ) ).
% insertE
thf(fact_384_insertE,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A3 ) ) ) ).
% insertE
thf(fact_385_insertI1,axiom,
! [A: lambda_lambda,B3: set_lambda_lambda] : ( member_lambda_lambda @ A @ ( insert_lambda_lambda @ A @ B3 ) ) ).
% insertI1
thf(fact_386_insertI1,axiom,
! [A: nat,B3: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B3 ) ) ).
% insertI1
thf(fact_387_insertI2,axiom,
! [A: lambda_lambda,B3: set_lambda_lambda,B: lambda_lambda] :
( ( member_lambda_lambda @ A @ B3 )
=> ( member_lambda_lambda @ A @ ( insert_lambda_lambda @ B @ B3 ) ) ) ).
% insertI2
thf(fact_388_insertI2,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( member_nat @ A @ B3 )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertI2
thf(fact_389_Set_Oset__insert,axiom,
! [X5: lambda_lambda,A3: set_lambda_lambda] :
( ( member_lambda_lambda @ X5 @ A3 )
=> ~ ! [B6: set_lambda_lambda] :
( ( A3
= ( insert_lambda_lambda @ X5 @ B6 ) )
=> ( member_lambda_lambda @ X5 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_390_Set_Oset__insert,axiom,
! [X5: nat,A3: set_nat] :
( ( member_nat @ X5 @ A3 )
=> ~ ! [B6: set_nat] :
( ( A3
= ( insert_nat @ X5 @ B6 ) )
=> ( member_nat @ X5 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_391_insert__ident,axiom,
! [X5: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X5 @ A3 )
=> ( ~ ( member_lambda_lambda @ X5 @ B3 )
=> ( ( ( insert_lambda_lambda @ X5 @ A3 )
= ( insert_lambda_lambda @ X5 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_392_insert__ident,axiom,
! [X5: nat,A3: set_nat,B3: set_nat] :
( ~ ( member_nat @ X5 @ A3 )
=> ( ~ ( member_nat @ X5 @ B3 )
=> ( ( ( insert_nat @ X5 @ A3 )
= ( insert_nat @ X5 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_393_insert__absorb,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ A3 )
=> ( ( insert_lambda_lambda @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_394_insert__absorb,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( insert_nat @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_395_insert__eq__iff,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda,B: lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A @ A3 )
=> ( ~ ( member_lambda_lambda @ B @ B3 )
=> ( ( ( insert_lambda_lambda @ A @ A3 )
= ( insert_lambda_lambda @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_lambda_lambda] :
( ( A3
= ( insert_lambda_lambda @ B @ C3 ) )
& ~ ( member_lambda_lambda @ B @ C3 )
& ( B3
= ( insert_lambda_lambda @ A @ C3 ) )
& ~ ( member_lambda_lambda @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_396_insert__eq__iff,axiom,
! [A: nat,A3: set_nat,B: nat,B3: set_nat] :
( ~ ( member_nat @ A @ A3 )
=> ( ~ ( member_nat @ B @ B3 )
=> ( ( ( insert_nat @ A @ A3 )
= ( insert_nat @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A3
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B3
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_397_insert__commute,axiom,
! [X5: lambda_lambda,Y5: lambda_lambda,A3: set_lambda_lambda] :
( ( insert_lambda_lambda @ X5 @ ( insert_lambda_lambda @ Y5 @ A3 ) )
= ( insert_lambda_lambda @ Y5 @ ( insert_lambda_lambda @ X5 @ A3 ) ) ) ).
% insert_commute
thf(fact_398_insert__commute,axiom,
! [X5: nat,Y5: nat,A3: set_nat] :
( ( insert_nat @ X5 @ ( insert_nat @ Y5 @ A3 ) )
= ( insert_nat @ Y5 @ ( insert_nat @ X5 @ A3 ) ) ) ).
% insert_commute
thf(fact_399_mk__disjoint__insert,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ A3 )
=> ? [B6: set_lambda_lambda] :
( ( A3
= ( insert_lambda_lambda @ A @ B6 ) )
& ~ ( member_lambda_lambda @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_400_mk__disjoint__insert,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ? [B6: set_nat] :
( ( A3
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_401_singletonD,axiom,
! [B: lambda_lambda,A: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_402_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_403_singleton__iff,axiom,
! [B: lambda_lambda,A: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_404_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_405_doubleton__eq__iff,axiom,
! [A: lambda_lambda,B: lambda_lambda,C: lambda_lambda,D: lambda_lambda] :
( ( ( insert_lambda_lambda @ A @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
= ( insert_lambda_lambda @ C @ ( insert_lambda_lambda @ D @ bot_bo1271416176687173302lambda ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_406_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_407_insert__not__empty,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda] :
( ( insert_lambda_lambda @ A @ A3 )
!= bot_bo1271416176687173302lambda ) ).
% insert_not_empty
thf(fact_408_insert__not__empty,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat @ A @ A3 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_409_singleton__inject,axiom,
! [A: lambda_lambda,B: lambda_lambda] :
( ( ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_410_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_411_Int__insert__right,axiom,
! [A: nat,A3: set_nat,B3: set_nat] :
( ( ( member_nat @ A @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A @ B3 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) )
& ( ~ ( member_nat @ A @ A3 )
=> ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A @ B3 ) )
= ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_412_Int__insert__right,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( member_lambda_lambda @ A @ A3 )
=> ( ( inf_in1766959559283430712lambda @ A3 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) )
& ( ~ ( member_lambda_lambda @ A @ A3 )
=> ( ( inf_in1766959559283430712lambda @ A3 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_413_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B3: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B3 ) @ C2 )
= ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_414_Int__insert__left,axiom,
! [A: lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( member_lambda_lambda @ A @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ B3 ) @ C2 )
= ( insert_lambda_lambda @ A @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) )
& ( ~ ( member_lambda_lambda @ A @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A @ B3 ) @ C2 )
= ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_415_transformation_Opreserves__trg,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,F2: lambda_lambda] :
( ( transf2236888680007955973lambda @ A3 @ B3 @ F @ G @ Tau )
=> ( ( ide_lambda_lambda @ A3 @ F2 )
=> ( ( trg_lambda_lambda @ B3 @ ( Tau @ F2 ) )
= ( G @ ( trg_lambda_lambda @ A3 @ F2 ) ) ) ) ) ).
% transformation.preserves_trg
thf(fact_416_transformation_Opreserves__src,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,F2: lambda_lambda] :
( ( transf2236888680007955973lambda @ A3 @ B3 @ F @ G @ Tau )
=> ( ( ide_lambda_lambda @ A3 @ F2 )
=> ( ( weakly4554751870655413501lambda @ B3 @ ( Tau @ F2 ) )
= ( F @ ( weakly4554751870655413501lambda @ A3 @ F2 ) ) ) ) ) ).
% transformation.preserves_src
thf(fact_417_simulation__to__weakly__extensional__rts_Opreserves__src,axiom,
! [A3: nat > nat > nat,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: nat > lambda_lambda,A: nat,T: nat] :
( ( simula2844598643962246286lambda @ A3 @ B3 @ F )
=> ( ( member_nat @ A @ ( sources_nat @ A3 @ T ) )
=> ( ( weakly4554751870655413501lambda @ B3 @ ( F @ T ) )
= ( F @ A ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_src
thf(fact_418_simulation__to__weakly__extensional__rts_Opreserves__src,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( simula5637266663281507978lambda @ A3 @ B3 @ F )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ A3 @ T ) )
=> ( ( weakly4554751870655413501lambda @ B3 @ ( F @ T ) )
= ( F @ A ) ) ) ) ).
% simulation_to_weakly_extensional_rts.preserves_src
thf(fact_419_the__elem__eq,axiom,
! [X5: lambda_lambda] :
( ( the_el5431214018478934239lambda @ ( insert_lambda_lambda @ X5 @ bot_bo1271416176687173302lambda ) )
= X5 ) ).
% the_elem_eq
thf(fact_420_the__elem__eq,axiom,
! [X5: nat] :
( ( the_elem_nat @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
= X5 ) ).
% the_elem_eq
thf(fact_421_is__singletonI,axiom,
! [X5: lambda_lambda] : ( is_sin2795064709772026974lambda @ ( insert_lambda_lambda @ X5 @ bot_bo1271416176687173302lambda ) ) ).
% is_singletonI
thf(fact_422_is__singletonI,axiom,
! [X5: nat] : ( is_singleton_nat @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_423_Set_Ois__empty__def,axiom,
( is_emp4698403878761571132lambda
= ( ^ [A7: set_lambda_lambda] : ( A7 = bot_bo1271416176687173302lambda ) ) ) ).
% Set.is_empty_def
thf(fact_424_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A7: set_nat] : ( A7 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_425_residuation__axioms,axiom,
residu6197316955523523210lambda @ lambda_resid ).
% residuation_axioms
thf(fact_426_is__singleton__def,axiom,
( is_sin2795064709772026974lambda
= ( ^ [A7: set_lambda_lambda] :
? [X3: lambda_lambda] :
( A7
= ( insert_lambda_lambda @ X3 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% is_singleton_def
thf(fact_427_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A7: set_nat] :
? [X3: nat] :
( A7
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_428_is__singletonE,axiom,
! [A3: set_lambda_lambda] :
( ( is_sin2795064709772026974lambda @ A3 )
=> ~ ! [X: lambda_lambda] :
( A3
!= ( insert_lambda_lambda @ X @ bot_bo1271416176687173302lambda ) ) ) ).
% is_singletonE
thf(fact_429_is__singletonE,axiom,
! [A3: set_nat] :
( ( is_singleton_nat @ A3 )
=> ~ ! [X: nat] :
( A3
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_430_rts__with__joins_Ohas__joins,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_wi2791606683991334555lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( joinab2519788767067900224lambda @ Resid @ T @ U ) ) ) ).
% rts_with_joins.has_joins
thf(fact_431_residuation_Ocube,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_432_residuation_Ocon__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_433_residuation_Otrg__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( trg_lambda_lambda @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_434_residuation_Oresid__arr__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_435_residuation_Oaxioms_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( partia3924075026459072775lambda @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_436_is__singleton__the__elem,axiom,
( is_sin2795064709772026974lambda
= ( ^ [A7: set_lambda_lambda] :
( A7
= ( insert_lambda_lambda @ ( the_el5431214018478934239lambda @ A7 ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% is_singleton_the_elem
thf(fact_437_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A7: set_nat] :
( A7
= ( insert_nat @ ( the_elem_nat @ A7 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_438_residuation_OideE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ~ ( ( con_lambda_lambda @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_439_residuation_OideI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_lambda_lambda @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_440_residuation_Oide__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
= ( ( con_lambda_lambda @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_441_residuation_Oide__implies__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( arr_lambda_lambda @ Resid @ A ) ) ) ).
% residuation.ide_implies_arr
thf(fact_442_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_443_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_444_residuation_OarrE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( con_lambda_lambda @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_445_residuation_OarrI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ T )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_446_residuation_Oarr__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
= ( con_lambda_lambda @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_447_residuation_Oarr__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_448_residuation_Oarr__resid__iff__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
= ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_449_is__singletonI_H,axiom,
! [A3: set_lambda_lambda] :
( ( A3 != bot_bo1271416176687173302lambda )
=> ( ! [X: lambda_lambda,Y3: lambda_lambda] :
( ( member_lambda_lambda @ X @ A3 )
=> ( ( member_lambda_lambda @ Y3 @ A3 )
=> ( X = Y3 ) ) )
=> ( is_sin2795064709772026974lambda @ A3 ) ) ) ).
% is_singletonI'
thf(fact_450_is__singletonI_H,axiom,
! [A3: set_nat] :
( ( A3 != bot_bot_set_nat )
=> ( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_nat @ Y3 @ A3 )
=> ( X = Y3 ) ) )
=> ( is_singleton_nat @ A3 ) ) ) ).
% is_singletonI'
thf(fact_451_residuation_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ Resid )
=> ( ( residu6479163625180095271lambda @ Resid )
=> ( residu6197316955523523210lambda @ Resid ) ) ) ).
% residuation.intro
thf(fact_452_residuation__def,axiom,
( residu6197316955523523210lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ Resid2 )
& ( residu6479163625180095271lambda @ Resid2 ) ) ) ) ).
% residuation_def
thf(fact_453_identities__form__normal__sub__rts,axiom,
normal1556980139574411093lambda @ lambda_resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ).
% identities_form_normal_sub_rts
thf(fact_454_not__arr__null,axiom,
~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% not_arr_null
thf(fact_455_con__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% con_def
thf(fact_456_conE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% conE
thf(fact_457_con__imp__arr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% con_imp_arr_resid
thf(fact_458_con__sym__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ U @ T )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% con_sym_ax
thf(fact_459_cube__ax,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ).
% cube_ax
thf(fact_460_null__eqI,axiom,
! [N: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( lambda_resid @ N @ T3 )
= N )
& ( ( lambda_resid @ T3 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% null_eqI
thf(fact_461_null__is__zero_I2_J,axiom,
! [T: lambda_lambda] :
( ( lambda_resid @ T @ ( partia1461432607296028304lambda @ lambda_resid ) )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% null_is_zero(2)
thf(fact_462_null__is__zero_I1_J,axiom,
! [T: lambda_lambda] :
( ( lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% null_is_zero(1)
thf(fact_463_conI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% conI
thf(fact_464_partial__magma_Onull_Ocong,axiom,
partia1461432607296028304lambda = partia1461432607296028304lambda ).
% partial_magma.null.cong
thf(fact_465_residuation__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( ( Resid @ T3 @ U4 )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ U4 @ T3 )
!= ( partia1461432607296028304lambda @ Resid ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( ( Resid @ T3 @ U4 )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ T3 @ U4 ) )
!= ( partia1461432607296028304lambda @ Resid ) ) )
=> ( ! [V3: lambda_lambda,T3: lambda_lambda,U4: lambda_lambda] :
( ( ( Resid @ ( Resid @ V3 @ T3 ) @ ( Resid @ U4 @ T3 ) )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ V3 @ T3 ) @ ( Resid @ U4 @ T3 ) )
= ( Resid @ ( Resid @ V3 @ U4 ) @ ( Resid @ T3 @ U4 ) ) ) )
=> ( residu6479163625180095271lambda @ Resid ) ) ) ) ).
% residuation_axioms.intro
thf(fact_466_normal__sub__rts_Obackward__stable,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( member_nat @ ( Resid @ T @ U ) @ NN )
=> ( member_nat @ T @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_467_normal__sub__rts_Obackward__stable,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T @ U ) @ NN )
=> ( member_lambda_lambda @ T @ NN ) ) ) ) ).
% normal_sub_rts.backward_stable
thf(fact_468_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( member_nat @ ( Resid @ T5 @ T ) @ NN )
& ( member_nat @ ( Resid @ T @ T5 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_469_normal__sub__rts_OCong_092_060_094sub_0620__symmetric,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_symmetric
thf(fact_470_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,T6: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( ( member_nat @ ( Resid @ T5 @ T6 ) @ NN )
& ( member_nat @ ( Resid @ T6 @ T5 ) @ NN ) )
=> ( ( member_nat @ ( Resid @ T @ T6 ) @ NN )
& ( member_nat @ ( Resid @ T6 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_471_normal__sub__rts_OCong_092_060_094sub_0620__transitive,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,T6: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( ( member_lambda_lambda @ ( Resid @ T5 @ T6 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T6 @ T5 ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ T @ T6 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T6 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_transitive
thf(fact_472_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ U ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( ( member_nat @ U @ NN )
=> ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_473_normal__sub__rts_OResid__along__normal__reflects__Cong_092_060_094sub_0620,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ U ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_reflects_Cong\<^sub>0
thf(fact_474_residuation__axioms__def,axiom,
( residu6479163625180095271lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T4: lambda_lambda,U2: lambda_lambda] :
( ( ( Resid2 @ T4 @ U2 )
!= ( partia1461432607296028304lambda @ Resid2 ) )
=> ( ( Resid2 @ U2 @ T4 )
!= ( partia1461432607296028304lambda @ Resid2 ) ) )
& ! [T4: lambda_lambda,U2: lambda_lambda] :
( ( ( Resid2 @ T4 @ U2 )
!= ( partia1461432607296028304lambda @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ T4 @ U2 ) @ ( Resid2 @ T4 @ U2 ) )
!= ( partia1461432607296028304lambda @ Resid2 ) ) )
& ! [V4: lambda_lambda,T4: lambda_lambda,U2: lambda_lambda] :
( ( ( Resid2 @ ( Resid2 @ V4 @ T4 ) @ ( Resid2 @ U2 @ T4 ) )
!= ( partia1461432607296028304lambda @ Resid2 ) )
=> ( ( Resid2 @ ( Resid2 @ V4 @ T4 ) @ ( Resid2 @ U2 @ T4 ) )
= ( Resid2 @ ( Resid2 @ V4 @ U2 ) @ ( Resid2 @ T4 @ U2 ) ) ) ) ) ) ) ).
% residuation_axioms_def
thf(fact_475_normal__sub__rts_Oide__closed,axiom,
! [Resid: nat > nat > nat,NN: set_nat,A: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ide_nat @ Resid @ A )
=> ( member_nat @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_476_normal__sub__rts_Oide__closed,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,A: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( member_lambda_lambda @ A @ NN ) ) ) ).
% normal_sub_rts.ide_closed
thf(fact_477_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( ide_nat @ Resid @ ( Resid @ T @ U ) )
=> ( member_nat @ T @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_478_normal__sub__rts_Oprfx__closed,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
=> ( member_lambda_lambda @ T @ NN ) ) ) ) ).
% normal_sub_rts.prfx_closed
thf(fact_479_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,U: nat,U3: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( ( member_nat @ ( Resid @ U @ U3 ) @ NN )
& ( member_nat @ ( Resid @ U3 @ U ) @ NN ) )
=> ( ( con_nat @ Resid @ T @ U )
= ( con_nat @ Resid @ T5 @ U3 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_480_normal__sub__rts_OCong_092_060_094sub_0620__subst__Con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( ( member_lambda_lambda @ ( Resid @ U @ U3 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ U3 @ U ) @ NN ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
= ( con_lambda_lambda @ Resid @ T5 @ U3 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_Con
thf(fact_481_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( con_nat @ Resid @ T @ T5 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_482_normal__sub__rts_OCong_092_060_094sub_0620__imp__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( con_lambda_lambda @ Resid @ T @ T5 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_con
thf(fact_483_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,U3: nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ U @ U3 ) @ NN )
& ( member_nat @ ( Resid @ U3 @ U ) @ NN ) )
=> ( ( con_nat @ Resid @ T @ U )
=> ( con_nat @ Resid @ T @ U3 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_484_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ U @ U3 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ U3 @ U ) @ NN ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ T @ U3 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(1)
thf(fact_485_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,U3: nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ U @ U3 ) @ NN )
& ( member_nat @ ( Resid @ U3 @ U ) @ NN ) )
=> ( ( con_nat @ Resid @ T @ U )
=> ( ( member_nat @ ( Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U3 @ U ) ) @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ U @ U3 ) ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ U @ U3 ) ) @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U3 @ U ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_486_normal__sub__rts_OCong_092_060_094sub_0620__subst__right_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ U @ U3 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ U3 @ U ) @ NN ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U3 @ U ) ) @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ U @ U3 ) ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ U @ U3 ) ) @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ U3 @ U ) ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_right(2)
thf(fact_487_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,U: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( con_nat @ Resid @ T @ U )
=> ( con_nat @ Resid @ T5 @ U ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_488_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ T5 @ U ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(1)
thf(fact_489_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,U: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( con_nat @ Resid @ T @ U )
=> ( ( member_nat @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_490_normal__sub__rts_OCong_092_060_094sub_0620__subst__left_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_subst_left(2)
thf(fact_491_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( arr_nat @ Resid @ T )
=> ( ( member_nat @ ( Resid @ T @ T ) @ NN )
& ( member_nat @ ( Resid @ T @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_492_normal__sub__rts_OCong_092_060_094sub_0620__reflexive,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ ( Resid @ T @ T ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_reflexive
thf(fact_493_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ T @ NN )
=> ( arr_nat @ Resid @ T ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_494_normal__sub__rts_Oelements__are__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ T @ NN )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% normal_sub_rts.elements_are_arr
thf(fact_495_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,U: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( member_nat @ U @ NN )
=> ( ( ( sources_nat @ Resid @ T )
= ( sources_nat @ Resid @ U ) )
=> ( ( member_nat @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_496_normal__sub__rts_OResid__along__normal__preserves__Cong_092_060_094sub_0620,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ U ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_Cong\<^sub>0
thf(fact_497_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( sources_nat @ Resid @ T )
= ( sources_nat @ Resid @ T5 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_498_normal__sub__rts_OCong_092_060_094sub_0620__imp__coinitial,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ T5 ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_imp_coinitial
thf(fact_499_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,V: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( composite_of_nat @ Resid @ T @ U @ V )
=> ( ( member_nat @ V @ NN )
=> ( member_nat @ U @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_500_normal__sub__rts_Ofactor__closed_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ V )
=> ( ( member_lambda_lambda @ V @ NN )
=> ( member_lambda_lambda @ U @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(2)
thf(fact_501_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,V: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( composite_of_nat @ Resid @ T @ U @ V )
=> ( ( member_nat @ V @ NN )
=> ( member_nat @ T @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_502_normal__sub__rts_Ofactor__closed_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ V )
=> ( ( member_lambda_lambda @ V @ NN )
=> ( member_lambda_lambda @ T @ NN ) ) ) ) ).
% normal_sub_rts.factor_closed(1)
thf(fact_503_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
= ( ? [U2: nat,U5: nat,V4: nat,V5: nat] :
( ( member_nat @ U2 @ NN )
& ( member_nat @ U5 @ NN )
& ( member_nat @ ( Resid @ V4 @ V5 ) @ NN )
& ( member_nat @ ( Resid @ V5 @ V4 ) @ NN )
& ( composite_of_nat @ Resid @ T @ U2 @ V4 )
& ( composite_of_nat @ Resid @ T5 @ U5 @ V5 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_504_normal__sub__rts_OCong_092_060_094sub_0620__iff,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
= ( ? [U2: lambda_lambda,U5: lambda_lambda,V4: lambda_lambda,V5: lambda_lambda] :
( ( member_lambda_lambda @ U2 @ NN )
& ( member_lambda_lambda @ U5 @ NN )
& ( member_lambda_lambda @ ( Resid @ V4 @ V5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ V5 @ V4 ) @ NN )
& ( compos9183582412961930448lambda @ Resid @ T @ U2 @ V4 )
& ( compos9183582412961930448lambda @ Resid @ T5 @ U5 @ V5 ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_iff
thf(fact_505_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,V: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ T @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( composite_of_nat @ Resid @ T @ U @ V )
=> ( member_nat @ V @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_506_normal__sub__rts_Ocomposite__closed,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ T @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ V )
=> ( member_lambda_lambda @ V @ NN ) ) ) ) ) ).
