TPTP Problem File: SLH0694^1.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_06896_305059__14917594_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1435 ( 467 unt; 156 typ; 0 def)
% Number of atoms : 3910 (1725 equ; 0 cnn)
% Maximal formula atoms : 61 ( 3 avg)
% Number of connectives : 14489 ( 616 ~; 42 |; 460 &;11671 @)
% ( 0 <=>;1700 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 674 ( 674 >; 0 *; 0 +; 0 <<)
% Number of symbols : 142 ( 139 usr; 11 con; 0-6 aty)
% Number of variables : 3538 ( 157 ^;3240 !; 141 ?;3538 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:49:07.351
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
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% Explicit typings (139)
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thf(sy_c_LambdaCalculus_Olambda__calculus_OSubst,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_OTrg,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Obottom__up__redex,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Ocontains__head__reduction,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Oelementary__reduction,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Ohead__redex,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Ohgt,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OApp,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OBeta,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_Oun__App2,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_Oun__Beta1,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_Oun__Beta2,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_Oun__Lam,type,
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thf(sy_c_LambdaCalculus_Olambda__calculus_Ored,type,
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thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OArr_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
paths_2015741907369688696lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_l5168835686021156086lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OIde_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
paths_7717490669901197737lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
paths_8607357831768207192lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OResid_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
paths_3312059784463760222lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OSrcs_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
paths_2897053694880971696lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > set_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OSrcs_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
paths_3172353640142224310lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_l5168835686021156086lambda > set_li7878742357367608272lambda ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__rts_OTrgs_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
paths_7594662086398683701lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > set_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__weakly__extensional__rts_OSrc_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
paths_1651982994781619320lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Opaths__in__weakly__extensional__rts_OTrg_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
paths_2695323870737867955lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > list_lambda_lambda > lambda_lambda ).
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__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
arr_li3691443969012319837lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > $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__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
con_li79537494059641180lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $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__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
ide_li6773656263178250516lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > $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_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__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
coinit1063259437155584316lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
compos7127019122090555255lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposable_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
compos3359172601056258045lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $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__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
coterm81126318419254786lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoin__of_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
join_o6956785509312998601lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > list_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_Ojoinable_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
joinab677607494065406278lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_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__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
seq_li8093550137657879445lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $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__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
source6119337877621565212lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).
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__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
target4026644918049360146lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_wi3086218056445523938lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
rts_wi1553966559393445608lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_Set_OCollect_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
collec621444948832755541lambda: ( lambda_lambda > $o ) > set_lambda_lambda ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
collec8830145823237851355lambda: ( list_lambda_lambda > $o ) > set_li7878742357367608272lambda ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
collec4999033578922223073lambda: ( list_l5168835686021156086lambda > $o ) > set_li846743354098393302lambda ).
thf(sy_c_Set_Oimage_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
image_2356535832909760517lambda: ( lambda_lambda > lambda_lambda ) > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Set_Oimage_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
image_3692823078739859851lambda: ( lambda_lambda > list_lambda_lambda ) > set_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > 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_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
member_lambda_lambda: lambda_lambda > set_lambda_lambda > $o ).
thf(sy_c_member_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
member2667661554319814297lambda: list_lambda_lambda > set_li7878742357367608272lambda > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
member9126999148103521439lambda: list_l5168835686021156086lambda > set_li846743354098393302lambda > $o ).
thf(sy_v_t1____,type,
t1: lambda_lambda ).
thf(sy_v_t2____,type,
t2: lambda_lambda ).
% Relevant facts (1275)
thf(fact_0_finite__developments,axiom,
! [T: lambda_lambda] : ( reduction_FD @ T ) ).
% finite_developments
thf(fact_1_development__imp__Arr,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U )
=> ( lambda_Arr @ T ) ) ).
% development_imp_Arr
thf(fact_2_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_3_lambda__calculus_OArr_Osimps_I5_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Beta @ T @ U2 ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U2 ) ) ) ).
% lambda_calculus.Arr.simps(5)
thf(fact_4_ind,axiom,
( ( ( lambda_Arr @ t1 )
& ( lambda_Arr @ t2 ) )
=> ( ( lambda_Arr @ ( lambda_Subst @ zero_zero_nat @ t2 @ t1 ) )
=> ( reduct9208763521642324371opment @ ( lambda_Subst @ zero_zero_nat @ t2 @ t1 ) @ ( reduct1603535164923377302opment @ ( lambda_Subst @ zero_zero_nat @ t2 @ t1 ) ) ) ) ) ).
% ind
thf(fact_5_reduction__paths_Odevelopment_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ nil_lambda_lambda )
= ( lambda_Arr @ T ) ) ).
% reduction_paths.development.simps(1)
thf(fact_6_complete__development__bottom__up__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( reduct7282684938351565978opment @ T @ ( reduct6613843825294780173opment @ T ) ) ) ).
% complete_development_bottom_up_development
thf(fact_7_lambda__calculus_Ocontains__head__reduction_Osimps_I8_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_Beta @ T @ U2 ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U2 ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(8)
thf(fact_8_lambda__calculus_Olambda_Osimps_I29_J,axiom,
! [F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ ( lambda_Beta @ X51 @ X52 ) )
= ( F5 @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.simps(29)
thf(fact_9_lambda__calculus_Olambda_Osel_I6_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_un_Beta2 @ ( lambda_Beta @ X51 @ X52 ) )
= X52 ) ).
% lambda_calculus.lambda.sel(6)
thf(fact_10_lambda__calculus_Olambda_Osel_I5_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_un_Beta1 @ ( lambda_Beta @ X51 @ X52 ) )
= X51 ) ).
% lambda_calculus.lambda.sel(5)
thf(fact_11_Ide__iff__standard__development__empty,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( reduct1603535164923377302opment @ T )
= nil_lambda_lambda ) ) ) ).
% Ide_iff_standard_development_empty
thf(fact_12_lambda__calculus_Olambda_Odisc_I5_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
~ ( lambda_is_Var @ ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.disc(5)
thf(fact_13_lambda__calculus_Olambda_Odisc_I20_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] : ( lambda_is_Beta @ ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.disc(20)
thf(fact_14_lambda__calculus_Olambda_OdiscI_I4_J,axiom,
! [Lambda: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
( ( Lambda
= ( lambda_Beta @ X51 @ X52 ) )
=> ( lambda_is_Beta @ Lambda ) ) ).
% lambda_calculus.lambda.discI(4)
thf(fact_15_development__Ide,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( lambda_Ide @ T )
=> ( ( reduct9208763521642324371opment @ T @ U )
= ( U = nil_lambda_lambda ) ) ) ).
% development_Ide
thf(fact_16_complete__development__Ide__iff,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ U )
=> ( ( lambda_Ide @ T )
= ( U = nil_lambda_lambda ) ) ) ).
% complete_development_Ide_iff
thf(fact_17_lambda__calculus_Olambda_Ocollapse_I4_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Beta @ Lambda )
=> ( ( lambda_Beta @ ( lambda_un_Beta1 @ Lambda ) @ ( lambda_un_Beta2 @ Lambda ) )
= Lambda ) ) ).
% lambda_calculus.lambda.collapse(4)
thf(fact_18_lambda__calculus_Olambda_Odistinct__disc_I14_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Beta @ Lambda )
=> ~ ( lambda_is_Var @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(14)
thf(fact_19_lambda__calculus_Olambda_Odistinct__disc_I13_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Var @ Lambda )
=> ~ ( lambda_is_Beta @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(13)
thf(fact_20_lambda__calculus_OIde__Subst,axiom,
! [T: lambda_lambda,V: lambda_lambda,N: nat] :
( ( lambda_Ide @ T )
=> ( ( lambda_Ide @ V )
=> ( lambda_Ide @ ( lambda_Subst @ N @ V @ T ) ) ) ) ).
% lambda_calculus.Ide_Subst
thf(fact_21_lambda__calculus_OArr__Subst,axiom,
! [V: lambda_lambda,T: lambda_lambda,N: nat] :
( ( lambda_Arr @ V )
=> ( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Subst @ N @ V @ T ) ) ) ) ).
% lambda_calculus.Arr_Subst
thf(fact_22_lambda__calculus_OIde_Osimps_I5_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
~ ( lambda_Ide @ ( lambda_Beta @ T @ U2 ) ) ).
% lambda_calculus.Ide.simps(5)
thf(fact_23_lambda__calculus_OIde__implies__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Ide_implies_Arr
thf(fact_24_lambda__calculus_Ois__Beta__def,axiom,
( lambda_is_Beta
= ( ^ [Lambda2: lambda_lambda] :
? [X512: lambda_lambda,X522: lambda_lambda] :
( Lambda2
= ( lambda_Beta @ X512 @ X522 ) ) ) ) ).
% lambda_calculus.is_Beta_def
thf(fact_25_lambda__calculus_Ocontains__head__reduction__iff,axiom,
( lambda8135948770199036598uction
= ( ^ [T2: lambda_lambda] :
( ( lambda_Arr @ T2 )
& ~ ( lambda_Ide @ ( lambda_head_redex @ T2 ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction_iff
thf(fact_26_lambda__calculus_Ocontains__head__reduction_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U2: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U2 ) )
= ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V @ Va ) )
& ( lambda_Arr @ U2 ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(7)
thf(fact_27_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_28_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_29_reduction__paths_Ostandard__development_Osimps_I5_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U2 ) )
=> ( ( reduct1603535164923377302opment @ ( lambda_Beta @ T @ U2 ) )
= ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U2 ) ) @ ( reduct1603535164923377302opment @ ( lambda_Subst @ zero_zero_nat @ U2 @ T ) ) ) ) )
& ( ~ ( ( lambda_Arr @ T )
& ( lambda_Arr @ U2 ) )
=> ( ( reduct1603535164923377302opment @ ( lambda_Beta @ T @ U2 ) )
= nil_lambda_lambda ) ) ) ).
% reduction_paths.standard_development.simps(5)
thf(fact_30_reduction__paths_Oapply__strategy_Osimps_I1_J,axiom,
! [F: lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( reduct6880286942256454073rategy @ F @ A @ zero_zero_nat )
= nil_lambda_lambda ) ).
% reduction_paths.apply_strategy.simps(1)
thf(fact_31_lambda__calculus_OTrg_Osimps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Trg @ ( lambda_Beta @ T @ U2 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U2 ) @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Trg.simps(4)
thf(fact_32_complete__development__bottom__up__development__ind,axiom,
! [T: lambda_lambda,N: nat] :
( ( lambda_Arr @ T )
=> ( ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ ( reduct6613843825294780173opment @ T ) ) @ N )
=> ( reduct7282684938351565978opment @ T @ ( reduct6613843825294780173opment @ T ) ) ) ) ).
% complete_development_bottom_up_development_ind
thf(fact_33_list_Oinject,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda,Y21: list_lambda_lambda,Y22: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X21 @ X22 )
= ( cons_l4558873032622161958lambda @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_34_list_Oinject,axiom,
! [X21: lambda_lambda,X22: list_lambda_lambda,Y21: lambda_lambda,Y22: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X21 @ X22 )
= ( cons_lambda_lambda @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_35_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_36_FD__def,axiom,
( reduction_FD
= ( ^ [T2: lambda_lambda] :
? [N2: nat] :
! [U3: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T2 @ U3 )
=> ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ U3 ) @ N2 ) ) ) ) ).
% FD_def
thf(fact_37_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_38_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_39_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_40_lambda__calculus_OSrc__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_Src @ T ) )
= ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Src_Src
thf(fact_41_mem__Collect__eq,axiom,
! [A: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
( ( member9126999148103521439lambda @ A @ ( collec4999033578922223073lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( member2667661554319814297lambda @ A @ ( collec8830145823237851355lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ A @ ( collec621444948832755541lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_li846743354098393302lambda] :
( ( collec4999033578922223073lambda
@ ^ [X2: list_l5168835686021156086lambda] : ( member9126999148103521439lambda @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( collec8830145823237851355lambda
@ ^ [X2: list_lambda_lambda] : ( member2667661554319814297lambda @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X2: lambda_lambda] : ( member_lambda_lambda @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [P: list_l5168835686021156086lambda > $o,Q: list_l5168835686021156086lambda > $o] :
( ! [X3: list_l5168835686021156086lambda] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec4999033578922223073lambda @ P )
= ( collec4999033578922223073lambda @ Q ) ) ) ).
% Collect_cong
thf(fact_48_Collect__cong,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ! [X3: list_lambda_lambda] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec8830145823237851355lambda @ P )
= ( collec8830145823237851355lambda @ Q ) ) ) ).
% Collect_cong
thf(fact_49_Collect__cong,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ! [X3: lambda_lambda] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec621444948832755541lambda @ P )
= ( collec621444948832755541lambda @ Q ) ) ) ).
% Collect_cong
thf(fact_50_lambda__calculus_OTrg__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Trg @ ( lambda_Trg @ T ) )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Trg_Trg
thf(fact_51_length__0__conv,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( ( size_s6298197441921507170lambda @ Xs )
= zero_zero_nat )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% length_0_conv
thf(fact_52_length__0__conv,axiom,
! [Xs: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs )
= zero_zero_nat )
= ( Xs = nil_lambda_lambda ) ) ).
% length_0_conv
thf(fact_53_lambda__calculus_OSrc__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_Trg @ T ) )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Src_Trg
thf(fact_54_lambda__calculus_OTrg__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Trg @ ( lambda_Src @ T ) )
= ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Trg_Src
thf(fact_55_lambda__calculus_OSrc__Subst,axiom,
! [T: lambda_lambda,U2: lambda_lambda,D: nat] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( lambda_Src @ ( lambda_Subst @ D @ T @ U2 ) )
= ( lambda_Subst @ D @ ( lambda_Src @ T ) @ ( lambda_Src @ U2 ) ) ) ) ) ).
% lambda_calculus.Src_Subst
thf(fact_56_lambda__calculus_OTrg__Subst,axiom,
! [T: lambda_lambda,U2: lambda_lambda,D: nat] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( lambda_Trg @ ( lambda_Subst @ D @ T @ U2 ) )
= ( lambda_Subst @ D @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U2 ) ) ) ) ) ).
% lambda_calculus.Trg_Subst
thf(fact_57_transpose_Ocases,axiom,
! [X: list_l4928420112573288188lambda] :
( ( X != nil_li6069905736445159420lambda )
=> ( ! [Xss: list_l4928420112573288188lambda] :
( X
!= ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ Xss ) )
=> ~ ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Xss: list_l4928420112573288188lambda] :
( X
!= ( cons_l4910729503725484972lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_58_transpose_Ocases,axiom,
! [X: list_l5168835686021156086lambda] :
( ( X != nil_li8379291187143545974lambda )
=> ( ! [Xss: list_l5168835686021156086lambda] :
( X
!= ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ Xss ) )
=> ~ ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Xss: list_l5168835686021156086lambda] :
( X
!= ( cons_l4558873032622161958lambda @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_59_list__induct2,axiom,
! [Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda,P: list_lambda_lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_60_list__induct2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda,P: list_l5168835686021156086lambda > list_lambda_lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: lambda_lambda,Ys2: list_lambda_lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_61_list__induct2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_62_list__induct2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_63_list__induct3,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_l5168835686021156086lambda,P: list_lambda_lambda > list_lambda_lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s6298197441921507170lambda @ Zs ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s6298197441921507170lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_l4558873032622161958lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_64_list__induct3,axiom,
! [Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Zs: list_lambda_lambda,P: list_lambda_lambda > list_l5168835686021156086lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda,Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_65_list__induct3,axiom,
! [Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,P: list_lambda_lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys )
= ( size_s6298197441921507170lambda @ Zs ) )
=> ( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda,Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys2 )
= ( size_s6298197441921507170lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) @ ( cons_l4558873032622161958lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_66_list__induct3,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,P: list_l5168835686021156086lambda > list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_67_list__induct3,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda,Zs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > list_lambda_lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s6298197441921507170lambda @ Zs ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s6298197441921507170lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_l4558873032622161958lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_68_list__induct3,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_lambda_lambda,P: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_lambda_lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda,Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_69_list__induct3,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys )
= ( size_s6298197441921507170lambda @ Zs ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda,Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys2 )
= ( size_s6298197441921507170lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) @ ( cons_l4558873032622161958lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_70_list__induct3,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_71_list__induct4,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ws: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs )
= ( size_s4208335251162658524lambda @ Ws ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: lambda_lambda,Zs2: list_lambda_lambda,W: lambda_lambda,Ws2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs2 )
= ( size_s4208335251162658524lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) @ ( cons_lambda_lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_72_list__induct4,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ws: list_l5168835686021156086lambda,P: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs )
= ( size_s6298197441921507170lambda @ Ws ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: lambda_lambda,Zs2: list_lambda_lambda,W: list_lambda_lambda,Ws2: list_l5168835686021156086lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs2 )
= ( size_s6298197441921507170lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) @ ( cons_l4558873032622161958lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_73_list__induct4,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_l5168835686021156086lambda,Ws: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > list_l5168835686021156086lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s6298197441921507170lambda @ Zs ) )
=> ( ( ( size_s6298197441921507170lambda @ Zs )
= ( size_s4208335251162658524lambda @ Ws ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda,W: lambda_lambda,Ws2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s6298197441921507170lambda @ Zs2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Zs2 )
= ( size_s4208335251162658524lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_l4558873032622161958lambda @ Z @ Zs2 ) @ ( cons_lambda_lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_74_list__induct4,axiom,
! [Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Zs: list_lambda_lambda,Ws: list_lambda_lambda,P: list_lambda_lambda > list_l5168835686021156086lambda > list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs )
= ( size_s4208335251162658524lambda @ Ws ) )
=> ( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda,Z: lambda_lambda,Zs2: list_lambda_lambda,W: lambda_lambda,Ws2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs2 )
= ( size_s4208335251162658524lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) @ ( cons_lambda_lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_75_list__induct4,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ws: list_lambda_lambda,P: list_l5168835686021156086lambda > list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs )
= ( size_s4208335251162658524lambda @ Ws ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: lambda_lambda,Zs2: list_lambda_lambda,W: lambda_lambda,Ws2: list_lambda_lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs2 )
= ( size_s4208335251162658524lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) @ ( cons_lambda_lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_76_list__induct4,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_l5168835686021156086lambda,Ws: list_l5168835686021156086lambda,P: list_lambda_lambda > list_lambda_lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s6298197441921507170lambda @ Zs ) )
=> ( ( ( size_s6298197441921507170lambda @ Zs )
= ( size_s6298197441921507170lambda @ Ws ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda,W: list_lambda_lambda,Ws2: list_l5168835686021156086lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s6298197441921507170lambda @ Zs2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Zs2 )
= ( size_s6298197441921507170lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_l4558873032622161958lambda @ Z @ Zs2 ) @ ( cons_l4558873032622161958lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_77_list__induct4,axiom,
! [Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Zs: list_lambda_lambda,Ws: list_l5168835686021156086lambda,P: list_lambda_lambda > list_l5168835686021156086lambda > list_lambda_lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs )
= ( size_s6298197441921507170lambda @ Ws ) )
=> ( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda,Z: lambda_lambda,Zs2: list_lambda_lambda,W: list_lambda_lambda,Ws2: list_l5168835686021156086lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs2 )
= ( size_s6298197441921507170lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) @ ( cons_l4558873032622161958lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_78_list__induct4,axiom,
! [Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ws: list_lambda_lambda,P: list_lambda_lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys )
= ( size_s6298197441921507170lambda @ Zs ) )
=> ( ( ( size_s6298197441921507170lambda @ Zs )
= ( size_s4208335251162658524lambda @ Ws ) )
=> ( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda,Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda,W: lambda_lambda,Ws2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s6298197441921507170lambda @ Ys2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Ys2 )
= ( size_s6298197441921507170lambda @ Zs2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Zs2 )
= ( size_s4208335251162658524lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) @ ( cons_l4558873032622161958lambda @ Z @ Zs2 ) @ ( cons_lambda_lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_79_list__induct4,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ws: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > list_lambda_lambda > list_lambda_lambda > list_l5168835686021156086lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs )
= ( size_s6298197441921507170lambda @ Ws ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: lambda_lambda,Zs2: list_lambda_lambda,W: list_lambda_lambda,Ws2: list_l5168835686021156086lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs2 )
= ( size_s6298197441921507170lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) @ ( cons_l4558873032622161958lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_80_list__induct4,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda,Zs: list_l5168835686021156086lambda,Ws: list_lambda_lambda,P: list_l5168835686021156086lambda > list_lambda_lambda > list_l5168835686021156086lambda > list_lambda_lambda > $o] :
( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s6298197441921507170lambda @ Zs ) )
=> ( ( ( size_s6298197441921507170lambda @ Zs )
= ( size_s4208335251162658524lambda @ Ws ) )
=> ( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: lambda_lambda,Ys2: list_lambda_lambda,Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda,W: lambda_lambda,Ws2: list_lambda_lambda] :
( ( ( size_s6298197441921507170lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s6298197441921507170lambda @ Zs2 ) )
=> ( ( ( size_s6298197441921507170lambda @ Zs2 )
= ( size_s4208335251162658524lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) @ ( cons_l4558873032622161958lambda @ Z @ Zs2 ) @ ( cons_lambda_lambda @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_81_size__neq__size__imp__neq,axiom,
! [X: list_l5168835686021156086lambda,Y2: list_l5168835686021156086lambda] :
( ( ( size_s6298197441921507170lambda @ X )
!= ( size_s6298197441921507170lambda @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_82_size__neq__size__imp__neq,axiom,
! [X: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ X )
!= ( size_s4208335251162658524lambda @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_83_size__neq__size__imp__neq,axiom,
! [X: char,Y2: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_84_size__neq__size__imp__neq,axiom,
! [X: lambda_lambda,Y2: lambda_lambda] :
( ( ( size_s1768714712973771222lambda @ X )
!= ( size_s1768714712973771222lambda @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_85_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_86_neq__if__length__neq,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( size_s6298197441921507170lambda @ Xs )
!= ( size_s6298197441921507170lambda @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_87_neq__if__length__neq,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs )
!= ( size_s4208335251162658524lambda @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_88_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_l5168835686021156086lambda] :
( ( size_s6298197441921507170lambda @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_89_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_lambda_lambda] :
( ( size_s4208335251162658524lambda @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_90_impossible__Cons,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,X: list_lambda_lambda] :
( ( ord_less_eq_nat @ ( size_s6298197441921507170lambda @ Xs ) @ ( size_s6298197441921507170lambda @ Ys ) )
=> ( Xs
!= ( cons_l4558873032622161958lambda @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_91_impossible__Cons,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,X: lambda_lambda] :
( ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ Xs ) @ ( size_s4208335251162658524lambda @ Ys ) )
=> ( Xs
!= ( cons_lambda_lambda @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_92_not__Cons__self2,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( cons_l4558873032622161958lambda @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_93_not__Cons__self2,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( cons_lambda_lambda @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_94_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_95_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_96_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_97_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_98_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_99_lambda__calculus_Ohead__redex_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U2: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U2 ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_App @ V @ Va ) ) @ ( lambda_Src @ U2 ) ) ) ).
% lambda_calculus.head_redex.simps(7)
thf(fact_100_lambda__calculus_Ohead__redex_Osimps_I8_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U2: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U2 ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Beta @ V @ Va ) ) @ ( lambda_Src @ U2 ) ) ) ).
% lambda_calculus.head_redex.simps(8)
thf(fact_101_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,T3: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T @ U2 ) )
= ( lambda_Src @ ( lambda_App @ T3 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T3 ) )
& ( ( lambda_Src @ U2 )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_102_lambda__calculus_OTrg_Osimps_I3_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Trg @ ( lambda_App @ T @ U2 ) )
= ( lambda_App @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U2 ) ) ) ).
% lambda_calculus.Trg.simps(3)
thf(fact_103_lambda__calculus_OSrc_Osimps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Src @ ( lambda_App @ T @ U2 ) )
= ( lambda_App @ ( lambda_Src @ T ) @ ( lambda_Src @ U2 ) ) ) ).
% lambda_calculus.Src.simps(4)
thf(fact_104_lambda__calculus_Ohead__redex_Osimps_I9_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_Beta @ T @ U2 ) )
= ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U2 ) ) ) ).
% lambda_calculus.head_redex.simps(9)
thf(fact_105_lambda__calculus_OSrc__head__redex,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_head_redex @ T ) )
= ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Src_head_redex
thf(fact_106_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_107_list__nonempty__induct,axiom,
! [Xs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_108_list__nonempty__induct,axiom,
! [Xs: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( Xs != nil_lambda_lambda )
=> ( ! [X3: lambda_lambda] : ( P @ ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_109_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_l5168835686021156086lambda > $o,Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ nil_li8379291187143545974lambda )
=> ( ! [Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_lambda_lambda @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_110_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_lambda_lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ nil_lambda_lambda )
=> ( ! [Y: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_lambda_lambda @ Y @ Ys2 ) )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_111_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ nil_li8379291187143545974lambda )
=> ( ! [Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_112_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_lambda_lambda > $o,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ nil_lambda_lambda )
=> ( ! [Y: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_lambda_lambda @ ( cons_lambda_lambda @ Y @ Ys2 ) )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_113_neq__Nil__conv,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
= ( ? [Y4: list_lambda_lambda,Ys3: list_l5168835686021156086lambda] :
( Xs
= ( cons_l4558873032622161958lambda @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_114_neq__Nil__conv,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
= ( ? [Y4: lambda_lambda,Ys3: list_lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_115_remdups__adj_Ocases,axiom,
! [X: list_l5168835686021156086lambda] :
( ( X != nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda] :
( X
!= ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) )
=> ~ ! [X3: list_lambda_lambda,Y: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( X
!= ( cons_l4558873032622161958lambda @ X3 @ ( cons_l4558873032622161958lambda @ Y @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_116_remdups__adj_Ocases,axiom,
! [X: list_lambda_lambda] :
( ( X != nil_lambda_lambda )
=> ( ! [X3: lambda_lambda] :
( X
!= ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) )
=> ~ ! [X3: lambda_lambda,Y: lambda_lambda,Xs2: list_lambda_lambda] :
( X
!= ( cons_lambda_lambda @ X3 @ ( cons_lambda_lambda @ Y @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_117_list_Oexhaust,axiom,
! [Y2: list_l5168835686021156086lambda] :
( ( Y2 != nil_li8379291187143545974lambda )
=> ~ ! [X212: list_lambda_lambda,X222: list_l5168835686021156086lambda] :
( Y2
!= ( cons_l4558873032622161958lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_118_list_Oexhaust,axiom,
! [Y2: list_lambda_lambda] :
( ( Y2 != nil_lambda_lambda )
=> ~ ! [X212: lambda_lambda,X222: list_lambda_lambda] :
( Y2
!= ( cons_lambda_lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_119_list_OdiscI,axiom,
! [List: list_l5168835686021156086lambda,X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( List
= ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
=> ( List != nil_li8379291187143545974lambda ) ) ).
% list.discI
thf(fact_120_list_OdiscI,axiom,
! [List: list_lambda_lambda,X21: lambda_lambda,X22: list_lambda_lambda] :
( ( List
= ( cons_lambda_lambda @ X21 @ X22 ) )
=> ( List != nil_lambda_lambda ) ) ).
% list.discI
thf(fact_121_list_Odistinct_I1_J,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( nil_li8379291187143545974lambda
!= ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_122_list_Odistinct_I1_J,axiom,
! [X21: lambda_lambda,X22: list_lambda_lambda] :
( nil_lambda_lambda
!= ( cons_lambda_lambda @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_123_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_124_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_125_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_126_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_127_lambda__calculus_OArr__Trg,axiom,
! [X4: lambda_lambda] :
( ( lambda_Arr @ X4 )
=> ( lambda_Arr @ ( lambda_Trg @ X4 ) ) ) ).
% lambda_calculus.Arr_Trg
thf(fact_128_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,T3: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_Beta @ T @ U2 ) )
= ( lambda_Src @ ( lambda_Beta @ T3 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T3 ) )
& ( ( lambda_Src @ U2 )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(3)
thf(fact_129_lambda__calculus_OArr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Arr_Src
thf(fact_130_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_131_reduction__paths_OStd_Ocases,axiom,
! [X: list_lambda_lambda] :
( ( X != nil_lambda_lambda )
=> ( ! [T4: lambda_lambda] :
( X
!= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( X
!= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ).
% reduction_paths.Std.cases
thf(fact_132_lambda__calculus_OIde_Osimps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Ide @ ( lambda_App @ T @ U2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U2 ) ) ) ).
% lambda_calculus.Ide.simps(4)
thf(fact_133_lambda__calculus_OArr_Osimps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ ( lambda_App @ T @ U2 ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U2 ) ) ) ).
% lambda_calculus.Arr.simps(4)
thf(fact_134_list_Osize_I3_J,axiom,
( ( size_s6298197441921507170lambda @ nil_li8379291187143545974lambda )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_135_list_Osize_I3_J,axiom,
( ( size_s4208335251162658524lambda @ nil_lambda_lambda )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_136_lambda__calculus_OSubst_Osimps_I4_J,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_App @ T @ U2 ) )
= ( lambda_App @ ( lambda_Subst @ N @ V @ T ) @ ( lambda_Subst @ N @ V @ U2 ) ) ) ).
% lambda_calculus.Subst.simps(4)
thf(fact_137_lambda__calculus_OArr__head__redex,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_head_redex @ T ) ) ) ).
% lambda_calculus.Arr_head_redex
thf(fact_138_lambda__calculus_Olambda_Odisc_I19_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
~ ( lambda_is_Beta @ ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.disc(19)
thf(fact_139_lambda__calculus_Olambda_Odisc_I4_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
~ ( lambda_is_Var @ ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.disc(4)
thf(fact_140_lambda__calculus_Olambda_Osimps_I28_J,axiom,
! [F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ ( lambda_App @ X41 @ X42 ) )
= ( F4 @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.simps(28)
thf(fact_141_lambda__calculus_OIde__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Ide_Trg
thf(fact_142_lambda__calculus_OIde__iff__Trg__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( lambda_Trg @ T )
= T ) ) ) ).
% lambda_calculus.Ide_iff_Trg_self
thf(fact_143_lambda__calculus_OIde__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Ide_Src
thf(fact_144_lambda__calculus_OIde__iff__Src__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( lambda_Src @ T )
= T ) ) ) ).
% lambda_calculus.Ide_iff_Src_self
thf(fact_145_lambda__calculus_Ocontains__head__reduction_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U2: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U2 ) )
= ( ( lambda8135948770199036598uction @ ( lambda_App @ V @ Va ) )
& ( lambda_Arr @ U2 ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(6)
thf(fact_146_length__devel__le__hgt,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U )
=> ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ U ) @ ( lambda_hgt @ T ) ) ) ).
% length_devel_le_hgt
thf(fact_147_Trg__last__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ~ ( lambda_Ide @ T )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ ( reduct1603535164923377302opment @ T ) ) )
= ( lambda_Trg @ T ) ) ) ) ).
% Trg_last_standard_development
thf(fact_148_Src__hd__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ~ ( lambda_Ide @ T )
=> ( ( lambda_Src @ ( hd_lambda_lambda @ ( reduct1603535164923377302opment @ T ) ) )
= ( lambda_Src @ T ) ) ) ) ).
% Src_hd_standard_development
thf(fact_149_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_150_dual__order_Orefl,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A @ A ) ).
% dual_order.refl
thf(fact_151_dual__order_Orefl,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ A @ A ) ).
% dual_order.refl
thf(fact_152_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_153_order__refl,axiom,
! [X: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ X @ X ) ).
% order_refl
thf(fact_154_order__refl,axiom,
! [X: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ X @ X ) ).
% order_refl
thf(fact_155_complete__development__cons,axiom,
! [T: lambda_lambda,U2: lambda_lambda,U: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ ( cons_lambda_lambda @ U2 @ U ) )
=> ( reduct7282684938351565978opment @ ( lambda_resid @ T @ U2 ) @ U ) ) ).
% complete_development_cons
thf(fact_156_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_157_Resid_Ocases,axiom,
! [X: produc3148917353989188007lambda] :
( ! [Uu: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ nil_lambda_lambda @ Uu ) )
=> ( ! [V2: lambda_lambda,Va2: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ ( cons_lambda_lambda @ V2 @ Va2 ) @ nil_lambda_lambda ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) )
=> ( ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda,U5: lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) @ ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda,U5: lambda_lambda,Vb: lambda_lambda,Vc: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) @ ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ Vb @ Vc ) ) ) ) ) ) ) ) ) ).
% Resid.cases
thf(fact_158_Resid1x_Ocases,axiom,
! [X: produc2030461353491475885lambda] :
( ! [T4: lambda_lambda] :
( X
!= ( produc8847644316606783005lambda @ T4 @ nil_lambda_lambda ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( X
!= ( produc8847644316606783005lambda @ T4 @ ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) ) )
=> ~ ! [T4: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( X
!= ( produc8847644316606783005lambda @ T4 @ ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ).
% Resid1x.cases
thf(fact_159_Residx1_Ocases,axiom,
! [X: produc3002779106298095521lambda] :
( ! [U5: lambda_lambda] :
( X
!= ( produc7613787137529623313lambda @ nil_lambda_lambda @ U5 ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( X
!= ( produc7613787137529623313lambda @ ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) @ U5 ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda,U5: lambda_lambda] :
( X
!= ( produc7613787137529623313lambda @ ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) @ U5 ) ) ) ) ).
% Residx1.cases
thf(fact_160__092_060Lambda_062_Oex__un__null,axiom,
? [X3: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( lambda_resid @ X3 @ T5 )
= X3 )
& ( ( lambda_resid @ T5 @ X3 )
= X3 ) )
& ! [Y3: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( lambda_resid @ Y3 @ T4 )
= Y3 )
& ( ( lambda_resid @ T4 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) ) ).
% \<Lambda>.ex_un_null
thf(fact_161__092_060Lambda_062_Ocube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U2 @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U2 ) @ ( lambda_resid @ T @ U2 ) ) ) ).
% \<Lambda>.cube
thf(fact_162_lambda__calculus_Oresid__Arr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ ( lambda_Src @ T ) )
= T ) ) ).
% lambda_calculus.resid_Arr_Src
thf(fact_163_lambda__calculus_Oresid__Src__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ ( lambda_Src @ T ) @ T )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.resid_Src_Arr
thf(fact_164_paths__in__confluent__rts__axioms,axiom,
paths_9040249923598999359lambda @ lambda_resid ).
% paths_in_confluent_rts_axioms
thf(fact_165_hd__Nil__eq__last,axiom,
( ( hd_lis2183642875561147995lambda @ nil_li8379291187143545974lambda )
= ( last_l1874196399739661071lambda @ nil_li8379291187143545974lambda ) ) ).
% hd_Nil_eq_last
thf(fact_166_hd__Nil__eq__last,axiom,
( ( hd_lambda_lambda @ nil_lambda_lambda )
= ( last_lambda_lambda @ nil_lambda_lambda ) ) ).
% hd_Nil_eq_last
thf(fact_167_list_Osel_I1_J,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( hd_lis2183642875561147995lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_168_list_Osel_I1_J,axiom,
! [X21: lambda_lambda,X22: list_lambda_lambda] :
( ( hd_lambda_lambda @ ( cons_lambda_lambda @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_169_shuffles_Ocases,axiom,
! [X: produc2428337531523351207lambda] :
( ! [Ys2: list_l5168835686021156086lambda] :
( X
!= ( produc4761665061662602647lambda @ nil_li8379291187143545974lambda @ Ys2 ) )
=> ( ! [Xs2: list_l5168835686021156086lambda] :
( X
!= ( produc4761665061662602647lambda @ Xs2 @ nil_li8379291187143545974lambda ) )
=> ~ ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( X
!= ( produc4761665061662602647lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_170_shuffles_Ocases,axiom,
! [X: produc3148917353989188007lambda] :
( ! [Ys2: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ nil_lambda_lambda @ Ys2 ) )
=> ( ! [Xs2: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ Xs2 @ nil_lambda_lambda ) )
=> ~ ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Y: lambda_lambda,Ys2: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ ( cons_lambda_lambda @ Y @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_171_splice_Ocases,axiom,
! [X: produc2428337531523351207lambda] :
( ! [Ys2: list_l5168835686021156086lambda] :
( X
!= ( produc4761665061662602647lambda @ nil_li8379291187143545974lambda @ Ys2 ) )
=> ~ ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Ys2: list_l5168835686021156086lambda] :
( X
!= ( produc4761665061662602647lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_172_splice_Ocases,axiom,
! [X: produc3148917353989188007lambda] :
( ! [Ys2: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ nil_lambda_lambda @ Ys2 ) )
=> ~ ! [X3: lambda_lambda,Xs2: list_lambda_lambda,Ys2: list_lambda_lambda] :
( X
!= ( produc1182943629000560279lambda @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_173_reduction__paths_Odevelopment_Ocases,axiom,
! [X: produc2030461353491475885lambda] :
( ! [T4: lambda_lambda] :
( X
!= ( produc8847644316606783005lambda @ T4 @ nil_lambda_lambda ) )
=> ~ ! [T4: lambda_lambda,U5: lambda_lambda,U6: list_lambda_lambda] :
( X
!= ( produc8847644316606783005lambda @ T4 @ ( cons_lambda_lambda @ U5 @ U6 ) ) ) ) ).
% reduction_paths.development.cases
thf(fact_174_last__ConsR,axiom,
! [Xs: list_l5168835686021156086lambda,X: list_lambda_lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) )
= ( last_l1874196399739661071lambda @ Xs ) ) ) ).
% last_ConsR
thf(fact_175_last__ConsR,axiom,
! [Xs: list_lambda_lambda,X: lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_ConsR
thf(fact_176_last__ConsL,axiom,
! [Xs: list_l5168835686021156086lambda,X: list_lambda_lambda] :
( ( Xs = nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_177_last__ConsL,axiom,
! [Xs: list_lambda_lambda,X: lambda_lambda] :
( ( Xs = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_178_last_Osimps,axiom,
! [Xs: list_l5168835686021156086lambda,X: list_lambda_lambda] :
( ( ( Xs = nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) )
= ( last_l1874196399739661071lambda @ Xs ) ) ) ) ).
% last.simps
thf(fact_179_last_Osimps,axiom,
! [Xs: list_lambda_lambda,X: lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ) ).
% last.simps
thf(fact_180_lambda__calculus_Oresid__Arr__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ T )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.resid_Arr_self
thf(fact_181_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_182_le__cases3,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_183_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_184_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_lambda_lambda,Z3: set_lambda_lambda] : ( Y5 = Z3 ) )
= ( ^ [X2: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y4 )
& ( ord_le1926335974945931754lambda @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_185_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] : ( Y5 = Z3 ) )
= ( ^ [X2: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y4 )
& ( ord_le3064231726521830256lambda @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_186_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_187_ord__eq__le__trans,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A = B )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_188_ord__eq__le__trans,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A = B )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_189_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_190_ord__le__eq__trans,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( B = C )
=> ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_191_ord__le__eq__trans,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( B = C )
=> ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_192_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_193_order__antisym,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_194_order__antisym,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_195_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_196_order_Otrans,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ A @ C ) ) ) ).
% order.trans
thf(fact_197_order_Otrans,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% order.trans
thf(fact_198_order__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_199_order__trans,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ Y2 @ Z2 )
=> ( ord_le1926335974945931754lambda @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_200_order__trans,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ Y2 @ Z2 )
=> ( ord_le3064231726521830256lambda @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_201_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_202_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_203_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_lambda_lambda,Z3: set_lambda_lambda] : ( Y5 = Z3 ) )
= ( ^ [A4: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ A4 )
& ( ord_le1926335974945931754lambda @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_204_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] : ( Y5 = Z3 ) )
= ( ^ [A4: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ A4 )
& ( ord_le3064231726521830256lambda @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_205_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_206_dual__order_Oantisym,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( ord_le1926335974945931754lambda @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_207_dual__order_Oantisym,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A )
=> ( ( ord_le3064231726521830256lambda @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_208_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_209_dual__order_Otrans,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( ord_le1926335974945931754lambda @ C @ B )
=> ( ord_le1926335974945931754lambda @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_210_dual__order_Otrans,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A )
=> ( ( ord_le3064231726521830256lambda @ C @ B )
=> ( ord_le3064231726521830256lambda @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_211_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_212_antisym,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_213_antisym,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_214_order__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order_eq_iff
thf(fact_215_order__eq__iff,axiom,
( ( ^ [Y5: set_lambda_lambda,Z3: set_lambda_lambda] : ( Y5 = Z3 ) )
= ( ^ [A4: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A4 @ B3 )
& ( ord_le1926335974945931754lambda @ B3 @ A4 ) ) ) ) ).
% order_eq_iff
thf(fact_216_order__eq__iff,axiom,
( ( ^ [Y5: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] : ( Y5 = Z3 ) )
= ( ^ [A4: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A4 @ B3 )
& ( ord_le3064231726521830256lambda @ B3 @ A4 ) ) ) ) ).
% order_eq_iff
thf(fact_217_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_218_order__subst1,axiom,
! [A: nat,F: set_lambda_lambda > nat,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_219_order__subst1,axiom,
! [A: nat,F: set_li7878742357367608272lambda > nat,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_220_order__subst1,axiom,
! [A: set_lambda_lambda,F: nat > set_lambda_lambda,B: nat,C: nat] :
( ( ord_le1926335974945931754lambda @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_221_order__subst1,axiom,
! [A: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_222_order__subst1,axiom,
! [A: set_lambda_lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_223_order__subst1,axiom,
! [A: set_li7878742357367608272lambda,F: nat > set_li7878742357367608272lambda,B: nat,C: nat] :
( ( ord_le3064231726521830256lambda @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_224_order__subst1,axiom,
! [A: set_li7878742357367608272lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_225_order__subst1,axiom,
! [A: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_226_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_227_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_228_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_229_order__subst2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > nat,C: nat] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_230_order__subst2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ ( F @ B ) @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_231_order__subst2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ ( F @ B ) @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_232_order__subst2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > nat,C: nat] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_233_order__subst2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ ( F @ B ) @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_234_order__subst2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ ( F @ B ) @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_235_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_236_order__eq__refl,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( X = Y2 )
=> ( ord_le1926335974945931754lambda @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_237_order__eq__refl,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( X = Y2 )
=> ( ord_le3064231726521830256lambda @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_238_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_239_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_240_ord__eq__le__subst,axiom,
! [A: set_lambda_lambda,F: nat > set_lambda_lambda,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_241_ord__eq__le__subst,axiom,
! [A: set_li7878742357367608272lambda,F: nat > set_li7878742357367608272lambda,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_242_ord__eq__le__subst,axiom,
! [A: nat,F: set_lambda_lambda > nat,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_243_ord__eq__le__subst,axiom,
! [A: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_244_ord__eq__le__subst,axiom,
! [A: set_li7878742357367608272lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_245_ord__eq__le__subst,axiom,
! [A: nat,F: set_li7878742357367608272lambda > nat,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_246_ord__eq__le__subst,axiom,
! [A: set_lambda_lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_247_ord__eq__le__subst,axiom,
! [A: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_248_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_249_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_250_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_251_ord__le__eq__subst,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > nat,C: nat] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_252_ord__le__eq__subst,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_253_ord__le__eq__subst,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_254_ord__le__eq__subst,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > nat,C: nat] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_255_ord__le__eq__subst,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_le1926335974945931754lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_256_ord__le__eq__subst,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ord_le3064231726521830256lambda @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_257_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_258_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_259_order__antisym__conv,axiom,
! [Y2: set_lambda_lambda,X: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y2 @ X )
=> ( ( ord_le1926335974945931754lambda @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_260_order__antisym__conv,axiom,
! [Y2: set_li7878742357367608272lambda,X: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ Y2 @ X )
=> ( ( ord_le3064231726521830256lambda @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_261_lambda__calculus_Oresid__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 ) ) ) ).
% lambda_calculus.resid_Arr_Ide
thf(fact_262_lambda__calculus_Oresid__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 ) ) ) ) ).
% lambda_calculus.resid_Ide_Arr
thf(fact_263__092_060Lambda_062_Opartial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% \<Lambda>.partial_magma_axioms
thf(fact_264_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list7041979136695072894lambda @ N @ nil_li8379291187143545974lambda )
= ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ nil_li6069905736445159420lambda ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list7041979136695072894lambda @ N @ nil_li8379291187143545974lambda )
= nil_li6069905736445159420lambda ) ) ) ).
% n_lists_Nil
thf(fact_265_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list4163717777911954552lambda @ N @ nil_lambda_lambda )
= ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list4163717777911954552lambda @ N @ nil_lambda_lambda )
= nil_li8379291187143545974lambda ) ) ) ).
% n_lists_Nil
thf(fact_266_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_267_n__lists_Osimps_I1_J,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( n_list7041979136695072894lambda @ zero_zero_nat @ Xs )
= ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ nil_li6069905736445159420lambda ) ) ).
% n_lists.simps(1)
thf(fact_268_n__lists_Osimps_I1_J,axiom,
! [Xs: list_lambda_lambda] :
( ( n_list4163717777911954552lambda @ zero_zero_nat @ Xs )
= ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ nil_li8379291187143545974lambda ) ) ).
% n_lists.simps(1)
thf(fact_269_prod_Oinject,axiom,
! [X1: lambda_lambda > lambda_lambda > $o,X23: list_lambda_lambda,Y1: lambda_lambda > lambda_lambda > $o,Y23: list_lambda_lambda] :
( ( ( produc8587549711031851289lambda @ X1 @ X23 )
= ( produc8587549711031851289lambda @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_270_prod_Oinject,axiom,
! [X1: lambda_lambda > lambda_lambda,X23: produc3148917353989188007lambda,Y1: lambda_lambda > lambda_lambda,Y23: produc3148917353989188007lambda] :
( ( ( produc1362658507006291625lambda @ X1 @ X23 )
= ( produc1362658507006291625lambda @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_271_prod_Oinject,axiom,
! [X1: list_lambda_lambda,X23: list_lambda_lambda,Y1: list_lambda_lambda,Y23: list_lambda_lambda] :
( ( ( produc1182943629000560279lambda @ X1 @ X23 )
= ( produc1182943629000560279lambda @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_272_prod_Oinject,axiom,
! [X1: lambda_lambda,X23: list_lambda_lambda,Y1: lambda_lambda,Y23: list_lambda_lambda] :
( ( ( produc8847644316606783005lambda @ X1 @ X23 )
= ( produc8847644316606783005lambda @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_273_prod_Oinject,axiom,
! [X1: list_lambda_lambda,X23: lambda_lambda,Y1: list_lambda_lambda,Y23: lambda_lambda] :
( ( ( produc7613787137529623313lambda @ X1 @ X23 )
= ( produc7613787137529623313lambda @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_274_old_Oprod_Oinject,axiom,
! [A: lambda_lambda > lambda_lambda > $o,B: list_lambda_lambda,A5: lambda_lambda > lambda_lambda > $o,B4: list_lambda_lambda] :
( ( ( produc8587549711031851289lambda @ A @ B )
= ( produc8587549711031851289lambda @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_275_old_Oprod_Oinject,axiom,
! [A: lambda_lambda > lambda_lambda,B: produc3148917353989188007lambda,A5: lambda_lambda > lambda_lambda,B4: produc3148917353989188007lambda] :
( ( ( produc1362658507006291625lambda @ A @ B )
= ( produc1362658507006291625lambda @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_276_old_Oprod_Oinject,axiom,
! [A: list_lambda_lambda,B: list_lambda_lambda,A5: list_lambda_lambda,B4: list_lambda_lambda] :
( ( ( produc1182943629000560279lambda @ A @ B )
= ( produc1182943629000560279lambda @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_277_old_Oprod_Oinject,axiom,
! [A: lambda_lambda,B: list_lambda_lambda,A5: lambda_lambda,B4: list_lambda_lambda] :
( ( ( produc8847644316606783005lambda @ A @ B )
= ( produc8847644316606783005lambda @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_278_old_Oprod_Oinject,axiom,
! [A: list_lambda_lambda,B: lambda_lambda,A5: list_lambda_lambda,B4: lambda_lambda] :
( ( ( produc7613787137529623313lambda @ A @ B )
= ( produc7613787137529623313lambda @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_279_length__Resid,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( size_s4208335251162658524lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( size_s4208335251162658524lambda @ T6 ) ) ) ).
% length_Resid
thf(fact_280_reduction__paths_Obottom__up__development_Oelims,axiom,
! [X: lambda_lambda,Y2: list_lambda_lambda] :
( ( ( reduct6613843825294780173opment @ X )
= Y2 )
=> ( ( ( ~ ( lambda_Arr @ X )
| ( lambda_Ide @ X ) )
=> ( Y2 = nil_lambda_lambda ) )
& ( ~ ( ~ ( lambda_Arr @ X )
| ( lambda_Ide @ X ) )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda5654762366400029632_redex @ X ) @ ( reduct6613843825294780173opment @ ( lambda_resid @ X @ ( lambda5654762366400029632_redex @ X ) ) ) ) ) ) ) ) ).
% reduction_paths.bottom_up_development.elims
thf(fact_281_reduction__paths_Obottom__up__development_Osimps,axiom,
( reduct6613843825294780173opment
= ( ^ [T2: lambda_lambda] :
( if_lis7195930416592267574lambda
@ ( ~ ( lambda_Arr @ T2 )
| ( lambda_Ide @ T2 ) )
@ nil_lambda_lambda
@ ( cons_lambda_lambda @ ( lambda5654762366400029632_redex @ T2 ) @ ( reduct6613843825294780173opment @ ( lambda_resid @ T2 @ ( lambda5654762366400029632_redex @ T2 ) ) ) ) ) ) ) ).
% reduction_paths.bottom_up_development.simps
thf(fact_282_ex__un__null,axiom,
? [X3: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X3 @ T5 )
= X3 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ X3 )
= X3 ) )
& ! [Y3: list_lambda_lambda] :
( ! [T4: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ Y3 @ T4 )
= Y3 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) ) ).
% ex_un_null
thf(fact_283_cube,axiom,
! [V: list_lambda_lambda,T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) ) ) ).
% cube
thf(fact_284_Con__sym,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 )
!= nil_lambda_lambda ) ) ).
% Con_sym
thf(fact_285_Resid_Osimps_I1_J,axiom,
! [Uu2: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ nil_lambda_lambda @ Uu2 )
= nil_lambda_lambda ) ).
% Resid.simps(1)
thf(fact_286_Cube_I1_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V3 ) )
!= nil_lambda_lambda ) ) ).
% Cube(1)
thf(fact_287_Cube_I2_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V3 ) ) ) ) ).
% Cube(2)
thf(fact_288_Con__cons_I2_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,U2: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ U ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) @ U )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_cons(2)
thf(fact_289_Con__cons_I1_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,T: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_cons(1)
thf(fact_290_Resid_Osimps_I2_J,axiom,
! [V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% Resid.simps(2)
thf(fact_291_Resid__cons_I2_J,axiom,
! [U: list_lambda_lambda,T6: list_lambda_lambda,U2: lambda_lambda] :
( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ U ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ U ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) @ U ) ) ) ) ).
% Resid_cons(2)
thf(fact_292_Resid__rec_I3_J,axiom,
! [U: list_lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ U ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ U ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) @ U ) ) ) ) ).
% Resid_rec(3)
thf(fact_293_Resid__rec_I2_J,axiom,
! [T6: list_lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U2 @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ).
% Resid_rec(2)
thf(fact_294_Con__initial__left,axiom,
! [T: lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
!= nil_lambda_lambda ) ) ).
% Con_initial_left
thf(fact_295_Con__initial__right,axiom,
! [T6: list_lambda_lambda,U2: lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ U ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ).
% Con_initial_right
thf(fact_296_Con__consI_I2_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,U2: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) @ U )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ U ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_consI(2)
thf(fact_297_Con__consI_I1_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,T: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_consI(1)
thf(fact_298_Resid__rec_I1_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) ) ) ).