% normal_sub_rts.composite_closed
thf(fact_507_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: nat > nat > nat,NN: set_nat,Arr: nat > $o,T: nat,U: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( Arr @ T )
=> ( ( member_nat @ U @ NN )
=> ( ( composite_of_nat @ Resid @ T @ U @ T5 )
=> ( ( member_nat @ ( Resid @ T5 @ T ) @ NN )
& ( member_nat @ ( Resid @ T @ T5 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_508_normal__sub__rts_Ocomposite__of__arr__normal,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,Arr: lambda_lambda > $o,T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( Arr @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ T5 )
=> ( ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.composite_of_arr_normal
thf(fact_509_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,V: nat,U3: nat,V2: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( composite_of_nat @ Resid @ T @ U @ V )
=> ( ( composite_of_nat @ Resid @ T @ U3 @ V2 )
=> ( ( ( member_nat @ ( Resid @ V @ V2 ) @ NN )
& ( member_nat @ ( Resid @ V2 @ V ) @ NN ) )
=> ( ( member_nat @ ( Resid @ U @ U3 ) @ NN )
& ( member_nat @ ( Resid @ U3 @ U ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_510_normal__sub__rts_OCong_092_060_094sub_0620__cancel__left,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,U3: lambda_lambda,V2: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U3 @ V2 )
=> ( ( ( member_lambda_lambda @ ( Resid @ V @ V2 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ V2 @ V ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ U @ U3 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ U3 @ U ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_cancel_left
thf(fact_511_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ T5 ) @ NN )
& ( member_nat @ ( Resid @ T5 @ T ) @ NN ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_512_normal__sub__rts_OCong_092_060_094sub_0620__implies__Cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 ) ) ) ).
% normal_sub_rts.Cong\<^sub>0_implies_Cong
thf(fact_513_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ T @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 )
=> ( member_nat @ T5 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_514_normal__sub__rts_Onormal__is__Cong__closed,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ T @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( member_lambda_lambda @ T5 @ NN ) ) ) ) ).
% normal_sub_rts.normal_is_Cong_closed
thf(fact_515_normal__sub__rts_OCong__transitive,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T6: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T6 )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T6 @ T5 )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 ) ) ) ) ).
% normal_sub_rts.Cong_transitive
thf(fact_516_normal__sub__rts_OCong__symmetric,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T5 @ T ) ) ) ).
% normal_sub_rts.Cong_symmetric
thf(fact_517_normal__sub__rts_OCong__def,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 )
= ( ? [U2: nat,U5: nat] :
( ( member_nat @ U2 @ NN )
& ( member_nat @ U5 @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T5 @ U5 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ U5 ) @ ( Resid @ T @ U2 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_518_normal__sub__rts_OCong__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
= ( ? [U2: lambda_lambda,U5: lambda_lambda] :
( ( member_lambda_lambda @ U2 @ NN )
& ( member_lambda_lambda @ U5 @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T5 @ U5 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ U5 ) @ ( Resid @ T @ U2 ) ) @ NN ) ) ) ) ) ).
% normal_sub_rts.Cong_def
thf(fact_519_normal__sub__rts_OCongI,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,U3: nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( member_nat @ U3 @ NN )
=> ( ( ( member_nat @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U3 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_520_normal__sub__rts_OCongI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U3 @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 ) ) ) ) ) ).
% normal_sub_rts.CongI
thf(fact_521_normal__sub__rts_OCongE,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 )
=> ~ ! [U4: nat] :
( ( member_nat @ U4 @ NN )
=> ! [U6: nat] :
( ( member_nat @ U6 @ NN )
=> ~ ( ( member_nat @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ T5 @ U6 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ U6 ) @ ( Resid @ T @ U4 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_522_normal__sub__rts_OCongE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ~ ! [U4: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ! [U6: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN )
=> ~ ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ T5 @ U6 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ U6 ) @ ( Resid @ T @ U4 ) ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.CongE
thf(fact_523_normal__sub__rts_OCong__closure__props_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ U )
=> ( normal3156941126761417472lambda @ Resid @ NN @ U @ T ) ) ) ).
% normal_sub_rts.Cong_closure_props(1)
thf(fact_524_normal__sub__rts_OCong__closure__props_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ U )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ U @ V )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ V ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(2)
thf(fact_525_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ U ) @ NN )
& ( member_nat @ ( Resid @ U @ T ) @ NN ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_526_normal__sub__rts_OCong__closure__props_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ U ) @ NN )
& ( member_lambda_lambda @ ( Resid @ U @ T ) @ NN ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong_closure_props(3)
thf(fact_527_normal__sub__rts_Oforward__stable,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( coinitial_nat @ Resid @ T @ U )
=> ( member_nat @ ( Resid @ U @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_528_normal__sub__rts_Oforward__stable,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( coinit1392077014771647542lambda @ Resid @ T @ U )
=> ( member_lambda_lambda @ ( Resid @ U @ T ) @ NN ) ) ) ) ).
% normal_sub_rts.forward_stable
thf(fact_529_residuation_Ocube__ax,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_530_residuation_Ocon__sym__ax,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_531_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_532_coherent__normal__sub__rts_Oaxioms_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( normal1556980139574411093lambda @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(1)
thf(fact_533_partial__magma_Onull__eqI,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,N: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ! [T3: lambda_lambda] :
( ( ( OP @ N @ T3 )
= N )
& ( ( OP @ T3 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_534_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ( OP @ ( partia1461432607296028304lambda @ OP ) @ T )
= ( partia1461432607296028304lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_535_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ( OP @ T @ ( partia1461432607296028304lambda @ OP ) )
= ( partia1461432607296028304lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_536_normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( normal5162919049924913650lambda @ Resid @ NN ) ) ).
% normal_sub_rts.axioms(2)
thf(fact_537_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( ( sources_nat @ Resid @ T )
= ( sources_nat @ Resid @ U ) )
=> ( ( con_nat @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) )
= ( con_nat @ Resid @ T @ T5 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_538_normal__sub__rts_OResid__along__normal__preserves__reflects__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( con_lambda_lambda @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) )
= ( con_lambda_lambda @ Resid @ T @ T5 ) ) ) ) ) ).
% normal_sub_rts.Resid_along_normal_preserves_reflects_con
thf(fact_539_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,V: nat,V2: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( con_nat @ Resid @ T @ U )
=> ( ( composite_of_nat @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( composite_of_nat @ Resid @ U @ ( Resid @ T @ U ) @ V2 )
=> ( ( member_nat @ ( Resid @ V @ V2 ) @ NN )
& ( member_nat @ ( Resid @ V2 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_540_normal__sub__rts_Odiamond__commutes__upto__Cong_092_060_094sub_0620,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V2: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ ( Resid @ T @ U ) @ V2 )
=> ( ( member_lambda_lambda @ ( Resid @ V @ V2 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ V2 @ V ) @ NN ) ) ) ) ) ) ).
% normal_sub_rts.diamond_commutes_upto_Cong\<^sub>0
thf(fact_541_normal__sub__rts_OCong__imp__arr_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( arr_lambda_lambda @ Resid @ T5 ) ) ) ).
% normal_sub_rts.Cong_imp_arr(2)
thf(fact_542_normal__sub__rts_OCong__imp__arr_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% normal_sub_rts.Cong_imp_arr(1)
thf(fact_543_normal__sub__rts_OCong__reflexive,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ T ) ) ) ).
% normal_sub_rts.Cong_reflexive
thf(fact_544_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( ( sources_nat @ Resid @ T )
= ( sources_nat @ Resid @ U ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_545_normal__sub__rts_OCong__closure__props_I4_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ).
% normal_sub_rts.Cong_closure_props(4)
thf(fact_546_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: nat > nat > nat,NN: set_nat,A: nat,T: nat,A2: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ A @ ( sources_nat @ Resid @ T ) )
=> ( ( member_nat @ A2 @ ( sources_nat @ Resid @ T ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ A @ A2 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_547_normal__sub__rts_Osources__are__Cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,A: lambda_lambda,T: lambda_lambda,A2: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ A @ A2 ) ) ) ) ).
% normal_sub_rts.sources_are_Cong
thf(fact_548_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,A: nat,A2: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 )
=> ( ( member_nat @ A @ ( sources_nat @ Resid @ T ) )
=> ( ( member_nat @ A2 @ ( sources_nat @ Resid @ T5 ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ A @ A2 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_549_normal__sub__rts_Oin__sources__respects__Cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,A: lambda_lambda,A2: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ Resid @ T5 ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ A @ A2 ) ) ) ) ) ).
% normal_sub_rts.in_sources_respects_Cong
thf(fact_550_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,U: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( con_nat @ Resid @ T @ T5 )
=> ( ( coinitial_nat @ Resid @ T @ U )
=> ( ( member_nat @ U @ NN )
=> ( con_nat @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_551_normal__sub__rts_Oresid__along__elem__preserves__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( con_lambda_lambda @ Resid @ T @ T5 )
=> ( ( coinit1392077014771647542lambda @ Resid @ T @ U )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( con_lambda_lambda @ Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) ) ) ) ) ).
% normal_sub_rts.resid_along_elem_preserves_con
thf(fact_552_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: nat > nat > nat,NN: set_nat,B: nat,T: nat,B2: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ B @ ( targets_nat @ Resid @ T ) )
=> ( ( member_nat @ B2 @ ( targets_nat @ Resid @ T ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ B @ B2 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_553_normal__sub__rts_Otargets__are__Cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ B @ B2 ) ) ) ) ).
% normal_sub_rts.targets_are_Cong
thf(fact_554_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,T5: nat,B: nat,B2: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 )
=> ( ( member_nat @ B @ ( targets_nat @ Resid @ T ) )
=> ( ( member_nat @ B2 @ ( targets_nat @ Resid @ T5 ) )
=> ( normal_sub_Cong_nat @ Resid @ NN @ B @ B2 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_555_normal__sub__rts_Oin__targets__respects__Cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda,B: lambda_lambda,B2: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ Resid @ T5 ) )
=> ( normal3156941126761417472lambda @ Resid @ NN @ B @ B2 ) ) ) ) ) ).
% normal_sub_rts.in_targets_respects_Cong
thf(fact_556_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( seq_nat @ Resid @ T @ U )
=> ? [X_12: nat] : ( composite_of_nat @ Resid @ T @ U @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_557_normal__sub__rts_Ocomposite__closed__right,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( seq_lambda_lambda @ Resid @ T @ U )
=> ? [X_12: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ T @ U @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_right
thf(fact_558_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( seq_nat @ Resid @ U @ T )
=> ? [X_12: nat] : ( composite_of_nat @ Resid @ U @ T @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_559_normal__sub__rts_Ocomposite__closed__left,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( seq_lambda_lambda @ Resid @ U @ T )
=> ? [X_12: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ U @ T @ X_12 ) ) ) ) ).
% normal_sub_rts.composite_closed_left
thf(fact_560_residuation_Ocon__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_561_residuation_OconI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_562_residuation_OconE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_563_residuation_Onot__arr__null,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ~ ( arr_lambda_lambda @ Resid @ ( partia1461432607296028304lambda @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_564_transformation_Oextensional,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,F2: lambda_lambda] :
( ( transf2236888680007955973lambda @ A3 @ B3 @ F @ G @ Tau )
=> ( ~ ( arr_lambda_lambda @ A3 @ F2 )
=> ( ( Tau @ F2 )
= ( partia1461432607296028304lambda @ B3 ) ) ) ) ).
% transformation.extensional
thf(fact_565_coherent__normal__sub__rts__def,axiom,
( cohere1690309856004283859lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda,NN2: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid2 @ NN2 )
& ( cohere8844529169994681456lambda @ Resid2 @ NN2 ) ) ) ) ).
% coherent_normal_sub_rts_def
thf(fact_566_coherent__normal__sub__rts_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( cohere8844529169994681456lambda @ Resid @ NN )
=> ( cohere1690309856004283859lambda @ Resid @ NN ) ) ) ).
% coherent_normal_sub_rts.intro
thf(fact_567_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( arr_nat @ Resid @ T )
=> ( ( member_nat @ U @ NN )
=> ( ( composite_of_nat @ Resid @ U @ T @ T5 )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T5 @ T ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_568_normal__sub__rts_Ocomposite__of__normal__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ T5 )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T5 @ T ) ) ) ) ) ).
% normal_sub_rts.composite_of_normal_arr
thf(fact_569_residuation_Oaxioms_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( residu6479163625180095271lambda @ Resid ) ) ).
% residuation.axioms(2)
thf(fact_570_transformation__axioms_Ointro,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [F3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A3 @ F3 )
=> ( ( Tau @ F3 )
= ( partia1461432607296028304lambda @ B3 ) ) )
=> ( ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A3 @ F3 )
=> ( ( weakly4554751870655413501lambda @ B3 @ ( Tau @ F3 ) )
= ( F @ ( weakly4554751870655413501lambda @ A3 @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A3 @ F3 )
=> ( ( trg_lambda_lambda @ B3 @ ( Tau @ F3 ) )
= ( G @ ( trg_lambda_lambda @ A3 @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A3 @ F3 )
=> ( ( B3 @ ( Tau @ ( weakly4554751870655413501lambda @ A3 @ F3 ) ) @ ( F @ F3 ) )
= ( Tau @ ( trg_lambda_lambda @ A3 @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A3 @ F3 )
=> ( ( B3 @ ( F @ F3 ) @ ( Tau @ ( weakly4554751870655413501lambda @ A3 @ F3 ) ) )
= ( G @ F3 ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A3 @ F3 )
=> ( join_o1386658102433174083lambda @ B3 @ ( Tau @ ( weakly4554751870655413501lambda @ A3 @ F3 ) ) @ ( F @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf356316610583137064lambda @ A3 @ B3 @ F @ G @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_571_transformation__axioms__def,axiom,
( transf356316610583137064lambda
= ( ^ [A7: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,G2: lambda_lambda > lambda_lambda,Tau2: lambda_lambda > lambda_lambda] :
( ! [F5: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A7 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia1461432607296028304lambda @ B4 ) ) )
& ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A7 @ F5 )
=> ( ( weakly4554751870655413501lambda @ B4 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly4554751870655413501lambda @ A7 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A7 @ F5 )
=> ( ( trg_lambda_lambda @ B4 @ ( Tau2 @ F5 ) )
= ( G2 @ ( trg_lambda_lambda @ A7 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F5 )
=> ( ( B4 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A7 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_lambda_lambda @ A7 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F5 )
=> ( ( B4 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly4554751870655413501lambda @ A7 @ F5 ) ) )
= ( G2 @ F5 ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F5 )
=> ( join_o1386658102433174083lambda @ B4 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A7 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_572_simulation__axioms__def,axiom,
( simula826895837143331878lambda
= ( ^ [A7: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda] :
( ! [T4: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A7 @ T4 )
=> ( ( F4 @ T4 )
= ( partia1461432607296028304lambda @ B4 ) ) )
& ! [T4: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ A7 @ T4 @ U2 )
=> ( con_lambda_lambda @ B4 @ ( F4 @ T4 ) @ ( F4 @ U2 ) ) )
& ! [T4: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ A7 @ T4 @ U2 )
=> ( ( F4 @ ( A7 @ T4 @ U2 ) )
= ( B4 @ ( F4 @ T4 ) @ ( F4 @ U2 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_573_simulation__axioms_Ointro,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A3 @ T3 )
=> ( ( F @ T3 )
= ( partia1461432607296028304lambda @ B3 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ A3 @ T3 @ U4 )
=> ( con_lambda_lambda @ B3 @ ( F @ T3 ) @ ( F @ U4 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ A3 @ T3 @ U4 )
=> ( ( F @ ( A3 @ T3 @ U4 ) )
= ( B3 @ ( F @ T3 ) @ ( F @ U4 ) ) ) )
=> ( simula826895837143331878lambda @ A3 @ B3 @ F ) ) ) ) ).
% simulation_axioms.intro
thf(fact_574_lambda__calculus_Otargets__simps_I1_J,axiom,
( ( target5490931411331770124lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.targets_simps(1)
thf(fact_575_lambda__calculus_Osources__simps_I1_J,axiom,
( ( source5627170679743844886lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.sources_simps(1)
thf(fact_576_Cube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ).
% Cube
thf(fact_577_Con__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
= ( ( lambda_resid @ U @ T )
!= lambda_Nil ) ) ).
% Con_sym
thf(fact_578_con__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ).
% con_char
thf(fact_579_null__char,axiom,
( ( partia1461432607296028304lambda @ lambda_resid )
= lambda_Nil ) ).
% null_char
thf(fact_580_lambda__calculus_Oresid_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(7)
thf(fact_581_lambda__calculus_Oresid_Osimps_I26_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_resid @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(26)
thf(fact_582_lambda__calculus_Olambda_Odistinct_I1_J,axiom,
! [X22: nat] :
( lambda_Nil
!= ( lambda_Var @ X22 ) ) ).
% lambda_calculus.lambda.distinct(1)
thf(fact_583_lambda__calculus_OJoin_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_Join @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(7)
thf(fact_584_lambda__calculus_OJoin_Osimps_I38_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_Join @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Join.simps(38)
thf(fact_585_lambda__calculus_OIde_Osimps_I1_J,axiom,
~ ( lambda_Ide @ lambda_Nil ) ).
% lambda_calculus.Ide.simps(1)
thf(fact_586_lambda__calculus_Oresid_Osimps_I1_J,axiom,
! [I: nat,I2: nat] :
( ( ( I = I2 )
=> ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I2 ) )
= ( lambda_Var @ I ) ) )
& ( ( I != I2 )
=> ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I2 ) )
= lambda_Nil ) ) ) ).
% lambda_calculus.resid.simps(1)
thf(fact_587_lambda__calculus_OJoin_Osimps_I1_J,axiom,
! [X5: nat,X6: nat] :
( ( ( X5 = X6 )
=> ( ( lambda_Join @ ( lambda_Var @ X5 ) @ ( lambda_Var @ X6 ) )
= ( lambda_Var @ X5 ) ) )
& ( ( X5 != X6 )
=> ( ( lambda_Join @ ( lambda_Var @ X5 ) @ ( lambda_Var @ X6 ) )
= lambda_Nil ) ) ) ).
% lambda_calculus.Join.simps(1)
thf(fact_588_Con__implies__is__Lam__iff__is__Lam,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_is_Lam @ T )
= ( lambda_is_Lam @ U ) ) ) ).
% Con_implies_is_Lam_iff_is_Lam
thf(fact_589_resid__Subst,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda,N: nat] :
( ( ( lambda_resid @ T @ T5 )
!= lambda_Nil )
=> ( ( ( lambda_resid @ U @ U3 )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Subst @ N @ T @ U ) @ ( lambda_Subst @ N @ T5 @ U3 ) )
= ( lambda_Subst @ N @ ( lambda_resid @ T @ T5 ) @ ( lambda_resid @ U @ U3 ) ) ) ) ) ).
% resid_Subst
thf(fact_590_Con__Raise,axiom,
! [T: lambda_lambda,U: lambda_lambda,D: nat,N: nat] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Raise @ D @ N @ T ) @ ( lambda_Raise @ D @ N @ U ) )
!= lambda_Nil ) ) ).
% Con_Raise
thf(fact_591_Raise__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda,K: nat,N: nat] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_Raise @ K @ N @ ( lambda_resid @ T @ U ) )
= ( lambda_resid @ ( lambda_Raise @ K @ N @ T ) @ ( lambda_Raise @ K @ N @ U ) ) ) ) ).
% Raise_resid
thf(fact_592_Arr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ T @ U ) ) ) ).
% Arr_resid
thf(fact_593_Raise__inj,axiom,
! [D: nat,N: nat,T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Raise @ D @ N @ T )
= ( lambda_Raise @ D @ N @ U ) )
=> ( T = U ) ) ).
% Raise_inj
thf(fact_594_Arr__not__Nil,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( T != lambda_Nil ) ) ).
% Arr_not_Nil
thf(fact_595_Raise__not__Nil,axiom,
! [T: lambda_lambda,D: nat,N: nat] :
( ( T != lambda_Nil )
=> ( ( lambda_Raise @ D @ N @ T )
!= lambda_Nil ) ) ).
% Raise_not_Nil
thf(fact_596_Subst__not__Nil,axiom,
! [V: lambda_lambda,T: lambda_lambda,N: nat] :
( ( V != lambda_Nil )
=> ( ( T != lambda_Nil )
=> ( ( T != lambda_Nil )
=> ( ( lambda_Subst @ N @ V @ T )
!= lambda_Nil ) ) ) ) ).
% Subst_not_Nil
thf(fact_597_Subst__Nil,axiom,
! [N: nat,V: lambda_lambda] :
( ( lambda_Subst @ N @ V @ lambda_Nil )
= lambda_Nil ) ).
% Subst_Nil
thf(fact_598_Arr__Raise,axiom,
! [D: nat,N: nat] :
( lambda_Arr
= ( ^ [T4: lambda_lambda] : ( lambda_Arr @ ( lambda_Raise @ D @ N @ T4 ) ) ) ) ).
% Arr_Raise
thf(fact_599_Ide__implies__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( lambda_Arr @ T ) ) ).
% Ide_implies_Arr
thf(fact_600_Arr__Subst,axiom,
! [V: lambda_lambda,T: lambda_lambda,N: nat] :
( ( lambda_Arr @ V )
=> ( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Subst @ N @ V @ T ) ) ) ) ).
% Arr_Subst
thf(fact_601_Ide__Raise,axiom,
! [D: nat,N: nat] :
( lambda_Ide
= ( ^ [T4: lambda_lambda] : ( lambda_Ide @ ( lambda_Raise @ D @ N @ T4 ) ) ) ) ).