% Resid_rec(1)
thf(fact_299_is__confluent__rts,axiom,
conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% is_confluent_rts
thf(fact_300_sorted__wrt_Ocases,axiom,
! [X: produc9138666002258319407lambda] :
( ! [P2: list_lambda_lambda > list_lambda_lambda > $o] :
( X
!= ( produc5138638581814799519lambda @ P2 @ nil_li8379291187143545974lambda ) )
=> ~ ! [P2: list_lambda_lambda > list_lambda_lambda > $o,X3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( X
!= ( produc5138638581814799519lambda @ P2 @ ( cons_l4558873032622161958lambda @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_301_sorted__wrt_Ocases,axiom,
! [X: produc1137487172120676137lambda] :
( ! [P2: lambda_lambda > lambda_lambda > $o] :
( X
!= ( produc8587549711031851289lambda @ P2 @ nil_lambda_lambda ) )
=> ~ ! [P2: lambda_lambda > lambda_lambda > $o,X3: lambda_lambda,Ys2: list_lambda_lambda] :
( X
!= ( produc8587549711031851289lambda @ P2 @ ( cons_lambda_lambda @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_302_successively_Ocases,axiom,
! [X: produc9138666002258319407lambda] :
( ! [P2: list_lambda_lambda > list_lambda_lambda > $o] :
( X
!= ( produc5138638581814799519lambda @ P2 @ nil_li8379291187143545974lambda ) )
=> ( ! [P2: list_lambda_lambda > list_lambda_lambda > $o,X3: list_lambda_lambda] :
( X
!= ( produc5138638581814799519lambda @ P2 @ ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) ) )
=> ~ ! [P2: list_lambda_lambda > list_lambda_lambda > $o,X3: list_lambda_lambda,Y: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( X
!= ( produc5138638581814799519lambda @ P2 @ ( cons_l4558873032622161958lambda @ X3 @ ( cons_l4558873032622161958lambda @ Y @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_303_successively_Ocases,axiom,
! [X: produc1137487172120676137lambda] :
( ! [P2: lambda_lambda > lambda_lambda > $o] :
( X
!= ( produc8587549711031851289lambda @ P2 @ nil_lambda_lambda ) )
=> ( ! [P2: lambda_lambda > lambda_lambda > $o,X3: lambda_lambda] :
( X
!= ( produc8587549711031851289lambda @ P2 @ ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) ) )
=> ~ ! [P2: lambda_lambda > lambda_lambda > $o,X3: lambda_lambda,Y: lambda_lambda,Xs2: list_lambda_lambda] :
( X
!= ( produc8587549711031851289lambda @ P2 @ ( cons_lambda_lambda @ X3 @ ( cons_lambda_lambda @ Y @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_304_map__tailrec__rev_Ocases,axiom,
! [X: produc7929126107877444343lambda] :
( ! [F6: lambda_lambda > lambda_lambda,Bs: list_lambda_lambda] :
( X
!= ( produc1362658507006291625lambda @ F6 @ ( produc1182943629000560279lambda @ nil_lambda_lambda @ Bs ) ) )
=> ~ ! [F6: lambda_lambda > lambda_lambda,A3: lambda_lambda,As: list_lambda_lambda,Bs: list_lambda_lambda] :
( X
!= ( produc1362658507006291625lambda @ F6 @ ( produc1182943629000560279lambda @ ( cons_lambda_lambda @ A3 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_305_prod__induct3,axiom,
! [P: produc7929126107877444343lambda > $o,X: produc7929126107877444343lambda] :
( ! [A3: lambda_lambda > lambda_lambda,B2: list_lambda_lambda,C3: list_lambda_lambda] : ( P @ ( produc1362658507006291625lambda @ A3 @ ( produc1182943629000560279lambda @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_306_prod__cases3,axiom,
! [Y2: produc7929126107877444343lambda] :
~ ! [A3: lambda_lambda > lambda_lambda,B2: list_lambda_lambda,C3: list_lambda_lambda] :
( Y2
!= ( produc1362658507006291625lambda @ A3 @ ( produc1182943629000560279lambda @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_307_Pair__inject,axiom,
! [A: lambda_lambda > lambda_lambda > $o,B: list_lambda_lambda,A5: lambda_lambda > lambda_lambda > $o,B4: list_lambda_lambda] :
( ( ( produc8587549711031851289lambda @ A @ B )
= ( produc8587549711031851289lambda @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_308_Pair__inject,axiom,
! [A: lambda_lambda > lambda_lambda,B: produc3148917353989188007lambda,A5: lambda_lambda > lambda_lambda,B4: produc3148917353989188007lambda] :
( ( ( produc1362658507006291625lambda @ A @ B )
= ( produc1362658507006291625lambda @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_309_Pair__inject,axiom,
! [A: list_lambda_lambda,B: list_lambda_lambda,A5: list_lambda_lambda,B4: list_lambda_lambda] :
( ( ( produc1182943629000560279lambda @ A @ B )
= ( produc1182943629000560279lambda @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_310_Pair__inject,axiom,
! [A: lambda_lambda,B: list_lambda_lambda,A5: lambda_lambda,B4: list_lambda_lambda] :
( ( ( produc8847644316606783005lambda @ A @ B )
= ( produc8847644316606783005lambda @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_311_Pair__inject,axiom,
! [A: list_lambda_lambda,B: lambda_lambda,A5: list_lambda_lambda,B4: lambda_lambda] :
( ( ( produc7613787137529623313lambda @ A @ B )
= ( produc7613787137529623313lambda @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_312_prod__cases,axiom,
! [P: produc1137487172120676137lambda > $o,P3: produc1137487172120676137lambda] :
( ! [A3: lambda_lambda > lambda_lambda > $o,B2: list_lambda_lambda] : ( P @ ( produc8587549711031851289lambda @ A3 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_313_prod__cases,axiom,
! [P: produc7929126107877444343lambda > $o,P3: produc7929126107877444343lambda] :
( ! [A3: lambda_lambda > lambda_lambda,B2: produc3148917353989188007lambda] : ( P @ ( produc1362658507006291625lambda @ A3 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_314_prod__cases,axiom,
! [P: produc3148917353989188007lambda > $o,P3: produc3148917353989188007lambda] :
( ! [A3: list_lambda_lambda,B2: list_lambda_lambda] : ( P @ ( produc1182943629000560279lambda @ A3 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_315_prod__cases,axiom,
! [P: produc2030461353491475885lambda > $o,P3: produc2030461353491475885lambda] :
( ! [A3: lambda_lambda,B2: list_lambda_lambda] : ( P @ ( produc8847644316606783005lambda @ A3 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_316_prod__cases,axiom,
! [P: produc3002779106298095521lambda > $o,P3: produc3002779106298095521lambda] :
( ! [A3: list_lambda_lambda,B2: lambda_lambda] : ( P @ ( produc7613787137529623313lambda @ A3 @ B2 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_317_surj__pair,axiom,
! [P3: produc1137487172120676137lambda] :
? [X3: lambda_lambda > lambda_lambda > $o,Y: list_lambda_lambda] :
( P3
= ( produc8587549711031851289lambda @ X3 @ Y ) ) ).
% surj_pair
thf(fact_318_surj__pair,axiom,
! [P3: produc7929126107877444343lambda] :
? [X3: lambda_lambda > lambda_lambda,Y: produc3148917353989188007lambda] :
( P3
= ( produc1362658507006291625lambda @ X3 @ Y ) ) ).
% surj_pair
thf(fact_319_surj__pair,axiom,
! [P3: produc3148917353989188007lambda] :
? [X3: list_lambda_lambda,Y: list_lambda_lambda] :
( P3
= ( produc1182943629000560279lambda @ X3 @ Y ) ) ).
% surj_pair
thf(fact_320_surj__pair,axiom,
! [P3: produc2030461353491475885lambda] :
? [X3: lambda_lambda,Y: list_lambda_lambda] :
( P3
= ( produc8847644316606783005lambda @ X3 @ Y ) ) ).
% surj_pair
thf(fact_321_surj__pair,axiom,
! [P3: produc3002779106298095521lambda] :
? [X3: list_lambda_lambda,Y: lambda_lambda] :
( P3
= ( produc7613787137529623313lambda @ X3 @ Y ) ) ).
% surj_pair
thf(fact_322_old_Oprod_Oexhaust,axiom,
! [Y2: produc1137487172120676137lambda] :
~ ! [A3: lambda_lambda > lambda_lambda > $o,B2: list_lambda_lambda] :
( Y2
!= ( produc8587549711031851289lambda @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_323_old_Oprod_Oexhaust,axiom,
! [Y2: produc7929126107877444343lambda] :
~ ! [A3: lambda_lambda > lambda_lambda,B2: produc3148917353989188007lambda] :
( Y2
!= ( produc1362658507006291625lambda @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_324_old_Oprod_Oexhaust,axiom,
! [Y2: produc3148917353989188007lambda] :
~ ! [A3: list_lambda_lambda,B2: list_lambda_lambda] :
( Y2
!= ( produc1182943629000560279lambda @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_325_old_Oprod_Oexhaust,axiom,
! [Y2: produc2030461353491475885lambda] :
~ ! [A3: lambda_lambda,B2: list_lambda_lambda] :
( Y2
!= ( produc8847644316606783005lambda @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_326_old_Oprod_Oexhaust,axiom,
! [Y2: produc3002779106298095521lambda] :
~ ! [A3: list_lambda_lambda,B2: lambda_lambda] :
( Y2
!= ( produc7613787137529623313lambda @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_327_lambda__calculus_Obottom__up__redex_Osimps_I4_J,axiom,
! [M2: lambda_lambda,N3: lambda_lambda] :
( ( ~ ( lambda_Ide @ M2 )
=> ( ( lambda5654762366400029632_redex @ ( lambda_App @ M2 @ N3 ) )
= ( lambda_App @ ( lambda5654762366400029632_redex @ M2 ) @ ( lambda_Src @ N3 ) ) ) )
& ( ( lambda_Ide @ M2 )
=> ( ( lambda5654762366400029632_redex @ ( lambda_App @ M2 @ N3 ) )
= ( lambda_App @ M2 @ ( lambda5654762366400029632_redex @ N3 ) ) ) ) ) ).
% lambda_calculus.bottom_up_redex.simps(4)
thf(fact_328_reduction__paths_Oorthogonal__App__single__single_I2_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ U2 ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ U2 ) @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Trg @ U2 ) ) @ nil_lambda_lambda ) ) ) ) ).
% reduction_paths.orthogonal_App_single_single(2)
thf(fact_329_reduction__paths_Oorthogonal__App__single__single_I1_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ U2 ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ U2 ) ) @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ T ) @ U2 ) @ nil_lambda_lambda ) ) ) ) ).
% reduction_paths.orthogonal_App_single_single(1)
thf(fact_330_partial__magma__axioms,axiom,
partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% partial_magma_axioms
thf(fact_331_rts__with__composites__axioms,axiom,
rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% rts_with_composites_axioms
thf(fact_332_complete__development__def,axiom,
( reduct7282684938351565978opment
= ( ^ [T2: lambda_lambda,U3: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T2 @ U3 )
& ( ( lambda_Ide @ T2 )
| ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T2 @ nil_lambda_lambda ) @ U3 ) ) ) ) ) ) ).
% complete_development_def
thf(fact_333_cong__standard__development,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ~ ( lambda_Ide @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( reduct1603535164923377302opment @ T ) ) ) ) ) ).
% cong_standard_development
thf(fact_334_Con__sym__ind,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T6 ) @ ( size_s4208335251162658524lambda @ U ) ) @ N )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 )
!= nil_lambda_lambda ) ) ) ).
% Con_sym_ind
thf(fact_335_Cube__ind,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda,N: nat] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T6 )
!= nil_lambda_lambda )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T6 ) @ ( size_s4208335251162658524lambda @ U ) ) @ ( size_s4208335251162658524lambda @ V3 ) ) @ N )
=> ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T6 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
!= nil_lambda_lambda ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T6 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T6 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) ) ) ) ) ) ) ) ).
% Cube_ind
thf(fact_336_length__Resid__ind,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T6 ) @ ( size_s4208335251162658524lambda @ U ) ) @ N )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( size_s4208335251162658524lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( size_s4208335251162658524lambda @ T6 ) ) ) ) ).
% length_Resid_ind
thf(fact_337_complete__development__cong,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ U )
=> ( ~ ( lambda_Ide @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).
% complete_development_cong
thf(fact_338_cong__symmetric,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) ) ) ) ).
% cong_symmetric
thf(fact_339_cong__transitive,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) ) ) ) ) ).
% cong_transitive
thf(fact_340_ide__backward__stable,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ A ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% ide_backward_stable
thf(fact_341_prfx__transitive,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) ) ) ) ).
% prfx_transitive
thf(fact_342_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_343_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_344_Src__hd__eqI,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) )
=> ( ( lambda_Src @ ( hd_lambda_lambda @ T6 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U ) ) ) ) ).
% Src_hd_eqI
thf(fact_345_Trg__last__eqI,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ T6 ) )
= ( lambda_Trg @ ( last_lambda_lambda @ U ) ) ) ) ).
% Trg_last_eqI
thf(fact_346_complete__developments__cong,axiom,
! [T: lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda] :
( ~ ( lambda_Ide @ T )
=> ( ( reduct7282684938351565978opment @ T @ U )
=> ( ( reduct7282684938351565978opment @ T @ V3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U ) ) ) ) ) ) ).
% complete_developments_cong
thf(fact_347_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_348_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_349_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_350_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_351_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_352_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_353_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_354_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_355_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_356_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_357_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_358_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_359_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_360_development__implies,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U )
=> ( ( reduct8314398545252454212n_path @ U )
& ( ( U != nil_lambda_lambda )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).
% development_implies
thf(fact_361_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_362_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_363_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_364_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_365_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_366_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_367_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_368_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_369_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_370_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_371_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_372_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_373_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_374_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_375_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_376_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_377_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_378_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_379_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_380_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_381_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C2: nat] :
( B3
= ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_382_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_383_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_384_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_385_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_386_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_387_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_388_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_389_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_390_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_391_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_392_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_393_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_394_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_395_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_396_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_397_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_398_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_399_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_400_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_401_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_402_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_403_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_404_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_405_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_406_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_407_lambda__calculus_Ohgt_Osimps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_hgt @ ( lambda_App @ T @ U2 ) )
= ( plus_plus_nat @ ( lambda_hgt @ T ) @ ( lambda_hgt @ U2 ) ) ) ).
% lambda_calculus.hgt.simps(4)
thf(fact_408_cong__implies__coterminal,axiom,
! [U2: list_lambda_lambda,U4: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U2 ) ) )
=> ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 @ U4 ) ) ).
% cong_implies_coterminal
thf(fact_409_identities__form__coherent__normal__sub__rts,axiom,
cohere6622258358606131289lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( collec8830145823237851355lambda @ ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% identities_form_coherent_normal_sub_rts
thf(fact_410_Resid__cons__ind,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,N: nat] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T6 ) @ ( size_s4208335251162658524lambda @ U ) ) @ N )
=> ( ! [T5: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T5 @ T6 ) @ U )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T5 @ nil_lambda_lambda ) @ U )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T5 @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) )
& ! [U7: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U7 @ U ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U7 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U7 @ nil_lambda_lambda ) ) @ U )
!= nil_lambda_lambda ) ) )
& ! [T5: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T5 @ T6 ) @ U )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T5 @ T6 ) @ U )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T5 @ nil_lambda_lambda ) @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T5 @ nil_lambda_lambda ) ) ) ) ) )
& ! [U7: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U7 @ U ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U7 @ U ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U7 @ nil_lambda_lambda ) ) @ U ) ) ) ) ) ) ) ).
% Resid_cons_ind
thf(fact_411_head__redex__decomp,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_head_redex @ T ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ T @ ( lambda_head_redex @ T ) ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_head_redex @ T ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ T @ ( lambda_head_redex @ T ) ) @ nil_lambda_lambda ) ) ) ) ) ) ).
% head_redex_decomp
thf(fact_412_paths__in__confluent__rts_Ois__confluent__rts,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( conflu1940213901339057510lambda @ ( paths_3312059784463760222lambda @ Resid ) ) ) ).
% paths_in_confluent_rts.is_confluent_rts
thf(fact_413_paths__in__confluent__rts_Ois__confluent__rts,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ Resid ) ) ) ).
% paths_in_confluent_rts.is_confluent_rts
thf(fact_414_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_415_Resid__append__ind,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T6 ) @ U )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V3 ) )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ V3 @ U ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 ) @ U )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T6 ) @ U )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T6 ) @ U )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V3 ) ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ V3 @ U ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ V3 @ U ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 ) @ U ) ) ) ) ) ) ) ).
% Resid_append_ind
thf(fact_416_Con__append_I1_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ V3 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T6 ) )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(1)
thf(fact_417_Con__append_I2_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ U @ V3 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) @ V3 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(2)
thf(fact_418_append_Oassoc,axiom,
! [A: list_l5168835686021156086lambda,B: list_l5168835686021156086lambda,C: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( append1017360404025248779lambda @ A @ B ) @ C )
= ( append1017360404025248779lambda @ A @ ( append1017360404025248779lambda @ B @ C ) ) ) ).
% append.assoc
thf(fact_419_append_Oassoc,axiom,
! [A: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ A @ B ) @ C )
= ( append_lambda_lambda @ A @ ( append_lambda_lambda @ B @ C ) ) ) ).
% append.assoc
thf(fact_420_append__assoc,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) @ Zs )
= ( append1017360404025248779lambda @ Xs @ ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_421_append__assoc,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) @ Zs )
= ( append_lambda_lambda @ Xs @ ( append_lambda_lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_422_append__same__eq,axiom,
! [Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Xs )
= ( append1017360404025248779lambda @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_423_append__same__eq,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Xs )
= ( append_lambda_lambda @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_424_same__append__eq,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_425_same__append__eq,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_426_Resid__cons_I1_J,axiom,
! [U: list_lambda_lambda,T: lambda_lambda,T6: list_lambda_lambda] :
( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% Resid_cons(1)
thf(fact_427_Resid__rec_I4_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U2 @ U ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U2 @ U ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U2 @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) ) ) ) ) ) ) ) ).
% Resid_rec(4)
thf(fact_428_prfx__decomp,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ U2 @ T ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ U2 @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ).
% prfx_decomp
thf(fact_429_append__is__Nil__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= nil_li8379291187143545974lambda )
= ( ( Xs = nil_li8379291187143545974lambda )
& ( Ys = nil_li8379291187143545974lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_430_append__is__Nil__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= nil_lambda_lambda )
= ( ( Xs = nil_lambda_lambda )
& ( Ys = nil_lambda_lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_431_Nil__is__append__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( nil_li8379291187143545974lambda
= ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( ( Xs = nil_li8379291187143545974lambda )
& ( Ys = nil_li8379291187143545974lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_432_Nil__is__append__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( append_lambda_lambda @ Xs @ Ys ) )
= ( ( Xs = nil_lambda_lambda )
& ( Ys = nil_lambda_lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_433_self__append__conv2,axiom,
! [Y2: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda] :
( ( Y2
= ( append1017360404025248779lambda @ Xs @ Y2 ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% self_append_conv2
thf(fact_434_self__append__conv2,axiom,
! [Y2: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Y2
= ( append_lambda_lambda @ Xs @ Y2 ) )
= ( Xs = nil_lambda_lambda ) ) ).
% self_append_conv2
thf(fact_435_append__self__conv2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= Ys )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% append_self_conv2
thf(fact_436_append__self__conv2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= Ys )
= ( Xs = nil_lambda_lambda ) ) ).
% append_self_conv2
thf(fact_437_self__append__conv,axiom,
! [Y2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Y2
= ( append1017360404025248779lambda @ Y2 @ Ys ) )
= ( Ys = nil_li8379291187143545974lambda ) ) ).
% self_append_conv
thf(fact_438_self__append__conv,axiom,
! [Y2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Y2
= ( append_lambda_lambda @ Y2 @ Ys ) )
= ( Ys = nil_lambda_lambda ) ) ).
% self_append_conv
thf(fact_439_append__self__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= Xs )
= ( Ys = nil_li8379291187143545974lambda ) ) ).
% append_self_conv
thf(fact_440_append__self__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= Xs )
= ( Ys = nil_lambda_lambda ) ) ).
% append_self_conv
thf(fact_441_append__Nil2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ Xs @ nil_li8379291187143545974lambda )
= Xs ) ).
% append_Nil2
thf(fact_442_append__Nil2,axiom,
! [Xs: list_lambda_lambda] :
( ( append_lambda_lambda @ Xs @ nil_lambda_lambda )
= Xs ) ).
% append_Nil2
thf(fact_443_append_Oright__neutral,axiom,
! [A: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ A @ nil_li8379291187143545974lambda )
= A ) ).
% append.right_neutral
thf(fact_444_append_Oright__neutral,axiom,
! [A: list_lambda_lambda] :
( ( append_lambda_lambda @ A @ nil_lambda_lambda )
= A ) ).
% append.right_neutral
thf(fact_445_append__eq__append__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Us: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
| ( ( size_s6298197441921507170lambda @ Us )
= ( size_s6298197441921507170lambda @ Vs ) ) )
=> ( ( ( append1017360404025248779lambda @ Xs @ Us )
= ( append1017360404025248779lambda @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_446_append__eq__append__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
| ( ( size_s4208335251162658524lambda @ Us )
= ( size_s4208335251162658524lambda @ Vs ) ) )
=> ( ( ( append_lambda_lambda @ Xs @ Us )
= ( append_lambda_lambda @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_447_append1__eq__conv,axiom,
! [Xs: list_l5168835686021156086lambda,X: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Y2: list_lambda_lambda] :
( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
= ( append1017360404025248779lambda @ Ys @ ( cons_l4558873032622161958lambda @ Y2 @ nil_li8379291187143545974lambda ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_448_append1__eq__conv,axiom,
! [Xs: list_lambda_lambda,X: lambda_lambda,Ys: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ Y2 @ nil_lambda_lambda ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_449_length__append,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( size_s6298197441921507170lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s6298197441921507170lambda @ Xs ) @ ( size_s6298197441921507170lambda @ Ys ) ) ) ).
% length_append
thf(fact_450_length__append,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( size_s4208335251162658524lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s4208335251162658524lambda @ Xs ) @ ( size_s4208335251162658524lambda @ Ys ) ) ) ).
% length_append
thf(fact_451_hd__append2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Xs ) ) ) ).
% hd_append2
thf(fact_452_hd__append2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Xs ) ) ) ).
% hd_append2
thf(fact_453_last__appendL,axiom,
! [Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda] :
( ( Ys = nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( last_l1874196399739661071lambda @ Xs ) ) ) ).
% last_appendL
thf(fact_454_last__appendL,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Ys = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_appendL
thf(fact_455_last__appendR,axiom,
! [Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda] :
( ( Ys != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( last_l1874196399739661071lambda @ Ys ) ) ) ).
% last_appendR
thf(fact_456_last__appendR,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Ys != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Ys ) ) ) ).
% last_appendR
thf(fact_457_last__snoc,axiom,
! [Xs: list_l5168835686021156086lambda,X: list_lambda_lambda] :
( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) ) )
= X ) ).
% last_snoc
thf(fact_458_last__snoc,axiom,
! [Xs: list_lambda_lambda,X: lambda_lambda] :
( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) ) )
= X ) ).
% last_snoc
thf(fact_459_Resid__append2,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda,W2: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( W2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ ( append_lambda_lambda @ V3 @ W2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ ( append_lambda_lambda @ V3 @ W2 ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 ) @ W2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T6 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 ) ) ) ) ) ) ) ) ) ) ).
% Resid_append2
thf(fact_460_Resid__append_I1_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ V3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ V3 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T6 ) ) ) ) ) ) ).
% Resid_append(1)
thf(fact_461_Resid__append_I2_J,axiom,
! [U: list_lambda_lambda,V3: list_lambda_lambda,T6: list_lambda_lambda] :
( ( U != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ U @ V3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ U @ V3 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) @ V3 ) ) ) ) ) ).
% Resid_append(2)
thf(fact_462_Con__appendI_I1_J,axiom,
! [T6: list_lambda_lambda,V3: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V3 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T6 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ V3 )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(1)
thf(fact_463_Con__appendI_I2_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) @ V3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ U @ V3 ) )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(2)
thf(fact_464_append__eq__appendI,axiom,
! [Xs: list_l5168835686021156086lambda,Xs1: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Us: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append1017360404025248779lambda @ Xs1 @ Us ) )
=> ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_465_append__eq__appendI,axiom,
! [Xs: list_lambda_lambda,Xs1: list_lambda_lambda,Zs: list_lambda_lambda,Ys: list_lambda_lambda,Us: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_lambda_lambda @ Xs1 @ Us ) )
=> ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_466_append__eq__append__conv2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ts: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Zs @ Ts ) )
= ( ? [Us2: list_l5168835686021156086lambda] :
( ( ( Xs
= ( append1017360404025248779lambda @ Zs @ Us2 ) )
& ( ( append1017360404025248779lambda @ Us2 @ Ys )
= Ts ) )
| ( ( ( append1017360404025248779lambda @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append1017360404025248779lambda @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_467_append__eq__append__conv2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ts: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs @ Ts ) )
= ( ? [Us2: list_lambda_lambda] :
( ( ( Xs
= ( append_lambda_lambda @ Zs @ Us2 ) )
& ( ( append_lambda_lambda @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_lambda_lambda @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_lambda_lambda @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_468_Cons__eq__appendI,axiom,
! [X: list_lambda_lambda,Xs1: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append1017360404025248779lambda @ Xs1 @ Zs ) )
=> ( ( cons_l4558873032622161958lambda @ X @ Xs )
= ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_469_Cons__eq__appendI,axiom,
! [X: lambda_lambda,Xs1: list_lambda_lambda,Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_lambda_lambda @ Xs1 @ Zs ) )
=> ( ( cons_lambda_lambda @ X @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_470_append__Cons,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) @ Ys )
= ( cons_l4558873032622161958lambda @ X @ ( append1017360404025248779lambda @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_471_append__Cons,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ ( cons_lambda_lambda @ X @ Xs ) @ Ys )
= ( cons_lambda_lambda @ X @ ( append_lambda_lambda @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_472_eq__Nil__appendI,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs = Ys )
=> ( Xs
= ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_473_eq__Nil__appendI,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs = Ys )
=> ( Xs
= ( append_lambda_lambda @ nil_lambda_lambda @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_474_append_Oleft__neutral,axiom,
! [A: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ A )
= A ) ).
% append.left_neutral
thf(fact_475_append_Oleft__neutral,axiom,
! [A: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ A )
= A ) ).
% append.left_neutral
thf(fact_476_append__Nil,axiom,
! [Ys: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_477_append__Nil,axiom,
! [Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_478_rev__induct,axiom,
! [P: list_l5168835686021156086lambda > $o,Xs: list_l5168835686021156086lambda] :
( ( P @ nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( P @ Xs2 )
=> ( P @ ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_479_rev__induct,axiom,
! [P: list_lambda_lambda > $o,Xs: list_lambda_lambda] :
( ( P @ nil_lambda_lambda )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( P @ Xs2 )
=> ( P @ ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_480_rev__exhaust,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ~ ! [Ys2: list_l5168835686021156086lambda,Y: list_lambda_lambda] :
( Xs
!= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ Y @ nil_li8379291187143545974lambda ) ) ) ) ).
% rev_exhaust
thf(fact_481_rev__exhaust,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ~ ! [Ys2: list_lambda_lambda,Y: lambda_lambda] :
( Xs
!= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ Y @ nil_lambda_lambda ) ) ) ) ).
% rev_exhaust
thf(fact_482_Cons__eq__append__conv,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X @ Xs )
= ( append1017360404025248779lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_li8379291187143545974lambda )
& ( ( cons_l4558873032622161958lambda @ X @ Xs )
= Zs ) )
| ? [Ys4: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X @ Ys4 )
= Ys )
& ( Xs
= ( append1017360404025248779lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_483_Cons__eq__append__conv,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( ( cons_lambda_lambda @ X @ Xs )
= Zs ) )
| ? [Ys4: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_lambda_lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_484_append__eq__Cons__conv,axiom,
! [Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Zs )
= ( cons_l4558873032622161958lambda @ X @ Xs ) )
= ( ( ( Ys = nil_li8379291187143545974lambda )
& ( Zs
= ( cons_l4558873032622161958lambda @ X @ Xs ) ) )
| ? [Ys4: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ X @ Ys4 ) )
& ( ( append1017360404025248779lambda @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_485_append__eq__Cons__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,X: lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( cons_lambda_lambda @ X @ Xs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( Zs
= ( cons_lambda_lambda @ X @ Xs ) ) )
| ? [Ys4: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ X @ Ys4 ) )
& ( ( append_lambda_lambda @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_486_rev__nonempty__induct,axiom,
! [Xs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ! [X3: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) )
=> ( ! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_487_rev__nonempty__induct,axiom,
! [Xs: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( Xs != nil_lambda_lambda )
=> ( ! [X3: lambda_lambda] : ( P @ ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) )
=> ( ! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_488_hd__append,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( Xs = nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Ys ) ) )
& ( ( Xs != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Xs ) ) ) ) ).
% hd_append
thf(fact_489_hd__append,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Ys ) ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Xs ) ) ) ) ).
% hd_append
thf(fact_490_longest__common__prefix,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
? [Ps: list_l5168835686021156086lambda,Xs3: list_l5168835686021156086lambda,Ys5: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ps @ Xs3 ) )
& ( Ys
= ( append1017360404025248779lambda @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_li8379291187143545974lambda )
| ( Ys5 = nil_li8379291187143545974lambda )
| ( ( hd_lis2183642875561147995lambda @ Xs3 )
!= ( hd_lis2183642875561147995lambda @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_491_longest__common__prefix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ps: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ps @ Xs3 ) )
& ( Ys
= ( append_lambda_lambda @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_lambda_lambda )
| ( Ys5 = nil_lambda_lambda )
| ( ( hd_lambda_lambda @ Xs3 )
!= ( hd_lambda_lambda @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_492_last__append,axiom,
! [Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda] :
( ( ( Ys = nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( last_l1874196399739661071lambda @ Xs ) ) )
& ( ( Ys != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( last_l1874196399739661071lambda @ Ys ) ) ) ) ).
% last_append
thf(fact_493_last__append,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( ( Ys = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Xs ) ) )
& ( ( Ys != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Ys ) ) ) ) ).
% last_append
thf(fact_494_longest__common__suffix,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
? [Ss: list_l5168835686021156086lambda,Xs3: list_l5168835686021156086lambda,Ys5: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Xs3 @ Ss ) )
& ( Ys
= ( append1017360404025248779lambda @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_li8379291187143545974lambda )
| ( Ys5 = nil_li8379291187143545974lambda )
| ( ( last_l1874196399739661071lambda @ Xs3 )
!= ( last_l1874196399739661071lambda @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_495_longest__common__suffix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ss: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Xs3 @ Ss ) )
& ( Ys
= ( append_lambda_lambda @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_lambda_lambda )
| ( Ys5 = nil_lambda_lambda )
| ( ( last_lambda_lambda @ Xs3 )
!= ( last_lambda_lambda @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_496_same__length__different,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs != Ys )
=> ( ( ( size_s6298197441921507170lambda @ Xs )
= ( size_s6298197441921507170lambda @ Ys ) )
=> ? [Pre: list_l5168835686021156086lambda,X3: list_lambda_lambda,Xs3: list_l5168835686021156086lambda,Y: list_lambda_lambda,Ys5: list_l5168835686021156086lambda] :
( ( X3 != Y )
& ( Xs
= ( append1017360404025248779lambda @ Pre @ ( append1017360404025248779lambda @ ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) @ Xs3 ) ) )
& ( Ys
= ( append1017360404025248779lambda @ Pre @ ( append1017360404025248779lambda @ ( cons_l4558873032622161958lambda @ Y @ nil_li8379291187143545974lambda ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_497_same__length__different,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs != Ys )
=> ( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ? [Pre: list_lambda_lambda,X3: lambda_lambda,Xs3: list_lambda_lambda,Y: lambda_lambda,Ys5: list_lambda_lambda] :
( ( X3 != Y )
& ( Xs
= ( append_lambda_lambda @ Pre @ ( append_lambda_lambda @ ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) @ Xs3 ) ) )
& ( Ys
= ( append_lambda_lambda @ Pre @ ( append_lambda_lambda @ ( cons_lambda_lambda @ Y @ nil_lambda_lambda ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_498_paths__in__rts_OResid_Ocong,axiom,
paths_3312059784463760222lambda = paths_3312059784463760222lambda ).
% paths_in_rts.Resid.cong
thf(fact_499_paths__in__rts_OResid_Ocong,axiom,
paths_8607357831768207192lambda = paths_8607357831768207192lambda ).
% paths_in_rts.Resid.cong
thf(fact_500_residuation_Oide_Ocong,axiom,
ide_li6773656263178250516lambda = ide_li6773656263178250516lambda ).
% residuation.ide.cong
thf(fact_501_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_502_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X2: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ X2 @ T2 )
= X2 )
& ( ( OP @ T2 @ X2 )
= X2 ) )
& ! [Y4: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ Y4 @ T2 )
= Y4 )
& ( ( OP @ T2 @ Y4 )
= Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_503_partial__magma__def,axiom,
( partia748546660539056525lambda
= ( ^ [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
? [X2: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP @ X2 @ T2 )
= X2 )
& ( ( OP @ T2 @ X2 )
= X2 ) )
& ! [Y4: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP @ Y4 @ T2 )
= Y4 )
& ( ( OP @ T2 @ Y4 )
= Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_504_partial__magma_Oex__un__null,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ? [X3: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP2 @ X3 @ T5 )
= X3 )
& ( ( OP2 @ T5 @ X3 )
= X3 ) )
& ! [Y3: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( OP2 @ Y3 @ T4 )
= Y3 )
& ( ( OP2 @ T4 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_505_partial__magma_Oex__un__null,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ? [X3: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( OP2 @ X3 @ T5 )
= X3 )
& ( ( OP2 @ T5 @ X3 )
= X3 ) )
& ! [Y3: list_lambda_lambda] :
( ! [T4: list_lambda_lambda] :
( ( ( OP2 @ Y3 @ T4 )
= Y3 )
& ( ( OP2 @ T4 @ Y3 )
= Y3 ) )
=> ( Y3 = X3 ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_506_partial__magma_Ointro,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ? [X5: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( OP2 @ X5 @ T4 )
= X5 )
& ( ( OP2 @ T4 @ X5 )
= X5 ) )
& ! [Y: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP2 @ Y @ T5 )
= Y )
& ( ( OP2 @ T5 @ Y )
= Y ) )
=> ( Y = X5 ) ) )
=> ( partia3924075026459072775lambda @ OP2 ) ) ).
% partial_magma.intro
thf(fact_507_partial__magma_Ointro,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ? [X5: list_lambda_lambda] :
( ! [T4: list_lambda_lambda] :
( ( ( OP2 @ X5 @ T4 )
= X5 )
& ( ( OP2 @ T4 @ X5 )
= X5 ) )
& ! [Y: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( OP2 @ Y @ T5 )
= Y )
& ( ( OP2 @ T5 @ Y )
= Y ) )
=> ( Y = X5 ) ) )
=> ( partia748546660539056525lambda @ OP2 ) ) ).
% partial_magma.intro
thf(fact_508_paths__in__confluent__rts_Oaxioms_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( conflu4036547884742428384lambda @ Resid ) ) ).
% paths_in_confluent_rts.axioms(2)
thf(fact_509_paths__in__confluent__rts_Oaxioms_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( conflu4927472611300549594lambda @ Resid ) ) ).
% paths_in_confluent_rts.axioms(2)
thf(fact_510_reduction__paths_OBeta__decomp_I1_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U2 ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ U2 @ T ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U2 ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U2 ) @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U2 ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ U2 @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% reduction_paths.Beta_decomp(1)
thf(fact_511_Trg__last__Src__hd__eqI,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ T6 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U ) ) ) ) ).
% Trg_last_Src_hd_eqI
thf(fact_512_cong__map__App1,axiom,
! [X: lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( lambda_Ide @ X )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X ) @ T6 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X ) @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X ) @ U ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X ) @ T6 ) ) ) ) ) ) ).
% cong_map_App1
thf(fact_513_reduction__paths_OBeta__decomp_I2_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ T ) @ U2 ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U2 ) ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U2 ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U2 ) @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ T ) @ U2 ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U2 ) ) @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% reduction_paths.Beta_decomp(2)
thf(fact_514_Resid__parallel,axiom,
! [T: list_lambda_lambda,T3: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ T ) ) )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T3 ) ) ) ) ).
% Resid_parallel
thf(fact_515_coinitial__ide__are__cong,axiom,
! [A: list_lambda_lambda,A5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A5 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A5 @ A ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_516__092_060Lambda_062_Ocong__symmetric,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) ) ) ) ).
% \<Lambda>.cong_symmetric
thf(fact_517__092_060Lambda_062_Ocong__transitive,axiom,
! [T: lambda_lambda,U2: lambda_lambda,V: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U2 ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) ) ) ) ) ).
% \<Lambda>.cong_transitive
thf(fact_518__092_060Lambda_062_Oide__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 ) ) ) ).
% \<Lambda>.ide_backward_stable
thf(fact_519__092_060Lambda_062_Oprfx__transitive,axiom,
! [T: lambda_lambda,U2: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ V ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) ) ) ) ).
% \<Lambda>.prfx_transitive
thf(fact_520__092_060Lambda_062_Oweak__extensionality,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ U2 )
=> ( T = U2 ) ) ) ) ).
% \<Lambda>.weak_extensionality
thf(fact_521_development__map__Lam,axiom,
! [T: lambda_lambda,T6: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ T6 )
=> ( reduct9208763521642324371opment @ ( lambda_Lam @ T ) @ ( map_la336867333220232516lambda @ lambda_Lam @ T6 ) ) ) ).
% development_map_Lam
thf(fact_522_cong__respects__seq_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,T7: list_lambda_lambda,U8: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T6 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U8 @ U ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 @ U8 ) ) ) ) ).
% cong_respects_seq\<^sub>P
thf(fact_523_cong__respects__seq,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,T3: list_lambda_lambda,U4: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 @ U4 ) ) ) ) ).
% cong_respects_seq
thf(fact_524_cong__implies__coinitial,axiom,
! [U2: list_lambda_lambda,U4: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U2 ) ) )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 @ U4 ) ) ).
% cong_implies_coinitial
thf(fact_525_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_526_cong__map__Lam,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T6 ) @ ( map_la336867333220232516lambda @ lambda_Lam @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ U ) @ ( map_la336867333220232516lambda @ lambda_Lam @ T6 ) ) ) ) ) ).
% cong_map_Lam
thf(fact_527_map__Lam__Resid,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( map_la336867333220232516lambda @ lambda_Lam @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T6 ) @ ( map_la336867333220232516lambda @ lambda_Lam @ U ) ) ) ) ).
% map_Lam_Resid
thf(fact_528_map__App1__Resid,axiom,
! [X: lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( lambda_Ide @ X )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( map_la336867333220232516lambda @ ( lambda_App @ X ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X ) @ T6 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X ) @ U ) ) ) ) ) ).
% map_App1_Resid
thf(fact_529_seq__imp__cong__Trg__last__Src__hd,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ ( lambda_Src @ ( hd_lambda_lambda @ U ) ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Src @ ( hd_lambda_lambda @ U ) ) @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) ) ) ) ) ).
% seq_imp_cong_Trg_last_Src_hd
thf(fact_530_list_Omap__disc__iff,axiom,
! [F: list_lambda_lambda > lambda_lambda,A: list_l5168835686021156086lambda] :
( ( ( map_li5201084558375397438lambda @ F @ A )
= nil_lambda_lambda )
= ( A = nil_li8379291187143545974lambda ) ) ).
% list.map_disc_iff
thf(fact_531_list_Omap__disc__iff,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,A: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ A )
= nil_li8379291187143545974lambda )
= ( A = nil_li8379291187143545974lambda ) ) ).
% list.map_disc_iff
thf(fact_532_list_Omap__disc__iff,axiom,
! [F: lambda_lambda > list_lambda_lambda,A: list_lambda_lambda] :
( ( ( map_la6434941737452557130lambda @ F @ A )
= nil_li8379291187143545974lambda )
= ( A = nil_lambda_lambda ) ) ).
% list.map_disc_iff
thf(fact_533_list_Omap__disc__iff,axiom,
! [F: lambda_lambda > lambda_lambda,A: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ A )
= nil_lambda_lambda )
= ( A = nil_lambda_lambda ) ) ).
% list.map_disc_iff
thf(fact_534_Nil__is__map__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda] :
( ( nil_li8379291187143545974lambda
= ( map_la6434941737452557130lambda @ F @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% Nil_is_map_conv
thf(fact_535_Nil__is__map__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( map_la336867333220232516lambda @ F @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% Nil_is_map_conv
thf(fact_536_map__is__Nil__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= nil_lambda_lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% map_is_Nil_conv
thf(fact_537_lambda__calculus_Oide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.ide_char
thf(fact_538_lambda__calculus_Oprfx__char,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
= ( lambda_Ide @ ( lambda_resid @ T @ U2 ) ) ) ).
% lambda_calculus.prfx_char
thf(fact_539_length__map,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( size_s4208335251162658524lambda @ ( map_la336867333220232516lambda @ F @ Xs ) )
= ( size_s4208335251162658524lambda @ Xs ) ) ).
% length_map
thf(fact_540_map__append,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( append_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) @ ( map_la336867333220232516lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_541_Resid__single__ide_I2_J,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
= T6 ) ) ) ).
% Resid_single_ide(2)
thf(fact_542_cong__append__ideI_I4_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ T6 @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ T6 ) ) ) ) ) ).
% cong_append_ideI(4)
thf(fact_543_cong__append__ideI_I3_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( append_lambda_lambda @ T6 @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ U ) ) ) ) ) ).
% cong_append_ideI(3)
thf(fact_544_cong__append__ideI_I2_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ T6 @ U ) ) ) ) ) ) ).
% cong_append_ideI(2)
thf(fact_545_cong__append__ideI_I1_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( append_lambda_lambda @ T6 @ U ) ) ) ) ) ) ).
% cong_append_ideI(1)
thf(fact_546_cong__append,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,T7: list_lambda_lambda,U8: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T6 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U8 @ U ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) @ ( append_lambda_lambda @ T7 @ U8 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T7 @ U8 ) @ ( append_lambda_lambda @ T6 @ U ) ) ) ) ) ) ) ).
% cong_append
thf(fact_547_cong__cons__ideI_I2_J,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U ) @ U ) ) ) ) ) ).
% cong_cons_ideI(2)
thf(fact_548_cong__cons__ideI_I1_J,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U ) @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ T @ U ) ) ) ) ) ) ).
% cong_cons_ideI(1)
thf(fact_549_cong__cons,axiom,
! [T: lambda_lambda,U: list_lambda_lambda,T3: lambda_lambda,U8: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T3 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U8 @ U ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U ) @ ( cons_lambda_lambda @ T3 @ U8 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T3 @ U8 ) @ ( cons_lambda_lambda @ T @ U ) ) ) ) ) ) ) ).
% cong_cons
thf(fact_550_reduction__paths_Ostandard__development_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( reduct1603535164923377302opment @ ( lambda_Lam @ T ) )
= ( map_la336867333220232516lambda @ lambda_Lam @ ( reduct1603535164923377302opment @ T ) ) ) ).
% reduction_paths.standard_development.simps(3)
thf(fact_551_list_Osimps_I9_J,axiom,
! [F: lambda_lambda > lambda_lambda,X21: lambda_lambda,X22: list_lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ ( cons_lambda_lambda @ X21 @ X22 ) )
= ( cons_lambda_lambda @ ( F @ X21 ) @ ( map_la336867333220232516lambda @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_552_Cons__eq__map__D,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X @ Xs )
= ( map_la336867333220232516lambda @ F @ Ys ) )
=> ? [Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_553_map__eq__Cons__D,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Y2: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( cons_lambda_lambda @ Y2 @ Ys ) )
=> ? [Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_554_Cons__eq__map__conv,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X @ Xs )
= ( map_la336867333220232516lambda @ F @ Ys ) )
= ( ? [Z4: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_555_map__eq__Cons__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Y2: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( cons_lambda_lambda @ Y2 @ Ys ) )
= ( ? [Z4: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_556_list_Osimps_I8_J,axiom,
! [F: lambda_lambda > lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.simps(8)
thf(fact_557_map__eq__imp__length__eq,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,G: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Ys ) )
=> ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_558_map__eq__append__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ? [Us2: list_lambda_lambda,Vs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us2 @ Vs2 ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us2 ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_559_append__eq__map__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( map_la336867333220232516lambda @ F @ Xs ) )
= ( ? [Us2: list_lambda_lambda,Vs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us2 @ Vs2 ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us2 ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_560_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_561_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_562_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_563_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_564_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_565_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_566_lambda__calculus_Ocontains__head__reduction_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_Lam @ T ) )
= ( lambda8135948770199036598uction @ T ) ) ).
% lambda_calculus.contains_head_reduction.simps(2)
thf(fact_567_lambda__calculus_Ohead__redex_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_head_redex @ T ) ) ) ).
% lambda_calculus.head_redex.simps(3)
thf(fact_568_lambda__calculus_Ohgt_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_hgt @ ( lambda_Lam @ T ) )
= ( lambda_hgt @ T ) ) ).
% lambda_calculus.hgt.simps(3)
thf(fact_569_lambda__calculus_Obottom__up__redex_Osimps_I3_J,axiom,
! [M2: lambda_lambda] :
( ( lambda5654762366400029632_redex @ ( lambda_Lam @ M2 ) )
= ( lambda_Lam @ ( lambda5654762366400029632_redex @ M2 ) ) ) ).
% lambda_calculus.bottom_up_redex.simps(3)
thf(fact_570_lambda__calculus_Olambda_Odisc_I18_J,axiom,
! [X32: lambda_lambda] :
~ ( lambda_is_Beta @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(18)
thf(fact_571_lambda__calculus_Olambda_Odisc_I3_J,axiom,
! [X32: lambda_lambda] :
~ ( lambda_is_Var @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(3)
thf(fact_572_lambda__calculus_Olambda_Osimps_I27_J,axiom,
! [F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o,X32: lambda_lambda] :
( ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ ( lambda_Lam @ X32 ) )
= ( F3 @ X32 ) ) ).
% lambda_calculus.lambda.simps(27)
thf(fact_573_hd__map,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) )
= ( F @ ( hd_lambda_lambda @ Xs ) ) ) ) ).
% hd_map
thf(fact_574_list_Omap__sel_I1_J,axiom,
! [A: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ F @ A ) )
= ( F @ ( hd_lambda_lambda @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_575_lambda__calculus_Ocong__Ide__are__eq,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) )
=> ( ( lambda_Ide @ T )
=> ( ( lambda_Ide @ U2 )
=> ( T = U2 ) ) ) ) ).
% lambda_calculus.cong_Ide_are_eq
thf(fact_576_lambda__calculus_Oeq__Ide__are__cong,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( T = U2 )
=> ( ( lambda_Ide @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) ) ) ) ).
% lambda_calculus.eq_Ide_are_cong
thf(fact_577_last__map,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) )
= ( F @ ( last_lambda_lambda @ Xs ) ) ) ) ).
% last_map
thf(fact_578_lambda__calculus_Ois__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% lambda_calculus.is_confluent_rts
thf(fact_579_lambda__calculus_Ocontains__head__reduction_Osimps_I3_J,axiom,
! [Uv: lambda_lambda,Uw: lambda_lambda] :
~ ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Lam @ Uv ) @ Uw ) ) ).
% lambda_calculus.contains_head_reduction.simps(3)
thf(fact_580_lambda__calculus_OSrc_Osimps_I5_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Src @ ( lambda_Beta @ T @ U2 ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U2 ) ) ) ).
% lambda_calculus.Src.simps(5)
thf(fact_581_lambda__calculus_OSrc__eq__iff_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,T3: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ ( lambda_Lam @ T ) @ U2 ) )
= ( lambda_Src @ ( lambda_Beta @ T3 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T3 ) )
& ( ( lambda_Src @ U2 )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(4)
thf(fact_582_lambda__calculus_Ohead__redex_Osimps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Lam @ T ) @ U2 ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U2 ) ) ) ).
% lambda_calculus.head_redex.simps(4)
thf(fact_583_lambda__calculus_Obottom__up__redex_Osimps_I5_J,axiom,
! [M2: lambda_lambda,N3: lambda_lambda] :
( ( ~ ( lambda_Ide @ M2 )
=> ( ( lambda5654762366400029632_redex @ ( lambda_Beta @ M2 @ N3 ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda5654762366400029632_redex @ M2 ) ) @ ( lambda_Src @ N3 ) ) ) )
& ( ( lambda_Ide @ M2 )
=> ( ( ~ ( lambda_Ide @ N3 )
=> ( ( lambda5654762366400029632_redex @ ( lambda_Beta @ M2 @ N3 ) )
= ( lambda_App @ ( lambda_Lam @ M2 ) @ ( lambda5654762366400029632_redex @ N3 ) ) ) )
& ( ( lambda_Ide @ N3 )
=> ( ( lambda5654762366400029632_redex @ ( lambda_Beta @ M2 @ N3 ) )
= ( lambda_Beta @ M2 @ N3 ) ) ) ) ) ) ).
% lambda_calculus.bottom_up_redex.simps(5)
thf(fact_584_seq__map__un__Lam,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Lam @ T ) @ nil_lambda_lambda ) @ U )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( map_la336867333220232516lambda @ lambda_un_Lam @ U ) ) ) ).
% seq_map_un_Lam
thf(fact_585_orthogonal__App__single__Arr,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ T ) ) @ U ) @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ ( hd_lambda_lambda @ U ) ) ) @ nil_lambda_lambda ) )
= ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ T ) ) @ U ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ ( hd_lambda_lambda @ U ) ) ) @ nil_lambda_lambda ) @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ T ) ) @ U ) )
= ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Trg @ ( last_lambda_lambda @ U ) ) ) @ nil_lambda_lambda ) ) ) ) ) ).
% orthogonal_App_single_Arr
thf(fact_586_joinable__implies__coinitial,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% joinable_implies_coinitial
thf(fact_587_has__composites,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% has_composites
thf(fact_588_composable__imp__seq,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% composable_imp_seq
thf(fact_589_Arr_Osimps_I1_J,axiom,
~ ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Arr.simps(1)
thf(fact_590_Con__imp__Arr__Resid,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) ) ) ).
% Con_imp_Arr_Resid
thf(fact_591_Con__Arr__self,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T6 )
!= nil_lambda_lambda ) ) ).
% Con_Arr_self
thf(fact_592_Arr__iff__Con__self,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T6 )
!= nil_lambda_lambda ) ) ).
% Arr_iff_Con_self
thf(fact_593_Con__implies__Arr_I1_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ T6 ) ) ).
% Con_implies_Arr(1)
thf(fact_594_Con__implies__Arr_I2_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ U ) ) ).
% Con_implies_Arr(2)
thf(fact_595_Arr__map__Lam,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T6 ) ) ) ).
% Arr_map_Lam
thf(fact_596_composable__iff__seq,axiom,
! [G: list_lambda_lambda,F: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F )
= ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F ) ) ).
% composable_iff_seq
thf(fact_597_Arr__Resid__single,axiom,
! [T6: list_lambda_lambda,U2: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) ) ) ).
% Arr_Resid_single
thf(fact_598_Arr__map__App2,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ A ) @ T6 ) ) ) ) ).
% Arr_map_App2
thf(fact_599_seqI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( lambda_Trg @ ( last_lambda_lambda @ T6 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U ) ) ) ) ).
% seqI\<^sub>\<Lambda>\<^sub>P
thf(fact_600_lambda__calculus_Olambda_Osel_I2_J,axiom,
! [X32: lambda_lambda] :
( ( lambda_un_Lam @ ( lambda_Lam @ X32 ) )
= X32 ) ).
% lambda_calculus.lambda.sel(2)
thf(fact_601_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,G: lambda_lambda,F: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( compos7127019122090555255lambda @ Resid @ G @ F )
= ( seq_lambda_lambda @ Resid @ G @ F ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_602_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda,F: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( compos3359172601056258045lambda @ Resid @ G @ F )
= ( seq_li8093550137657879445lambda @ Resid @ G @ F ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_603_rts__with__composites_Ohas__composites,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( seq_lambda_lambda @ Resid @ T @ U2 )
=> ( compos7127019122090555255lambda @ Resid @ T @ U2 ) ) ) ).
% rts_with_composites.has_composites
thf(fact_604_rts__with__composites_Ohas__composites,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ Resid @ T @ U2 )
=> ( compos3359172601056258045lambda @ Resid @ T @ U2 ) ) ) ).
% rts_with_composites.has_composites
thf(fact_605_in__sources__iff,axiom,
! [T6: list_lambda_lambda,A2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ A2 ) ) ) ) ) ).
% in_sources_iff
thf(fact_606_in__targets__iff,axiom,
! [T6: list_lambda_lambda,B5: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member2667661554319814297lambda @ B5 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B5 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ B5 ) ) ) ) ) ).
% in_targets_iff
thf(fact_607__092_060Lambda_062_Oidentities__form__coherent__normal__sub__rts,axiom,
cohere1690309856004283859lambda @ lambda_resid @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ).
% \<Lambda>.identities_form_coherent_normal_sub_rts
thf(fact_608_single__Src__hd__in__sources,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% single_Src_hd_in_sources
thf(fact_609_single__Trg__last__in__targets,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% single_Trg_last_in_targets
thf(fact_610__092_060Lambda_062_Ocomposable__imp__seq,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U2 )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.composable_imp_seq
thf(fact_611__092_060Lambda_062_Ocong__respects__seq,axiom,
! [T: lambda_lambda,U2: lambda_lambda,T3: lambda_lambda,U4: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T3 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U2 ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T3 @ U4 ) ) ) ) ).
% \<Lambda>.cong_respects_seq
thf(fact_612_sources__cong__closed,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A5 @ A ) ) )
=> ( member2667661554319814297lambda @ A5 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% sources_cong_closed
thf(fact_613_sources__are__cong,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A5 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A5 @ A ) ) ) ) ) ).
% sources_are_cong
thf(fact_614_source__is__ide,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A ) ) ).
% source_is_ide
thf(fact_615_targets__cong__closed,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ B4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B4 @ B ) ) )
=> ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% targets_cong_closed
thf(fact_616_targets__are__cong,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ B4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B4 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_617_target__is__ide,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A ) ) ).
% target_is_ide
thf(fact_618_resid__source__in__targets,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% resid_source_in_targets
thf(fact_619_composableD_I3_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ) ).
% composableD(3)
thf(fact_620_sources__cons,axiom,
! [T: lambda_lambda,T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ T6 ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% sources_cons
thf(fact_621_lambda__calculus_OseqI_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( ( lambda_Trg @ T )
= ( lambda_Src @ U2 ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U2 ) ) ) ) ).
% lambda_calculus.seqI\<^sub>\<Lambda>
thf(fact_622_sources__append,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T6 @ U ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% sources_append
thf(fact_623_targets__append,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T6 @ U ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).