% Ide_Raise
thf(fact_602_Ide__Subst,axiom,
! [T: lambda_lambda,V: lambda_lambda,N: nat] :
( ( lambda_Ide @ T )
=> ( ( lambda_Ide @ V )
=> ( lambda_Ide @ ( lambda_Subst @ N @ V @ T ) ) ) ) ).
% Ide_Subst
thf(fact_603_Con__implies__Arr2,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ U ) ) ).
% Con_implies_Arr2
thf(fact_604_Con__implies__Arr1,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ T ) ) ).
% Con_implies_Arr1
thf(fact_605_arr__char,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( lambda_Arr @ T ) ) ).
% arr_char
thf(fact_606_assms,axiom,
( ( lambda_Arr @ t )
& ( lambda_Arr @ u )
& ( ( lambda_Src @ t )
= ( lambda_Src @ u ) ) ) ).
% assms
thf(fact_607_lambda__calculus_OArr_Osimps_I1_J,axiom,
~ ( lambda_Arr @ lambda_Nil ) ).
% lambda_calculus.Arr.simps(1)
thf(fact_608_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_609_lambda__calculus_OArr_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Arr @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Arr.simps(2)
thf(fact_610_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_611_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_612_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_613_lambda__calculus_Olambda_Odisc_I6_J,axiom,
~ ( lambda_is_Lam @ lambda_Nil ) ).
% lambda_calculus.lambda.disc(6)
thf(fact_614_lambda__calculus_Olambda_Odisc_I7_J,axiom,
! [X22: nat] :
~ ( lambda_is_Lam @ ( lambda_Var @ X22 ) ) ).
% lambda_calculus.lambda.disc(7)
thf(fact_615_targets__Raise,axiom,
! [T: lambda_lambda,D: nat,N: nat] :
( ( lambda_Arr @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Raise @ D @ N @ T ) )
= ( insert_lambda_lambda @ ( lambda_Raise @ D @ N @ ( lambda_Trg @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% targets_Raise
thf(fact_616_targets__char_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( lambda_Arr @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= bot_bo1271416176687173302lambda ) ) ) ).
% targets_char\<^sub>\<Lambda>
thf(fact_617_sources__Raise,axiom,
! [T: lambda_lambda,D: nat,N: nat] :
( ( lambda_Arr @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Raise @ D @ N @ T ) )
= ( insert_lambda_lambda @ ( lambda_Raise @ D @ N @ ( lambda_Src @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% sources_Raise
thf(fact_618_sources__char_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( lambda_Arr @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= bot_bo1271416176687173302lambda ) ) ) ).
% sources_char\<^sub>\<Lambda>
thf(fact_619_Ide__Subst__iff,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda] :
( ( lambda_Ide @ ( lambda_Subst @ N @ V @ T ) )
= ( ( lambda_Ide @ T )
& ( ( member_nat @ N @ ( lambda_FV @ T ) )
=> ( lambda_Ide @ V ) ) ) ) ).
% Ide_Subst_iff
thf(fact_620_Arr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Src @ T ) ) ) ).
% Arr_Src
thf(fact_621_Src__Raise,axiom,
! [D: nat,N: nat,T: lambda_lambda] :
( ( lambda_Src @ ( lambda_Raise @ D @ N @ T ) )
= ( lambda_Raise @ D @ N @ ( lambda_Src @ T ) ) ) ).
% Src_Raise
thf(fact_622_Arr__Trg,axiom,
! [X7: lambda_lambda] :
( ( lambda_Arr @ X7 )
=> ( lambda_Arr @ ( lambda_Trg @ X7 ) ) ) ).
% Arr_Trg
thf(fact_623_resid__Arr__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ T )
= ( lambda_Trg @ T ) ) ) ).
% resid_Arr_self
thf(fact_624_Src__Join,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
=> ( ( lambda_Src @ ( lambda_Join @ T @ U ) )
= ( lambda_Src @ T ) ) ) ).
% Src_Join
thf(fact_625_Coinitial__Raise__Raise,axiom,
! [T: lambda_lambda,U: lambda_lambda,D: nat,N: nat] :
( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
=> ( ( lambda_Arr @ ( lambda_Raise @ D @ N @ T ) )
& ( lambda_Arr @ ( lambda_Raise @ D @ N @ U ) )
& ( ( lambda_Src @ ( lambda_Raise @ D @ N @ T ) )
= ( lambda_Src @ ( lambda_Raise @ D @ N @ U ) ) ) ) ) ).
% Coinitial_Raise_Raise
thf(fact_626_Ide__iff__Src__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( lambda_Src @ T )
= T ) ) ) ).
% Ide_iff_Src_self
thf(fact_627_Ide__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Src @ T ) ) ) ).
% Ide_Src
thf(fact_628_Ide__iff__Trg__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( lambda_Trg @ T )
= T ) ) ) ).
% Ide_iff_Trg_self
thf(fact_629_Ide__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Trg @ T ) ) ) ).
% Ide_Trg
thf(fact_630_Coinitial__iff__Con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
= ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ).
% Coinitial_iff_Con
thf(fact_631_Coinitial__resid__resid,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ V )
!= lambda_Nil )
=> ( ( ( lambda_resid @ U @ V )
!= lambda_Nil )
=> ( ( lambda_Arr @ ( lambda_resid @ T @ V ) )
& ( lambda_Arr @ ( lambda_resid @ U @ V ) )
& ( ( lambda_Src @ ( lambda_resid @ T @ V ) )
= ( lambda_Src @ ( lambda_resid @ U @ V ) ) ) ) ) ) ).
% Coinitial_resid_resid
thf(fact_632_resid__Join,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
=> ( ( lambda_resid @ ( lambda_Join @ T @ U ) @ U )
= ( lambda_resid @ T @ U ) ) ) ).
% resid_Join
thf(fact_633_Src__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_Src @ ( lambda_resid @ T @ U ) )
= ( lambda_Trg @ U ) ) ) ).
% Src_resid
thf(fact_634_resid__Arr__Ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( ( lambda_Arr @ T )
& ( lambda_Arr @ A )
& ( ( lambda_Src @ T )
= ( lambda_Src @ A ) ) )
=> ( ( lambda_resid @ T @ A )
= T ) ) ) ).
% resid_Arr_Ide
thf(fact_635_resid__Ide__Arr,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( ( lambda_Arr @ A )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ A )
= ( lambda_Src @ T ) ) )
=> ( lambda_Ide @ ( lambda_resid @ A @ T ) ) ) ) ).
% resid_Ide_Arr
thf(fact_636_prfx__Join,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Join @ T @ U ) ) ) ) ).
% prfx_Join
thf(fact_637_Ide__resid__Join,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
=> ( lambda_Ide @ ( lambda_resid @ U @ ( lambda_Join @ T @ U ) ) ) ) ).
% Ide_resid_Join
thf(fact_638_seq__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Trg @ T )
= ( lambda_Src @ U ) ) ) ) ).
% seq_char
thf(fact_639_seqE_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ~ ( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( lambda_Trg @ T )
!= ( lambda_Src @ U ) ) ) ) ) ).
% seqE\<^sub>\<Lambda>
thf(fact_640_Src__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_Src @ T ) )
= ( lambda_Src @ T ) ) ) ).
% Src_Src
thf(fact_641_Trg__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Trg @ ( lambda_Trg @ T ) )
= ( lambda_Trg @ T ) ) ) ).
% Trg_Trg
thf(fact_642_resid__Arr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ ( lambda_Src @ T ) )
= T ) ) ).
% resid_Arr_Src
thf(fact_643_Src__Subst,axiom,
! [T: lambda_lambda,U: lambda_lambda,D: nat] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( lambda_Src @ ( lambda_Subst @ D @ T @ U ) )
= ( lambda_Subst @ D @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ) ) ).
% Src_Subst
thf(fact_644_Trg__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Trg @ ( lambda_Src @ T ) )
= ( lambda_Src @ T ) ) ) ).
% Trg_Src
thf(fact_645_Src__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_Trg @ T ) )
= ( lambda_Trg @ T ) ) ) ).
% Src_Trg
thf(fact_646_Trg__Subst,axiom,
! [T: lambda_lambda,U: lambda_lambda,D: nat] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( lambda_Trg @ ( lambda_Subst @ D @ T @ U ) )
= ( lambda_Subst @ D @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) ) ) ) ).
% Trg_Subst
thf(fact_647_ConI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Src @ T )
= ( lambda_Src @ U ) )
=> ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ) ) ).
% ConI
thf(fact_648_resid__Src__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ ( lambda_Src @ T ) @ T )
= ( lambda_Trg @ T ) ) ) ).
% resid_Src_Arr
thf(fact_649_coinitial__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) ) ) ).
% coinitial_char
thf(fact_650_src__char,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( lambda_Src @ T ) ) )
& ( ~ ( lambda_Arr @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= lambda_Nil ) ) ) ).
% src_char
thf(fact_651_trg__char,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_Trg @ T ) ) )
& ( ~ ( lambda_Arr @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= lambda_Nil ) ) ) ).
% trg_char
thf(fact_652_seqI_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Trg @ T )
= ( lambda_Src @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% seqI\<^sub>\<Lambda>
thf(fact_653_sources__simp,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) ) ).
% sources_simp
thf(fact_654_targets__simp,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ).
% targets_simp
thf(fact_655_lambda__calculus_OSrc__eq__iff_I1_J,axiom,
! [I: nat,I2: nat] :
( ( ( lambda_Src @ ( lambda_Var @ I ) )
= ( lambda_Src @ ( lambda_Var @ I2 ) ) )
= ( I = I2 ) ) ).
% lambda_calculus.Src_eq_iff(1)
thf(fact_656_lambda__calculus_OSrc_Osimps_I2_J,axiom,
! [I: nat] :
( ( lambda_Src @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Src.simps(2)
thf(fact_657_lambda__calculus_OSrc_Osimps_I1_J,axiom,
( ( lambda_Src @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Src.simps(1)
thf(fact_658_lambda__calculus_OTrg_Osimps_I5_J,axiom,
( ( lambda_Trg @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Trg.simps(5)
thf(fact_659_lambda__calculus_OTrg_Osimps_I1_J,axiom,
! [I: nat] :
( ( lambda_Trg @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Trg.simps(1)
thf(fact_660_lambda__calculus_OFV_Osimps_I1_J,axiom,
( ( lambda_FV @ lambda_Nil )
= bot_bot_set_nat ) ).
% lambda_calculus.FV.simps(1)
thf(fact_661_lambda__calculus_OFV_Osimps_I2_J,axiom,
! [I: nat] :
( ( lambda_FV @ ( lambda_Var @ I ) )
= ( insert_nat @ I @ bot_bot_set_nat ) ) ).
% lambda_calculus.FV.simps(2)
thf(fact_662_lambda__calculus_OCon__implies__Coinitial3_I1_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ 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_663_lambda__calculus_OCon__implies__Coinitial3_I2_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ V )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ V )
= ( lambda_Src @ T ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(2)
thf(fact_664_lambda__calculus_OCon__implies__Coinitial3_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ U )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ U )
= ( lambda_Src @ T ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(3)
thf(fact_665_lambda__calculus_OCon__Arr__Src_I1_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ ( lambda_Src @ T ) )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_Src(1)
thf(fact_666_lambda__calculus_OCon__Arr__Src_I2_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ ( lambda_Src @ T ) @ T )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_Src(2)
thf(fact_667_vacuous__Subst,axiom,
! [V: lambda_lambda,I: nat,T: lambda_lambda] :
( ( lambda_Arr @ V )
=> ( ~ ( member_nat @ I @ ( lambda_FV @ T ) )
=> ( ( lambda_Raise @ I @ one_one_nat @ ( lambda_Subst @ I @ V @ T ) )
= T ) ) ) ).
% vacuous_Subst
thf(fact_668_targets__subst,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Subst @ zero_zero_nat @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% targets_subst
thf(fact_669_sources__subst,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Subst @ zero_zero_nat @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% sources_subst
thf(fact_670_lambda__calculus_Otargets__simps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_App @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.targets_simps(4)
thf(fact_671_Raise__0,axiom,
! [D: nat,T: lambda_lambda] :
( ( lambda_Raise @ D @ zero_zero_nat @ T )
= T ) ).
% Raise_0
thf(fact_672_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_673_lambda__calculus_Osources__simps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_App @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_simps(4)
thf(fact_674_lambda__calculus_OSrc_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_App @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.Src.simps(4)
thf(fact_675_lambda__calculus_OTrg_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Trg @ ( lambda_App @ T @ U ) )
= ( lambda_App @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) ) ).
% lambda_calculus.Trg.simps(3)
thf(fact_676_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_Src @ ( lambda_App @ T5 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T5 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_677_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_678_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_679_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_680_lambda__calculus_OIde_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Ide @ ( lambda_App @ T @ U ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.Ide.simps(4)
thf(fact_681_lambda__calculus_OJoin_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ T @ U ) @ ( lambda_App @ T5 @ U3 ) )
= ( lambda_App @ ( lambda_Join @ T @ T5 ) @ ( lambda_Join @ U @ U3 ) ) ) ).
% lambda_calculus.Join.simps(5)
thf(fact_682_lambda__calculus_OArr_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_App @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(4)
thf(fact_683_lambda__calculus_ORaise_Osimps_I4_J,axiom,
! [D: nat,N: nat,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_App @ T @ U ) )
= ( lambda_App @ ( lambda_Raise @ D @ N @ T ) @ ( lambda_Raise @ D @ N @ U ) ) ) ).
% lambda_calculus.Raise.simps(4)
thf(fact_684_lambda__calculus_OSubst_Osimps_I4_J,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_App @ T @ U ) )
= ( lambda_App @ ( lambda_Subst @ N @ V @ T ) @ ( lambda_Subst @ N @ V @ U ) ) ) ).
% lambda_calculus.Subst.simps(4)
thf(fact_685_lambda__calculus_Oresid_Osimps_I3_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T5 @ U3 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T5 @ U3 ) )
= ( lambda_App @ ( lambda_resid @ T @ T5 ) @ ( lambda_resid @ U @ U3 ) ) ) ) ) ).
% lambda_calculus.resid.simps(3)
thf(fact_686_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_687_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_688_lambda__calculus_OConD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T5 @ U3 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ T5 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U3 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(1)
thf(fact_689_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_690_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_691_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_692_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_693_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_694_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_695_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_696_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_697_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_698_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_699_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_700_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_701_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_702_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_703_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_704_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_705_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_706_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_707_raise__Subst,axiom,
! [T: lambda_lambda,V: lambda_lambda,P2: nat,N: nat] :
( ( T != lambda_Nil )
=> ( ( V != lambda_Nil )
=> ( ( lambda_Raise @ zero_zero_nat @ P2 @ ( lambda_Subst @ N @ V @ T ) )
= ( lambda_Subst @ ( plus_plus_nat @ P2 @ N ) @ V @ ( lambda_Raise @ zero_zero_nat @ P2 @ T ) ) ) ) ) ).
% raise_Subst
thf(fact_708_lambda__calculus_Otargets__simps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U ) @ ( lambda_Trg @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.targets_simps(5)
thf(fact_709_Raise__subst,axiom,
! [N: nat,K: nat,V: lambda_lambda,T: lambda_lambda] :
( ( lambda_Raise @ N @ K @ ( lambda_Subst @ zero_zero_nat @ V @ T ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Raise @ N @ K @ V ) @ ( lambda_Raise @ ( suc @ N ) @ K @ T ) ) ) ).
% Raise_subst
thf(fact_710_substitution__lemma,axiom,
! [V: lambda_lambda,W: lambda_lambda,N: nat,T: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( W != lambda_Nil )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Subst @ zero_zero_nat @ W @ T ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Subst @ N @ V @ W ) @ ( lambda_Subst @ ( suc @ N ) @ V @ T ) ) ) ) ) ).
% substitution_lemma
thf(fact_711_Raise__plus,axiom,
! [D: nat,M: nat,N: nat,T: lambda_lambda] :
( ( lambda_Raise @ D @ ( plus_plus_nat @ M @ N ) @ T )
= ( lambda_Raise @ ( plus_plus_nat @ D @ M ) @ N @ ( lambda_Raise @ D @ M @ T ) ) ) ).
% Raise_plus
thf(fact_712_raise__Raise,axiom,
! [P2: nat,N: nat,K: nat,T: lambda_lambda] :
( ( lambda_Raise @ zero_zero_nat @ P2 @ ( lambda_Raise @ N @ K @ T ) )
= ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Raise @ zero_zero_nat @ P2 @ T ) ) ) ).
% raise_Raise
thf(fact_713_Raise__Suc,axiom,
! [D: nat,N: nat,T: lambda_lambda] :
( ( lambda_Raise @ D @ ( suc @ N ) @ T )
= ( lambda_Raise @ D @ one_one_nat @ ( lambda_Raise @ D @ N @ T ) ) ) ).
% Raise_Suc
thf(fact_714_Subst__Subst,axiom,
! [V: lambda_lambda,W: lambda_lambda,M: nat,N: nat,T: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( W != lambda_Nil )
=> ( ( lambda_Subst @ ( plus_plus_nat @ M @ N ) @ W @ ( lambda_Subst @ M @ V @ T ) )
= ( lambda_Subst @ M @ ( lambda_Subst @ N @ W @ V ) @ ( lambda_Subst @ ( suc @ ( plus_plus_nat @ M @ N ) ) @ W @ T ) ) ) ) ) ).
% Subst_Subst
thf(fact_715_Raise__Subst,axiom,
! [P2: nat,N: nat,K: nat,V: lambda_lambda,T: lambda_lambda] :
( ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Subst @ P2 @ V @ T ) )
= ( lambda_Subst @ P2 @ ( lambda_Raise @ N @ K @ V ) @ ( lambda_Raise @ ( suc @ ( plus_plus_nat @ P2 @ N ) ) @ K @ T ) ) ) ).
% Raise_Subst
thf(fact_716_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_717_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_718_lambda__calculus_OSubst_Osimps_I5_J,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_Beta @ T @ U ) )
= ( lambda_Beta @ ( lambda_Subst @ ( suc @ N ) @ V @ T ) @ ( lambda_Subst @ N @ V @ U ) ) ) ).
% lambda_calculus.Subst.simps(5)
thf(fact_719_lambda__calculus_ORaise_Osimps_I5_J,axiom,
! [D: nat,N: nat,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_Beta @ T @ U ) )
= ( lambda_Beta @ ( lambda_Raise @ ( suc @ D ) @ N @ T ) @ ( lambda_Raise @ D @ N @ U ) ) ) ).
% lambda_calculus.Raise.simps(5)
thf(fact_720_lambda__calculus_OArr_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Beta @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(5)
thf(fact_721_lambda__calculus_OIde_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
~ ( lambda_Ide @ ( lambda_Beta @ T @ U ) ) ).
% lambda_calculus.Ide.simps(5)
thf(fact_722_lambda__calculus_OJoin_Osimps_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ T5 @ U3 ) )
= ( lambda_Beta @ ( lambda_Join @ T @ T5 ) @ ( lambda_Join @ U @ U3 ) ) ) ).
% lambda_calculus.Join.simps(6)
thf(fact_723_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_724_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_725_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_726_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T5 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T5 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(3)
thf(fact_727_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_728_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_729_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_730_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_731_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_732_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_733_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_734_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_735_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_736_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_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_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_761_lambda__calculus_OTrg_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Trg @ ( lambda_Beta @ T @ U ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U ) @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Trg.simps(4)
thf(fact_762_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_763_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_764_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_765_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_766_lambda__calculus_Oresid_Osimps_I4_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ T5 @ U3 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ T5 @ U3 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U @ U3 ) @ ( lambda_resid @ T @ T5 ) ) ) ) ) ).
% lambda_calculus.resid.simps(4)
thf(fact_767_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_768_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_769_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_770_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_771_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_772_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_773_Subst__Raise,axiom,
! [V: lambda_lambda,D: nat,M: nat,N: nat,T: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ D @ M )
=> ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ D ) )
=> ( ( lambda_Subst @ M @ V @ ( lambda_Raise @ D @ ( suc @ N ) @ T ) )
= ( lambda_Raise @ D @ N @ T ) ) ) ) ) ).
% Subst_Raise
thf(fact_774_Subst__raise,axiom,
! [V: lambda_lambda,M: nat,N: nat,T: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( lambda_Subst @ M @ V @ ( lambda_Raise @ zero_zero_nat @ ( suc @ N ) @ T ) )
= ( lambda_Raise @ zero_zero_nat @ N @ T ) ) ) ) ).
% Subst_raise
thf(fact_775_Coinitial__cases,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ T5 )
=> ( ( ( lambda_Src @ T )
= ( lambda_Src @ T5 ) )
=> ( ( ( T = lambda_Nil )
& ( T5 = lambda_Nil ) )
| ? [X: nat] :
( ( T
= ( lambda_Var @ X ) )
& ( T5
= ( lambda_Var @ X ) ) )
| ? [U4: lambda_lambda,U6: lambda_lambda] :
( ( T
= ( lambda_Lam @ U4 ) )
& ( T5
= ( lambda_Lam @ U6 ) ) )
| ? [U4: lambda_lambda,V3: lambda_lambda,U6: lambda_lambda,V6: lambda_lambda] :
( ( T
= ( lambda_App @ U4 @ V3 ) )
& ( T5
= ( lambda_App @ U6 @ V6 ) ) )
| ? [U4: lambda_lambda,V3: lambda_lambda,U6: lambda_lambda,V6: lambda_lambda] :
( ( T
= ( lambda_Beta @ U4 @ V3 ) )
& ( T5
= ( lambda_Beta @ U6 @ V6 ) ) )
| ? [U4: lambda_lambda,V3: lambda_lambda,U6: lambda_lambda,V6: lambda_lambda] :
( ( T
= ( lambda_App @ ( lambda_Lam @ U4 ) @ V3 ) )
& ( T5
= ( lambda_Beta @ U6 @ V6 ) ) )
| ? [U4: lambda_lambda,V3: lambda_lambda,U6: lambda_lambda,V6: lambda_lambda] :
( ( T
= ( lambda_Beta @ U4 @ V3 ) )
& ( T5
= ( lambda_App @ ( lambda_Lam @ U6 ) @ V6 ) ) ) ) ) ) ) ).
% Coinitial_cases
thf(fact_776_Raise__Subst_H,axiom,
! [T: lambda_lambda,V: lambda_lambda,K: nat,N: nat,P2: nat] :
( ( T != lambda_Nil )
=> ( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( lambda_Raise @ K @ P2 @ ( lambda_Subst @ N @ V @ T ) )
= ( lambda_Subst @ ( plus_plus_nat @ P2 @ N ) @ V @ ( lambda_Raise @ K @ P2 @ T ) ) ) ) ) ) ).
% Raise_Subst'
thf(fact_777_Raise__Raise,axiom,
! [I: nat,N: nat,P2: nat,K: nat,T: lambda_lambda] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( lambda_Raise @ I @ P2 @ ( lambda_Raise @ N @ K @ T ) )
= ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Raise @ I @ P2 @ T ) ) ) ) ).
% Raise_Raise
thf(fact_778_Raise__plus_H,axiom,
! [D2: nat,D: nat,N: nat,M: nat,T: lambda_lambda] :
( ( ord_less_eq_nat @ D2 @ ( plus_plus_nat @ D @ N ) )
=> ( ( ord_less_eq_nat @ D @ D2 )
=> ( ( lambda_Raise @ D @ ( plus_plus_nat @ M @ N ) @ T )
= ( lambda_Raise @ D2 @ M @ ( lambda_Raise @ D @ N @ T ) ) ) ) ) ).
% Raise_plus'
thf(fact_779_raise__plus,axiom,
! [D: nat,N: nat,M: nat,T: lambda_lambda] :
( ( ord_less_eq_nat @ D @ N )
=> ( ( lambda_Raise @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) @ T )
= ( lambda_Raise @ D @ M @ ( lambda_Raise @ zero_zero_nat @ N @ T ) ) ) ) ).
% raise_plus
thf(fact_780_lambda__calculus_Olambda_Oinject_I2_J,axiom,
! [X32: lambda_lambda,Y32: lambda_lambda] :
( ( ( lambda_Lam @ X32 )
= ( lambda_Lam @ Y32 ) )
= ( X32 = Y32 ) ) ).
% lambda_calculus.lambda.inject(2)
thf(fact_781_ArrE,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ! [I3: nat] :
( T
!= ( lambda_Var @ I3 ) )
=> ( ! [U4: lambda_lambda] :
( T
!= ( lambda_Lam @ U4 ) )
=> ( ! [U4: lambda_lambda,V3: lambda_lambda] :
( T
!= ( lambda_App @ U4 @ V3 ) )
=> ~ ! [U4: lambda_lambda,V3: lambda_lambda] :
( T
!= ( lambda_Beta @ U4 @ V3 ) ) ) ) ) ) ).
% ArrE
thf(fact_782_inf_Obounded__iff,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) )
= ( ( ord_le1926335974945931754lambda @ A @ B )
& ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_783_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_784_le__inf__iff,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda,Z: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X5 @ ( inf_in1766959559283430712lambda @ Y5 @ Z ) )
= ( ( ord_le1926335974945931754lambda @ X5 @ Y5 )
& ( ord_le1926335974945931754lambda @ X5 @ Z ) ) ) ).
% le_inf_iff
thf(fact_785_le__inf__iff,axiom,
! [X5: nat,Y5: nat,Z: nat] :
( ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ Y5 @ Z ) )
= ( ( ord_less_eq_nat @ X5 @ Y5 )
& ( ord_less_eq_nat @ X5 @ Z ) ) ) ).
% le_inf_iff
thf(fact_786_lambda__calculus_Osources__simps_I3_J,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Lam @ T ) )
= ( insert_lambda_lambda @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.sources_simps(3)
thf(fact_787_lambda__calculus_Otargets__simps_I3_J,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Lam @ T ) )
= ( insert_lambda_lambda @ ( lambda_Lam @ ( lambda_Trg @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.targets_simps(3)
thf(fact_788_lambda__calculus_Osources__simps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_simps(5)
thf(fact_789_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_790_lambda__calculus_OSrc_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Src @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Src.simps(3)
thf(fact_791_lambda__calculus_OTrg_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Trg @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Trg.simps(2)
thf(fact_792_lambda__calculus_Olambda_Odisc_I8_J,axiom,
! [X32: lambda_lambda] : ( lambda_is_Lam @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(8)
thf(fact_793_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_794_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_795_lambda__calculus_Olambda_Odistinct_I3_J,axiom,
! [X32: lambda_lambda] :
( lambda_Nil
!= ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.distinct(3)
thf(fact_796_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_797_lambda__calculus_OJoin_Osimps_I2_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( lambda_Join @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T5 ) )
= ( lambda_Lam @ ( lambda_Join @ T @ T5 ) ) ) ).
% lambda_calculus.Join.simps(2)
thf(fact_798_inf_OcoboundedI2,axiom,
! [B: set_lambda_lambda,C: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_799_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_800_inf_OcoboundedI1,axiom,
! [A: set_lambda_lambda,C: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_801_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_802_inf_Oabsorb__iff2,axiom,
( ord_le1926335974945931754lambda
= ( ^ [B5: set_lambda_lambda,A6: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_803_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A6: nat] :
( ( inf_inf_nat @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_804_inf_Oabsorb__iff1,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_805_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
( ( inf_inf_nat @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_806_inf_Ocobounded2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_807_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_808_inf_Ocobounded1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_809_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_810_inf_Oorder__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( A6
= ( inf_in1766959559283430712lambda @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_811_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
( A6
= ( inf_inf_nat @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_812_inf__greatest,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda,Z: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X5 @ Y5 )
=> ( ( ord_le1926335974945931754lambda @ X5 @ Z )
=> ( ord_le1926335974945931754lambda @ X5 @ ( inf_in1766959559283430712lambda @ Y5 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_813_inf__greatest,axiom,
! [X5: nat,Y5: nat,Z: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( ord_less_eq_nat @ X5 @ Z )
=> ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ Y5 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_814_inf_OboundedI,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ A @ C )
=> ( ord_le1926335974945931754lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_815_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_816_inf_OboundedE,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) )
=> ~ ( ( ord_le1926335974945931754lambda @ A @ B )
=> ~ ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_817_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_818_inf__absorb2,axiom,
! [Y5: set_lambda_lambda,X5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y5 @ X5 )
=> ( ( inf_in1766959559283430712lambda @ X5 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_819_inf__absorb2,axiom,
! [Y5: nat,X5: nat] :
( ( ord_less_eq_nat @ Y5 @ X5 )
=> ( ( inf_inf_nat @ X5 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_820_inf__absorb1,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X5 @ Y5 )
=> ( ( inf_in1766959559283430712lambda @ X5 @ Y5 )
= X5 ) ) ).
% inf_absorb1
thf(fact_821_inf__absorb1,axiom,
! [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
=> ( ( inf_inf_nat @ X5 @ Y5 )
= X5 ) ) ).
% inf_absorb1
thf(fact_822_inf_Oabsorb2,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_823_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_824_inf_Oabsorb1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_825_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_826_le__iff__inf,axiom,
( ord_le1926335974945931754lambda
= ( ^ [X3: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_827_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_828_inf__unique,axiom,
! [F2: set_lambda_lambda > set_lambda_lambda > set_lambda_lambda,X5: set_lambda_lambda,Y5: set_lambda_lambda] :
( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ( ord_le1926335974945931754lambda @ X @ Z2 )
=> ( ord_le1926335974945931754lambda @ X @ ( F2 @ Y3 @ Z2 ) ) ) )
=> ( ( inf_in1766959559283430712lambda @ X5 @ Y5 )
= ( F2 @ X5 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_829_inf__unique,axiom,
! [F2: nat > nat > nat,X5: nat,Y5: nat] :
( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X @ Y3 ) @ X )
=> ( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X @ Y3 ) @ Y3 )
=> ( ! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( F2 @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X5 @ Y5 )
= ( F2 @ X5 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_830_inf_OorderI,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A
= ( inf_in1766959559283430712lambda @ A @ B ) )
=> ( ord_le1926335974945931754lambda @ A @ B ) ) ).
% inf.orderI
thf(fact_831_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_832_inf_OorderE,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( A
= ( inf_in1766959559283430712lambda @ A @ B ) ) ) ).
% inf.orderE
thf(fact_833_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_834_le__infI2,axiom,
! [B: set_lambda_lambda,X5: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ X5 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ X5 ) ) ).
% le_infI2
thf(fact_835_le__infI2,axiom,
! [B: nat,X5: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X5 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X5 ) ) ).
% le_infI2
thf(fact_836_le__infI1,axiom,
! [A: set_lambda_lambda,X5: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ X5 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ X5 ) ) ).
% le_infI1
thf(fact_837_le__infI1,axiom,
! [A: nat,X5: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X5 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X5 ) ) ).
% le_infI1
thf(fact_838_inf__mono,axiom,
! [A: set_lambda_lambda,C: set_lambda_lambda,B: set_lambda_lambda,D: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ C )
=> ( ( ord_le1926335974945931754lambda @ B @ D )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ ( inf_in1766959559283430712lambda @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_839_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_840_le__infI,axiom,
! [X5: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X5 @ A )
=> ( ( ord_le1926335974945931754lambda @ X5 @ B )
=> ( ord_le1926335974945931754lambda @ X5 @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% le_infI
thf(fact_841_le__infI,axiom,
! [X5: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X5 @ A )
=> ( ( ord_less_eq_nat @ X5 @ B )
=> ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_842_le__infE,axiom,
! [X5: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X5 @ ( inf_in1766959559283430712lambda @ A @ B ) )
=> ~ ( ( ord_le1926335974945931754lambda @ X5 @ A )
=> ~ ( ord_le1926335974945931754lambda @ X5 @ B ) ) ) ).
% le_infE
thf(fact_843_le__infE,axiom,
! [X5: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X5 @ A )
=> ~ ( ord_less_eq_nat @ X5 @ B ) ) ) ).
% le_infE
thf(fact_844_inf__le2,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_845_inf__le2,axiom,
! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_846_inf__le1,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) @ X5 ) ).
% inf_le1
thf(fact_847_inf__le1,axiom,
! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ X5 ) ).
% inf_le1
thf(fact_848_inf__sup__ord_I1_J,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_849_inf__sup__ord_I1_J,axiom,
! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_850_inf__sup__ord_I2_J,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X5 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_851_inf__sup__ord_I2_J,axiom,
! [X5: nat,Y5: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_852_lambda__calculus_OIde_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ ( lambda_Lam @ T ) )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.Ide.simps(3)
thf(fact_853_lambda__calculus_OArr_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Lam @ T ) )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.Arr.simps(3)
thf(fact_854_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_855_bot_Oextremum,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A ) ).
% bot.extremum
thf(fact_856_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_857_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_858_bot_Oextremum__unique,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_unique
thf(fact_859_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_860_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_861_bot_Oextremum__uniqueI,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
=> ( A = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_862_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_863_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_864_lambda__calculus_Oresid_Osimps_I2_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T5 ) )
= lambda_Nil ) )
& ( ( ( lambda_resid @ T @ T5 )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T5 ) )
= ( lambda_Lam @ ( lambda_resid @ T @ T5 ) ) ) ) ) ).
% lambda_calculus.resid.simps(2)
thf(fact_865_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_866_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_867_lambda__calculus_ORaise_Osimps_I3_J,axiom,
! [D: nat,N: nat,T: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_Raise @ ( suc @ D ) @ N @ T ) ) ) ).
% lambda_calculus.Raise.simps(3)
thf(fact_868_lambda__calculus_OSubst_Osimps_I3_J,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_Subst @ ( suc @ N ) @ V @ T ) ) ) ).
% lambda_calculus.Subst.simps(3)
thf(fact_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_lambda__calculus_OConD_I2_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,V2: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_Beta @ V2 @ U3 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Lam @ V2 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U3 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(2)
thf(fact_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_lambda__calculus_OSrc__eq__iff_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T5 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T5 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(4)
thf(fact_891_lambda__calculus_OSrc_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.Src.simps(5)
thf(fact_892_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_893_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_894_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_895_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_896_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_897_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_898_lambda__calculus_OJoin_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( lambda_Join @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
= ( lambda_Beta @ ( lambda_Join @ T @ T5 ) @ ( lambda_Join @ U @ U3 ) ) ) ).
% lambda_calculus.Join.simps(4)
thf(fact_899_lambda__calculus_OJoin_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( lambda_Join @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( lambda_Beta @ T5 @ U3 ) )
= ( lambda_Beta @ ( lambda_Join @ T @ T5 ) @ ( lambda_Join @ U @ U3 ) ) ) ).
% lambda_calculus.Join.simps(3)
thf(fact_900_lambda__calculus_OIde_Oelims_I2_J,axiom,
! [X5: lambda_lambda] :
( ( lambda_Ide @ X5 )
=> ( ! [Uu2: nat] :
( X5
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Ide @ T3 ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U4 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(2)
thf(fact_901_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_902_lambda__calculus_Oresid_Osimps_I6_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T5 ) @ U3 ) )
= ( lambda_Beta @ ( lambda_resid @ T @ T5 ) @ ( lambda_resid @ U @ U3 ) ) ) ) ) ).
% lambda_calculus.resid.simps(6)
thf(fact_903_lambda__calculus_OConD_I3_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_App @ T5 @ U3 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ T5 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U3 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(3)
thf(fact_904_lambda__calculus_OConD_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V2: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_Beta @ V2 @ U3 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ ( lambda_Lam @ V2 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U3 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(4)
thf(fact_905_lambda__calculus_OTrg_Ocases,axiom,
! [X5: lambda_lambda] :
( ! [I3: nat] :
( X5
!= ( lambda_Var @ I3 ) )
=> ( ! [T3: lambda_lambda] :
( X5
!= ( lambda_Lam @ T3 ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( X5
!= ( lambda_App @ T3 @ U4 ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( X5
!= ( lambda_Beta @ T3 @ U4 ) )
=> ( X5 = lambda_Nil ) ) ) ) ) ).
% lambda_calculus.Trg.cases
thf(fact_906_lambda__calculus_Osize_Ocases,axiom,
! [X5: lambda_lambda] :
( ( X5 != lambda_Nil )
=> ( ! [Uu2: nat] :
( X5
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( X5
!= ( lambda_Lam @ T3 ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( X5
!= ( lambda_App @ T3 @ U4 ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( X5
!= ( lambda_Beta @ T3 @ U4 ) ) ) ) ) ) ).
% lambda_calculus.size.cases
thf(fact_907_lambda__calculus_Olambda_Oexhaust,axiom,
! [Y5: lambda_lambda] :
( ( Y5 != lambda_Nil )
=> ( ! [X23: nat] :
( Y5
!= ( lambda_Var @ X23 ) )
=> ( ! [X34: lambda_lambda] :
( Y5
!= ( lambda_Lam @ X34 ) )
=> ( ! [X412: lambda_lambda,X422: lambda_lambda] :
( Y5
!= ( lambda_App @ X412 @ X422 ) )
=> ~ ! [X512: lambda_lambda,X522: lambda_lambda] :
( Y5
!= ( lambda_Beta @ X512 @ X522 ) ) ) ) ) ) ).
% lambda_calculus.lambda.exhaust
thf(fact_908_lambda__calculus_OArr_Oelims_I3_J,axiom,
! [X5: lambda_lambda] :
( ~ ( lambda_Arr @ X5 )
=> ( ( X5 != lambda_Nil )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( lambda_Arr @ T3 ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(3)
thf(fact_909_lambda__calculus_OArr_Oelims_I2_J,axiom,
! [X5: lambda_lambda] :
( ( lambda_Arr @ X5 )
=> ( ! [Uu2: nat] :
( X5
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(2)
thf(fact_910_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_911_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_912_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_913_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_914_lambda__calculus_OIde_Oelims_I3_J,axiom,
! [X5: lambda_lambda] :
( ~ ( lambda_Ide @ X5 )
=> ( ( X5 != lambda_Nil )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( lambda_Ide @ T3 ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U4 ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( X5
!= ( lambda_Beta @ T3 @ U4 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(3)
thf(fact_915_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_916_lambda__calculus_OArr_Oelims_I1_J,axiom,
! [X5: lambda_lambda,Y5: $o] :
( ( ( lambda_Arr @ X5 )
= Y5 )
=> ( ( ( X5 = lambda_Nil )
=> Y5 )
=> ( ( ? [Uu2: nat] :
( X5
= ( lambda_Var @ Uu2 ) )
=> ~ Y5 )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( Y5
= ( ~ ( lambda_Arr @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( Y5
= ( ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( Y5
= ( ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(1)
thf(fact_917_lambda__calculus_OSrc_Oelims,axiom,
! [X5: lambda_lambda,Y5: lambda_lambda] :
( ( ( lambda_Src @ X5 )
= Y5 )
=> ( ( ( X5 = lambda_Nil )
=> ( Y5 != lambda_Nil ) )
=> ( ! [I3: nat] :
( ( X5
= ( lambda_Var @ I3 ) )
=> ( Y5
!= ( lambda_Var @ I3 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( Y5
!= ( lambda_Lam @ ( lambda_Src @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( Y5
!= ( lambda_App @ ( lambda_Src @ T3 ) @ ( lambda_Src @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( Y5
!= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T3 ) ) @ ( lambda_Src @ U4 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Src.elims
thf(fact_918_lambda__calculus_OIde_Oelims_I1_J,axiom,
! [X5: lambda_lambda,Y5: $o] :
( ( ( lambda_Ide @ X5 )
= Y5 )
=> ( ( ( X5 = lambda_Nil )
=> Y5 )
=> ( ( ? [Uu2: nat] :
( X5
= ( lambda_Var @ Uu2 ) )
=> ~ Y5 )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( Y5
= ( ~ ( lambda_Ide @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( Y5
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U4 ) ) ) ) )
=> ~ ( ? [T3: lambda_lambda,U4: lambda_lambda] :
( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> Y5 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(1)
thf(fact_919_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_920_lambda__calculus_Oresid_Osimps_I5_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( lambda_Beta @ T5 @ U3 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T5 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U3 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( lambda_Beta @ T5 @ U3 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U @ U3 ) @ ( lambda_resid @ T @ T5 ) ) ) ) ) ).
% lambda_calculus.resid.simps(5)
thf(fact_921_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_922_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_923_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_924_lambda__calculus_OTrg_Oelims,axiom,
! [X5: lambda_lambda,Y5: lambda_lambda] :
( ( ( lambda_Trg @ X5 )
= Y5 )
=> ( ! [I3: nat] :
( ( X5
= ( lambda_Var @ I3 ) )
=> ( Y5
!= ( lambda_Var @ I3 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( Y5
!= ( lambda_Lam @ ( lambda_Trg @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( Y5
!= ( lambda_App @ ( lambda_Trg @ T3 ) @ ( lambda_Trg @ U4 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( Y5
!= ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U4 ) @ ( lambda_Trg @ T3 ) ) ) )
=> ~ ( ( X5 = lambda_Nil )
=> ( Y5 != lambda_Nil ) ) ) ) ) ) ) ).
% lambda_calculus.Trg.elims
thf(fact_925_prfx__Lam__iff,axiom,
! [U: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Lam @ T ) ) )
= ( ( lambda_is_Lam @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_Lam @ U ) @ T ) ) ) ) ).
% prfx_Lam_iff
thf(fact_926_Con__implies__Coinitial__ind,axiom,
! [T: lambda_lambda,U: lambda_lambda,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( lambda_size @ T ) @ ( lambda_size @ U ) ) @ N )
=> ( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) ) ) ) ).
% Con_implies_Coinitial_ind
thf(fact_927_Coinitial__implies__Con__ind,axiom,
! [T: lambda_lambda,N: nat,U: lambda_lambda] :
( ( ord_less_eq_nat @ ( lambda_size @ ( lambda_Src @ T ) ) @ N )
=> ( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
=> ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ) ).
% Coinitial_implies_Con_ind
thf(fact_928_size__Raise,axiom,
! [D: nat,N: nat,T: lambda_lambda] :
( ( lambda_size @ ( lambda_Raise @ D @ N @ T ) )
= ( lambda_size @ T ) ) ).
% size_Raise
thf(fact_929_subset__empty,axiom,
! [A3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ bot_bo1271416176687173302lambda )
= ( A3 = bot_bo1271416176687173302lambda ) ) ).
% subset_empty
thf(fact_930_subset__empty,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_931_empty__subsetI,axiom,
! [A3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A3 ) ).
% empty_subsetI
thf(fact_932_empty__subsetI,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% empty_subsetI
thf(fact_933_insert__subset,axiom,
! [X5: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ X5 @ A3 ) @ B3 )
= ( ( member_lambda_lambda @ X5 @ B3 )
& ( ord_le1926335974945931754lambda @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_934_insert__subset,axiom,
! [X5: nat,A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X5 @ A3 ) @ B3 )
= ( ( member_nat @ X5 @ B3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_935_Int__subset__iff,axiom,
! [C2: set_lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) )
= ( ( ord_le1926335974945931754lambda @ C2 @ A3 )
& ( ord_le1926335974945931754lambda @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_936_Con__Src,axiom,
! [T: lambda_lambda,U: lambda_lambda,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( lambda_size @ T ) @ ( lambda_size @ U ) ) @ N )
=> ( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) )
!= lambda_Nil ) ) ) ).
% Con_Src
thf(fact_937_singleton__insert__inj__eq,axiom,
! [B: lambda_lambda,A: lambda_lambda,A3: set_lambda_lambda] :
( ( ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ A @ A3 ) )
= ( ( A = B )
& ( ord_le1926335974945931754lambda @ A3 @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_938_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A3: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_939_singleton__insert__inj__eq_H,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda,B: lambda_lambda] :
( ( ( insert_lambda_lambda @ A @ A3 )
= ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
= ( ( A = B )
& ( ord_le1926335974945931754lambda @ A3 @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_940_singleton__insert__inj__eq_H,axiom,
! [A: nat,A3: set_nat,B: nat] :
( ( ( insert_nat @ A @ A3 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_941_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_942_Int__mono,axiom,
! [A3: set_lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda,D3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ C2 )
=> ( ( ord_le1926335974945931754lambda @ B3 @ D3 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) @ ( inf_in1766959559283430712lambda @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_943_Int__lower1,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) @ A3 ) ).
% Int_lower1
thf(fact_944_Int__lower2,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_945_Int__absorb1,axiom,
! [B3: set_lambda_lambda,A3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ A3 )
=> ( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_946_Int__absorb2,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ B3 )
=> ( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= A3 ) ) ).