% targets_append
thf(fact_624_sources__single__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Src @ T ) @ nil_lambda_lambda ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% sources_single_Src
thf(fact_625_targets__single__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Trg @ T ) @ nil_lambda_lambda ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% targets_single_Trg
thf(fact_626_targets__single__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Src @ T ) @ nil_lambda_lambda ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% targets_single_Src
thf(fact_627_sources__single__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Trg @ T ) @ nil_lambda_lambda ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% sources_single_Trg
thf(fact_628_lambda__calculus_Oseq__char,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U2 )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U2 )
& ( ( lambda_Trg @ T )
= ( lambda_Src @ U2 ) ) ) ) ).
% lambda_calculus.seq_char
thf(fact_629_lambda__calculus_OseqE_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U2 )
=> ~ ( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( lambda_Trg @ T )
!= ( lambda_Src @ U2 ) ) ) ) ) ).
% lambda_calculus.seqE\<^sub>\<Lambda>
thf(fact_630__092_060Lambda_062_Ocong__implies__coterminal,axiom,
! [U2: lambda_lambda,U4: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U2 ) ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ U2 @ U4 ) ) ).
% \<Lambda>.cong_implies_coterminal
thf(fact_631_Con__single__ide__ind,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% Con_single_ide_ind
thf(fact_632_Std__imp__Arr,axiom,
! [T6: list_lambda_lambda] :
( ( reduction_Std @ T6 )
=> ( ( T6 != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ T6 ) ) ) ).
% Std_imp_Arr
thf(fact_633_seqE,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
!= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ) ) ) ).
% seqE
thf(fact_634_Srcs__are__ide,axiom,
! [T6: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).
% Srcs_are_ide
thf(fact_635_ex__un__Src,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
& ! [Y3: lambda_lambda] :
( ( member_lambda_lambda @ Y3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( Y3 = X3 ) ) ) ) ).
% ex_un_Src
thf(fact_636_Std__map__Lam,axiom,
! [T6: list_lambda_lambda] :
( ( reduction_Std @ T6 )
=> ( reduction_Std @ ( map_la336867333220232516lambda @ lambda_Lam @ T6 ) ) ) ).
% Std_map_Lam
thf(fact_637_Con__imp__eq__Srcs,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U ) ) ) ).
% Con_imp_eq_Srcs
thf(fact_638_cong__reflexive,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ) ).
% cong_reflexive
thf(fact_639_ide__implies__arr,axiom,
! [A: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A ) ) ).
% ide_implies_arr
thf(fact_640_prfx__reflexive,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ).
% prfx_reflexive
thf(fact_641_arr__char,axiom,
! [T6: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( paths_366302053746739058lambda @ lambda_resid @ T6 ) ) ).
% arr_char
thf(fact_642_composableD_I2_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ).
% composableD(2)
thf(fact_643_composableD_I1_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% composableD(1)
thf(fact_644_confluence__ind,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda ) ) ) ) ).
% confluence_ind
thf(fact_645_coinitialE,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
!= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ) ) ) ).
% coinitialE
thf(fact_646_coinitial__iff,axiom,
! [T: list_lambda_lambda,T3: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T3 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 )
& ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 ) ) ) ) ).
% coinitial_iff
thf(fact_647_coterminalE,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
!= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ) ) ) ).
% coterminalE
thf(fact_648_coterminal__iff,axiom,
! [T: list_lambda_lambda,T3: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T3 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 ) ) ) ) ).
% coterminal_iff
thf(fact_649_Resid__Arr__Src,axiom,
! [T6: list_lambda_lambda,A: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
= T6 ) ) ) ).
% Resid_Arr_Src
thf(fact_650_arr__append__imp__seq,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T6 @ U ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U ) ) ) ) ).
% arr_append_imp_seq
thf(fact_651_seq__def,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ) ) ).
% seq_def
thf(fact_652_arrI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ).
% arrI\<^sub>P
thf(fact_653_Srcs__append,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
= ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ).
% Srcs_append
thf(fact_654_arr__append,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T6 @ U ) ) ) ).
% arr_append
thf(fact_655_coinitialI,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ) ).
% coinitialI
thf(fact_656_coterminalI,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) )
=> ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ) ).
% coterminalI
thf(fact_657_seqI,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ) ) ).
% seqI
thf(fact_658_Con__single__ideI_I1_J,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(1)
thf(fact_659_Con__single__ideI_I2_J,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(2)
thf(fact_660_reduction__paths_OStd_Osimps_I1_J,axiom,
reduction_Std @ nil_lambda_lambda ).
% reduction_paths.Std.simps(1)
thf(fact_661_paths__in__confluent__rts_Oconfluence__ind,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T6 )
=> ( ( paths_366302053746739058lambda @ Resid @ U )
=> ( ( ( paths_2897053694880971696lambda @ Resid @ T6 )
= ( paths_2897053694880971696lambda @ Resid @ U ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ T6 @ U )
!= nil_lambda_lambda ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_ind
thf(fact_662_arr__iff__has__target,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
!= bot_bo1397037886704484924lambda ) ) ).
% arr_iff_has_target
thf(fact_663_arr__iff__has__source,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
!= bot_bo1397037886704484924lambda ) ) ).
% arr_iff_has_source
thf(fact_664_Std__imp__sseq__last__hd,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( reduction_Std @ ( append_lambda_lambda @ T6 @ U ) )
=> ( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( lambda_sseq @ ( last_lambda_lambda @ T6 ) @ ( hd_lambda_lambda @ U ) ) ) ) ) ).
% Std_imp_sseq_last_hd
thf(fact_665_Std__append,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( reduction_Std @ T6 )
=> ( ( reduction_Std @ U )
=> ( ( ( T6 = nil_lambda_lambda )
| ( U = nil_lambda_lambda )
| ( lambda_sseq @ ( last_lambda_lambda @ T6 ) @ ( hd_lambda_lambda @ U ) ) )
=> ( reduction_Std @ ( append_lambda_lambda @ T6 @ U ) ) ) ) ) ).
% Std_append
thf(fact_666_Std__consE,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ U ) )
=> ~ ( ( lambda_Arr @ T )
=> ( ( ( U != nil_lambda_lambda )
=> ( lambda_sseq @ T @ ( hd_lambda_lambda @ U ) ) )
=> ~ ( reduction_Std @ U ) ) ) ) ).
% Std_consE
thf(fact_667_Std__append__single,axiom,
! [T6: list_lambda_lambda,U2: lambda_lambda] :
( ( reduction_Std @ T6 )
=> ( ( T6 != nil_lambda_lambda )
=> ( ( lambda_sseq @ ( last_lambda_lambda @ T6 ) @ U2 )
=> ( reduction_Std @ ( append_lambda_lambda @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) ) ) ) ) ).
% Std_append_single
thf(fact_668_lambda__calculus_Osseq_Osimps_I3_J,axiom,
! [T: lambda_lambda,T3: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T3 ) )
= ( lambda_sseq @ T @ T3 ) ) ).
% lambda_calculus.sseq.simps(3)
thf(fact_669_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_670_bot_Oextremum__uniqueI,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
=> ( A = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_671_bot_Oextremum__uniqueI,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ bot_bo1397037886704484924lambda )
=> ( A = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_672_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_673_bot_Oextremum__unique,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_unique
thf(fact_674_bot_Oextremum__unique,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ bot_bo1397037886704484924lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_unique
thf(fact_675_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_676_bot_Oextremum,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A ) ).
% bot.extremum
thf(fact_677_bot_Oextremum,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A ) ).
% bot.extremum
thf(fact_678_lambda__calculus_Osseq_Osimps_I33_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc2: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc2 ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(33)
thf(fact_679_lambda__calculus_Osseq_Osimps_I23_J,axiom,
! [Vc2: lambda_lambda,Vb2: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Lam @ Vc2 ) @ Vb2 ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(23)
thf(fact_680_lambda__calculus_Osseq_Osimps_I24_J,axiom,
! [Vc2: lambda_lambda,Vd: lambda_lambda,Vb2: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vc2 @ Vd ) @ Vb2 ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(24)
thf(fact_681_lambda__calculus_Osseq_Osimps_I27_J,axiom,
! [Vb2: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Lam @ Vb2 ) @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(27)
thf(fact_682_lambda__calculus_Osseq_Osimps_I30_J,axiom,
! [Vb2: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Lam @ Vb2 ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(30)
thf(fact_683_lambda__calculus_Osseq__imp__seq,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_sseq @ T @ U2 )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% lambda_calculus.sseq_imp_seq
thf(fact_684_lambda__calculus_Osseq_Osimps_I5_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,T3: lambda_lambda,U4: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Lam @ T ) @ U2 ) @ ( lambda_Beta @ T3 @ U4 ) ) ).
% lambda_calculus.sseq.simps(5)
thf(fact_685_reduction__paths_OStd_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( ( lambda_sseq @ T @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
& ( reduction_Std @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ).
% reduction_paths.Std.simps(3)
thf(fact_686_coterminal__def,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
= ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) )
!= bot_bo1397037886704484924lambda ) ) ).
% coterminal_def
thf(fact_687_coinitial__def,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
= ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) )
!= bot_bo1397037886704484924lambda ) ) ).
% coinitial_def
thf(fact_688_Arr__append__iff_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( paths_366302053746739058lambda @ lambda_resid @ U )
& ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P
thf(fact_689_targets__eqI,axiom,
! [T: list_lambda_lambda,T3: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 ) ) ) ).
% targets_eqI
thf(fact_690_Srcs_Osimps_I1_J,axiom,
( ( paths_2897053694880971696lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Srcs.simps(1)
thf(fact_691_Trgs_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ).
% Trgs.simps(3)
thf(fact_692_Trgs_Osimps_I1_J,axiom,
( ( paths_7594662086398683701lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Trgs.simps(1)
thf(fact_693_Arr__has__Src,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
!= bot_bo1271416176687173302lambda ) ) ).
% Arr_has_Src
thf(fact_694_Arr__has__Trg,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
!= bot_bo1271416176687173302lambda ) ) ).
% Arr_has_Trg
thf(fact_695_ex__un__Trg,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
& ! [Y3: lambda_lambda] :
( ( member_lambda_lambda @ Y3 @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
=> ( Y3 = X3 ) ) ) ) ).
% ex_un_Trg
thf(fact_696_Trgs__are__ide,axiom,
! [T6: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).
% Trgs_are_ide
thf(fact_697_Trgs__Resid__sym__Arr__single,axiom,
! [T6: list_lambda_lambda,U2: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) @ T6 ) ) ) ) ).
% Trgs_Resid_sym_Arr_single
thf(fact_698_seq__implies__Trgs__eq__Srcs,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U ) )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U ) ) ) ) ) ).
% seq_implies_Trgs_eq_Srcs
thf(fact_699_sources__eqI,axiom,
! [T: list_lambda_lambda,T3: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 ) ) ) ).
% sources_eqI
thf(fact_700_Srcs__Resid__single__Arr,axiom,
! [U2: lambda_lambda,T6: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) @ T6 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ).
% Srcs_Resid_single_Arr
thf(fact_701_Arr__appendE_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
=> ( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
!= ( paths_2897053694880971696lambda @ lambda_resid @ U ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P
thf(fact_702_seq__char,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( paths_366302053746739058lambda @ lambda_resid @ U )
& ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U ) ) ) ) ).
% seq_char
thf(fact_703_Trgs__Resid__sym,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) ) ) ).
% Trgs_Resid_sym
thf(fact_704_Trgs__append,axiom,
! [U: list_lambda_lambda,T6: list_lambda_lambda] :
( ( U != nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ U ) ) ) ).
% Trgs_append
thf(fact_705_Srcs__Resid,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ U ) ) ) ).
% Srcs_Resid
thf(fact_706_Arr__appendI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) ) ) ) ) ).
% Arr_appendI\<^sub>P
thf(fact_707_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_708_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_709_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_710_inf_Obounded__iff,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C ) )
= ( ( ord_le3064231726521830256lambda @ A @ B )
& ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_711_le__inf__iff,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_712_le__inf__iff,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ ( inf_in1766959559283430712lambda @ Y2 @ Z2 ) )
= ( ( ord_le1926335974945931754lambda @ X @ Y2 )
& ( ord_le1926335974945931754lambda @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_713_le__inf__iff,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ ( inf_in3354698378728726462lambda @ Y2 @ Z2 ) )
= ( ( ord_le3064231726521830256lambda @ X @ Y2 )
& ( ord_le3064231726521830256lambda @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_714_seq__char_H,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( paths_366302053746739058lambda @ lambda_resid @ U )
& ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ) ).
% seq_char'
thf(fact_715_Trgs__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( ( reduct1603535164923377302opment @ T )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% Trgs_standard_development
thf(fact_716_Srcs__eqI,axiom,
! [T6: list_lambda_lambda,T7: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ T7 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T7 ) ) ) ).
% Srcs_eqI
thf(fact_717_Trgs__eqI,axiom,
! [T6: list_lambda_lambda,T7: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_7594662086398683701lambda @ lambda_resid @ T7 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T7 ) ) ) ).
% Trgs_eqI
thf(fact_718_Srcs__simp_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Srcs_simp\<^sub>\<Lambda>\<^sub>P
thf(fact_719_Trgs__simp_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Trgs_simp\<^sub>\<Lambda>\<^sub>P
thf(fact_720_Trgs__complete__development,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ U )
=> ( ~ ( lambda_Ide @ T )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ U )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% Trgs_complete_development
thf(fact_721_Srcs__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( ( reduct1603535164923377302opment @ T )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) )
= ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% Srcs_standard_development
thf(fact_722_seqI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U ) ) ) ) ).
% seqI\<^sub>P
thf(fact_723_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_724_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_725_inf_OcoboundedI2,axiom,
! [B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_726_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_727_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_728_inf_OcoboundedI1,axiom,
! [A: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_729_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_730_inf_Oabsorb__iff2,axiom,
( ord_le1926335974945931754lambda
= ( ^ [B3: set_lambda_lambda,A4: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_731_inf_Oabsorb__iff2,axiom,
( ord_le3064231726521830256lambda
= ( ^ [B3: set_li7878742357367608272lambda,A4: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A4 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_732_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( inf_inf_nat @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_733_inf_Oabsorb__iff1,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A4: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_734_inf_Oabsorb__iff1,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A4: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A4 @ B3 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_735_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_736_inf_Ocobounded2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_737_inf_Ocobounded2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_738_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_739_inf_Ocobounded1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_740_inf_Ocobounded1,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_741_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( A4
= ( inf_inf_nat @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_742_inf_Oorder__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A4: set_lambda_lambda,B3: set_lambda_lambda] :
( A4
= ( inf_in1766959559283430712lambda @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_743_inf_Oorder__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A4: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( A4
= ( inf_in3354698378728726462lambda @ A4 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_744_inf__greatest,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_745_inf__greatest,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ X @ Z2 )
=> ( ord_le1926335974945931754lambda @ X @ ( inf_in1766959559283430712lambda @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_746_inf__greatest,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ X @ Z2 )
=> ( ord_le3064231726521830256lambda @ X @ ( inf_in3354698378728726462lambda @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_747_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_748_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_749_inf_OboundedI,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ A @ C )
=> ( ord_le3064231726521830256lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_750_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_751_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_752_inf_OboundedE,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C ) )
=> ~ ( ( ord_le3064231726521830256lambda @ A @ B )
=> ~ ( ord_le3064231726521830256lambda @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_753_inf__absorb2,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( inf_inf_nat @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_754_inf__absorb2,axiom,
! [Y2: set_lambda_lambda,X: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y2 @ X )
=> ( ( inf_in1766959559283430712lambda @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_755_inf__absorb2,axiom,
! [Y2: set_li7878742357367608272lambda,X: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ Y2 @ X )
=> ( ( inf_in3354698378728726462lambda @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_756_inf__absorb1,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( inf_inf_nat @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_757_inf__absorb1,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ( inf_in1766959559283430712lambda @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_758_inf__absorb1,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ( inf_in3354698378728726462lambda @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_759_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_760_inf_Oabsorb2,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_761_inf_Oabsorb2,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_762_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_763_inf_Oabsorb1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_764_inf_Oabsorb1,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_765_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( inf_inf_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_766_le__iff__inf,axiom,
( ord_le1926335974945931754lambda
= ( ^ [X2: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_767_le__iff__inf,axiom,
( ord_le3064231726521830256lambda
= ( ^ [X2: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_768_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y2: nat] :
( ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y @ Z ) ) ) )
=> ( ( inf_inf_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_769_inf__unique,axiom,
! [F: set_lambda_lambda > set_lambda_lambda > set_lambda_lambda,X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_lambda_lambda,Y: set_lambda_lambda,Z: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y )
=> ( ( ord_le1926335974945931754lambda @ X3 @ Z )
=> ( ord_le1926335974945931754lambda @ X3 @ ( F @ Y @ Z ) ) ) )
=> ( ( inf_in1766959559283430712lambda @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_770_inf__unique,axiom,
! [F: set_li7878742357367608272lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda,X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_li7878742357367608272lambda,Y: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X3 @ Y )
=> ( ( ord_le3064231726521830256lambda @ X3 @ Z )
=> ( ord_le3064231726521830256lambda @ X3 @ ( F @ Y @ Z ) ) ) )
=> ( ( inf_in3354698378728726462lambda @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_771_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_772_inf_OorderI,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A
= ( inf_in1766959559283430712lambda @ A @ B ) )
=> ( ord_le1926335974945931754lambda @ A @ B ) ) ).
% inf.orderI
thf(fact_773_inf_OorderI,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A
= ( inf_in3354698378728726462lambda @ A @ B ) )
=> ( ord_le3064231726521830256lambda @ A @ B ) ) ).
% inf.orderI
thf(fact_774_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_775_inf_OorderE,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( A
= ( inf_in1766959559283430712lambda @ A @ B ) ) ) ).
% inf.orderE
thf(fact_776_inf_OorderE,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( A
= ( inf_in3354698378728726462lambda @ A @ B ) ) ) ).
% inf.orderE
thf(fact_777_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_778_le__infI2,axiom,
! [B: set_lambda_lambda,X: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ X )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_779_le__infI2,axiom,
! [B: set_li7878742357367608272lambda,X: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ X )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_780_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_781_le__infI1,axiom,
! [A: set_lambda_lambda,X: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ X )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_782_le__infI1,axiom,
! [A: set_li7878742357367608272lambda,X: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ X )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_783_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_784_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_785_inf__mono,axiom,
! [A: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,D: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ C )
=> ( ( ord_le3064231726521830256lambda @ B @ D )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ ( inf_in3354698378728726462lambda @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_786_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_787_le__infI,axiom,
! [X: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ A )
=> ( ( ord_le1926335974945931754lambda @ X @ B )
=> ( ord_le1926335974945931754lambda @ X @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% le_infI
thf(fact_788_le__infI,axiom,
! [X: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ A )
=> ( ( ord_le3064231726521830256lambda @ X @ B )
=> ( ord_le3064231726521830256lambda @ X @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% le_infI
thf(fact_789_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_790_le__infE,axiom,
! [X: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ ( inf_in1766959559283430712lambda @ A @ B ) )
=> ~ ( ( ord_le1926335974945931754lambda @ X @ A )
=> ~ ( ord_le1926335974945931754lambda @ X @ B ) ) ) ).
% le_infE
thf(fact_791_le__infE,axiom,
! [X: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ ( inf_in3354698378728726462lambda @ A @ B ) )
=> ~ ( ( ord_le3064231726521830256lambda @ X @ A )
=> ~ ( ord_le3064231726521830256lambda @ X @ B ) ) ) ).
% le_infE
thf(fact_792_inf__le2,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_793_inf__le2,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_794_inf__le2,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_795_inf__le1,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_796_inf__le1,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_797_inf__le1,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_798_inf__sup__ord_I1_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_799_inf__sup__ord_I1_J,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_800_inf__sup__ord_I1_J,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_801_inf__sup__ord_I2_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_802_inf__sup__ord_I2_J,axiom,
! [X: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_803_inf__sup__ord_I2_J,axiom,
! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_804_Srcs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( insert_lambda_lambda @ ( paths_1651982994781619320lambda @ lambda_resid @ T6 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Srcs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_805_con__imp__common__source,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) )
!= bot_bo1397037886704484924lambda ) ) ).
% con_imp_common_source
thf(fact_806_Ide__append__iff_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
= ( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
& ( paths_7717490669901197737lambda @ lambda_resid @ U )
& ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P
thf(fact_807_join__of__arr__src_I1_J,axiom,
! [T: list_lambda_lambda,A: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ T @ T ) ) ) ).
% join_of_arr_src(1)
thf(fact_808_resid__reflects__con,axiom,
! [T: list_lambda_lambda,V: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ V )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 @ V )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ) ) ).
% resid_reflects_con
thf(fact_809_con__sym,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 @ T ) ) ).
% con_sym
thf(fact_810_Ide_Osimps_I1_J,axiom,
~ ( paths_7717490669901197737lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Ide.simps(1)
thf(fact_811_Ide__implies__Arr,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( paths_366302053746739058lambda @ lambda_resid @ T6 ) ) ).
% Ide_implies_Arr
thf(fact_812_join__of__symmetric,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ V )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 @ T @ V ) ) ).
% join_of_symmetric
thf(fact_813_con__char,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda ) ) ).
% con_char
thf(fact_814_Resid__Ide__Arr__ind,axiom,
! [A2: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T6 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T6 ) ) ) ) ).
% Resid_Ide_Arr_ind
thf(fact_815_Resid__Arr__Ide__ind,axiom,
! [A2: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ A2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ A2 )
= T6 ) ) ) ).
% Resid_Arr_Ide_ind
thf(fact_816_Resid__Ide_I1_J,axiom,
! [A2: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T6 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ A2 )
= T6 ) ) ) ).
% Resid_Ide(1)
thf(fact_817_cong__subst__left_I2_J,axiom,
! [T: list_lambda_lambda,T3: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_818_cong__subst__left_I1_J,axiom,
! [T: list_lambda_lambda,T3: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T3 @ U2 ) ) ) ).
% cong_subst_left(1)
thf(fact_819_cong__subst__right_I2_J,axiom,
! [U2: list_lambda_lambda,U4: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U2 ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U4 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U4 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_820_cong__subst__right_I1_J,axiom,
! [U2: list_lambda_lambda,U4: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U2 ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U4 ) ) ) ).
% cong_subst_right(1)
thf(fact_821_con__imp__coinitial__ax,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ? [A3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 @ T )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 @ U2 ) ) ) ).
% con_imp_coinitial_ax
thf(fact_822_con__target,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U2 ) ) ) ) ).
% con_target
thf(fact_823_con__transitive__on__ide,axiom,
! [A: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ C )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ B )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ C )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ C ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_824_ideE,axiom,
! [A: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ~ ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ A )
!= A ) ) ) ).
% ideE
thf(fact_825_ide__def,axiom,
! [A: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
= ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ A )
= A ) ) ) ).
% ide_def
thf(fact_826_ide__imp__con__iff__cong,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) ) ) ) ) ) ).
% ide_imp_con_iff_cong
thf(fact_827_prfx__implies__con,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% prfx_implies_con
thf(fact_828_resid__arr__ide,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ A )
= T ) ) ) ).
% resid_arr_ide
thf(fact_829_resid__ide__arr,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ T )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ T ) ) ) ) ).
% resid_ide_arr
thf(fact_830_ide__char,axiom,
! [T6: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( paths_7717490669901197737lambda @ lambda_resid @ T6 ) ) ).
% ide_char
thf(fact_831_Resid__Arr__self,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T6 ) ) ) ).
% Resid_Arr_self
thf(fact_832_sources__are__con,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A5 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A5 ) ) ) ).
% sources_are_con
thf(fact_833_targets__resid__sym,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) ) ) ) ).
% targets_resid_sym
thf(fact_834_targets__are__con,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ B4 ) ) ) ).
% targets_are_con
thf(fact_835_arr__resid__iff__con,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% arr_resid_iff_con
thf(fact_836_arr__resid,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) ) ) ).
% arr_resid
thf(fact_837_arr__def,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).
% arr_def
thf(fact_838_arrE,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).
% arrE
thf(fact_839_con__implies__arr_I1_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% con_implies_arr(1)
thf(fact_840_con__implies__arr_I2_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ).
% con_implies_arr(2)
thf(fact_841_join__of__un__upto__cong,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ V )
=> ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ V4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ V4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ V ) ) ) ) ) ).
% join_of_un_upto_cong
thf(fact_842_Ide__imp__Ide__hd,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) ) ) ).
% Ide_imp_Ide_hd
thf(fact_843_join__of__resid,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,W3: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ W3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V @ W3 )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W3 @ V ) ) ) ) ).
% join_of_resid
thf(fact_844_con__with__join__of__iff_I1_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,W3: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ W3 )
=> ( ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 @ V )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W3 @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_845_con__with__join__of__iff_I2_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,W3: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ W3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W3 @ V )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ V )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_846_Ide__imp__Ide__last,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ide_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T6 ) ) ) ).
% Ide_imp_Ide_last
thf(fact_847_sources__join__of_I2_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ V )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).
% sources_join_of(2)
thf(fact_848_sources__join__of_I1_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ V )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).
% sources_join_of(1)
thf(fact_849_targets__join__of_I2_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ V )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).
% targets_join_of(2)
thf(fact_850_targets__join__of_I1_J,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ V )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).
% targets_join_of(1)
thf(fact_851_join__of__arr__self,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T @ T ) ) ).
% join_of_arr_self
thf(fact_852_con__imp__coinitial,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% con_imp_coinitial
thf(fact_853_confluence,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% confluence
thf(fact_854_confluence_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U ) ) ).
% confluence\<^sub>P
thf(fact_855_joinable__implies__con,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% joinable_implies_con
thf(fact_856_joinable__def,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
= ( ? [X6: list_lambda_lambda] : ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 @ X6 ) ) ) ).
% joinable_def
thf(fact_857_Ide_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( ide_lambda_lambda @ lambda_resid @ T ) ) ).
% Ide.simps(2)
thf(fact_858_sources__con__closed,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A5 )
=> ( member2667661554319814297lambda @ A5 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% sources_con_closed
thf(fact_859_in__sourcesE,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A ) ) ) ).
% in_sourcesE
thf(fact_860_targets__con__closed,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B4 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ B4 )
=> ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% targets_con_closed
thf(fact_861_Ide__imp__sources__eq__targets,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% Ide_imp_sources_eq_targets
thf(fact_862_join__of__arr__src_I2_J,axiom,
! [T: list_lambda_lambda,A: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A @ T ) ) ) ).
% join_of_arr_src(2)
thf(fact_863_Con__Ide__iff,axiom,
! [A2: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T6 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) ) ) ) ) ).
% Con_Ide_iff
thf(fact_864_Con__IdeI_I1_J,axiom,
! [A2: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T6 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(1)
thf(fact_865_Con__IdeI_I2_J,axiom,
! [A2: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ A2 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(2)
thf(fact_866_conI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U ) ) ).
% conI\<^sub>P
thf(fact_867_ideI,axiom,
! [A: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ A )
= A )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A ) ) ) ).
% ideI
thf(fact_868_arrI,axiom,
! [T: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% arrI
thf(fact_869_in__sourcesI,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A )
=> ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% in_sourcesI
thf(fact_870_sources__resid,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ) ).