% Int_absorb2
thf(fact_947_Int__greatest,axiom,
! [C2: set_lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ A3 )
=> ( ( ord_le1926335974945931754lambda @ C2 @ B3 )
=> ( ord_le1926335974945931754lambda @ C2 @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_948_Int__Collect__mono,axiom,
! [A3: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_949_Int__Collect__mono,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ A3 @ B3 )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A3 )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A3 @ ( collec621444948832755541lambda @ P ) ) @ ( inf_in1766959559283430712lambda @ B3 @ ( collec621444948832755541lambda @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_950_insert__mono,axiom,
! [C2: set_lambda_lambda,D3: set_lambda_lambda,A: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ D3 )
=> ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ A @ C2 ) @ ( insert_lambda_lambda @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_951_insert__mono,axiom,
! [C2: set_nat,D3: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D3 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_952_subset__insert,axiom,
! [X5: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X5 @ A3 )
=> ( ( ord_le1926335974945931754lambda @ A3 @ ( insert_lambda_lambda @ X5 @ B3 ) )
= ( ord_le1926335974945931754lambda @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_953_subset__insert,axiom,
! [X5: nat,A3: set_nat,B3: set_nat] :
( ~ ( member_nat @ X5 @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X5 @ B3 ) )
= ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_954_subset__insertI,axiom,
! [B3: set_lambda_lambda,A: lambda_lambda] : ( ord_le1926335974945931754lambda @ B3 @ ( insert_lambda_lambda @ A @ B3 ) ) ).
% subset_insertI
thf(fact_955_subset__insertI,axiom,
! [B3: set_nat,A: nat] : ( ord_less_eq_set_nat @ B3 @ ( insert_nat @ A @ B3 ) ) ).
% subset_insertI
thf(fact_956_subset__insertI2,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,B: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ B3 )
=> ( ord_le1926335974945931754lambda @ A3 @ ( insert_lambda_lambda @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_957_subset__insertI2,axiom,
! [A3: set_nat,B3: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_958_lambda__calculus_Olambda_Osel_I2_J,axiom,
! [X32: lambda_lambda] :
( ( lambda_un_Lam @ ( lambda_Lam @ X32 ) )
= X32 ) ).
% lambda_calculus.lambda.sel(2)
thf(fact_959_subset__singletonD,axiom,
! [A3: set_lambda_lambda,X5: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ ( insert_lambda_lambda @ X5 @ bot_bo1271416176687173302lambda ) )
=> ( ( A3 = bot_bo1271416176687173302lambda )
| ( A3
= ( insert_lambda_lambda @ X5 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singletonD
thf(fact_960_subset__singletonD,axiom,
! [A3: set_nat,X5: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
=> ( ( A3 = bot_bot_set_nat )
| ( A3
= ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_961_subset__singleton__iff,axiom,
! [X7: set_lambda_lambda,A: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X7 @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) )
= ( ( X7 = bot_bo1271416176687173302lambda )
| ( X7
= ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_962_subset__singleton__iff,axiom,
! [X7: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X7 = bot_bot_set_nat )
| ( X7
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_963_lambda__calculus_Osize_Osimps_I1_J,axiom,
( ( lambda_size @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.size.simps(1)
thf(fact_964_lambda__calculus_Osize_Osimps_I2_J,axiom,
! [Uu: nat] :
( ( lambda_size @ ( lambda_Var @ Uu ) )
= one_one_nat ) ).
% lambda_calculus.size.simps(2)
thf(fact_965_lambda__calculus_Osize_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_size @ ( lambda_App @ T @ U ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T ) @ ( lambda_size @ U ) ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(4)
thf(fact_966_lambda__calculus_Osize_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_size @ ( lambda_Beta @ T @ U ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T ) @ one_one_nat ) @ ( lambda_size @ U ) ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(5)
thf(fact_967_lambda__calculus_Osize_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_size @ ( lambda_Lam @ T ) )
= ( plus_plus_nat @ ( lambda_size @ T ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(3)
thf(fact_968_lambda__calculus_Osize_Oelims,axiom,
! [X5: lambda_lambda,Y5: nat] :
( ( ( lambda_size @ X5 )
= Y5 )
=> ( ( ( X5 = lambda_Nil )
=> ( Y5 != zero_zero_nat ) )
=> ( ( ? [Uu2: nat] :
( X5
= ( lambda_Var @ Uu2 ) )
=> ( Y5 != one_one_nat ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( Y5
!= ( plus_plus_nat @ ( lambda_size @ T3 ) @ one_one_nat ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( Y5
!= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T3 ) @ ( lambda_size @ U4 ) ) @ one_one_nat ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( Y5
!= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T3 ) @ one_one_nat ) @ ( lambda_size @ U4 ) ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% lambda_calculus.size.elims
thf(fact_969_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_970_sub__rts__axioms_Ointro,axiom,
! [Arr2: lambda_lambda > $o,R: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( arr_lambda_lambda @ R @ T3 ) )
=> ( ! [T3: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R @ T3 ) @ ( collec621444948832755541lambda @ Arr2 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ( Arr2 @ U4 )
=> ( ( con_lambda_lambda @ R @ T3 @ U4 )
=> ( Arr2 @ ( R @ T3 @ U4 ) ) ) ) )
=> ( sub_rt3575465509250962105lambda @ R @ Arr2 ) ) ) ) ).
% sub_rts_axioms.intro
thf(fact_971_sub__rts__axioms__def,axiom,
( sub_rt3575465509250962105lambda
= ( ^ [R2: lambda_lambda > lambda_lambda > lambda_lambda,Arr3: lambda_lambda > $o] :
( ! [T4: lambda_lambda] :
( ( Arr3 @ T4 )
=> ( arr_lambda_lambda @ R2 @ T4 ) )
& ! [T4: lambda_lambda] :
( ( Arr3 @ T4 )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R2 @ T4 ) @ ( collec621444948832755541lambda @ Arr3 ) ) )
& ! [T4: lambda_lambda,U2: lambda_lambda] :
( ( Arr3 @ T4 )
=> ( ( Arr3 @ U2 )
=> ( ( con_lambda_lambda @ R2 @ T4 @ U2 )
=> ( Arr3 @ ( R2 @ T4 @ U2 ) ) ) ) ) ) ) ) ).
% sub_rts_axioms_def
thf(fact_972_subsetI,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A3 )
=> ( member_lambda_lambda @ X @ B3 ) )
=> ( ord_le1926335974945931754lambda @ A3 @ B3 ) ) ).
% subsetI
thf(fact_973_subsetI,axiom,
! [A3: set_nat,B3: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B3 ) )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% subsetI
thf(fact_974_in__mono,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,X5: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ B3 )
=> ( ( member_lambda_lambda @ X5 @ A3 )
=> ( member_lambda_lambda @ X5 @ B3 ) ) ) ).
% in_mono
thf(fact_975_in__mono,axiom,
! [A3: set_nat,B3: set_nat,X5: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ X5 @ A3 )
=> ( member_nat @ X5 @ B3 ) ) ) ).
% in_mono
thf(fact_976_subsetD,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,C: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ B3 )
=> ( ( member_lambda_lambda @ C @ A3 )
=> ( member_lambda_lambda @ C @ B3 ) ) ) ).
% subsetD
thf(fact_977_subsetD,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_978_subset__eq,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A7: set_lambda_lambda,B4: set_lambda_lambda] :
! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A7 )
=> ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_979_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A7 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_980_subset__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A7: set_lambda_lambda,B4: set_lambda_lambda] :
! [T4: lambda_lambda] :
( ( member_lambda_lambda @ T4 @ A7 )
=> ( member_lambda_lambda @ T4 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_981_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A7: set_nat,B4: set_nat] :
! [T4: nat] :
( ( member_nat @ T4 @ A7 )
=> ( member_nat @ T4 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_982_Collect__mono,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ! [X: lambda_lambda] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le1926335974945931754lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) ) ) ).
% Collect_mono
thf(fact_983_Collect__mono__iff,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) )
= ( ! [X3: lambda_lambda] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_984_inf_Ostrict__coboundedI2,axiom,
! [B: set_lambda_lambda,C: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ B @ C )
=> ( ord_le4648583965110040310lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_985_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_986_inf_Ostrict__coboundedI1,axiom,
! [A: set_lambda_lambda,C: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ A @ C )
=> ( ord_le4648583965110040310lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_987_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_988_inf_Ostrict__order__iff,axiom,
( ord_le4648583965110040310lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( ( A6
= ( inf_in1766959559283430712lambda @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_989_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A6: nat,B5: nat] :
( ( A6
= ( inf_inf_nat @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_990_inf_Ostrict__boundedE,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C ) )
=> ~ ( ( ord_le4648583965110040310lambda @ A @ B )
=> ~ ( ord_le4648583965110040310lambda @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_991_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_992_inf_Oabsorb4,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ B @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_993_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_994_inf_Oabsorb3,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ A @ B )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_995_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_996_less__infI2,axiom,
! [B: set_lambda_lambda,X5: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ B @ X5 )
=> ( ord_le4648583965110040310lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ X5 ) ) ).
% less_infI2
thf(fact_997_less__infI2,axiom,
! [B: nat,X5: nat,A: nat] :
( ( ord_less_nat @ B @ X5 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X5 ) ) ).
% less_infI2
thf(fact_998_less__infI1,axiom,
! [A: set_lambda_lambda,X5: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ A @ X5 )
=> ( ord_le4648583965110040310lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ X5 ) ) ).
% less_infI1
thf(fact_999_less__infI1,axiom,
! [A: nat,X5: nat,B: nat] :
( ( ord_less_nat @ A @ X5 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X5 ) ) ).
% less_infI1
thf(fact_1000_bot_Oextremum__strict,axiom,
! [A: set_lambda_lambda] :
~ ( ord_le4648583965110040310lambda @ A @ bot_bo1271416176687173302lambda ) ).
% bot.extremum_strict
thf(fact_1001_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1002_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1003_bot_Onot__eq__extremum,axiom,
! [A: set_lambda_lambda] :
( ( A != bot_bo1271416176687173302lambda )
= ( ord_le4648583965110040310lambda @ bot_bo1271416176687173302lambda @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1004_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1005_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1006_subterm__implies__smaller,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T @ U )
=> ( ord_less_nat @ ( lambda_size @ T ) @ ( lambda_size @ U ) ) ) ).
% subterm_implies_smaller
thf(fact_1007_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_1008_not__psubset__empty,axiom,
! [A3: set_lambda_lambda] :
~ ( ord_le4648583965110040310lambda @ A3 @ bot_bo1271416176687173302lambda ) ).
% not_psubset_empty
thf(fact_1009_not__psubset__empty,axiom,
! [A3: set_nat] :
~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1010_lambda__calculus_Osubterm__lemmas_I1_J,axiom,
! [T: lambda_lambda] : ( lambda_subterm @ T @ ( lambda_Lam @ T ) ) ).
% lambda_calculus.subterm_lemmas(1)
thf(fact_1011_lambda__calculus_Osubterm_Ointros_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( lambda_subterm @ T @ U )
=> ( ( lambda_subterm @ U @ V )
=> ( lambda_subterm @ T @ V ) ) ) ).
% lambda_calculus.subterm.intros(6)
thf(fact_1012_lambda__calculus_Osubterm__lemmas_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T @ ( lambda_App @ T @ U ) )
& ( lambda_subterm @ U @ ( lambda_App @ T @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(3)
thf(fact_1013_lambda__calculus_Osubterm_Ointros_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] : ( lambda_subterm @ T @ ( lambda_App @ T @ U ) ) ).
% lambda_calculus.subterm.intros(2)
thf(fact_1014_lambda__calculus_Osubterm_Ointros_I3_J,axiom,
! [U: lambda_lambda,T: lambda_lambda] : ( lambda_subterm @ U @ ( lambda_App @ T @ U ) ) ).
% lambda_calculus.subterm.intros(3)
thf(fact_1015_lambda__calculus_Osubterm__lemmas_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T @ ( lambda_Beta @ T @ U ) )
& ( lambda_subterm @ U @ ( lambda_Beta @ T @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(4)
thf(fact_1016_lambda__calculus_Osubterm_Ointros_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] : ( lambda_subterm @ T @ ( lambda_Beta @ T @ U ) ) ).
% lambda_calculus.subterm.intros(4)
thf(fact_1017_lambda__calculus_Osubterm_Ointros_I5_J,axiom,
! [U: lambda_lambda,T: lambda_lambda] : ( lambda_subterm @ U @ ( lambda_Beta @ T @ U ) ) ).
% lambda_calculus.subterm.intros(5)
thf(fact_1018_lambda__calculus_Osubterm__lemmas_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
& ( lambda_subterm @ U @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(2)
thf(fact_1019_lambda__calculus_Olambda_Osize__gen_I1_J,axiom,
( ( lambda_size_lambda @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size_gen(1)
thf(fact_1020_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_1021_lambda__calculus_Osubterm_Ocases,axiom,
! [A1: lambda_lambda,A22: lambda_lambda] :
( ( lambda_subterm @ A1 @ A22 )
=> ( ( A22
!= ( lambda_Lam @ A1 ) )
=> ( ! [U4: lambda_lambda] :
( A22
!= ( lambda_App @ A1 @ U4 ) )
=> ( ! [T3: lambda_lambda] :
( A22
!= ( lambda_App @ T3 @ A1 ) )
=> ( ! [U4: lambda_lambda] :
( A22
!= ( lambda_Beta @ A1 @ U4 ) )
=> ( ! [T3: lambda_lambda] :
( A22
!= ( lambda_Beta @ T3 @ A1 ) )
=> ~ ! [U4: lambda_lambda] :
( ( lambda_subterm @ A1 @ U4 )
=> ~ ( lambda_subterm @ U4 @ A22 ) ) ) ) ) ) ) ) ).
% lambda_calculus.subterm.cases
thf(fact_1022_lambda__calculus_Osubterm_Osimps,axiom,
( lambda_subterm
= ( ^ [A12: lambda_lambda,A23: lambda_lambda] :
( ? [T4: lambda_lambda] :
( ( A12 = T4 )
& ( A23
= ( lambda_Lam @ T4 ) ) )
| ? [T4: lambda_lambda,U2: lambda_lambda] :
( ( A12 = T4 )
& ( A23
= ( lambda_App @ T4 @ U2 ) ) )
| ? [T4: lambda_lambda,U2: lambda_lambda] :
( ( A12 = U2 )
& ( A23
= ( lambda_App @ T4 @ U2 ) ) )
| ? [T4: lambda_lambda,U2: lambda_lambda] :
( ( A12 = T4 )
& ( A23
= ( lambda_Beta @ T4 @ U2 ) ) )
| ? [T4: lambda_lambda,U2: lambda_lambda] :
( ( A12 = U2 )
& ( A23
= ( lambda_Beta @ T4 @ U2 ) ) )
| ? [T4: lambda_lambda,U2: lambda_lambda,V4: lambda_lambda] :
( ( A12 = T4 )
& ( A23 = V4 )
& ( lambda_subterm @ T4 @ U2 )
& ( lambda_subterm @ U2 @ V4 ) ) ) ) ) ).
% lambda_calculus.subterm.simps
thf(fact_1023_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_1024_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_1025_subset__emptyI,axiom,
! [A3: set_lambda_lambda] :
( ! [X: lambda_lambda] :
~ ( member_lambda_lambda @ X @ A3 )
=> ( ord_le1926335974945931754lambda @ A3 @ bot_bo1271416176687173302lambda ) ) ).
% subset_emptyI
thf(fact_1026_subset__emptyI,axiom,
! [A3: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1027_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_1028_psubsetD,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,C: lambda_lambda] :
( ( ord_le4648583965110040310lambda @ A3 @ B3 )
=> ( ( member_lambda_lambda @ C @ A3 )
=> ( member_lambda_lambda @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_1029_psubsetD,axiom,
! [A3: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_1030_lambda__calculus_Olambda_Osize_I6_J,axiom,
( ( size_s1768714712973771222lambda @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size(6)
thf(fact_1031_lambda__calculus_Olambda_Osize_I7_J,axiom,
! [X22: nat] :
( ( size_s1768714712973771222lambda @ ( lambda_Var @ X22 ) )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size(7)
thf(fact_1032_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_1033_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_1034_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_1035_lambda__calculus_Osize_Opelims,axiom,
! [X5: lambda_lambda,Y5: nat] :
( ( ( lambda_size @ X5 )
= Y5 )
=> ( ( accp_lambda_lambda @ lambda_size_rel @ X5 )
=> ( ( ( X5 = lambda_Nil )
=> ( ( Y5 = zero_zero_nat )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X5
= ( lambda_Var @ Uu2 ) )
=> ( ( Y5 = one_one_nat )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( Y5
= ( plus_plus_nat @ ( lambda_size @ T3 ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Lam @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( Y5
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T3 ) @ ( lambda_size @ U4 ) ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_App @ T3 @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( ( Y5
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T3 ) @ one_one_nat ) @ ( lambda_size @ U4 ) ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Beta @ T3 @ U4 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.size.pelims
thf(fact_1036_diff__shunt__var,axiom,
! [X5: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( ( minus_5489901412966541873lambda @ X5 @ Y5 )
= bot_bo1271416176687173302lambda )
= ( ord_le1926335974945931754lambda @ X5 @ Y5 ) ) ).
% diff_shunt_var
thf(fact_1037_diff__shunt__var,axiom,
! [X5: set_nat,Y5: set_nat] :
( ( ( minus_minus_set_nat @ X5 @ Y5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X5 @ Y5 ) ) ).
% diff_shunt_var
thf(fact_1038_lambda__calculus_OTrg_Opelims,axiom,
! [X5: lambda_lambda,Y5: lambda_lambda] :
( ( ( lambda_Trg @ X5 )
= Y5 )
=> ( ( accp_lambda_lambda @ lambda_Trg_rel @ X5 )
=> ( ! [I3: nat] :
( ( X5
= ( lambda_Var @ I3 ) )
=> ( ( Y5
= ( lambda_Var @ I3 ) )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ ( lambda_Var @ I3 ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( Y5
= ( lambda_Lam @ ( lambda_Trg @ T3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ ( lambda_Lam @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( Y5
= ( lambda_App @ ( lambda_Trg @ T3 ) @ ( lambda_Trg @ U4 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ ( lambda_App @ T3 @ U4 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( ( Y5
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U4 ) @ ( lambda_Trg @ T3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ ( lambda_Beta @ T3 @ U4 ) ) ) )
=> ~ ( ( X5 = lambda_Nil )
=> ( ( Y5 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Trg_rel @ lambda_Nil ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Trg.pelims
thf(fact_1039_lambda__calculus_OSrc_Opelims,axiom,
! [X5: lambda_lambda,Y5: lambda_lambda] :
( ( ( lambda_Src @ X5 )
= Y5 )
=> ( ( accp_lambda_lambda @ lambda_Src_rel @ X5 )
=> ( ( ( X5 = lambda_Nil )
=> ( ( Y5 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ lambda_Nil ) ) )
=> ( ! [I3: nat] :
( ( X5
= ( lambda_Var @ I3 ) )
=> ( ( Y5
= ( lambda_Var @ I3 ) )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ ( lambda_Var @ I3 ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( Y5
= ( lambda_Lam @ ( lambda_Src @ T3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ ( lambda_Lam @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( Y5
= ( lambda_App @ ( lambda_Src @ T3 ) @ ( lambda_Src @ U4 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ ( lambda_App @ T3 @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( ( Y5
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T3 ) ) @ ( lambda_Src @ U4 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Src_rel @ ( lambda_Beta @ T3 @ U4 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Src.pelims
thf(fact_1040_Diff__cancel,axiom,
! [A3: set_lambda_lambda] :
( ( minus_5489901412966541873lambda @ A3 @ A3 )
= bot_bo1271416176687173302lambda ) ).
% Diff_cancel
thf(fact_1041_Diff__cancel,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ A3 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1042_empty__Diff,axiom,
! [A3: set_lambda_lambda] :
( ( minus_5489901412966541873lambda @ bot_bo1271416176687173302lambda @ A3 )
= bot_bo1271416176687173302lambda ) ).
% empty_Diff
thf(fact_1043_empty__Diff,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1044_Diff__empty,axiom,
! [A3: set_lambda_lambda] :
( ( minus_5489901412966541873lambda @ A3 @ bot_bo1271416176687173302lambda )
= A3 ) ).
% Diff_empty
thf(fact_1045_Diff__empty,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Diff_empty
thf(fact_1046_Diff__insert0,axiom,
! [X5: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X5 @ A3 )
=> ( ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ X5 @ B3 ) )
= ( minus_5489901412966541873lambda @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1047_Diff__insert0,axiom,
! [X5: nat,A3: set_nat,B3: set_nat] :
( ~ ( member_nat @ X5 @ A3 )
=> ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ B3 ) )
= ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1048_insert__Diff1,axiom,
! [X5: lambda_lambda,B3: set_lambda_lambda,A3: set_lambda_lambda] :
( ( member_lambda_lambda @ X5 @ B3 )
=> ( ( minus_5489901412966541873lambda @ ( insert_lambda_lambda @ X5 @ A3 ) @ B3 )
= ( minus_5489901412966541873lambda @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1049_insert__Diff1,axiom,
! [X5: nat,B3: set_nat,A3: set_nat] :
( ( member_nat @ X5 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X5 @ A3 ) @ B3 )
= ( minus_minus_set_nat @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1050_Diff__eq__empty__iff,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( minus_5489901412966541873lambda @ A3 @ B3 )
= bot_bo1271416176687173302lambda )
= ( ord_le1926335974945931754lambda @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_1051_Diff__eq__empty__iff,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( minus_minus_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_1052_insert__Diff__single,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda] :
( ( insert_lambda_lambda @ A @ ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) )
= ( insert_lambda_lambda @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_1053_insert__Diff__single,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_1054_Diff__disjoint,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A3 @ ( minus_5489901412966541873lambda @ B3 @ A3 ) )
= bot_bo1271416176687173302lambda ) ).
% Diff_disjoint
thf(fact_1055_Diff__disjoint,axiom,
! [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1056_psubset__imp__ex__mem,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ A3 @ B3 )
=> ? [B7: lambda_lambda] : ( member_lambda_lambda @ B7 @ ( minus_5489901412966541873lambda @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1057_psubset__imp__ex__mem,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
=> ? [B7: nat] : ( member_nat @ B7 @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1058_subset__Diff__insert,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,X5: lambda_lambda,C2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ ( minus_5489901412966541873lambda @ B3 @ ( insert_lambda_lambda @ X5 @ C2 ) ) )
= ( ( ord_le1926335974945931754lambda @ A3 @ ( minus_5489901412966541873lambda @ B3 @ C2 ) )
& ~ ( member_lambda_lambda @ X5 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1059_subset__Diff__insert,axiom,
! [A3: set_nat,B3: set_nat,X5: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ ( insert_nat @ X5 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ C2 ) )
& ~ ( member_nat @ X5 @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_1060_insert__Diff__if,axiom,
! [X5: lambda_lambda,B3: set_lambda_lambda,A3: set_lambda_lambda] :
( ( ( member_lambda_lambda @ X5 @ B3 )
=> ( ( minus_5489901412966541873lambda @ ( insert_lambda_lambda @ X5 @ A3 ) @ B3 )
= ( minus_5489901412966541873lambda @ A3 @ B3 ) ) )
& ( ~ ( member_lambda_lambda @ X5 @ B3 )
=> ( ( minus_5489901412966541873lambda @ ( insert_lambda_lambda @ X5 @ A3 ) @ B3 )
= ( insert_lambda_lambda @ X5 @ ( minus_5489901412966541873lambda @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1061_insert__Diff__if,axiom,
! [X5: nat,B3: set_nat,A3: set_nat] :
( ( ( member_nat @ X5 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X5 @ A3 ) @ B3 )
= ( minus_minus_set_nat @ A3 @ B3 ) ) )
& ( ~ ( member_nat @ X5 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X5 @ A3 ) @ B3 )
= ( insert_nat @ X5 @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1062_Diff__Int__distrib2,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,C2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( minus_5489901412966541873lambda @ A3 @ B3 ) @ C2 )
= ( minus_5489901412966541873lambda @ ( inf_in1766959559283430712lambda @ A3 @ C2 ) @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1063_Diff__Int__distrib,axiom,
! [C2: set_lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ C2 @ ( minus_5489901412966541873lambda @ A3 @ B3 ) )
= ( minus_5489901412966541873lambda @ ( inf_in1766959559283430712lambda @ C2 @ A3 ) @ ( inf_in1766959559283430712lambda @ C2 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_1064_Diff__Diff__Int,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( minus_5489901412966541873lambda @ A3 @ ( minus_5489901412966541873lambda @ A3 @ B3 ) )
= ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_1065_Diff__Int2,axiom,
! [A3: set_lambda_lambda,C2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( minus_5489901412966541873lambda @ ( inf_in1766959559283430712lambda @ A3 @ C2 ) @ ( inf_in1766959559283430712lambda @ B3 @ C2 ) )
= ( minus_5489901412966541873lambda @ ( inf_in1766959559283430712lambda @ A3 @ C2 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_1066_Int__Diff,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda,C2: set_lambda_lambda] :
( ( minus_5489901412966541873lambda @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) @ C2 )
= ( inf_in1766959559283430712lambda @ A3 @ ( minus_5489901412966541873lambda @ B3 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1067_Diff__triv,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= bot_bo1271416176687173302lambda )
=> ( ( minus_5489901412966541873lambda @ A3 @ B3 )
= A3 ) ) ).
% Diff_triv
thf(fact_1068_Diff__triv,axiom,
! [A3: set_nat,B3: set_nat] :
( ( ( inf_inf_set_nat @ A3 @ B3 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A3 @ B3 )
= A3 ) ) ).
% Diff_triv
thf(fact_1069_Int__Diff__disjoint,axiom,
! [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A3 @ B3 ) @ ( minus_5489901412966541873lambda @ A3 @ B3 ) )
= bot_bo1271416176687173302lambda ) ).
% Int_Diff_disjoint
thf(fact_1070_Int__Diff__disjoint,axiom,
! [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ B3 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1071_Diff__insert__absorb,axiom,
! [X5: lambda_lambda,A3: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X5 @ A3 )
=> ( ( minus_5489901412966541873lambda @ ( insert_lambda_lambda @ X5 @ A3 ) @ ( insert_lambda_lambda @ X5 @ bot_bo1271416176687173302lambda ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1072_Diff__insert__absorb,axiom,
! [X5: nat,A3: set_nat] :
( ~ ( member_nat @ X5 @ A3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X5 @ A3 ) @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_1073_Diff__insert2,axiom,
! [A3: set_lambda_lambda,A: lambda_lambda,B3: set_lambda_lambda] :
( ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( minus_5489901412966541873lambda @ ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1074_Diff__insert2,axiom,
! [A3: set_nat,A: nat,B3: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B3 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1075_insert__Diff,axiom,
! [A: lambda_lambda,A3: set_lambda_lambda] :
( ( member_lambda_lambda @ A @ A3 )
=> ( ( insert_lambda_lambda @ A @ ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1076_insert__Diff,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_1077_Diff__insert,axiom,
! [A3: set_lambda_lambda,A: lambda_lambda,B3: set_lambda_lambda] :
( ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ A @ B3 ) )
= ( minus_5489901412966541873lambda @ ( minus_5489901412966541873lambda @ A3 @ B3 ) @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) ) ) ).
% Diff_insert
thf(fact_1078_Diff__insert,axiom,
! [A3: set_nat,A: nat,B3: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B3 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1079_Diff__single__insert,axiom,
! [A3: set_lambda_lambda,X5: lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ X5 @ bot_bo1271416176687173302lambda ) ) @ B3 )
=> ( ord_le1926335974945931754lambda @ A3 @ ( insert_lambda_lambda @ X5 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1080_Diff__single__insert,axiom,
! [A3: set_nat,X5: nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) @ B3 )
=> ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X5 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1081_subset__insert__iff,axiom,
! [A3: set_lambda_lambda,X5: lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ ( insert_lambda_lambda @ X5 @ B3 ) )
= ( ( ( member_lambda_lambda @ X5 @ A3 )
=> ( ord_le1926335974945931754lambda @ ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ X5 @ bot_bo1271416176687173302lambda ) ) @ B3 ) )
& ( ~ ( member_lambda_lambda @ X5 @ A3 )
=> ( ord_le1926335974945931754lambda @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1082_subset__insert__iff,axiom,
! [A3: set_nat,X5: nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X5 @ B3 ) )
= ( ( ( member_nat @ X5 @ A3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X5 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1083_psubset__insert__iff,axiom,
! [A3: set_lambda_lambda,X5: lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le4648583965110040310lambda @ A3 @ ( insert_lambda_lambda @ X5 @ B3 ) )
= ( ( ( member_lambda_lambda @ X5 @ B3 )
=> ( ord_le4648583965110040310lambda @ A3 @ B3 ) )
& ( ~ ( member_lambda_lambda @ X5 @ B3 )
=> ( ( ( member_lambda_lambda @ X5 @ A3 )
=> ( ord_le4648583965110040310lambda @ ( minus_5489901412966541873lambda @ A3 @ ( insert_lambda_lambda @ X5 @ bot_bo1271416176687173302lambda ) ) @ B3 ) )
& ( ~ ( member_lambda_lambda @ X5 @ A3 )
=> ( ord_le1926335974945931754lambda @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1084_psubset__insert__iff,axiom,
! [A3: set_nat,X5: nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ ( insert_nat @ X5 @ B3 ) )
= ( ( ( member_nat @ X5 @ B3 )
=> ( ord_less_set_nat @ A3 @ B3 ) )
& ( ~ ( member_nat @ X5 @ B3 )
=> ( ( ( member_nat @ X5 @ A3 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X5 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1085_lambda__calculus_OIde_Opelims_I1_J,axiom,
! [X5: lambda_lambda,Y5: $o] :
( ( ( lambda_Ide @ X5 )
= Y5 )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X5 )
=> ( ( ( X5 = lambda_Nil )
=> ( ~ Y5
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X5
= ( lambda_Var @ Uu2 ) )
=> ( Y5
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( Y5
= ( lambda_Ide @ T3 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( Y5
= ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U4 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T3 @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( ~ Y5
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Beta @ T3 @ U4 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(1)
thf(fact_1086_lambda__calculus_OArr_Opelims_I1_J,axiom,
! [X5: lambda_lambda,Y5: $o] :
( ( ( lambda_Arr @ X5 )
= Y5 )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X5 )
=> ( ( ( X5 = lambda_Nil )
=> ( ~ Y5
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X5
= ( lambda_Var @ Uu2 ) )
=> ( Y5
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( Y5
= ( lambda_Arr @ T3 ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( Y5
= ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T3 @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( ( Y5
= ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T3 @ U4 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(1)
thf(fact_1087_Diff__iff,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( minus_5489901412966541873lambda @ A3 @ B3 ) )
= ( ( member_lambda_lambda @ C @ A3 )
& ~ ( member_lambda_lambda @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_1088_Diff__iff,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_1089_DiffI,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ A3 )
=> ( ~ ( member_lambda_lambda @ C @ B3 )
=> ( member_lambda_lambda @ C @ ( minus_5489901412966541873lambda @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_1090_DiffI,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_1091_DiffD2,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( minus_5489901412966541873lambda @ A3 @ B3 ) )
=> ~ ( member_lambda_lambda @ C @ B3 ) ) ).
% DiffD2
thf(fact_1092_DiffD2,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_1093_DiffD1,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( minus_5489901412966541873lambda @ A3 @ B3 ) )
=> ( member_lambda_lambda @ C @ A3 ) ) ).
% DiffD1
thf(fact_1094_DiffD1,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ( member_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_1095_DiffE,axiom,
! [C: lambda_lambda,A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( minus_5489901412966541873lambda @ A3 @ B3 ) )
=> ~ ( ( member_lambda_lambda @ C @ A3 )
=> ( member_lambda_lambda @ C @ B3 ) ) ) ).
% DiffE
thf(fact_1096_DiffE,axiom,
! [C: nat,A3: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_1097_lambda__calculus_OIde_Opelims_I2_J,axiom,
! [X5: lambda_lambda] :
( ( lambda_Ide @ X5 )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X5 )
=> ( ! [Uu2: nat] :
( ( X5
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Ide @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T3 @ U4 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U4 ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(2)
thf(fact_1098_lambda__calculus_OArr_Opelims_I3_J,axiom,
! [X5: lambda_lambda] :
( ~ ( lambda_Arr @ X5 )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X5 )
=> ( ( ( X5 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ lambda_Nil ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T3 ) )
=> ( lambda_Arr @ T3 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T3 @ U4 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T3 @ U4 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(3)
thf(fact_1099_lambda__calculus_OArr_Opelims_I2_J,axiom,
! [X5: lambda_lambda] :
( ( lambda_Arr @ X5 )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X5 )
=> ( ! [Uu2: nat] :
( ( X5
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T3 @ U4 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T3 @ U4 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U4 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(2)
thf(fact_1100_lambda__calculus_OIde_Opelims_I3_J,axiom,
! [X5: lambda_lambda] :
( ~ ( lambda_Ide @ X5 )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X5 )
=> ( ( ( X5 = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ lambda_Nil ) )
=> ( ! [T3: lambda_lambda] :
( ( X5
= ( lambda_Lam @ T3 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T3 ) )
=> ( lambda_Ide @ T3 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_App @ T3 @ U4 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T3 @ U4 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X5
= ( lambda_Beta @ T3 @ U4 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Beta @ T3 @ U4 ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(3)
thf(fact_1101_remove__def,axiom,
( remove_lambda_lambda
= ( ^ [X3: lambda_lambda,A7: set_lambda_lambda] : ( minus_5489901412966541873lambda @ A7 @ ( insert_lambda_lambda @ X3 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% remove_def
thf(fact_1102_remove__def,axiom,
( remove_nat
= ( ^ [X3: nat,A7: set_nat] : ( minus_minus_set_nat @ A7 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_1103_pairwise__alt,axiom,
( pairwi8946569470594120789lambda
= ( ^ [R2: lambda_lambda > lambda_lambda > $o,S: set_lambda_lambda] :
! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ S )
=> ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( minus_5489901412966541873lambda @ S @ ( insert_lambda_lambda @ X3 @ bot_bo1271416176687173302lambda ) ) )
=> ( R2 @ X3 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1104_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R2: nat > nat > $o,S: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( minus_minus_set_nat @ S @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
=> ( R2 @ X3 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_1105_member__remove,axiom,
! [X5: lambda_lambda,Y5: lambda_lambda,A3: set_lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( remove_lambda_lambda @ Y5 @ A3 ) )
= ( ( member_lambda_lambda @ X5 @ A3 )
& ( X5 != Y5 ) ) ) ).
% member_remove
thf(fact_1106_member__remove,axiom,
! [X5: nat,Y5: nat,A3: set_nat] :
( ( member_nat @ X5 @ ( remove_nat @ Y5 @ A3 ) )
= ( ( member_nat @ X5 @ A3 )
& ( X5 != Y5 ) ) ) ).
% member_remove
thf(fact_1107_pairwiseD,axiom,
! [R: lambda_lambda > lambda_lambda > $o,S2: set_lambda_lambda,X5: lambda_lambda,Y5: lambda_lambda] :
( ( pairwi8946569470594120789lambda @ R @ S2 )
=> ( ( member_lambda_lambda @ X5 @ S2 )
=> ( ( member_lambda_lambda @ Y5 @ S2 )
=> ( ( X5 != Y5 )
=> ( R @ X5 @ Y5 ) ) ) ) ) ).
% pairwiseD
thf(fact_1108_pairwiseD,axiom,
! [R: nat > nat > $o,S2: set_nat,X5: nat,Y5: nat] :
( ( pairwise_nat @ R @ S2 )
=> ( ( member_nat @ X5 @ S2 )
=> ( ( member_nat @ Y5 @ S2 )
=> ( ( X5 != Y5 )
=> ( R @ X5 @ Y5 ) ) ) ) ) ).
% pairwiseD
thf(fact_1109_pairwiseI,axiom,
! [S2: set_lambda_lambda,R: lambda_lambda > lambda_lambda > $o] :
( ! [X: lambda_lambda,Y3: lambda_lambda] :
( ( member_lambda_lambda @ X @ S2 )
=> ( ( member_lambda_lambda @ Y3 @ S2 )
=> ( ( X != Y3 )
=> ( R @ X @ Y3 ) ) ) )
=> ( pairwi8946569470594120789lambda @ R @ S2 ) ) ).
% pairwiseI
thf(fact_1110_pairwiseI,axiom,
! [S2: set_nat,R: nat > nat > $o] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ S2 )
=> ( ( member_nat @ Y3 @ S2 )
=> ( ( X != Y3 )
=> ( R @ X @ Y3 ) ) ) )
=> ( pairwise_nat @ R @ S2 ) ) ).
% pairwiseI
thf(fact_1111_pairwise__insert,axiom,
! [R3: lambda_lambda > lambda_lambda > $o,X5: lambda_lambda,S3: set_lambda_lambda] :
( ( pairwi8946569470594120789lambda @ R3 @ ( insert_lambda_lambda @ X5 @ S3 ) )
= ( ! [Y4: lambda_lambda] :
( ( ( member_lambda_lambda @ Y4 @ S3 )
& ( Y4 != X5 ) )
=> ( ( R3 @ X5 @ Y4 )
& ( R3 @ Y4 @ X5 ) ) )
& ( pairwi8946569470594120789lambda @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_1112_pairwise__insert,axiom,
! [R3: nat > nat > $o,X5: nat,S3: set_nat] :
( ( pairwise_nat @ R3 @ ( insert_nat @ X5 @ S3 ) )
= ( ! [Y4: nat] :
( ( ( member_nat @ Y4 @ S3 )
& ( Y4 != X5 ) )
=> ( ( R3 @ X5 @ Y4 )
& ( R3 @ Y4 @ X5 ) ) )
& ( pairwise_nat @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_1113_pairwise__empty,axiom,
! [P: lambda_lambda > lambda_lambda > $o] : ( pairwi8946569470594120789lambda @ P @ bot_bo1271416176687173302lambda ) ).
% pairwise_empty
thf(fact_1114_pairwise__empty,axiom,
! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% pairwise_empty
thf(fact_1115_pairwise__singleton,axiom,
! [P: lambda_lambda > lambda_lambda > $o,A3: lambda_lambda] : ( pairwi8946569470594120789lambda @ P @ ( insert_lambda_lambda @ A3 @ bot_bo1271416176687173302lambda ) ) ).
% pairwise_singleton
thf(fact_1116_pairwise__singleton,axiom,
! [P: nat > nat > $o,A3: nat] : ( pairwise_nat @ P @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_1117_normal__sub__rts_OCong_H_Ointros_I4_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( arr_nat @ Resid @ T )
=> ( ( member_nat @ U @ NN )
=> ( ( ( sources_nat @ Resid @ T )
= ( sources_nat @ Resid @ U ) )
=> ( normal_sub_Cong_nat2 @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.intros(4)
thf(fact_1118_normal__sub__rts_OCong_H_Ointros_I4_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( normal7263485545378587287lambda @ Resid @ NN @ T @ ( Resid @ T @ U ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.intros(4)
thf(fact_1119_normal__sub__rts_OCong_H_Ocases,axiom,
! [Resid: nat > nat > nat,NN: set_nat,A1: nat,A22: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat2 @ Resid @ NN @ A1 @ A22 )
=> ( ~ ( normal_sub_Cong_nat2 @ Resid @ NN @ A22 @ A1 )
=> ( ! [U4: nat] :
( ( normal_sub_Cong_nat2 @ Resid @ NN @ A1 @ U4 )
=> ~ ( normal_sub_Cong_nat2 @ Resid @ NN @ U4 @ A22 ) )
=> ( ~ ( ( member_nat @ ( Resid @ A1 @ A22 ) @ NN )
& ( member_nat @ ( Resid @ A22 @ A1 ) @ NN ) )
=> ~ ! [U4: nat] :
( ( A22
= ( Resid @ A1 @ U4 ) )
=> ( ( arr_nat @ Resid @ A1 )
=> ( ( member_nat @ U4 @ NN )
=> ( ( sources_nat @ Resid @ A1 )
!= ( sources_nat @ Resid @ U4 ) ) ) ) ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.cases
thf(fact_1120_normal__sub__rts_OCong_H_Ocases,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,A1: lambda_lambda,A22: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7263485545378587287lambda @ Resid @ NN @ A1 @ A22 )
=> ( ~ ( normal7263485545378587287lambda @ Resid @ NN @ A22 @ A1 )
=> ( ! [U4: lambda_lambda] :
( ( normal7263485545378587287lambda @ Resid @ NN @ A1 @ U4 )
=> ~ ( normal7263485545378587287lambda @ Resid @ NN @ U4 @ A22 ) )
=> ( ~ ( ( member_lambda_lambda @ ( Resid @ A1 @ A22 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ A22 @ A1 ) @ NN ) )
=> ~ ! [U4: lambda_lambda] :
( ( A22
= ( Resid @ A1 @ U4 ) )
=> ( ( arr_lambda_lambda @ Resid @ A1 )
=> ( ( member_lambda_lambda @ U4 @ NN )
=> ( ( source5627170679743844886lambda @ Resid @ A1 )
!= ( source5627170679743844886lambda @ Resid @ U4 ) ) ) ) ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.cases
thf(fact_1121_normal__sub__rts_OCong_H_Ointros_I3_J,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat,U: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( ( member_nat @ ( Resid @ T @ U ) @ NN )
& ( member_nat @ ( Resid @ U @ T ) @ NN ) )
=> ( normal_sub_Cong_nat2 @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong'.intros(3)
thf(fact_1122_normal__sub__rts_OCong_H_Ointros_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ T @ U ) @ NN )
& ( member_lambda_lambda @ ( Resid @ U @ T ) @ NN ) )
=> ( normal7263485545378587287lambda @ Resid @ NN @ T @ U ) ) ) ).
% normal_sub_rts.Cong'.intros(3)
thf(fact_1123_normal__sub__rts_OCong_H_Ointros_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7263485545378587287lambda @ Resid @ NN @ T @ U )
=> ( ( normal7263485545378587287lambda @ Resid @ NN @ U @ V )
=> ( normal7263485545378587287lambda @ Resid @ NN @ T @ V ) ) ) ) ).
% normal_sub_rts.Cong'.intros(2)
thf(fact_1124_normal__sub__rts_OCong_H_Ointros_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7263485545378587287lambda @ Resid @ NN @ T @ U )
=> ( normal7263485545378587287lambda @ Resid @ NN @ U @ T ) ) ) ).
% normal_sub_rts.Cong'.intros(1)
thf(fact_1125_normal__sub__rts_OCong_H__if,axiom,
! [Resid: nat > nat > nat,NN: set_nat,U: nat,U3: nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( member_nat @ U @ NN )
=> ( ( member_nat @ U3 @ NN )
=> ( ( ( member_nat @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U3 ) ) @ NN )
& ( member_nat @ ( Resid @ ( Resid @ T5 @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal_sub_Cong_nat2 @ Resid @ NN @ T @ T5 ) ) ) ) ) ).
% normal_sub_rts.Cong'_if
thf(fact_1126_normal__sub__rts_OCong_H__if,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U3 @ NN )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) )
=> ( normal7263485545378587287lambda @ Resid @ NN @ T @ T5 ) ) ) ) ) ).
% normal_sub_rts.Cong'_if
thf(fact_1127_normal__sub__rts_OCong__char,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
= ( normal7263485545378587287lambda @ Resid @ NN @ T @ T5 ) ) ) ).