% sources_resid
thf(fact_871_Src__append,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
= ( paths_1651982994781619320lambda @ lambda_resid @ T6 ) ) ) ).
% Src_append
thf(fact_872_conI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
=> ( ( ( lambda_Src @ ( hd_lambda_lambda @ T6 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U ) ) ) ) ).
% conI\<^sub>\<Lambda>\<^sub>P
thf(fact_873_Ide__appendI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) ) ) ) ) ).
% Ide_appendI\<^sub>P
thf(fact_874_confluent__rts_Oconfluence,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( conflu4036547884742428384lambda @ Resid )
=> ( ( coinit1063259437155584316lambda @ Resid @ T @ U2 )
=> ( con_li79537494059641180lambda @ Resid @ T @ U2 ) ) ) ).
% confluent_rts.confluence
thf(fact_875_confluent__rts_Oconfluence,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( conflu4927472611300549594lambda @ Resid )
=> ( ( coinit1392077014771647542lambda @ Resid @ T @ U2 )
=> ( con_lambda_lambda @ Resid @ T @ U2 ) ) ) ).
% confluent_rts.confluence
thf(fact_876_paths__in__confluent__rts_Oconfluence_092_060_094sub_062P,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T6 @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T6 @ U ) ) ) ).
% paths_in_confluent_rts.confluence\<^sub>P
thf(fact_877_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_l5168835686021156086lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( ! [T4: list_lambda_lambda,U5: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ Resid @ T4 @ U5 )
=> ( con_li79537494059641180lambda @ Resid @ T4 @ U5 ) )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( paths_2015741907369688696lambda @ Resid @ U )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( paths_3172353640142224310lambda @ Resid @ U ) )
=> ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ U )
!= nil_li8379291187143545974lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_878_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid @ T4 @ U5 )
=> ( con_lambda_lambda @ Resid @ T4 @ U5 ) )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( paths_366302053746739058lambda @ Resid @ U )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( paths_2897053694880971696lambda @ Resid @ U ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_879_path__from__NF__is__Ide,axiom,
! [A: lambda_lambda,U: list_lambda_lambda] :
( ( lambda_NF @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( paths_1651982994781619320lambda @ lambda_resid @ U )
= A )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U ) ) ) ) ).
% path_from_NF_is_Ide
thf(fact_880_const__ide__is__Ide,axiom,
! [T6: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T6 ) @ bot_bo1271416176687173302lambda ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ T6 ) ) ) ) ).
% const_ide_is_Ide
thf(fact_881_set__Ide__subset__single__hd,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T6 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% set_Ide_subset_single_hd
thf(fact_882__092_060Lambda_062_Osources__are__con,axiom,
! [A: lambda_lambda,T: lambda_lambda,A5: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A5 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ A @ A5 ) ) ) ).
% \<Lambda>.sources_are_con
thf(fact_883__092_060Lambda_062_Oconfluence,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U2 )
=> ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.confluence
thf(fact_884__092_060Lambda_062_Ocon__imp__coinitial,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.con_imp_coinitial
thf(fact_885__092_060Lambda_062_Oarr__resid__iff__con,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
= ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.arr_resid_iff_con
thf(fact_886__092_060Lambda_062_Oarr__resid,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) ) ) ).
% \<Lambda>.arr_resid
thf(fact_887__092_060Lambda_062_Oarr__def,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( con_lambda_lambda @ lambda_resid @ T @ T ) ) ).
% \<Lambda>.arr_def
thf(fact_888__092_060Lambda_062_OarrE,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( con_lambda_lambda @ lambda_resid @ T @ T ) ) ).
% \<Lambda>.arrE
thf(fact_889__092_060Lambda_062_Ocon__implies__arr_I1_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.con_implies_arr(1)
thf(fact_890__092_060Lambda_062_Ocon__implies__arr_I2_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( arr_lambda_lambda @ lambda_resid @ U2 ) ) ).
% \<Lambda>.con_implies_arr(2)
thf(fact_891__092_060Lambda_062_Oarr__has__un__source,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
& ! [Y3: lambda_lambda] :
( ( member_lambda_lambda @ Y3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( Y3 = X3 ) ) ) ) ).
% \<Lambda>.arr_has_un_source
thf(fact_892__092_060Lambda_062_OcoinitialE,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U2 )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
!= ( source5627170679743844886lambda @ lambda_resid @ U2 ) ) ) ) ) ).
% \<Lambda>.coinitialE
thf(fact_893__092_060Lambda_062_Ocoinitial__iff,axiom,
! [T: lambda_lambda,T3: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T3 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) ) ) ).
% \<Lambda>.coinitial_iff
thf(fact_894__092_060Lambda_062_Ocon__sym,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( con_lambda_lambda @ lambda_resid @ U2 @ T ) ) ).
% \<Lambda>.con_sym
thf(fact_895__092_060Lambda_062_Oresid__reflects__con,axiom,
! [T: lambda_lambda,V: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ U2 @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U2 @ V ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ) ) ).
% \<Lambda>.resid_reflects_con
thf(fact_896__092_060Lambda_062_Osources__con__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A5: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A5 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A5 )
=> ( member_lambda_lambda @ A5 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.sources_con_closed
thf(fact_897__092_060Lambda_062_Oin__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 ) ) ) ).
% \<Lambda>.in_sourcesE
thf(fact_898__092_060Lambda_062_Osource__is__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ).
% \<Lambda>.source_is_ide
thf(fact_899__092_060Lambda_062_Osources__are__cong,axiom,
! [A: lambda_lambda,T: lambda_lambda,A5: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A5 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A5 @ A ) ) ) ) ) ).
% \<Lambda>.sources_are_cong
thf(fact_900__092_060Lambda_062_Osources__cong__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A5: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A5 @ A ) ) )
=> ( member_lambda_lambda @ A5 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.sources_cong_closed
thf(fact_901__092_060Lambda_062_Ocong__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 ) ) ) ) ).
% \<Lambda>.cong_reflexive
thf(fact_902__092_060Lambda_062_Oide__implies__arr,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( arr_lambda_lambda @ lambda_resid @ A ) ) ).
% \<Lambda>.ide_implies_arr
thf(fact_903__092_060Lambda_062_Oprfx__reflexive,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) ) ) ).
% \<Lambda>.prfx_reflexive
thf(fact_904__092_060Lambda_062_Ocong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T3 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U2 ) @ ( lambda_resid @ T3 @ U2 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T3 @ U2 ) @ ( lambda_resid @ T @ U2 ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_left(2)
thf(fact_905__092_060Lambda_062_Ocong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T3: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T3 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( con_lambda_lambda @ lambda_resid @ T3 @ U2 ) ) ) ).
% \<Lambda>.cong_subst_left(1)
thf(fact_906__092_060Lambda_062_Ocong__subst__right_I2_J,axiom,
! [U2: lambda_lambda,U4: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U2 ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U2 ) @ ( lambda_resid @ T @ U4 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U4 ) @ ( lambda_resid @ T @ U2 ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_right(2)
thf(fact_907__092_060Lambda_062_Ocong__subst__right_I1_J,axiom,
! [U2: lambda_lambda,U4: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U2 ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( con_lambda_lambda @ lambda_resid @ T @ U4 ) ) ) ).
% \<Lambda>.cong_subst_right(1)
thf(fact_908__092_060Lambda_062_Ocon__ide__are__eq,axiom,
! [A: lambda_lambda,A5: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A5 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A5 )
=> ( A = A5 ) ) ) ) ).
% \<Lambda>.con_ide_are_eq
thf(fact_909__092_060Lambda_062_Ocon__imp__coinitial__ax,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ? [A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A3 )
& ( con_lambda_lambda @ lambda_resid @ A3 @ T )
& ( con_lambda_lambda @ lambda_resid @ A3 @ U2 ) ) ) ).
% \<Lambda>.con_imp_coinitial_ax
thf(fact_910__092_060Lambda_062_Ocon__target,axiom,
! [T: lambda_lambda,U2: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
=> ( ( con_lambda_lambda @ lambda_resid @ U2 @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) @ ( lambda_resid @ V @ U2 ) ) ) ) ).
% \<Lambda>.con_target
thf(fact_911__092_060Lambda_062_Ocon__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 ) ) ) ) ) ) ).
% \<Lambda>.con_transitive_on_ide
thf(fact_912__092_060Lambda_062_OideE,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ~ ( ( con_lambda_lambda @ lambda_resid @ A @ A )
=> ( ( lambda_resid @ A @ A )
!= A ) ) ) ).
% \<Lambda>.ideE
thf(fact_913__092_060Lambda_062_Oide__def,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
= ( ( con_lambda_lambda @ lambda_resid @ A @ A )
& ( ( lambda_resid @ A @ A )
= A ) ) ) ).
% \<Lambda>.ide_def
thf(fact_914__092_060Lambda_062_Oide__imp__con__iff__cong,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ U2 )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) ) ) ) ) ).
% \<Lambda>.ide_imp_con_iff_cong
thf(fact_915__092_060Lambda_062_Oprfx__implies__con,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.prfx_implies_con
thf(fact_916__092_060Lambda_062_Oresid__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 ) ) ) ).
% \<Lambda>.resid_arr_ide
thf(fact_917__092_060Lambda_062_Oresid__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 ) ) ) ) ).
% \<Lambda>.resid_ide_arr
thf(fact_918__092_060Lambda_062_Ocong__implies__coinitial,axiom,
! [U2: lambda_lambda,U4: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U2 ) ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ U2 @ U4 ) ) ).
% \<Lambda>.cong_implies_coinitial
thf(fact_919__092_060Lambda_062_Ocoinitial__ide__are__eq,axiom,
! [A: lambda_lambda,A5: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A5 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A5 )
=> ( A = A5 ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_eq
thf(fact_920__092_060Lambda_062_Ocoinitial__ide__are__cong,axiom,
! [A: lambda_lambda,A5: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A5 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A5 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A5 @ A ) ) ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_cong
thf(fact_921__092_060Lambda_062_Oresid__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 ) ) ) ).
% \<Lambda>.resid_ide(1)
thf(fact_922_Srcs__are__con,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda,A5: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( member_lambda_lambda @ A5 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( con_lambda_lambda @ lambda_resid @ A @ A5 ) ) ) ).
% Srcs_are_con
thf(fact_923__092_060Lambda_062_Oarr__iff__has__source,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( ( source5627170679743844886lambda @ lambda_resid @ T )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>.arr_iff_has_source
thf(fact_924_Trgs__are__con,axiom,
! [B: lambda_lambda,T6: list_lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
=> ( ( member_lambda_lambda @ B4 @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
=> ( con_lambda_lambda @ lambda_resid @ B @ B4 ) ) ) ).
% Trgs_are_con
thf(fact_925__092_060Lambda_062_OcomposableD_I2_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U2 )
=> ( arr_lambda_lambda @ lambda_resid @ U2 ) ) ).
% \<Lambda>.composableD(2)
thf(fact_926__092_060Lambda_062_OcomposableD_I1_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U2 )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.composableD(1)
thf(fact_927_Srcs_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( source5627170679743844886lambda @ lambda_resid @ T ) ) ).
% Srcs.simps(3)
thf(fact_928_Srcs__con__closed,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda,A5: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A5 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A5 )
=> ( member_lambda_lambda @ A5 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_929_Trgs__con__closed,axiom,
! [B: lambda_lambda,T6: list_lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B4 )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ B4 )
=> ( member_lambda_lambda @ B4 @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% Trgs_con_closed
thf(fact_930_Arr__imp__arr__hd,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( arr_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) ) ) ).
% Arr_imp_arr_hd
thf(fact_931_Arr__imp__arr__last,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( arr_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T6 ) ) ) ).
% Arr_imp_arr_last
thf(fact_932__092_060Lambda_062_Ocon__imp__common__source,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ U2 ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>.con_imp_common_source
thf(fact_933__092_060Lambda_062_Ocoinitial__def,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U2 )
= ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ U2 ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>.coinitial_def
thf(fact_934__092_060Lambda_062_Osources__eqI,axiom,
! [T: lambda_lambda,T3: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T3 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) ) ).
% \<Lambda>.sources_eqI
thf(fact_935_Resid_Osimps_I3_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(3)
thf(fact_936_Con__rec_I1_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
= ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% Con_rec(1)
thf(fact_937_Con__rec_I2_J,axiom,
! [T6: list_lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U2 @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_rec(2)
thf(fact_938_Con__rec_I3_J,axiom,
! [U: list_lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U2 @ U ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) @ U )
!= nil_lambda_lambda ) ) ) ) ).
% Con_rec(3)
thf(fact_939_Con__rec_I4_J,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U2 @ U ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U2 @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) @ U )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U2 @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_rec(4)
thf(fact_940_Arr_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% Arr.simps(2)
thf(fact_941_Srcs_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( source5627170679743844886lambda @ lambda_resid @ T ) ) ).
% Srcs.simps(2)
thf(fact_942_Srcs__simp_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( source5627170679743844886lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) ) ) ) ).
% Srcs_simp\<^sub>P
thf(fact_943_set__Arr__subset__arr,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ lambda_resid ) ) ) ) ).
% set_Arr_subset_arr
thf(fact_944__092_060Lambda_062_OarrI,axiom,
! [T: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ T )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.arrI
thf(fact_945__092_060Lambda_062_OcoinitialI,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U2 ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U2 ) ) ) ).
% \<Lambda>.coinitialI
thf(fact_946_Srcs_Oelims,axiom,
! [X: list_lambda_lambda,Y2: set_lambda_lambda] :
( ( ( paths_2897053694880971696lambda @ lambda_resid @ X )
= Y2 )
=> ( ( ( X = nil_lambda_lambda )
=> ( Y2 != bot_bo1271416176687173302lambda ) )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( Y2
!= ( source5627170679743844886lambda @ lambda_resid @ T4 ) ) )
=> ~ ! [T4: lambda_lambda] :
( ? [V2: lambda_lambda,Va2: list_lambda_lambda] :
( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( Y2
!= ( source5627170679743844886lambda @ lambda_resid @ T4 ) ) ) ) ) ) ).
% Srcs.elims
thf(fact_947_set__Ide__subset__ide,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ).
% set_Ide_subset_ide
thf(fact_948_confluence__single,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T4 @ U5 )
=> ( con_lambda_lambda @ lambda_resid @ T4 @ U5 ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( paths_2897053694880971696lambda @ lambda_resid @ U ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
!= nil_lambda_lambda ) ) ) ) ) ).
% confluence_single
thf(fact_949_Ide__char,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ) ).
% Ide_char
thf(fact_950_lambda__calculus_Oarr__char,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.arr_char
thf(fact_951__092_060Lambda_062_Oin__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 ) ) ) ) ).
% \<Lambda>.in_sourcesI
thf(fact_952__092_060Lambda_062_OideI,axiom,
! [A: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ A @ A )
=> ( ( ( lambda_resid @ A @ A )
= A )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ) ).
% \<Lambda>.ideI
thf(fact_953_map__eq__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Xs ) )
= ( ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_954_set__empty,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( ( set_li3846194476672328965lambda @ Xs )
= bot_bo1397037886704484924lambda )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty
thf(fact_955_set__empty,axiom,
! [Xs: list_lambda_lambda] :
( ( ( set_lambda_lambda2 @ Xs )
= bot_bo1271416176687173302lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty
thf(fact_956_set__empty2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ Xs ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty2
thf(fact_957_set__empty2,axiom,
! [Xs: list_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty2
thf(fact_958_list_Osimps_I15_J,axiom,
! [X21: lambda_lambda,X22: list_lambda_lambda] :
( ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) )
= ( insert_lambda_lambda @ X21 @ ( set_lambda_lambda2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_959_lambda__calculus_Ocoinitial__char,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U2 )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U2 )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U2 ) ) ) ) ).
% lambda_calculus.coinitial_char
thf(fact_960_lambda__calculus_Osources__simps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_App @ T @ U2 ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ ( lambda_Src @ U2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_simps(4)
thf(fact_961_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_962_lambda__calculus_Osources__simp,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.sources_simp
thf(fact_963_IdeI,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ T6 ) ) ) ).
% IdeI
thf(fact_964_lambda__calculus_Osources__simps_I5_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Beta @ T @ U2 ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_simps(5)
thf(fact_965_lambda__calculus_Osources__subst,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Subst @ zero_zero_nat @ T @ U2 ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Src @ T ) @ ( lambda_Src @ U2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_subst
thf(fact_966_set__ConsD,axiom,
! [Y2: list_lambda_lambda,X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_967_set__ConsD,axiom,
! [Y2: lambda_lambda,X: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_968_list_Oset__cases,axiom,
! [E: list_lambda_lambda,A: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ A ) )
=> ( ! [Z22: list_l5168835686021156086lambda] :
( A
!= ( cons_l4558873032622161958lambda @ E @ Z22 ) )
=> ~ ! [Z1: list_lambda_lambda,Z22: list_l5168835686021156086lambda] :
( ( A
= ( cons_l4558873032622161958lambda @ Z1 @ Z22 ) )
=> ~ ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_969_list_Oset__cases,axiom,
! [E: lambda_lambda,A: list_lambda_lambda] :
( ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ A ) )
=> ( ! [Z22: list_lambda_lambda] :
( A
!= ( cons_lambda_lambda @ E @ Z22 ) )
=> ~ ! [Z1: lambda_lambda,Z22: list_lambda_lambda] :
( ( A
= ( cons_lambda_lambda @ Z1 @ Z22 ) )
=> ~ ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_970_list_Oset__intros_I1_J,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] : ( member2667661554319814297lambda @ X21 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_971_list_Oset__intros_I1_J,axiom,
! [X21: lambda_lambda,X22: list_lambda_lambda] : ( member_lambda_lambda @ X21 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_972_list_Oset__intros_I2_J,axiom,
! [Y2: list_lambda_lambda,X22: list_l5168835686021156086lambda,X21: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ X22 ) )
=> ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_973_list_Oset__intros_I2_J,axiom,
! [Y2: lambda_lambda,X22: list_lambda_lambda,X21: lambda_lambda] :
( ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ X22 ) )
=> ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_974_list_Omap__cong,axiom,
! [X: list_lambda_lambda,Ya: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( X = Ya )
=> ( ! [Z: lambda_lambda] :
( ( member_lambda_lambda @ Z @ ( set_lambda_lambda2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X )
= ( map_la336867333220232516lambda @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_975_list_Omap__cong0,axiom,
! [X: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [Z: lambda_lambda] :
( ( member_lambda_lambda @ Z @ ( set_lambda_lambda2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X )
= ( map_la336867333220232516lambda @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_976_list_Oinj__map__strong,axiom,
! [X: list_lambda_lambda,Xa: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Fa: lambda_lambda > lambda_lambda] :
( ! [Z: lambda_lambda,Za: lambda_lambda] :
( ( member_lambda_lambda @ Z @ ( set_lambda_lambda2 @ X ) )
=> ( ( member_lambda_lambda @ Za @ ( set_lambda_lambda2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_la336867333220232516lambda @ F @ X )
= ( map_la336867333220232516lambda @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_977_list_Omap__ident__strong,axiom,
! [T: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [Z: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z @ ( set_li3846194476672328965lambda @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li385728805257398596lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_978_list_Omap__ident__strong,axiom,
! [T: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [Z: lambda_lambda] :
( ( member_lambda_lambda @ Z @ ( set_lambda_lambda2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_la336867333220232516lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_979_map__ext,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Xs ) ) ) ).
% map_ext
thf(fact_980_map__idI,axiom,
! [Xs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li385728805257398596lambda @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_981_map__idI,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_982_map__cong,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( Xs = Ys )
=> ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_983_ex__map__conv,axiom,
! [Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( ? [Xs4: list_lambda_lambda] :
( Ys
= ( map_la336867333220232516lambda @ F @ Xs4 ) ) )
= ( ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys ) )
=> ? [Y4: lambda_lambda] :
( X2
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_984_lambda__calculus_Onot__NF__elim,axiom,
! [T: lambda_lambda] :
( ~ ( lambda_NF @ T )
=> ( ( lambda_Ide @ T )
=> ~ ! [U5: lambda_lambda] :
~ ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U5 )
& ~ ( lambda_Ide @ U5 ) ) ) ) ).
% lambda_calculus.not_NF_elim
thf(fact_985_subset__code_I1_J,axiom,
! [Xs: list_lambda_lambda,B5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ B5 )
= ( ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ( member_lambda_lambda @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_986_subset__code_I1_J,axiom,
! [Xs: list_l5168835686021156086lambda,B5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ B5 )
= ( ! [X2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( member2667661554319814297lambda @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_987_lambda__calculus_ONF__Lam__iff,axiom,
! [T: lambda_lambda] :
( ( lambda_NF @ ( lambda_Lam @ T ) )
= ( lambda_NF @ T ) ) ).
% lambda_calculus.NF_Lam_iff
thf(fact_988_empty__set,axiom,
( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ nil_li8379291187143545974lambda ) ) ).
% empty_set
thf(fact_989_empty__set,axiom,
( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ nil_lambda_lambda ) ) ).
% empty_set
thf(fact_990_set__subset__Cons,axiom,
! [Xs: list_lambda_lambda,X: lambda_lambda] : ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_991_set__subset__Cons,axiom,
! [Xs: list_l5168835686021156086lambda,X: list_lambda_lambda] : ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_992_split__list__first__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list_lambda_lambda,X2: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_993_split__list__last__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list_lambda_lambda,X2: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_994_in__set__conv__decomp__first,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_995_in__set__conv__decomp__first,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_996_in__set__conv__decomp__last,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_997_in__set__conv__decomp__last,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_998_split__list__first__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_lambda_lambda,X3: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_999_split__list__last__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_lambda_lambda,X3: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1000_split__list__first__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_lambda_lambda,X3: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1001_split__list__last__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_lambda_lambda,X3: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1002_in__set__conv__decomp,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1003_in__set__conv__decomp,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1004_append__Cons__eq__iff,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs5: list_l5168835686021156086lambda,Ys6: list_l5168835686021156086lambda] :
( ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Ys ) )
=> ( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X @ Ys ) )
= ( append1017360404025248779lambda @ Xs5 @ ( cons_l4558873032622161958lambda @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1005_append__Cons__eq__iff,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Xs5: list_lambda_lambda,Ys6: list_lambda_lambda] :
( ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
=> ( ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X @ Ys ) )
= ( append_lambda_lambda @ Xs5 @ ( cons_lambda_lambda @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1006_split__list__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_lambda_lambda,X3: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1007_split__list__first,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1008_split__list__first,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1009_split__list__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_lambda_lambda,X3: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1010_split__list__last,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1011_split__list__last,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1012_split__list,axiom,
! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1013_split__list,axiom,
! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1014_hd__in__set,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ Xs ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ).
% hd_in_set
thf(fact_1015_hd__in__set,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( member_lambda_lambda @ ( hd_lambda_lambda @ Xs ) @ ( set_lambda_lambda2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1016_list_Oset__sel_I1_J,axiom,
! [A: list_l5168835686021156086lambda] :
( ( A != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ A ) @ ( set_li3846194476672328965lambda @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1017_list_Oset__sel_I1_J,axiom,
! [A: list_lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( member_lambda_lambda @ ( hd_lambda_lambda @ A ) @ ( set_lambda_lambda2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1018_last__in__set,axiom,
! [As2: list_l5168835686021156086lambda] :
( ( As2 != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( last_l1874196399739661071lambda @ As2 ) @ ( set_li3846194476672328965lambda @ As2 ) ) ) ).
% last_in_set
thf(fact_1019_last__in__set,axiom,
! [As2: list_lambda_lambda] :
( ( As2 != nil_lambda_lambda )
=> ( member_lambda_lambda @ ( last_lambda_lambda @ As2 ) @ ( set_lambda_lambda2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1020_lambda__calculus_Osources__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 ) ) ) ).
% lambda_calculus.sources_char\<^sub>\<Lambda>
thf(fact_1021_lambda__calculus_ONF__def,axiom,
( lambda_NF
= ( ^ [A4: lambda_lambda] :
( ( lambda_Ide @ A4 )
& ! [T2: lambda_lambda] :
( ( ( lambda_Arr @ T2 )
& ( ( lambda_Src @ T2 )
= A4 ) )
=> ( lambda_Ide @ T2 ) ) ) ) ) ).
% lambda_calculus.NF_def
thf(fact_1022_seq__Lam__Arr__implies,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
=> ( ( lambda_is_Lam @ T )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) ) ) ) ).
% seq_Lam_Arr_implies
thf(fact_1023_Arr__map__un__Lam,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_Lam @ T6 ) ) ) ) ).
% Arr_map_un_Lam
thf(fact_1024_Std__map__un__Lam,axiom,
! [T6: list_lambda_lambda] :
( ( reduction_Std @ T6 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) )
=> ( reduction_Std @ ( map_la336867333220232516lambda @ lambda_un_Lam @ T6 ) ) ) ) ).
% Std_map_un_Lam
thf(fact_1025_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_1026_lambda__calculus_Olambda_Odisc_I8_J,axiom,
! [X32: lambda_lambda] : ( lambda_is_Lam @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(8)
thf(fact_1027_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_1028_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_1029_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_1030_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_1031_lambda__calculus_Olambda_Odistinct__disc_I10_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ~ ( lambda_is_Var @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(10)
thf(fact_1032_lambda__calculus_Olambda_Odistinct__disc_I9_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Var @ Lambda )
=> ~ ( lambda_is_Lam @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(9)
thf(fact_1033_lambda__calculus_Olambda_Odistinct__disc_I18_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Beta @ Lambda )
=> ~ ( lambda_is_Lam @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(18)
thf(fact_1034_lambda__calculus_Olambda_Odistinct__disc_I17_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ~ ( lambda_is_Beta @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(17)
thf(fact_1035_length__n__lists__elem,axiom,
! [Ys: list_lambda_lambda,N: nat,Xs: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Ys @ ( set_li3846194476672328965lambda @ ( n_list4163717777911954552lambda @ N @ Xs ) ) )
=> ( ( size_s4208335251162658524lambda @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1036_lambda__calculus_ONF__App__iff,axiom,
! [T1: lambda_lambda,T22: lambda_lambda] :
( ( lambda_NF @ ( lambda_App @ T1 @ T22 ) )
= ( ~ ( lambda_is_Lam @ T1 )
& ( lambda_NF @ T1 )
& ( lambda_NF @ T22 ) ) ) ).
% lambda_calculus.NF_App_iff
thf(fact_1037_lambda__calculus_Oprfx__Lam__iff,axiom,
! [U2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ ( lambda_Lam @ T ) ) )
= ( ( lambda_is_Lam @ U2 )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_Lam @ U2 ) @ T ) ) ) ) ).
% lambda_calculus.prfx_Lam_iff
thf(fact_1038_seq__App__Std__implies,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ U ) )
=> ( ( ( lambda_is_App @ T )
& ~ ( lambda8135948770199036598uction @ T ) )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U ) @ ( collec621444948832755541lambda @ lambda_is_App ) ) ) ) ).