% normal_sub_rts.Cong_char
thf(fact_1128_normal__sub__rts_OCong_H_Osimps,axiom,
! [Resid: nat > nat > nat,NN: set_nat,A1: nat,A22: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal_sub_Cong_nat2 @ Resid @ NN @ A1 @ A22 )
= ( ? [T4: nat,U2: nat] :
( ( A1 = U2 )
& ( A22 = T4 )
& ( normal_sub_Cong_nat2 @ Resid @ NN @ T4 @ U2 ) )
| ? [T4: nat,U2: nat,V4: nat] :
( ( A1 = T4 )
& ( A22 = V4 )
& ( normal_sub_Cong_nat2 @ Resid @ NN @ T4 @ U2 )
& ( normal_sub_Cong_nat2 @ Resid @ NN @ U2 @ V4 ) )
| ? [T4: nat,U2: nat] :
( ( A1 = T4 )
& ( A22 = U2 )
& ( member_nat @ ( Resid @ T4 @ U2 ) @ NN )
& ( member_nat @ ( Resid @ U2 @ T4 ) @ NN ) )
| ? [T4: nat,U2: nat] :
( ( A1 = T4 )
& ( A22
= ( Resid @ T4 @ U2 ) )
& ( arr_nat @ Resid @ T4 )
& ( member_nat @ U2 @ NN )
& ( ( sources_nat @ Resid @ T4 )
= ( sources_nat @ Resid @ U2 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_1129_normal__sub__rts_OCong_H_Osimps,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,A1: lambda_lambda,A22: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7263485545378587287lambda @ Resid @ NN @ A1 @ A22 )
= ( ? [T4: lambda_lambda,U2: lambda_lambda] :
( ( A1 = U2 )
& ( A22 = T4 )
& ( normal7263485545378587287lambda @ Resid @ NN @ T4 @ U2 ) )
| ? [T4: lambda_lambda,U2: lambda_lambda,V4: lambda_lambda] :
( ( A1 = T4 )
& ( A22 = V4 )
& ( normal7263485545378587287lambda @ Resid @ NN @ T4 @ U2 )
& ( normal7263485545378587287lambda @ Resid @ NN @ U2 @ V4 ) )
| ? [T4: lambda_lambda,U2: lambda_lambda] :
( ( A1 = T4 )
& ( A22 = U2 )
& ( member_lambda_lambda @ ( Resid @ T4 @ U2 ) @ NN )
& ( member_lambda_lambda @ ( Resid @ U2 @ T4 ) @ NN ) )
| ? [T4: lambda_lambda,U2: lambda_lambda] :
( ( A1 = T4 )
& ( A22
= ( Resid @ T4 @ U2 ) )
& ( arr_lambda_lambda @ Resid @ T4 )
& ( member_lambda_lambda @ U2 @ NN )
& ( ( source5627170679743844886lambda @ Resid @ T4 )
= ( source5627170679743844886lambda @ Resid @ U2 ) ) ) ) ) ) ).
% normal_sub_rts.Cong'.simps
thf(fact_1130_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 )
=> ~ ( ( T7 != bot_bot_set_nat )
=> ( ! [T2: nat] :
( ( member_nat @ T2 @ T7 )
=> ! [T8: nat] :
( ( member_nat @ T8 @ T7 )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T2 @ T8 ) ) )
=> ~ ! [T2: nat] :
( ( member_nat @ T2 @ T7 )
=> ! [T8: nat] :
( ( normal_sub_Cong_nat @ Resid @ NN @ T8 @ T2 )
=> ( member_nat @ T8 @ T7 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_1131_normal__sub__rts_Ois__Cong__classE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
=> ~ ( ( T7 != bot_bo1271416176687173302lambda )
=> ( ! [T2: lambda_lambda] :
( ( member_lambda_lambda @ T2 @ T7 )
=> ! [T8: lambda_lambda] :
( ( member_lambda_lambda @ T8 @ T7 )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T2 @ T8 ) ) )
=> ~ ! [T2: lambda_lambda] :
( ( member_lambda_lambda @ T2 @ T7 )
=> ! [T8: lambda_lambda] :
( ( normal3156941126761417472lambda @ Resid @ NN @ T8 @ T2 )
=> ( member_lambda_lambda @ T8 @ T7 ) ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classE
thf(fact_1132_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( T7 != bot_bot_set_nat )
=> ( ! [T3: nat,T9: nat] :
( ( member_nat @ T3 @ T7 )
=> ( ( member_nat @ T9 @ T7 )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T3 @ T9 ) ) )
=> ( ! [T3: nat,T9: nat] :
( ( member_nat @ T3 @ T7 )
=> ( ( normal_sub_Cong_nat @ Resid @ NN @ T9 @ T3 )
=> ( member_nat @ T9 @ T7 ) ) )
=> ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_1133_normal__sub__rts_Ois__Cong__classI_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( T7 != bot_bo1271416176687173302lambda )
=> ( ! [T3: lambda_lambda,T9: lambda_lambda] :
( ( member_lambda_lambda @ T3 @ T7 )
=> ( ( member_lambda_lambda @ T9 @ T7 )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T3 @ T9 ) ) )
=> ( ! [T3: lambda_lambda,T9: lambda_lambda] :
( ( member_lambda_lambda @ T3 @ T7 )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T9 @ T3 )
=> ( member_lambda_lambda @ T9 @ T7 ) ) )
=> ( normal7899599941841162628lambda @ Resid @ NN @ T7 ) ) ) ) ) ).
% normal_sub_rts.is_Cong_classI'
thf(fact_1134_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 )
=> ( T7 != bot_bot_set_nat ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_1135_normal__sub__rts_OCong__class__is__nonempty,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
=> ( T7 != bot_bo1271416176687173302lambda ) ) ) ).
% normal_sub_rts.Cong_class_is_nonempty
thf(fact_1136_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 )
=> ( ( member_nat @ T @ T7 )
=> ( arr_nat @ Resid @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_1137_normal__sub__rts_OCong__class__memb__is__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
=> ( ( member_lambda_lambda @ T @ T7 )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_is_arr
thf(fact_1138_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat,T: nat,T5: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 )
=> ( ( member_nat @ T @ T7 )
=> ( ( member_nat @ T5 @ T7 )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T @ T5 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_1139_normal__sub__rts_OCong__class__membs__are__Cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
=> ( ( member_lambda_lambda @ T @ T7 )
=> ( ( member_lambda_lambda @ T5 @ T7 )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_membs_are_Cong
thf(fact_1140_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat,U7: set_nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ U7 )
=> ( ( ( inf_inf_set_nat @ T7 @ U7 )
!= bot_bot_set_nat )
=> ( T7 = U7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_1141_normal__sub__rts_OCong__class__eqI_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda,U7: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ U7 )
=> ( ( ( inf_in1766959559283430712lambda @ T7 @ U7 )
!= bot_bo1271416176687173302lambda )
=> ( T7 = U7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI'
thf(fact_1142_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 )
=> ( ( member_nat @ T @ T7 )
=> ( normal_sub_Cong_nat @ Resid @ NN @ T @ ( normal3802779619712508127ep_nat @ T7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_1143_normal__sub__rts_OCong__class__memb__Cong__rep,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
=> ( ( member_lambda_lambda @ T @ T7 )
=> ( normal3156941126761417472lambda @ Resid @ NN @ T @ ( normal6971565589081524635lambda @ T7 ) ) ) ) ) ).
% normal_sub_rts.Cong_class_memb_Cong_rep
thf(fact_1144_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 )
=> ( member_nat @ ( normal3802779619712508127ep_nat @ T7 ) @ T7 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_1145_normal__sub__rts_Orep__in__Cong__class,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
=> ( member_lambda_lambda @ ( normal6971565589081524635lambda @ T7 ) @ T7 ) ) ) ).
% normal_sub_rts.rep_in_Cong_class
thf(fact_1146_normal__sub__rts_OCong__class__rep,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
=> ( ( normal348130670494188410lambda @ Resid @ NN @ ( normal6971565589081524635lambda @ T7 ) )
= T7 ) ) ) ).
% normal_sub_rts.Cong_class_rep
thf(fact_1147_normal__sub__rts_Ois__Cong__classI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( normal7899599941841162628lambda @ Resid @ NN @ ( normal348130670494188410lambda @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.is_Cong_classI
thf(fact_1148_normal__sub__rts_Oarr__in__Cong__class,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T: nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( arr_nat @ Resid @ T )
=> ( member_nat @ T @ ( normal4883526030722344896ss_nat @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.arr_in_Cong_class
thf(fact_1149_normal__sub__rts_Oarr__in__Cong__class,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( member_lambda_lambda @ T @ ( normal348130670494188410lambda @ Resid @ NN @ T ) ) ) ) ).
% normal_sub_rts.arr_in_Cong_class
thf(fact_1150_normal__sub__rts_OCong__class__eqI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal3156941126761417472lambda @ Resid @ NN @ T @ T5 )
=> ( ( normal348130670494188410lambda @ Resid @ NN @ T )
= ( normal348130670494188410lambda @ Resid @ NN @ T5 ) ) ) ) ).
% normal_sub_rts.Cong_class_eqI
thf(fact_1151_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: nat > nat > nat,NN: set_nat,T7: set_nat] :
( ( normal_sub_rts_nat @ Resid @ NN )
=> ( ( normal8642108778142466614ss_nat @ Resid @ NN @ T7 )
= ( ? [T4: nat] :
( ( member_nat @ T4 @ T7 )
& ( T7
= ( normal4883526030722344896ss_nat @ Resid @ NN @ T4 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_1152_normal__sub__rts_Ois__Cong__class__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T7: set_lambda_lambda] :
( ( normal1556980139574411093lambda @ Resid @ NN )
=> ( ( normal7899599941841162628lambda @ Resid @ NN @ T7 )
= ( ? [T4: lambda_lambda] :
( ( member_lambda_lambda @ T4 @ T7 )
& ( T7
= ( normal348130670494188410lambda @ Resid @ NN @ T4 ) ) ) ) ) ) ).
% normal_sub_rts.is_Cong_class_def
thf(fact_1153_weakly__extensional__rts_OseqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( seq_lambda_lambda @ Resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ Resid @ U )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( trg_lambda_lambda @ Resid @ T )
!= ( weakly4554751870655413501lambda @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.seqE\<^sub>W\<^sub>E
thf(fact_1154_weakly__extensional__rts_OseqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ U )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( ( trg_lambda_lambda @ Resid @ T )
= ( weakly4554751870655413501lambda @ Resid @ U ) )
=> ( seq_lambda_lambda @ Resid @ T @ U ) ) ) ) ) ).
% weakly_extensional_rts.seqI\<^sub>W\<^sub>E
thf(fact_1155_weakly__extensional__rts__axioms,axiom,
weakly7886464935474173793lambda @ lambda_resid ).
% weakly_extensional_rts_axioms
thf(fact_1156_weakly__extensional__rts_Osrc__join__of_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( join_o1386658102433174083lambda @ Resid @ T @ U @ V )
=> ( ( weakly4554751870655413501lambda @ Resid @ U )
= ( weakly4554751870655413501lambda @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.src_join_of(2)
thf(fact_1157_weakly__extensional__rts_Osrc__join__of_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( join_o1386658102433174083lambda @ Resid @ T @ U @ V )
=> ( ( weakly4554751870655413501lambda @ Resid @ T )
= ( weakly4554751870655413501lambda @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.src_join_of(1)
thf(fact_1158_weakly__extensional__rts_Osrc__src,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( weakly4554751870655413501lambda @ Resid @ ( weakly4554751870655413501lambda @ Resid @ T ) )
= ( weakly4554751870655413501lambda @ Resid @ T ) ) ) ).
% weakly_extensional_rts.src_src
thf(fact_1159_weakly__extensional__rts_Oapex__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( trg_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
= ( trg_lambda_lambda @ Resid @ ( Resid @ U @ T ) ) ) ) ).
% weakly_extensional_rts.apex_sym
thf(fact_1160_weakly__extensional__rts_Otrg__trg,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( trg_lambda_lambda @ Resid @ ( trg_lambda_lambda @ Resid @ T ) )
= ( trg_lambda_lambda @ Resid @ T ) ) ) ).
% weakly_extensional_rts.trg_trg
thf(fact_1161_weakly__extensional__rts_Oweak__extensionality,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ Resid @ T )
=> ( ( ide_lambda_lambda @ Resid @ U )
=> ( T = U ) ) ) ) ) ).
% weakly_extensional_rts.weak_extensionality
thf(fact_1162_weakly__extensional__rts_Otrg__join__of_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( join_o1386658102433174083lambda @ Resid @ T @ U @ V )
=> ( ( trg_lambda_lambda @ Resid @ ( Resid @ U @ T ) )
= ( trg_lambda_lambda @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.trg_join_of(2)
thf(fact_1163_weakly__extensional__rts_Otrg__join__of_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( join_o1386658102433174083lambda @ Resid @ T @ U @ V )
=> ( ( trg_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
= ( trg_lambda_lambda @ Resid @ V ) ) ) ) ).
% weakly_extensional_rts.trg_join_of(1)
thf(fact_1164_weakly__extensional__rts_Osrc__composite__of,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ V )
=> ( ( weakly4554751870655413501lambda @ Resid @ V )
= ( weakly4554751870655413501lambda @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.src_composite_of
thf(fact_1165_weakly__extensional__rts_Ocoinitial__ide__are__eq,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,A2: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( ide_lambda_lambda @ Resid @ A2 )
=> ( ( coinit1392077014771647542lambda @ Resid @ A @ A2 )
=> ( A = A2 ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_ide_are_eq
thf(fact_1166_weakly__extensional__rts_Oresid__ide_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( coinit1392077014771647542lambda @ Resid @ A @ T )
=> ( ( Resid @ T @ A )
= T ) ) ) ) ).
% weakly_extensional_rts.resid_ide(1)
thf(fact_1167_weakly__extensional__rts_Otrg__composite__of,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ V )
=> ( ( trg_lambda_lambda @ Resid @ V )
= ( trg_lambda_lambda @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.trg_composite_of
thf(fact_1168_weakly__extensional__rts_Osrc__trg,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( weakly4554751870655413501lambda @ Resid @ ( trg_lambda_lambda @ Resid @ T ) )
= ( trg_lambda_lambda @ Resid @ T ) ) ) ).
% weakly_extensional_rts.src_trg
thf(fact_1169_weakly__extensional__rts_Otrg__src,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( trg_lambda_lambda @ Resid @ ( weakly4554751870655413501lambda @ Resid @ T ) )
= ( weakly4554751870655413501lambda @ Resid @ T ) ) ) ).
% weakly_extensional_rts.trg_src
thf(fact_1170_weakly__extensional__rts_Oarr__src__iff__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ ( weakly4554751870655413501lambda @ Resid @ T ) )
= ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% weakly_extensional_rts.arr_src_iff_arr
thf(fact_1171_weakly__extensional__rts_Ocon__imp__eq__src,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( weakly4554751870655413501lambda @ Resid @ T )
= ( weakly4554751870655413501lambda @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.con_imp_eq_src
thf(fact_1172_weakly__extensional__rts_Oarr__trg__iff__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ ( trg_lambda_lambda @ Resid @ T ) )
= ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% weakly_extensional_rts.arr_trg_iff_arr
thf(fact_1173_weakly__extensional__rts_Otrg__resid__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( trg_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
= ( trg_lambda_lambda @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% weakly_extensional_rts.trg_resid_sym
thf(fact_1174_weakly__extensional__rts_Oarr__has__un__target,axiom,
! [Resid: nat > nat > nat,T: nat] :
( ( weakly1361546452046171801ts_nat @ Resid )
=> ( ( arr_nat @ Resid @ T )
=> ? [X: nat] :
( ( member_nat @ X @ ( targets_nat @ Resid @ T ) )
& ! [Y: nat] :
( ( member_nat @ Y @ ( targets_nat @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_target
thf(fact_1175_weakly__extensional__rts_Oarr__has__un__target,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( target5490931411331770124lambda @ Resid @ T ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_target
thf(fact_1176_weakly__extensional__rts_Osrc__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( weakly4554751870655413501lambda @ Resid @ A )
= A ) ) ) ).
% weakly_extensional_rts.src_ide
thf(fact_1177_weakly__extensional__rts_Oarr__has__un__source,axiom,
! [Resid: nat > nat > nat,T: nat] :
( ( weakly1361546452046171801ts_nat @ Resid )
=> ( ( arr_nat @ Resid @ T )
=> ? [X: nat] :
( ( member_nat @ X @ ( sources_nat @ Resid @ T ) )
& ! [Y: nat] :
( ( member_nat @ Y @ ( sources_nat @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_source
thf(fact_1178_weakly__extensional__rts_Oarr__has__un__source,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( source5627170679743844886lambda @ Resid @ T ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( Y = X ) ) ) ) ) ).
% weakly_extensional_rts.arr_has_un_source
thf(fact_1179_weakly__extensional__rts_Otrg__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( trg_lambda_lambda @ Resid @ A )
= A ) ) ) ).
% weakly_extensional_rts.trg_ide
thf(fact_1180_weakly__extensional__rts_Ocon__ide__are__eq,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,A2: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( ide_lambda_lambda @ Resid @ A2 )
=> ( ( con_lambda_lambda @ Resid @ A @ A2 )
=> ( A = A2 ) ) ) ) ) ).
% weakly_extensional_rts.con_ide_are_eq
thf(fact_1181_weakly__extensional__rts_Oide__iff__trg__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ A )
=> ( ( ide_lambda_lambda @ Resid @ A )
= ( ( trg_lambda_lambda @ Resid @ A )
= A ) ) ) ) ).
% weakly_extensional_rts.ide_iff_trg_self
thf(fact_1182_weakly__extensional__rts_Osrc__eqI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( con_lambda_lambda @ Resid @ A @ T )
=> ( ( weakly4554751870655413501lambda @ Resid @ T )
= A ) ) ) ) ).
% weakly_extensional_rts.src_eqI
thf(fact_1183_weakly__extensional__rts_Oide__src,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ide_lambda_lambda @ Resid @ ( weakly4554751870655413501lambda @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.ide_src
thf(fact_1184_weakly__extensional__rts_Oide__iff__src__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ A )
=> ( ( ide_lambda_lambda @ Resid @ A )
= ( ( weakly4554751870655413501lambda @ Resid @ A )
= A ) ) ) ) ).
% weakly_extensional_rts.ide_iff_src_self
thf(fact_1185_weakly__extensional__rts_Osrc__in__sources,axiom,
! [Resid: nat > nat > nat,T: nat] :
( ( weakly1361546452046171801ts_nat @ Resid )
=> ( ( arr_nat @ Resid @ T )
=> ( member_nat @ ( weakly1695633097759462397rc_nat @ Resid @ T ) @ ( sources_nat @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.src_in_sources
thf(fact_1186_weakly__extensional__rts_Osrc__in__sources,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( member_lambda_lambda @ ( weakly4554751870655413501lambda @ Resid @ T ) @ ( source5627170679743844886lambda @ Resid @ T ) ) ) ) ).
% weakly_extensional_rts.src_in_sources
thf(fact_1187_weakly__extensional__rts_Osrc__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( weakly4554751870655413501lambda @ Resid @ ( Resid @ T @ U ) )
= ( trg_lambda_lambda @ Resid @ U ) ) ) ) ).
% weakly_extensional_rts.src_resid
thf(fact_1188_weakly__extensional__rts_Oresid__ide_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( coinit1392077014771647542lambda @ Resid @ A @ T )
=> ( ( Resid @ A @ T )
= ( trg_lambda_lambda @ Resid @ T ) ) ) ) ) ).
% weakly_extensional_rts.resid_ide(2)
thf(fact_1189_weakly__extensional__rts_Ocoinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( coinit1392077014771647542lambda @ Resid @ T @ U )
= ( ( arr_lambda_lambda @ Resid @ T )
& ( arr_lambda_lambda @ Resid @ U )
& ( ( weakly4554751870655413501lambda @ Resid @ T )
= ( weakly4554751870655413501lambda @ Resid @ U ) ) ) ) ) ).
% weakly_extensional_rts.coinitial_iff\<^sub>W\<^sub>E
thf(fact_1190_weakly__extensional__rts_OcoinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( ( weakly4554751870655413501lambda @ Resid @ T )
= ( weakly4554751870655413501lambda @ Resid @ U ) )
=> ( coinit1392077014771647542lambda @ Resid @ T @ U ) ) ) ) ).
% weakly_extensional_rts.coinitialI\<^sub>W\<^sub>E
thf(fact_1191_weakly__extensional__rts_OcoinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( coinit1392077014771647542lambda @ Resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( arr_lambda_lambda @ Resid @ U )
=> ( ( weakly4554751870655413501lambda @ Resid @ T )
!= ( weakly4554751870655413501lambda @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.coinitialE\<^sub>W\<^sub>E
thf(fact_1192_weakly__extensional__rts_Ocoterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( coterm4142657152830665724lambda @ Resid @ T @ U )
= ( ( arr_lambda_lambda @ Resid @ T )
& ( arr_lambda_lambda @ Resid @ U )
& ( ( trg_lambda_lambda @ Resid @ T )
= ( trg_lambda_lambda @ Resid @ U ) ) ) ) ) ).
% weakly_extensional_rts.coterminal_iff\<^sub>W\<^sub>E
thf(fact_1193_weakly__extensional__rts_OcoterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( ( trg_lambda_lambda @ Resid @ T )
= ( trg_lambda_lambda @ Resid @ U ) )
=> ( coterm4142657152830665724lambda @ Resid @ T @ U ) ) ) ) ).
% weakly_extensional_rts.coterminalI\<^sub>W\<^sub>E
thf(fact_1194_weakly__extensional__rts_OcoterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( weakly7886464935474173793lambda @ Resid )
=> ( ( coterm4142657152830665724lambda @ Resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( arr_lambda_lambda @ Resid @ U )
=> ( ( trg_lambda_lambda @ Resid @ T )
!= ( trg_lambda_lambda @ Resid @ U ) ) ) ) ) ) ).
% weakly_extensional_rts.coterminalE\<^sub>W\<^sub>E
thf(fact_1195_transformation__def,axiom,
( transf2236888680007955973lambda
= ( ^ [A7: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,G2: lambda_lambda > lambda_lambda,Tau2: lambda_lambda > lambda_lambda] :
( ( weakly7886464935474173793lambda @ A7 )
& ( weakly7886464935474173793lambda @ B4 )
& ( simula5235733974575307523lambda @ A7 @ B4 @ F4 )
& ( simula5235733974575307523lambda @ A7 @ B4 @ G2 )
& ( transf356316610583137064lambda @ A7 @ B4 @ F4 @ G2 @ Tau2 ) ) ) ) ).
% transformation_def
thf(fact_1196_transformation_Ointro,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda] :
( ( weakly7886464935474173793lambda @ A3 )
=> ( ( weakly7886464935474173793lambda @ B3 )
=> ( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( simula5235733974575307523lambda @ A3 @ B3 @ G )
=> ( ( transf356316610583137064lambda @ A3 @ B3 @ F @ G @ Tau )
=> ( transf2236888680007955973lambda @ A3 @ B3 @ F @ G @ Tau ) ) ) ) ) ) ).
% transformation.intro
thf(fact_1197_simulation_Opreserves__trg,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( arr_lambda_lambda @ A3 @ T )
=> ( ( F @ ( trg_lambda_lambda @ A3 @ T ) )
= ( trg_lambda_lambda @ B3 @ ( F @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1198_simulation_Oextensional,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ~ ( arr_lambda_lambda @ A3 @ T )
=> ( ( F @ T )
= ( partia1461432607296028304lambda @ B3 ) ) ) ) ).
% simulation.extensional
thf(fact_1199_simulation_Opreserves__joins,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( join_o1386658102433174083lambda @ A3 @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B3 @ ( F @ T ) @ ( F @ U ) @ ( F @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1200_simulation_Opreserves__prfx,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( ide_lambda_lambda @ A3 @ ( A3 @ T @ U ) )
=> ( ide_lambda_lambda @ B3 @ ( B3 @ ( F @ T ) @ ( F @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1201_simulation_Opreserves__cong,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( ( ide_lambda_lambda @ A3 @ ( A3 @ T @ U ) )
& ( ide_lambda_lambda @ A3 @ ( A3 @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B3 @ ( B3 @ ( F @ T ) @ ( F @ U ) ) )
& ( ide_lambda_lambda @ B3 @ ( B3 @ ( F @ U ) @ ( F @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1202_simulation_Opreserves__ide,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( ide_lambda_lambda @ A3 @ A )
=> ( ide_lambda_lambda @ B3 @ ( F @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1203_simulation_Opreserves__con,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( con_lambda_lambda @ A3 @ T @ U )
=> ( con_lambda_lambda @ B3 @ ( F @ T ) @ ( F @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1204_simulation_Opreserves__reflects__arr,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( arr_lambda_lambda @ B3 @ ( F @ T ) )
= ( arr_lambda_lambda @ A3 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1205_simulation_Opreserves__composites,axiom,
! [A3: lambda_lambda > lambda_lambda > lambda_lambda,B3: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula5235733974575307523lambda @ A3 @ B3 @ F )
=> ( ( compos9183582412961930448lambda @ A3 @ T @ U @ V )
=> ( compos9183582412961930448lambda @ B3 @ ( F @ T ) @ ( F @ U ) @ ( F @ V ) ) ) ) ).
% simulation.preserves_composites
thf(fact_1206_extensional__rts__with__joins_Ojoinable__iff__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( extens4217917907808584361lambda @ Resid )
=> ( ( joinab2519788767067900224lambda @ Resid @ T @ U )
= ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% extensional_rts_with_joins.joinable_iff_con
thf(fact_1207_Gcd__0__iff,axiom,
! [A3: set_nat] :
( ( ( gcd_Gcd_nat @ A3 )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_0_iff
thf(fact_1208_Gcd__empty,axiom,
( ( gcd_Gcd_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Gcd_empty
thf(fact_1209_rts_Ocoterminal__def,axiom,
! [Resid: nat > nat > nat,T: nat,U: nat] :
( ( rts_nat @ Resid )
=> ( ( coterminal_nat @ Resid @ T @ U )
= ( ( inf_inf_set_nat @ ( targets_nat @ Resid @ T ) @ ( targets_nat @ Resid @ U ) )
!= bot_bot_set_nat ) ) ) ).
% rts.coterminal_def
thf(fact_1210_rts_Ocoterminal__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( coterm4142657152830665724lambda @ Resid @ T @ U )
= ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ Resid @ T ) @ ( target5490931411331770124lambda @ Resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ) ).
% rts.coterminal_def
thf(fact_1211_rts_OseqE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( seq_lambda_lambda @ Resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( arr_lambda_lambda @ Resid @ U )
=> ( ( target5490931411331770124lambda @ Resid @ T )
!= ( source5627170679743844886lambda @ Resid @ U ) ) ) ) ) ) ).
% rts.seqE
thf(fact_1212_rts__axioms,axiom,
rts_lambda_lambda @ lambda_resid ).
% rts_axioms
thf(fact_1213_rts_Ocomposite__of__source__arr,axiom,
! [Resid: nat > nat > nat,T: nat,A: nat] :
( ( rts_nat @ Resid )
=> ( ( arr_nat @ Resid @ T )
=> ( ( member_nat @ A @ ( sources_nat @ Resid @ T ) )
=> ( composite_of_nat @ Resid @ A @ T @ T ) ) ) ) ).
% rts.composite_of_source_arr
thf(fact_1214_rts_Ocomposite__of__source__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,A: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( compos9183582412961930448lambda @ Resid @ A @ T @ T ) ) ) ) ).
% rts.composite_of_source_arr
thf(fact_1215_rts_Otrg__in__targets,axiom,
! [Resid: nat > nat > nat,T: nat] :
( ( rts_nat @ Resid )
=> ( ( arr_nat @ Resid @ T )
=> ( member_nat @ ( trg_nat @ Resid @ T ) @ ( targets_nat @ Resid @ T ) ) ) ) ).
% rts.trg_in_targets
thf(fact_1216_rts_Otrg__in__targets,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( member_lambda_lambda @ ( trg_lambda_lambda @ Resid @ T ) @ ( target5490931411331770124lambda @ Resid @ T ) ) ) ) ).
% rts.trg_in_targets
thf(fact_1217_rts_Ocomposite__of__arr__target,axiom,
! [Resid: nat > nat > nat,T: nat,B: nat] :
( ( rts_nat @ Resid )
=> ( ( arr_nat @ Resid @ T )
=> ( ( member_nat @ B @ ( targets_nat @ Resid @ T ) )
=> ( composite_of_nat @ Resid @ T @ B @ T ) ) ) ) ).
% rts.composite_of_arr_target
thf(fact_1218_rts_Ocomposite__of__arr__target,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,B: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( compos9183582412961930448lambda @ Resid @ T @ B @ T ) ) ) ) ).
% rts.composite_of_arr_target
thf(fact_1219_rts_Osources__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( source5627170679743844886lambda @ Resid @ ( Resid @ T @ U ) )
= ( target5490931411331770124lambda @ Resid @ U ) ) ) ) ).
% rts.sources_resid
thf(fact_1220_rts_Osources__eqI,axiom,
! [Resid: nat > nat > nat,T: nat,T5: nat] :
( ( rts_nat @ Resid )
=> ( ( ( inf_inf_set_nat @ ( sources_nat @ Resid @ T ) @ ( sources_nat @ Resid @ T5 ) )
!= bot_bot_set_nat )
=> ( ( sources_nat @ Resid @ T )
= ( sources_nat @ Resid @ T5 ) ) ) ) ).
% rts.sources_eqI
thf(fact_1221_rts_Osources__eqI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ Resid @ T ) @ ( source5627170679743844886lambda @ Resid @ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ T5 ) ) ) ) ).
% rts.sources_eqI
thf(fact_1222_rts_Otargets__eqI,axiom,
! [Resid: nat > nat > nat,T: nat,T5: nat] :
( ( rts_nat @ Resid )
=> ( ( ( inf_inf_set_nat @ ( targets_nat @ Resid @ T ) @ ( targets_nat @ Resid @ T5 ) )
!= bot_bot_set_nat )
=> ( ( targets_nat @ Resid @ T )
= ( targets_nat @ Resid @ T5 ) ) ) ) ).
% rts.targets_eqI
thf(fact_1223_rts_Otargets__eqI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,T5: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ Resid @ T ) @ ( target5490931411331770124lambda @ Resid @ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ Resid @ T )
= ( target5490931411331770124lambda @ Resid @ T5 ) ) ) ) ).
% rts.targets_eqI
thf(fact_1224_rts_Oarr__iff__has__source,axiom,
! [Resid: nat > nat > nat,T: nat] :
( ( rts_nat @ Resid )
=> ( ( arr_nat @ Resid @ T )
= ( ( sources_nat @ Resid @ T )
!= bot_bot_set_nat ) ) ) ).
% rts.arr_iff_has_source
thf(fact_1225_rts_Oarr__iff__has__source,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
= ( ( source5627170679743844886lambda @ Resid @ T )
!= bot_bo1271416176687173302lambda ) ) ) ).
% rts.arr_iff_has_source
thf(fact_1226_rts__with__joins__def,axiom,
( rts_wi2791606683991334555lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( rts_lambda_lambda @ Resid2 )
& ( rts_wi6344169777088970040lambda @ Resid2 ) ) ) ) ).
% rts_with_joins_def
thf(fact_1227_rts__with__joins_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( rts_wi6344169777088970040lambda @ Resid )
=> ( rts_wi2791606683991334555lambda @ Resid ) ) ) ).
% rts_with_joins.intro
thf(fact_1228_rts_Ocong__subst__left_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ T5 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ T5 @ T ) ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T5 @ U ) ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ ( Resid @ T5 @ U ) @ ( Resid @ T @ U ) ) ) ) ) ) ) ).
% rts.cong_subst_left(2)
thf(fact_1229_rts_Ocong__subst__left_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ T5 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ T5 @ T ) ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ T5 @ U ) ) ) ) ).
% rts.cong_subst_left(1)
thf(fact_1230_rts_Ocong__subst__right_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ U @ U3 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ U3 @ U ) ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U3 ) ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T @ U ) ) ) ) ) ) ) ).
% rts.cong_subst_right(2)
thf(fact_1231_rts_Ocong__subst__right_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ U @ U3 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ U3 @ U ) ) )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ T @ U3 ) ) ) ) ).
% rts.cong_subst_right(1)
thf(fact_1232_rts_Ocon__target,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
=> ( ( con_lambda_lambda @ Resid @ U @ V )
=> ( con_lambda_lambda @ Resid @ ( Resid @ T @ U ) @ ( Resid @ V @ U ) ) ) ) ) ).
% rts.con_target
thf(fact_1233_rts_Oresid__arr__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( con_lambda_lambda @ Resid @ T @ A )
=> ( ( Resid @ T @ A )
= T ) ) ) ) ).
% rts.resid_arr_ide
thf(fact_1234_rts_Oresid__ide__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( con_lambda_lambda @ Resid @ A @ T )
=> ( ide_lambda_lambda @ Resid @ ( Resid @ A @ T ) ) ) ) ) ).
% rts.resid_ide_arr
thf(fact_1235_rts_Oprfx__implies__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
=> ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% rts.prfx_implies_con
thf(fact_1236_rts_Ocon__imp__coinitial__ax,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ? [A4: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A4 )
& ( con_lambda_lambda @ Resid @ A4 @ T )
& ( con_lambda_lambda @ Resid @ A4 @ U ) ) ) ) ).
% rts.con_imp_coinitial_ax
thf(fact_1237_rts_Oide__imp__con__iff__cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ T )
=> ( ( ide_lambda_lambda @ Resid @ U )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
= ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ U @ T ) ) ) ) ) ) ) ).
% rts.ide_imp_con_iff_cong
thf(fact_1238_rts_Ocon__transitive__on__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( ide_lambda_lambda @ Resid @ B )
=> ( ( ide_lambda_lambda @ Resid @ C )
=> ( ( con_lambda_lambda @ Resid @ A @ B )
=> ( ( con_lambda_lambda @ Resid @ B @ C )
=> ( con_lambda_lambda @ Resid @ A @ C ) ) ) ) ) ) ) ).
% rts.con_transitive_on_ide
thf(fact_1239_rts_Ocong__reflexive,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ T ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ T @ T ) ) ) ) ) ).
% rts.cong_reflexive
thf(fact_1240_rts_Oprfx__reflexive,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ide_lambda_lambda @ Resid @ ( Resid @ T @ T ) ) ) ) ).
% rts.prfx_reflexive
thf(fact_1241_rts_Osource__is__ide,axiom,
! [Resid: nat > nat > nat,A: nat,T: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ A @ ( sources_nat @ Resid @ T ) )
=> ( ide_nat @ Resid @ A ) ) ) ).
% rts.source_is_ide
thf(fact_1242_rts_Osource__is__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( ide_lambda_lambda @ Resid @ A ) ) ) ).
% rts.source_is_ide
thf(fact_1243_rts_Osources__are__cong,axiom,
! [Resid: nat > nat > nat,A: nat,T: nat,A2: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ A @ ( sources_nat @ Resid @ T ) )
=> ( ( member_nat @ A2 @ ( sources_nat @ Resid @ T ) )
=> ( ( ide_nat @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_nat @ Resid @ ( Resid @ A2 @ A ) ) ) ) ) ) ).
% rts.sources_are_cong
thf(fact_1244_rts_Osources__are__cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda,A2: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ A2 @ A ) ) ) ) ) ) ).
% rts.sources_are_cong
thf(fact_1245_rts_Osources__cong__closed,axiom,
! [Resid: nat > nat > nat,A: nat,T: nat,A2: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ A @ ( sources_nat @ Resid @ T ) )
=> ( ( ( ide_nat @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_nat @ Resid @ ( Resid @ A2 @ A ) ) )
=> ( member_nat @ A2 @ ( sources_nat @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_1246_rts_Osources__cong__closed,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda,A2: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ A @ A2 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ A2 @ A ) ) )
=> ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ Resid @ T ) ) ) ) ) ).
% rts.sources_cong_closed
thf(fact_1247_rts_Otarget__is__ide,axiom,
! [Resid: nat > nat > nat,A: nat,T: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ A @ ( targets_nat @ Resid @ T ) )
=> ( ide_nat @ Resid @ A ) ) ) ).
% rts.target_is_ide
thf(fact_1248_rts_Otarget__is__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ A @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( ide_lambda_lambda @ Resid @ A ) ) ) ).
% rts.target_is_ide
thf(fact_1249_rts_Otargets__are__cong,axiom,
! [Resid: nat > nat > nat,B: nat,T: nat,B2: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ B @ ( targets_nat @ Resid @ T ) )
=> ( ( member_nat @ B2 @ ( targets_nat @ Resid @ T ) )
=> ( ( ide_nat @ Resid @ ( Resid @ B @ B2 ) )
& ( ide_nat @ Resid @ ( Resid @ B2 @ B ) ) ) ) ) ) ).
% rts.targets_are_cong
thf(fact_1250_rts_Otargets__are__cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ B @ B2 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ B2 @ B ) ) ) ) ) ) ).
% rts.targets_are_cong
thf(fact_1251_rts_Otargets__cong__closed,axiom,
! [Resid: nat > nat > nat,B: nat,T: nat,B2: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ B @ ( targets_nat @ Resid @ T ) )
=> ( ( ( ide_nat @ Resid @ ( Resid @ B @ B2 ) )
& ( ide_nat @ Resid @ ( Resid @ B2 @ B ) ) )
=> ( member_nat @ B2 @ ( targets_nat @ Resid @ T ) ) ) ) ) ).
% rts.targets_cong_closed
thf(fact_1252_rts_Otargets__cong__closed,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ B @ B2 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ B2 @ B ) ) )
=> ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ Resid @ T ) ) ) ) ) ).
% rts.targets_cong_closed
thf(fact_1253_rts_Osources__are__con,axiom,
! [Resid: nat > nat > nat,A: nat,T: nat,A2: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ A @ ( sources_nat @ Resid @ T ) )
=> ( ( member_nat @ A2 @ ( sources_nat @ Resid @ T ) )
=> ( con_nat @ Resid @ A @ A2 ) ) ) ) ).
% rts.sources_are_con
thf(fact_1254_rts_Osources__are__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda,A2: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( con_lambda_lambda @ Resid @ A @ A2 ) ) ) ) ).
% rts.sources_are_con
thf(fact_1255_rts_Otargets__resid__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( target5490931411331770124lambda @ Resid @ ( Resid @ T @ U ) )
= ( target5490931411331770124lambda @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% rts.targets_resid_sym
thf(fact_1256_rts_Otargets__are__con,axiom,
! [Resid: nat > nat > nat,B: nat,T: nat,B2: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ B @ ( targets_nat @ Resid @ T ) )
=> ( ( member_nat @ B2 @ ( targets_nat @ Resid @ T ) )
=> ( con_nat @ Resid @ B @ B2 ) ) ) ) ).
% rts.targets_are_con
thf(fact_1257_rts_Otargets__are__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ Resid @ T ) )
=> ( con_lambda_lambda @ Resid @ B @ B2 ) ) ) ) ).
% rts.targets_are_con
thf(fact_1258_rts_Ocomposite__of__unq__upto__cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,T: lambda_lambda,V: lambda_lambda,V2: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ V )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ V2 )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ V @ V2 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ V2 @ V ) ) ) ) ) ) ).
% rts.composite_of_unq_upto_cong
thf(fact_1259_rts_Ocomposite__of__cancel__left,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,U3: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U3 @ V )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ U @ U3 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ U3 @ U ) ) ) ) ) ) ).
% rts.composite_of_cancel_left
thf(fact_1260_rts_Ocomposite__of__ide__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( compos9183582412961930448lambda @ Resid @ A @ A @ A ) ) ) ).
% rts.composite_of_ide_self
thf(fact_1261_rts_Ocomposite__of__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ V )
= ( ( ide_lambda_lambda @ Resid @ ( Resid @ U @ V ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) ) ) ) ).
% rts.composite_of_def
thf(fact_1262_rts_Ocomposite__ofI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,V: lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ U @ V ) )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) )
=> ( compos9183582412961930448lambda @ Resid @ U @ T @ V ) ) ) ) ).
% rts.composite_ofI
thf(fact_1263_rts_Ocomposite__ofE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ V )
=> ~ ( ( ide_lambda_lambda @ Resid @ ( Resid @ U @ V ) )
=> ~ ( ( ide_lambda_lambda @ Resid @ ( Resid @ ( Resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ T @ ( Resid @ V @ U ) ) ) ) ) ) ) ).
% rts.composite_ofE
thf(fact_1264_rts_Oidentities__form__normal__sub__rts,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( normal1556980139574411093lambda @ Resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ Resid ) ) ) ) ).
% rts.identities_form_normal_sub_rts
thf(fact_1265_rts_Oresid__composite__of_I4_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ Resid @ W @ V )
=> ( compos9183582412961930448lambda @ Resid @ ( Resid @ T @ V ) @ ( Resid @ U @ ( Resid @ V @ T ) ) @ ( Resid @ W @ V ) ) ) ) ) ).
% rts.resid_composite_of(4)
thf(fact_1266_rts_Oresid__composite__of_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ Resid @ W @ V )
=> ( con_lambda_lambda @ Resid @ ( Resid @ V @ T ) @ U ) ) ) ) ).
% rts.resid_composite_of(2)
thf(fact_1267_rts_Oresid__composite__of_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ Resid @ W @ V )
=> ( con_lambda_lambda @ Resid @ ( Resid @ V @ T ) @ ( Resid @ W @ T ) ) ) ) ) ).
% rts.resid_composite_of(1)
thf(fact_1268_rts_Ocon__prfx__composite__of_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ Resid @ W @ V )
=> ( con_lambda_lambda @ Resid @ T @ V ) ) ) ) ).
% rts.con_prfx_composite_of(2)
thf(fact_1269_rts_Ocon__prfx__composite__of_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,W: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ W )
=> ( con_lambda_lambda @ Resid @ T @ W ) ) ) ).
% rts.con_prfx_composite_of(1)
thf(fact_1270_rts_Obounded__imp__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ Resid @ T5 @ U3 @ V )
=> ( con_lambda_lambda @ Resid @ T @ T5 ) ) ) ) ).
% rts.bounded_imp_con
thf(fact_1271_rts_Ocon__composite__of__iff,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,W: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ V )
=> ( ( con_lambda_lambda @ Resid @ W @ V )
= ( con_lambda_lambda @ Resid @ ( Resid @ W @ T ) @ U ) ) ) ) ).
% rts.con_composite_of_iff
thf(fact_1272_rts_Oarr__composite__of,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ V )
=> ( arr_lambda_lambda @ Resid @ V ) ) ) ).
% rts.arr_composite_of
thf(fact_1273_rts_Oresid__source__in__targets,axiom,
! [Resid: nat > nat > nat,A: nat,T: nat] :
( ( rts_nat @ Resid )
=> ( ( member_nat @ A @ ( sources_nat @ Resid @ T ) )
=> ( member_nat @ ( Resid @ A @ T ) @ ( targets_nat @ Resid @ T ) ) ) ) ).
% rts.resid_source_in_targets
thf(fact_1274_rts_Oresid__source__in__targets,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ Resid @ T ) )
=> ( member_lambda_lambda @ ( Resid @ A @ T ) @ ( target5490931411331770124lambda @ Resid @ T ) ) ) ) ).
% rts.resid_source_in_targets
thf(fact_1275_rts_Ocong__respects__seq,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( seq_lambda_lambda @ Resid @ T @ U )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ T5 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ T5 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ Resid @ ( Resid @ U @ U3 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ U3 @ U ) ) )
=> ( seq_lambda_lambda @ Resid @ T5 @ U3 ) ) ) ) ) ).
% rts.cong_respects_seq
thf(fact_1276_rts_OCong__iff__cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( normal3156941126761417472lambda @ Resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ Resid ) ) @ T @ U )
= ( ( ide_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ U @ T ) ) ) ) ) ).
% rts.Cong_iff_cong
thf(fact_1277_rts_Ojoin__of__un__upto__cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V2: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( join_o1386658102433174083lambda @ Resid @ T @ U @ V )
=> ( ( join_o1386658102433174083lambda @ Resid @ T @ U @ V2 )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ V @ V2 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ V2 @ V ) ) ) ) ) ) ).
% rts.join_of_un_upto_cong
thf(fact_1278_rts_Osources__composite__of,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( rts_lambda_lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ T @ V )
=> ( ( source5627170679743844886lambda @ Resid @ V )
= ( source5627170679743844886lambda @ Resid @ U ) ) ) ) ).
% rts.sources_composite_of
% 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,
! [X5: nat,Y5: nat] :
( ( if_nat @ $false @ X5 @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X5: nat,Y5: nat] :
( ( if_nat @ $true @ X5 @ Y5 )
= X5 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ ( lambda_Join @ t @ u ) @ u ) @ ( lambda_resid @ t @ u ) ) ).
%------------------------------------------------------------------------------