% seq_App_Std_implies
thf(fact_1039_elementary__reduction__path__def,axiom,
( reduct8314398545252454212n_path
= ( ^ [T8: list_lambda_lambda] :
( ( T8 = nil_lambda_lambda )
| ( ( paths_366302053746739058lambda @ lambda_resid @ T8 )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T8 ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ) ) ) ).
% elementary_reduction_path_def
thf(fact_1040_Std__implies__set__subset__elementary__reduction,axiom,
! [U: list_lambda_lambda] :
( ( reduction_Std @ U )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ).
% Std_implies_set_subset_elementary_reduction
thf(fact_1041_set__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ ( reduct1603535164923377302opment @ T ) ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ).
% set_standard_development
thf(fact_1042_lambda__calculus_Osseq__BetaI,axiom,
! [T: lambda_lambda,U2: lambda_lambda,V: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( ( lambda_Ide @ U2 )
=> ( ( lambda2670285437103676873uction @ V )
=> ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U2 ) @ V )
=> ( lambda_sseq @ ( lambda_Beta @ T @ U2 ) @ V ) ) ) ) ) ).
% lambda_calculus.sseq_BetaI
thf(fact_1043_lambda__calculus_Oelementary__reduction__is__arr,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% lambda_calculus.elementary_reduction_is_arr
thf(fact_1044_lambda__calculus_Olambda_Odistinct__disc_I15_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ~ ( lambda_is_App @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(15)
thf(fact_1045_lambda__calculus_Olambda_Odistinct__disc_I16_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_App @ Lambda )
=> ~ ( lambda_is_Lam @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(16)
thf(fact_1046_lambda__calculus_Oelementary__reduction__Lam__iff,axiom,
! [T: lambda_lambda] :
( ( lambda_is_Lam @ T )
=> ( ( lambda2670285437103676873uction @ T )
= ( lambda2670285437103676873uction @ ( lambda_un_Lam @ T ) ) ) ) ).
% lambda_calculus.elementary_reduction_Lam_iff
thf(fact_1047_lambda__calculus_Oelementary__reduction_Osimps_I5_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Beta @ T @ U2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U2 ) ) ) ).
% lambda_calculus.elementary_reduction.simps(5)
thf(fact_1048_lambda__calculus_Oelementary__reduction__not__ide,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T ) ) ).
% lambda_calculus.elementary_reduction_not_ide
thf(fact_1049_lambda__calculus_Ocong__elementary__reductions__are__equal,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ( ( lambda2670285437103676873uction @ U2 )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) )
=> ( T = U2 ) ) ) ) ).
% lambda_calculus.cong_elementary_reductions_are_equal
thf(fact_1050_lambda__calculus_Oelementary__reduction_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Lam @ T ) )
= ( lambda2670285437103676873uction @ T ) ) ).
% lambda_calculus.elementary_reduction.simps(3)
thf(fact_1051_lambda__calculus_Olambda_Odisc_I13_J,axiom,
! [X32: lambda_lambda] :
~ ( lambda_is_App @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(13)
thf(fact_1052_lambda__calculus_Olambda_Odisc_I15_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
~ ( lambda_is_App @ ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.disc(15)
thf(fact_1053_lambda__calculus_Ois__App__def,axiom,
( lambda_is_App
= ( ^ [Lambda2: lambda_lambda] :
? [X412: lambda_lambda,X422: lambda_lambda] :
( Lambda2
= ( lambda_App @ X412 @ X422 ) ) ) ) ).
% lambda_calculus.is_App_def
thf(fact_1054_lambda__calculus_Olambda_OdiscI_I3_J,axiom,
! [Lambda: lambda_lambda,X41: lambda_lambda,X42: lambda_lambda] :
( ( Lambda
= ( lambda_App @ X41 @ X42 ) )
=> ( lambda_is_App @ Lambda ) ) ).
% lambda_calculus.lambda.discI(3)
thf(fact_1055_lambda__calculus_Olambda_Odisc_I14_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] : ( lambda_is_App @ ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.disc(14)
thf(fact_1056_lambda__calculus_Osseq__imp__elementary__reduction2,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_sseq @ T @ U2 )
=> ( lambda2670285437103676873uction @ U2 ) ) ).
% lambda_calculus.sseq_imp_elementary_reduction2
thf(fact_1057_lambda__calculus_Osseq__imp__elementary__reduction1,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_sseq @ T @ U2 )
=> ( lambda2670285437103676873uction @ T ) ) ).
% lambda_calculus.sseq_imp_elementary_reduction1
thf(fact_1058_lambda__calculus_Olambda_Odistinct__disc_I11_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Var @ Lambda )
=> ~ ( lambda_is_App @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(11)
thf(fact_1059_lambda__calculus_Olambda_Odistinct__disc_I12_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_App @ Lambda )
=> ~ ( lambda_is_Var @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(12)
thf(fact_1060_lambda__calculus_Olambda_Odistinct__disc_I19_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_App @ Lambda )
=> ~ ( lambda_is_Beta @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(19)
thf(fact_1061_lambda__calculus_Olambda_Odistinct__disc_I20_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Beta @ Lambda )
=> ~ ( lambda_is_App @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(20)
thf(fact_1062_lambda__calculus_Oelementary__reduction_Osimps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_App @ T @ U2 ) )
= ( ( ( lambda2670285437103676873uction @ T )
& ( lambda_Ide @ U2 ) )
| ( ( lambda_Ide @ T )
& ( lambda2670285437103676873uction @ U2 ) ) ) ) ).
% lambda_calculus.elementary_reduction.simps(4)
thf(fact_1063_lambda__calculus_Osseq__preserves__App__and__no__head__reduction,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_sseq @ T @ U2 )
=> ( ( ( lambda_is_App @ T )
& ~ ( lambda8135948770199036598uction @ T ) )
=> ( ( lambda_is_App @ U2 )
& ~ ( lambda8135948770199036598uction @ U2 ) ) ) ) ).
% lambda_calculus.sseq_preserves_App_and_no_head_reduction
thf(fact_1064_reduction__paths_OStd_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( lambda2670285437103676873uction @ T ) ) ).
% reduction_paths.Std.simps(2)
thf(fact_1065_lambda__calculus_Oelementary__reduction__head__redex,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ~ ( lambda_Ide @ ( lambda_head_redex @ T ) )
=> ( lambda2670285437103676873uction @ ( lambda_head_redex @ T ) ) ) ) ).
% lambda_calculus.elementary_reduction_head_redex
thf(fact_1066_lambda__calculus_Oelementary__reduction__bottom__up__redex,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ~ ( lambda_Ide @ T )
=> ( lambda2670285437103676873uction @ ( lambda5654762366400029632_redex @ T ) ) ) ) ).
% lambda_calculus.elementary_reduction_bottom_up_redex
thf(fact_1067_lambda__calculus_Oelementary__reduction_Oelims_I2_J,axiom,
! [X: lambda_lambda] :
( ( lambda2670285437103676873uction @ X )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( lambda_Lam @ T4 ) )
=> ~ ( lambda2670285437103676873uction @ T4 ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( X
= ( lambda_App @ T4 @ U5 ) )
=> ~ ( ( ( lambda2670285437103676873uction @ T4 )
& ( lambda_Ide @ U5 ) )
| ( ( lambda_Ide @ T4 )
& ( lambda2670285437103676873uction @ U5 ) ) ) )
=> ~ ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( X
= ( lambda_Beta @ T4 @ U5 ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.elims(2)
thf(fact_1068_lambda__calculus_Osseq__Beta,axiom,
! [T: lambda_lambda,U2: lambda_lambda,V: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U2 ) @ V )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U2 )
& ( lambda2670285437103676873uction @ V )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U2 ) @ V ) ) ) ).
% lambda_calculus.sseq_Beta
thf(fact_1069_lambda__calculus_Osseq_Osimps_I12_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,Va: lambda_lambda,Vb2: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U2 ) @ ( lambda_Beta @ Va @ Vb2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U2 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U2 ) @ ( lambda_Beta @ Va @ Vb2 ) ) ) ) ).
% lambda_calculus.sseq.simps(12)
thf(fact_1070_lambda__calculus_Osseq_Osimps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,T3: lambda_lambda,U4: lambda_lambda] :
( ( lambda_sseq @ ( lambda_App @ T @ U2 ) @ ( lambda_App @ T3 @ U4 ) )
= ( ( ( lambda_sseq @ T @ T3 )
& ( lambda_Ide @ U2 )
& ( U2 = U4 ) )
| ( ( lambda_Ide @ T )
& ( T = T3 )
& ( lambda_sseq @ U2 @ U4 ) )
| ( ( lambda2670285437103676873uction @ T )
& ( ( lambda_Trg @ T )
= T3 )
& ( U2
= ( lambda_Src @ U4 ) )
& ( lambda2670285437103676873uction @ U4 ) ) ) ) ).
% lambda_calculus.sseq.simps(4)
thf(fact_1071_reduction__paths_OStd_Oelims_I1_J,axiom,
! [X: list_lambda_lambda,Y2: $o] :
( ( ( reduction_Std @ X )
= Y2 )
=> ( ( ( X = nil_lambda_lambda )
=> ~ Y2 )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( Y2
= ( ~ ( lambda2670285437103676873uction @ T4 ) ) ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( Y2
= ( ~ ( ( lambda_sseq @ T4 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
& ( reduction_Std @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ).
% reduction_paths.Std.elims(1)
thf(fact_1072_reduction__paths_OStd_Oelims_I2_J,axiom,
! [X: list_lambda_lambda] :
( ( reduction_Std @ X )
=> ( ( X != nil_lambda_lambda )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ~ ( lambda2670285437103676873uction @ T4 ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ~ ( ( lambda_sseq @ T4 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
& ( reduction_Std @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ).
% reduction_paths.Std.elims(2)
thf(fact_1073_reduction__paths_OStd_Oelims_I3_J,axiom,
! [X: list_lambda_lambda] :
( ~ ( reduction_Std @ X )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( lambda2670285437103676873uction @ T4 ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( ( lambda_sseq @ T4 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
& ( reduction_Std @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ).
% reduction_paths.Std.elims(3)
thf(fact_1074_lambda__calculus_Oelementary__reduction__Beta__iff,axiom,
! [T: lambda_lambda] :
( ( lambda_is_Beta @ T )
=> ( ( lambda2670285437103676873uction @ T )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_un_Beta1 @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_un_Beta2 @ T ) ) ) ) ) ).
% lambda_calculus.elementary_reduction_Beta_iff
thf(fact_1075_lambda__calculus_Osseq_Osimps_I11_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,Va: lambda_lambda,Vb2: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U2 ) @ ( lambda_App @ Va @ Vb2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U2 )
& ( lambda2670285437103676873uction @ ( lambda_App @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U2 ) @ ( lambda_App @ Va @ Vb2 ) ) ) ) ).
% lambda_calculus.sseq.simps(11)
thf(fact_1076_lambda__calculus_Osseq_Osimps_I8_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,V: lambda_lambda,Va: lambda_lambda,Vb2: lambda_lambda] :
( ( lambda_sseq @ ( lambda_App @ ( lambda_Beta @ T @ U2 ) @ V ) @ ( lambda_Beta @ Va @ Vb2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U2 )
& ( lambda_Ide @ V )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T @ U2 ) @ V ) @ ( lambda_Beta @ Va @ Vb2 ) ) ) ) ).
% lambda_calculus.sseq.simps(8)
thf(fact_1077_lambda__calculus_Osseq_Osimps_I10_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,Va: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U2 ) @ ( lambda_Lam @ Va ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U2 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U2 ) @ ( lambda_Lam @ Va ) ) ) ) ).
% lambda_calculus.sseq.simps(10)
thf(fact_1078_lambda__calculus_Osseq_Osimps_I7_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_sseq @ ( lambda_App @ ( lambda_Beta @ T @ U2 ) @ V ) @ ( lambda_Lam @ Va ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U2 )
& ( lambda_Ide @ V )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T @ U2 ) @ V ) @ ( lambda_Lam @ Va ) ) ) ) ).
% lambda_calculus.sseq.simps(7)
thf(fact_1079_Arr__map__un__App2,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_App2 @ T6 ) ) ) ) ).
% Arr_map_un_App2
thf(fact_1080_Arr__map__un__App1,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_App1 @ T6 ) ) ) ) ).
% Arr_map_un_App1
thf(fact_1081_lambda__calculus_Olambda_Ocollapse_I3_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_App @ Lambda )
=> ( ( lambda_App @ ( lambda_un_App1 @ Lambda ) @ ( lambda_un_App2 @ Lambda ) )
= Lambda ) ) ).
% lambda_calculus.lambda.collapse(3)
thf(fact_1082_lambda__calculus_Olambda_Osel_I3_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_un_App1 @ ( lambda_App @ X41 @ X42 ) )
= X41 ) ).
% lambda_calculus.lambda.sel(3)
thf(fact_1083_lambda__calculus_Olambda_Osel_I4_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_un_App2 @ ( lambda_App @ X41 @ X42 ) )
= X42 ) ).
% lambda_calculus.lambda.sel(4)
thf(fact_1084_lambda__calculus_Oprfx__App__iff,axiom,
! [U2: lambda_lambda,T1: lambda_lambda,T22: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ ( lambda_App @ T1 @ T22 ) ) )
= ( ( lambda_is_App @ U2 )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_App1 @ U2 ) @ T1 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_App2 @ U2 ) @ T22 ) ) ) ) ).
% lambda_calculus.prfx_App_iff
thf(fact_1085_lambda__calculus_Oelementary__reduction__App__iff,axiom,
! [T: lambda_lambda] :
( ( lambda_is_App @ T )
=> ( ( lambda2670285437103676873uction @ T )
= ( ( ( lambda2670285437103676873uction @ ( lambda_un_App1 @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_un_App2 @ T ) ) )
| ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_un_App1 @ T ) )
& ( lambda2670285437103676873uction @ ( lambda_un_App2 @ T ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction_App_iff
thf(fact_1086_lambda__calculus_Oseq__cases,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( ( lambda_is_Var @ T )
& ( lambda_is_Var @ U2 ) )
| ( ( lambda_is_Lam @ T )
& ( lambda_is_Lam @ U2 ) )
| ( ( lambda_is_App @ T )
& ( lambda_is_App @ U2 ) )
| ( ( lambda_is_App @ T )
& ( lambda_is_Beta @ U2 )
& ( lambda_is_Lam @ ( lambda_un_App1 @ T ) ) )
| ( ( lambda_is_App @ T )
& ( lambda_is_Beta @ U2 )
& ( lambda_is_Beta @ ( lambda_un_App1 @ T ) ) )
| ( lambda_is_Beta @ T ) ) ) ).
% lambda_calculus.seq_cases
thf(fact_1087_map__App__map__un__App2,axiom,
! [U: list_lambda_lambda,A: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( lambda_Ide @ A )
=> ( ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ lambda_un_App1 @ ( set_lambda_lambda2 @ U ) ) @ ( insert_lambda_lambda @ A @ bot_bo1271416176687173302lambda ) )
=> ( ( map_la336867333220232516lambda @ ( lambda_App @ A ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U ) )
= U ) ) ) ) ) ).
% map_App_map_un_App2
thf(fact_1088_Ide__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
= ( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
& ( paths_7717490669901197737lambda @ lambda_resid @ U )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_1089_Trg_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ).
% Trg.simps(3)
thf(fact_1090_Trg__resid__sym,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) ) ) ).
% Trg_resid_sym
thf(fact_1091_Ide__imp__Src__eq__Trg,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ T6 )
= ( paths_2695323870737867955lambda @ lambda_resid @ T6 ) ) ) ).
% Ide_imp_Src_eq_Trg
thf(fact_1092_Arr__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( paths_366302053746739058lambda @ lambda_resid @ U )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_1093_Arr__appendE_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
=> ( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
!= ( paths_1651982994781619320lambda @ lambda_resid @ U ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P\<^sub>W\<^sub>E
thf(fact_1094_Ide__appendE,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
=> ( ( T6 != nil_lambda_lambda )
=> ( ( U != nil_lambda_lambda )
=> ~ ( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
!= ( paths_1651982994781619320lambda @ lambda_resid @ U ) ) ) ) ) ) ) ).
% Ide_appendE
thf(fact_1095_list_Oset__map,axiom,
! [F: lambda_lambda > list_lambda_lambda,V: list_lambda_lambda] :
( ( set_li3846194476672328965lambda @ ( map_la6434941737452557130lambda @ F @ V ) )
= ( image_3692823078739859851lambda @ F @ ( set_lambda_lambda2 @ V ) ) ) ).
% list.set_map
thf(fact_1096_list_Oset__map,axiom,
! [F: lambda_lambda > lambda_lambda,V: list_lambda_lambda] :
( ( set_lambda_lambda2 @ ( map_la336867333220232516lambda @ F @ V ) )
= ( image_2356535832909760517lambda @ F @ ( set_lambda_lambda2 @ V ) ) ) ).
% list.set_map
thf(fact_1097_Src__resid,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ U ) ) ) ).
% Src_resid
thf(fact_1098_Trg__append,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ U ) ) ) ).
% Trg_append
thf(fact_1099_Arr__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) ) ) ) ) ).
% Arr_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_1100_Ide__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U )
=> ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U ) ) ) ) ) ).
% Ide_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_1101_Trgs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( insert_lambda_lambda @ ( paths_2695323870737867955lambda @ lambda_resid @ T6 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Trgs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_1102_image__set,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda] :
( ( image_3692823078739859851lambda @ F @ ( set_lambda_lambda2 @ Xs ) )
= ( set_li3846194476672328965lambda @ ( map_la6434941737452557130lambda @ F @ Xs ) ) ) ).
% image_set
thf(fact_1103_image__set,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( image_2356535832909760517lambda @ F @ ( set_lambda_lambda2 @ Xs ) )
= ( set_lambda_lambda2 @ ( map_la336867333220232516lambda @ F @ Xs ) ) ) ).
% image_set
thf(fact_1104_red__iff,axiom,
( lambda_red
= ( ^ [A4: lambda_lambda,B3: lambda_lambda] :
? [T8: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T8 )
& ( ( paths_1651982994781619320lambda @ lambda_resid @ T8 )
= A4 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T8 )
= B3 ) ) ) ) ).
% red_iff
thf(fact_1105_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_1106_lambda__calculus_ONF__unique,axiom,
! [T: lambda_lambda,U2: lambda_lambda,U4: lambda_lambda] :
( ( lambda_red @ T @ U2 )
=> ( ( lambda_red @ T @ U4 )
=> ( ( lambda_NF @ U2 )
=> ( ( lambda_NF @ U4 )
=> ( U2 = U4 ) ) ) ) ) ).
% lambda_calculus.NF_unique
thf(fact_1107_lambda__calculus_ONF__reduct__is__trivial,axiom,
! [A: lambda_lambda,B: lambda_lambda] :
( ( lambda_NF @ A )
=> ( ( lambda_red @ A @ B )
=> ( A = B ) ) ) ).
% lambda_calculus.NF_reduct_is_trivial
thf(fact_1108_lambda__calculus_Oweak__diamond,axiom,
! [A: lambda_lambda,B: lambda_lambda,B4: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( ( lambda_red @ A @ B4 )
=> ~ ! [C3: lambda_lambda] :
( ( lambda_red @ B @ C3 )
=> ~ ( lambda_red @ B4 @ C3 ) ) ) ) ).
% lambda_calculus.weak_diamond
thf(fact_1109_lambda__calculus_Ored_Ointros_I2_J,axiom,
! [A: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( ( lambda_red @ B @ C )
=> ( lambda_red @ A @ C ) ) ) ).
% lambda_calculus.red.intros(2)
thf(fact_1110_lambda__calculus_Ored_Osimps,axiom,
( lambda_red
= ( ^ [A1: lambda_lambda,A22: lambda_lambda] :
( ? [T2: lambda_lambda] :
( ( A1
= ( lambda_Src @ T2 ) )
& ( A22
= ( lambda_Trg @ T2 ) )
& ( lambda_Arr @ T2 ) )
| ? [A4: lambda_lambda,B3: lambda_lambda,C2: lambda_lambda] :
( ( A1 = A4 )
& ( A22 = C2 )
& ( lambda_red @ A4 @ B3 )
& ( lambda_red @ B3 @ C2 ) ) ) ) ) ).
% lambda_calculus.red.simps
thf(fact_1111_lambda__calculus_Ored_Ocases,axiom,
! [A12: lambda_lambda,A23: lambda_lambda] :
( ( lambda_red @ A12 @ A23 )
=> ( ! [T4: lambda_lambda] :
( ( A12
= ( lambda_Src @ T4 ) )
=> ( ( A23
= ( lambda_Trg @ T4 ) )
=> ~ ( lambda_Arr @ T4 ) ) )
=> ~ ! [B2: lambda_lambda] :
( ( lambda_red @ A12 @ B2 )
=> ~ ( lambda_red @ B2 @ A23 ) ) ) ) ).
% lambda_calculus.red.cases
thf(fact_1112_lambda__calculus_Ored_Ointros_I1_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_red @ ( lambda_Src @ T ) @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.red.intros(1)
thf(fact_1113_Ide_Oelims_I3_J,axiom,
! [X: list_lambda_lambda] :
( ~ ( paths_7717490669901197737lambda @ lambda_resid @ X )
=> ( ( X != nil_lambda_lambda )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( ide_lambda_lambda @ lambda_resid @ T4 ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T4 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ) ).
% Ide.elims(3)
thf(fact_1114_Ide_Oelims_I2_J,axiom,
! [X: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ X )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T4 ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T4 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ).
% Ide.elims(2)
thf(fact_1115__092_060Lambda_062_Otarget__is__ide,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ide_lambda_lambda @ lambda_resid @ A ) ) ).
% \<Lambda>.target_is_ide
thf(fact_1116__092_060Lambda_062_Otargets__are__cong,axiom,
! [B: lambda_lambda,T: lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B4 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B4 @ B ) ) ) ) ) ).
% \<Lambda>.targets_are_cong
thf(fact_1117__092_060Lambda_062_Otargets__cong__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B4 @ B ) ) )
=> ( member_lambda_lambda @ B4 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.targets_cong_closed
thf(fact_1118__092_060Lambda_062_Otargets__resid__sym,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
= ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) ) ) ).
% \<Lambda>.targets_resid_sym
thf(fact_1119__092_060Lambda_062_Otargets__are__con,axiom,
! [B: lambda_lambda,T: lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B4 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ B @ B4 ) ) ) ).
% \<Lambda>.targets_are_con
thf(fact_1120__092_060Lambda_062_Oarr__has__un__target,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
& ! [Y3: lambda_lambda] :
( ( member_lambda_lambda @ Y3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( Y3 = X3 ) ) ) ) ).
% \<Lambda>.arr_has_un_target
thf(fact_1121__092_060Lambda_062_Oresid__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 ) ) ) ).
% \<Lambda>.resid_source_in_targets
thf(fact_1122__092_060Lambda_062_Otargets__con__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B4 )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ B4 )
=> ( member_lambda_lambda @ B4 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.targets_con_closed
thf(fact_1123__092_060Lambda_062_Oarr__iff__has__target,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( ( target5490931411331770124lambda @ lambda_resid @ T )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>.arr_iff_has_target
thf(fact_1124__092_060Lambda_062_OcoterminalE,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U2 )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
!= ( target5490931411331770124lambda @ lambda_resid @ U2 ) ) ) ) ) ).
% \<Lambda>.coterminalE
thf(fact_1125__092_060Lambda_062_Ocoterminal__iff,axiom,
! [T: lambda_lambda,T3: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T3 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T3 ) ) ) ) ).
% \<Lambda>.coterminal_iff
thf(fact_1126__092_060Lambda_062_Otargets__eqI,axiom,
! [T: lambda_lambda,T3: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T3 ) ) ) ).
% \<Lambda>.targets_eqI
thf(fact_1127__092_060Lambda_062_OcomposableD_I3_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U2 ) ) ) ).
% \<Lambda>.composableD(3)
thf(fact_1128_Trgs_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( target5490931411331770124lambda @ lambda_resid @ T ) ) ).
% Trgs.simps(2)
thf(fact_1129__092_060Lambda_062_OseqE,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U2 )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
!= ( source5627170679743844886lambda @ lambda_resid @ U2 ) ) ) ) ) ).
% \<Lambda>.seqE
thf(fact_1130__092_060Lambda_062_Oseq__def,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U2 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U2 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U2 ) ) ) ) ).
% \<Lambda>.seq_def
thf(fact_1131_Trgs__simp_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( target5490931411331770124lambda @ lambda_resid @ ( last_lambda_lambda @ T6 ) ) ) ) ).
% Trgs_simp\<^sub>P
thf(fact_1132__092_060Lambda_062_Ocoterminal__def,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U2 )
= ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ U2 ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>.coterminal_def
thf(fact_1133_Srcs__Resid__Arr__single,axiom,
! [T6: list_lambda_lambda,U2: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) ) )
= ( target5490931411331770124lambda @ lambda_resid @ U2 ) ) ) ).
% Srcs_Resid_Arr_single
thf(fact_1134_Trgs_Oelims,axiom,
! [X: list_lambda_lambda,Y2: set_lambda_lambda] :
( ( ( paths_7594662086398683701lambda @ lambda_resid @ X )
= Y2 )
=> ( ( ( X = nil_lambda_lambda )
=> ( Y2 != bot_bo1271416176687173302lambda ) )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( Y2
!= ( target5490931411331770124lambda @ lambda_resid @ T4 ) ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( Y2
!= ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ).
% Trgs.elims
thf(fact_1135_Arr_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ).
% Arr.simps(3)
thf(fact_1136_Ide_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( ( ide_lambda_lambda @ lambda_resid @ T )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ).
% Ide.simps(3)
thf(fact_1137_Arr_Oelims_I1_J,axiom,
! [X: list_lambda_lambda,Y2: $o] :
( ( ( paths_366302053746739058lambda @ lambda_resid @ X )
= Y2 )
=> ( ( ( X = nil_lambda_lambda )
=> Y2 )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( Y2
= ( ~ ( arr_lambda_lambda @ lambda_resid @ T4 ) ) ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( Y2
= ( ~ ( ( arr_lambda_lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% Arr.elims(1)
thf(fact_1138_Arr_Oelims_I2_J,axiom,
! [X: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ X )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ~ ( arr_lambda_lambda @ lambda_resid @ T4 ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ).
% Arr.elims(2)
thf(fact_1139_Arr_Oelims_I3_J,axiom,
! [X: list_lambda_lambda] :
( ~ ( paths_366302053746739058lambda @ lambda_resid @ X )
=> ( ( X != nil_lambda_lambda )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( arr_lambda_lambda @ lambda_resid @ T4 ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ) ).
% Arr.elims(3)
thf(fact_1140_Ide_Oelims_I1_J,axiom,
! [X: list_lambda_lambda,Y2: $o] :
( ( ( paths_7717490669901197737lambda @ lambda_resid @ X )
= Y2 )
=> ( ( ( X = nil_lambda_lambda )
=> Y2 )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( Y2
= ( ~ ( ide_lambda_lambda @ lambda_resid @ T4 ) ) ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( Y2
= ( ~ ( ( ide_lambda_lambda @ lambda_resid @ T4 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% Ide.elims(1)
thf(fact_1141__092_060Lambda_062_Osources__resid,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
= ( target5490931411331770124lambda @ lambda_resid @ U2 ) ) ) ).
% \<Lambda>.sources_resid
thf(fact_1142__092_060Lambda_062_OcoterminalI,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ U2 ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ T @ U2 ) ) ) ).
% \<Lambda>.coterminalI
thf(fact_1143__092_060Lambda_062_OseqI,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U2 ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U2 ) ) ) ) ).
% \<Lambda>.seqI
thf(fact_1144_lambda__calculus_Otargets__simp,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.targets_simp
thf(fact_1145_Arr__consI_092_060_094sub_062P,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U ) ) ) ) ) ).
% Arr_consI\<^sub>P
thf(fact_1146_lambda__calculus_Otargets__simps_I4_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_App @ T @ U2 ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.targets_simps(4)
thf(fact_1147_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_1148_lambda__calculus_Otargets__subst,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Subst @ zero_zero_nat @ T @ U2 ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U2 ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.targets_subst
thf(fact_1149_lambda__calculus_Otargets__simps_I5_J,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Beta @ T @ U2 ) )
= ( insert_lambda_lambda @ ( lambda_Subst @ zero_zero_nat @ ( lambda_Trg @ U2 ) @ ( lambda_Trg @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.targets_simps(5)
thf(fact_1150_lambda__calculus_Otargets__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 ) ) ) ).
% lambda_calculus.targets_char\<^sub>\<Lambda>
thf(fact_1151_lambda__calculus_Onormalizable__def,axiom,
( lambda_normalizable
= ( ^ [A4: lambda_lambda] :
( ( lambda_Ide @ A4 )
& ? [B3: lambda_lambda] :
( ( lambda_red @ A4 @ B3 )
& ( lambda_NF @ B3 ) ) ) ) ) ).
% lambda_calculus.normalizable_def
thf(fact_1152_Ide__consE,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ( U != nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U ) )
=> ~ ( ( U != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U ) ) ) ) ) ) ).
% Ide_consE
thf(fact_1153__092_060Lambda_062_Oapex__sym,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
= ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) ) ).
% \<Lambda>.apex_sym
thf(fact_1154__092_060Lambda_062_Otrg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_resid @ T @ T ) ) ).
% \<Lambda>.trg_def
thf(fact_1155__092_060Lambda_062_Otrg__resid__sym,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U2 ) )
= ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ T ) ) ) ) ).
% \<Lambda>.trg_resid_sym
thf(fact_1156__092_060Lambda_062_Oide__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 ) ) ) ).
% \<Lambda>.ide_iff_trg_self
thf(fact_1157__092_060Lambda_062_Oide__trg,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.ide_trg
thf(fact_1158__092_060Lambda_062_Otrg__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 ) ) ) ).
% \<Lambda>.trg_in_targets
thf(fact_1159__092_060Lambda_062_Oresid__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 ) ) ) ) ).
% \<Lambda>.resid_ide(2)
thf(fact_1160__092_060Lambda_062_Ocoterminal__iff__con__trg,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U2 )
= ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ ( trg_lambda_lambda @ lambda_resid @ U2 ) ) ) ).
% \<Lambda>.coterminal_iff_con_trg
thf(fact_1161__092_060Lambda_062_OcoterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U2 )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U2 )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
!= ( trg_lambda_lambda @ lambda_resid @ U2 ) ) ) ) ) ).
% \<Lambda>.coterminalE\<^sub>W\<^sub>E
thf(fact_1162__092_060Lambda_062_Ocoterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U2 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U2 )
& ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( trg_lambda_lambda @ lambda_resid @ U2 ) ) ) ) ).
% \<Lambda>.coterminal_iff\<^sub>W\<^sub>E
thf(fact_1163_Trg_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ).
% Trg.simps(2)
thf(fact_1164__092_060Lambda_062_Oin__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 ) ) ) ).
% \<Lambda>.in_targetsE
thf(fact_1165_Arr__consE,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( U != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ U ) )
=> ~ ( ( U != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U ) ) ) ) ) ) ).
% Arr_consE
thf(fact_1166__092_060Lambda_062_Otrg__trg,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.trg_trg
thf(fact_1167__092_060Lambda_062_Otrg__ide,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( trg_lambda_lambda @ lambda_resid @ A )
= A ) ) ).
% \<Lambda>.trg_ide
thf(fact_1168__092_060Lambda_062_Oarr__trg__iff__arr,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
= ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.arr_trg_iff_arr
thf(fact_1169__092_060Lambda_062_OcoterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( trg_lambda_lambda @ lambda_resid @ U2 ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ T @ U2 ) ) ) ).
% \<Lambda>.coterminalI\<^sub>W\<^sub>E
thf(fact_1170__092_060Lambda_062_Oin__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 ) ) ) ) ).
% \<Lambda>.in_targetsI
thf(fact_1171_Trg__simp,axiom,
! [T6: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( trg_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T6 ) ) ) ) ).
% Trg_simp
thf(fact_1172_Arr__consI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U ) ) ) ) ) ).
% Arr_consI\<^sub>P\<^sub>W\<^sub>E
thf(fact_1173_Ide__consI,axiom,
! [T: lambda_lambda,U: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U ) ) ) ) ) ).
% Ide_consI
thf(fact_1174_reflects__con,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) @ ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ U2 ) @ ( cons_lambda_lambda @ U2 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) )
!= nil_lambda_lambda )
=> ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% reflects_con
thf(fact_1175__092_060Lambda_062_Ojoinable__implies__coinitial,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U2 )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.joinable_implies_coinitial
thf(fact_1176_con__imp__arr__resid,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% con_imp_arr_resid
thf(fact_1177_con__sym__ax,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% con_sym_ax
thf(fact_1178_cube__ax,axiom,
! [V: list_lambda_lambda,T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) ) ) ) ).
% cube_ax
thf(fact_1179_null__eqI,axiom,
! [N: list_lambda_lambda] :
( ! [T4: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ N @ T4 )
= N )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ N )
= N ) )
=> ( N
= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% null_eqI
thf(fact_1180__092_060Lambda_062_Ojoinable__implies__con,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U2 )
=> ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.joinable_implies_con
thf(fact_1181__092_060Lambda_062_Ohas__joins,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( joinab2519788767067900224lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.has_joins
thf(fact_1182_null__char,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= nil_lambda_lambda ) ).
% null_char
thf(fact_1183_conE,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% conE
thf(fact_1184_con__def,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% con_def
thf(fact_1185_not__arr__null,axiom,
~ ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% not_arr_null
thf(fact_1186_null__is__zero_I2_J,axiom,
! [T: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ T @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% null_is_zero(2)
thf(fact_1187_null__is__zero_I1_J,axiom,
! [T: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) @ T )
= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% null_is_zero(1)
thf(fact_1188_conI,axiom,
! [T: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ).
% conI
thf(fact_1189_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ( ( OP2 @ T @ ( partia6873046715171720854lambda @ OP2 ) )
= ( partia6873046715171720854lambda @ OP2 ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_1190_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ( ( OP2 @ T @ ( partia1461432607296028304lambda @ OP2 ) )
= ( partia1461432607296028304lambda @ OP2 ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_1191_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ( ( OP2 @ ( partia6873046715171720854lambda @ OP2 ) @ T )
= ( partia6873046715171720854lambda @ OP2 ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_1192_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ( ( OP2 @ ( partia1461432607296028304lambda @ OP2 ) @ T )
= ( partia1461432607296028304lambda @ OP2 ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_1193_partial__magma_Onull__eqI,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,N: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ( ! [T4: list_lambda_lambda] :
( ( ( OP2 @ N @ T4 )
= N )
& ( ( OP2 @ T4 @ N )
= N ) )
=> ( N
= ( partia6873046715171720854lambda @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_1194_partial__magma_Onull__eqI,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda,N: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ( ! [T4: lambda_lambda] :
( ( ( OP2 @ N @ T4 )
= N )
& ( ( OP2 @ T4 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_1195_map__App__decomp,axiom,
! [U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ U ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ U ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U ) ) )
@ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid @ U
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ U ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ U ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U ) ) ) ) ) ) ) ) ).
% map_App_decomp
thf(fact_1196_lambda__calculus_Osseq_Oelims_I2_J,axiom,
! [X: lambda_lambda,Xa: lambda_lambda] :
( ( lambda_sseq @ X @ Xa )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( lambda_Lam @ T4 ) )
=> ! [T9: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T9 ) )
=> ~ ( lambda_sseq @ T4 @ T9 ) ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( X
= ( lambda_App @ T4 @ U5 ) )
=> ! [T9: lambda_lambda,U9: lambda_lambda] :
( ( Xa
= ( lambda_App @ T9 @ U9 ) )
=> ~ ( ( ( lambda_sseq @ T4 @ T9 )
& ( lambda_Ide @ U5 )
& ( U5 = U9 ) )
| ( ( lambda_Ide @ T4 )
& ( T4 = T9 )
& ( lambda_sseq @ U5 @ U9 ) )
| ( ( lambda2670285437103676873uction @ T4 )
& ( ( lambda_Trg @ T4 )
= T9 )
& ( U5
= ( lambda_Src @ U9 ) )
& ( lambda2670285437103676873uction @ U9 ) ) ) ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X
= ( lambda_App @ ( lambda_Beta @ T4 @ U5 ) @ V2 ) )
=> ! [Va2: nat] :
( ( Xa
= ( lambda_Var @ Va2 ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T4 @ U5 ) @ V2 ) @ ( lambda_Var @ Va2 ) ) ) ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X
= ( lambda_App @ ( lambda_Beta @ T4 @ U5 ) @ V2 ) )
=> ! [Va2: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ Va2 ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T4 @ U5 ) @ V2 ) @ ( lambda_Lam @ Va2 ) ) ) ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X
= ( lambda_App @ ( lambda_Beta @ T4 @ U5 ) @ V2 ) )
=> ! [Va2: lambda_lambda,Vb: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ Va2 @ Vb ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va2 @ Vb ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T4 @ U5 ) @ V2 ) @ ( lambda_Beta @ Va2 @ Vb ) ) ) ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( X
= ( lambda_Beta @ T4 @ U5 ) )
=> ! [Va2: nat] :
( ( Xa
= ( lambda_Var @ Va2 ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T4 @ U5 ) @ ( lambda_Var @ Va2 ) ) ) ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( X
= ( lambda_Beta @ T4 @ U5 ) )
=> ! [Va2: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ Va2 ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T4 @ U5 ) @ ( lambda_Lam @ Va2 ) ) ) ) )
=> ( ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( X
= ( lambda_Beta @ T4 @ U5 ) )
=> ! [Va2: lambda_lambda,Vb: lambda_lambda] :
( ( Xa
= ( lambda_App @ Va2 @ Vb ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_App @ Va2 @ Vb ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T4 @ U5 ) @ ( lambda_App @ Va2 @ Vb ) ) ) ) )
=> ~ ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( X
= ( lambda_Beta @ T4 @ U5 ) )
=> ! [Va2: lambda_lambda,Vb: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ Va2 @ Vb ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va2 @ Vb ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T4 @ U5 ) @ ( lambda_Beta @ Va2 @ Vb ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.sseq.elims(2)
thf(fact_1197__092_060Lambda_062_Onull__eqI,axiom,
! [N: lambda_lambda] :
( ! [T4: lambda_lambda] :
( ( ( lambda_resid @ N @ T4 )
= N )
& ( ( lambda_resid @ T4 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.null_eqI
thf(fact_1198__092_060Lambda_062_Ocube__ax,axiom,
! [V: lambda_lambda,T: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U2 @ T ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U2 @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U2 ) @ ( lambda_resid @ T @ U2 ) ) ) ) ).
% \<Lambda>.cube_ax
thf(fact_1199__092_060Lambda_062_Ocon__sym__ax,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ T @ U2 )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ U2 @ T )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.con_sym_ax
thf(fact_1200__092_060Lambda_062_Ocon__imp__arr__resid,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ T @ U2 )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ ( lambda_resid @ T @ U2 ) @ ( lambda_resid @ T @ U2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.con_imp_arr_resid
thf(fact_1201__092_060Lambda_062_Ocon__def,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
= ( ( lambda_resid @ T @ U2 )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.con_def
thf(fact_1202__092_060Lambda_062_OconE,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U2 )
=> ( ( lambda_resid @ T @ U2 )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.conE
thf(fact_1203__092_060Lambda_062_Onot__arr__null,axiom,
~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.not_arr_null
thf(fact_1204_Trg_Osimps_I1_J,axiom,
( ( paths_2695323870737867955lambda @ lambda_resid @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Trg.simps(1)
thf(fact_1205__092_060Lambda_062_Onull__is__zero_I1_J,axiom,
! [T: lambda_lambda] :
( ( lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.null_is_zero(1)
thf(fact_1206__092_060Lambda_062_Onull__is__zero_I2_J,axiom,
! [T: lambda_lambda] :
( ( lambda_resid @ T @ ( partia1461432607296028304lambda @ lambda_resid ) )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.null_is_zero(2)
thf(fact_1207__092_060Lambda_062_Otargets__char,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B3: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ T ) @ B3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ ( lambda_resid @ T @ T ) ) ) ) ) ) ).
% \<Lambda>.targets_char
thf(fact_1208__092_060Lambda_062_Osources__def,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A4 )
& ( con_lambda_lambda @ lambda_resid @ T @ A4 ) ) ) ) ).
% \<Lambda>.sources_def
thf(fact_1209_lambda__calculus_Olambda_Oinject_I1_J,axiom,
! [X23: nat,Y23: nat] :
( ( ( lambda_Var @ X23 )
= ( lambda_Var @ Y23 ) )
= ( X23 = Y23 ) ) ).
% lambda_calculus.lambda.inject(1)
thf(fact_1210_Std__map__App2,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( reduction_Std @ T6 )
=> ( reduction_Std @ ( map_la336867333220232516lambda @ ( lambda_App @ A ) @ T6 ) ) ) ) ).
% Std_map_App2
thf(fact_1211_Std__map__App1,axiom,
! [B: lambda_lambda,T6: list_lambda_lambda] :
( ( lambda_Ide @ B )
=> ( ( reduction_Std @ T6 )
=> ( reduction_Std
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ B )
@ T6 ) ) ) ) ).
% Std_map_App1
thf(fact_1212__092_060Lambda_062_Otargets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B3: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ( trg_lambda_lambda @ lambda_resid @ T )
= B3 ) ) ) ) ).
% \<Lambda>.targets_char\<^sub>W\<^sub>E
thf(fact_1213_Trg_Oelims,axiom,
! [X: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_2695323870737867955lambda @ lambda_resid @ X )
= Y2 )
=> ( ( ( X = nil_lambda_lambda )
=> ( Y2
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ nil_lambda_lambda ) )
=> ( Y2
!= ( trg_lambda_lambda @ lambda_resid @ T4 ) ) )
=> ~ ! [T4: lambda_lambda,V2: lambda_lambda,Va2: list_lambda_lambda] :
( ( X
= ( cons_lambda_lambda @ T4 @ ( cons_lambda_lambda @ V2 @ Va2 ) ) )
=> ( Y2
!= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va2 ) ) ) ) ) ) ) ).
% Trg.elims
thf(fact_1214_Arr__map__App1,axiom,
! [B: lambda_lambda,T6: list_lambda_lambda] :
( ( lambda_Ide @ B )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( paths_366302053746739058lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T2: lambda_lambda] : ( lambda_App @ T2 @ B )
@ T6 ) ) ) ) ).
% Arr_map_App1
thf(fact_1215_sources__def,axiom,
! [T: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [A4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A4 ) ) ) ) ).
% sources_def
thf(fact_1216_targets__char,axiom,
! [T: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [B3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) @ B3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ) ) ) ).
% targets_char
thf(fact_1217_development__map__App__2,axiom,
! [T: lambda_lambda,U2: lambda_lambda,U: list_lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( reduct9208763521642324371opment @ U2 @ U )
=> ( reduct9208763521642324371opment @ ( lambda_App @ T @ U2 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ T ) ) @ U ) ) ) ) ).
% development_map_App_2
thf(fact_1218_development__map__App__1,axiom,
! [T: lambda_lambda,T6: list_lambda_lambda,U2: lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ T6 )
=> ( ( lambda_Arr @ U2 )
=> ( reduct9208763521642324371opment @ ( lambda_App @ T @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X2: lambda_lambda] : ( lambda_App @ X2 @ ( lambda_Src @ U2 ) )
@ T6 ) ) ) ) ).
% development_map_App_1
thf(fact_1219__092_060Lambda_062_Otargets__def,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B3 )
& ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B3 ) ) ) ) ).
% \<Lambda>.targets_def
thf(fact_1220_map__ident,axiom,
( ( map_la336867333220232516lambda
@ ^ [X2: lambda_lambda] : X2 )
= ( ^ [Xs4: list_lambda_lambda] : Xs4 ) ) ).
% map_ident
thf(fact_1221_cong__map__App2,axiom,
! [X: lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( lambda_Ide @ X )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ X )
@ T6 )
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ X )
@ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ X )
@ U )
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ X )
@ T6 ) ) ) ) ) ) ).
% cong_map_App2
thf(fact_1222_Resid__single__ide_I1_J,axiom,
! [A: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda )
=> ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T6 )
@ ( image_3692823078739859851lambda
@ ^ [B3: lambda_lambda] : ( cons_lambda_lambda @ B3 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% Resid_single_ide(1)
thf(fact_1223_map__App2__Resid,axiom,
! [X: lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( lambda_Ide @ X )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U )
=> ( ( map_la336867333220232516lambda
@ ^ [T2: lambda_lambda] : ( lambda_App @ T2 @ X )
@ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T2: lambda_lambda] : ( lambda_App @ T2 @ X )
@ T6 )
@ ( map_la336867333220232516lambda
@ ^ [T2: lambda_lambda] : ( lambda_App @ T2 @ X )
@ U ) ) ) ) ) ).
% map_App2_Resid
thf(fact_1224_sources__char_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( collec8830145823237851355lambda
@ ^ [A6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A6 )
& ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ A6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% sources_char\<^sub>P
thf(fact_1225_img__Resid__Srcs,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ord_le3064231726521830256lambda
@ ( image_3692823078739859851lambda
@ ^ [A4: lambda_lambda] : ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A4 @ nil_lambda_lambda ) @ T6 )
@ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
@ ( image_3692823078739859851lambda
@ ^ [B3: lambda_lambda] : ( cons_lambda_lambda @ B3 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ) ).
% img_Resid_Srcs
thf(fact_1226_targets__char_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( collec8830145823237851355lambda
@ ^ [B6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ B6 )
& ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ B6 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% targets_char\<^sub>P
thf(fact_1227__092_060Lambda_062_OconI,axiom,
! [T: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ T @ U2 )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ).
% \<Lambda>.conI
thf(fact_1228_sources__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( collec8830145823237851355lambda
@ ^ [A6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A6 @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ A6 ) ) ) ) ) ).
% sources_char\<^sub>\<Lambda>\<^sub>P
thf(fact_1229_targets__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( collec8830145823237851355lambda
@ ^ [B6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B6 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ B6 ) ) ) ) ) ).
% targets_char\<^sub>\<Lambda>\<^sub>P
thf(fact_1230_orthogonal__App__Arr__Arr,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) ) @ U )
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Src @ ( hd_lambda_lambda @ U ) ) )
@ T6 ) )
= ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) ) @ U ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Src @ ( hd_lambda_lambda @ U ) ) )
@ T6 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) ) @ U ) )
= ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Trg @ ( last_lambda_lambda @ U ) ) )
@ T6 ) ) ) ) ) ).
% orthogonal_App_Arr_Arr
thf(fact_1231_orthogonal__App__cong,axiom,
! [T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Src @ ( hd_lambda_lambda @ U ) ) )
@ T6 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) ) @ U ) )
@ ( append_lambda_lambda @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) ) @ U )
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Trg @ ( last_lambda_lambda @ U ) ) )
@ T6 ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) ) @ U )
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Trg @ ( last_lambda_lambda @ U ) ) )
@ T6 ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X6: lambda_lambda] : ( lambda_App @ X6 @ ( lambda_Src @ ( hd_lambda_lambda @ U ) ) )
@ T6 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) ) @ U ) ) ) ) ) ) ) ).
% orthogonal_App_cong
thf(fact_1232_map__App__map__un__App1,axiom,
! [U: list_lambda_lambda,B: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( lambda_Ide @ B )
=> ( ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ lambda_un_App2 @ ( set_lambda_lambda2 @ U ) ) @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
=> ( ( map_la336867333220232516lambda
@ ^ [T2: lambda_lambda] : ( lambda_App @ T2 @ B )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U ) )
= U ) ) ) ) ) ).
% map_App_map_un_App1
thf(fact_1233_lambda__calculus_Otargets__simps_I2_J,axiom,
! [X: nat] :
( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Var @ X ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.targets_simps(2)
thf(fact_1234_lambda__calculus_Osources__simps_I2_J,axiom,
! [X: nat] :
( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Var @ X ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.sources_simps(2)
thf(fact_1235_lambda__calculus_Olambda_Odisc_I12_J,axiom,
! [X23: nat] :
~ ( lambda_is_App @ ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.disc(12)
thf(fact_1236_lambda__calculus_Oelementary__reduction_Osimps_I2_J,axiom,
! [Uu2: nat] :
~ ( lambda2670285437103676873uction @ ( lambda_Var @ Uu2 ) ) ).
% lambda_calculus.elementary_reduction.simps(2)
thf(fact_1237_lambda__calculus_Olambda_Odisc__eq__case_I4_J,axiom,
( lambda_is_App
= ( lambda_case_lambda_o @ $false
@ ^ [Uu3: nat] : $false
@ ^ [Uu3: lambda_lambda] : $false
@ ^ [Uu3: lambda_lambda,Uv2: lambda_lambda] : $true
@ ^ [Uu3: lambda_lambda,Uv2: lambda_lambda] : $false ) ) ).
% lambda_calculus.lambda.disc_eq_case(4)
thf(fact_1238_lambda__calculus_Osseq_Osimps_I2_J,axiom,
! [Uv: nat,Uw: nat] :
~ ( lambda_sseq @ ( lambda_Var @ Uv ) @ ( lambda_Var @ Uw ) ) ).
% lambda_calculus.sseq.simps(2)
thf(fact_1239_lambda__calculus_Olambda_Odisc_I17_J,axiom,
! [X23: nat] :
~ ( lambda_is_Beta @ ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.disc(17)
thf(fact_1240_lambda__calculus_Olambda_Odisc_I2_J,axiom,
! [X23: nat] : ( lambda_is_Var @ ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.disc(2)
thf(fact_1241_lambda__calculus_Olambda_OdiscI_I1_J,axiom,
! [Lambda: lambda_lambda,X23: nat] :
( ( Lambda
= ( lambda_Var @ X23 ) )
=> ( lambda_is_Var @ Lambda ) ) ).
% lambda_calculus.lambda.discI(1)
thf(fact_1242_lambda__calculus_Ois__Var__def,axiom,
( lambda_is_Var
= ( ^ [Lambda2: lambda_lambda] :
? [X24: nat] :
( Lambda2
= ( lambda_Var @ X24 ) ) ) ) ).
% lambda_calculus.is_Var_def
thf(fact_1243_lambda__calculus_Olambda_Osimps_I26_J,axiom,
! [F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o,X23: nat] :
( ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ ( lambda_Var @ X23 ) )
= ( F2 @ X23 ) ) ).
% lambda_calculus.lambda.simps(26)
thf(fact_1244_lambda__calculus_Ocontains__head__reduction_Osimps_I1_J,axiom,
! [Uu2: nat] :
~ ( lambda8135948770199036598uction @ ( lambda_Var @ Uu2 ) ) ).
% lambda_calculus.contains_head_reduction.simps(1)
thf(fact_1245_lambda__calculus_Olambda_Odisc__eq__case_I5_J,axiom,
( lambda_is_Beta
= ( lambda_case_lambda_o @ $false
@ ^ [Uu3: nat] : $false
@ ^ [Uu3: lambda_lambda] : $false
@ ^ [Uu3: lambda_lambda,Uv2: lambda_lambda] : $false
@ ^ [Uu3: lambda_lambda,Uv2: lambda_lambda] : $true ) ) ).
% lambda_calculus.lambda.disc_eq_case(5)
thf(fact_1246_lambda__calculus_Olambda_Odisc__eq__case_I2_J,axiom,
( lambda_is_Var
= ( lambda_case_lambda_o @ $false
@ ^ [Uu3: nat] : $true
@ ^ [Uu3: lambda_lambda] : $false
@ ^ [Uu3: lambda_lambda,Uv2: lambda_lambda] : $false
@ ^ [Uu3: lambda_lambda,Uv2: lambda_lambda] : $false ) ) ).
% lambda_calculus.lambda.disc_eq_case(2)
thf(fact_1247_lambda__calculus_Olambda_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F2: nat > $o,F3: lambda_lambda > $o,F4: lambda_lambda > lambda_lambda > $o,F5: lambda_lambda > lambda_lambda > $o,Lambda: lambda_lambda] :
( ( H @ ( lambda_case_lambda_o @ F1 @ F2 @ F3 @ F4 @ F5 @ Lambda ) )
= ( lambda_case_lambda_o @ ( H @ F1 )
@ ^ [X2: nat] : ( H @ ( F2 @ X2 ) )
@ ^ [X2: lambda_lambda] : ( H @ ( F3 @ X2 ) )
@ ^ [X12: lambda_lambda,X24: lambda_lambda] : ( H @ ( F4 @ X12 @ X24 ) )
@ ^ [X12: lambda_lambda,X24: lambda_lambda] : ( H @ ( F5 @ X12 @ X24 ) )
@ Lambda ) ) ).
% lambda_calculus.lambda.case_distrib
thf(fact_1248_lambda__calculus_OTrg_Osimps_I1_J,axiom,
! [I: nat] :
( ( lambda_Trg @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Trg.simps(1)
thf(fact_1249_lambda__calculus_OSrc_Osimps_I2_J,axiom,
! [I: nat] :
( ( lambda_Src @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Src.simps(2)
thf(fact_1250_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_1251_lambda__calculus_OArr_Osimps_I2_J,axiom,
! [Uu2: nat] : ( lambda_Arr @ ( lambda_Var @ Uu2 ) ) ).
% lambda_calculus.Arr.simps(2)
thf(fact_1252_lambda__calculus_OIde_Osimps_I2_J,axiom,
! [Uu2: nat] : ( lambda_Ide @ ( lambda_Var @ Uu2 ) ) ).
% lambda_calculus.Ide.simps(2)
thf(fact_1253_lambda__calculus_Olambda_Odistinct_I9_J,axiom,
! [X23: nat,X32: lambda_lambda] :
( ( lambda_Var @ X23 )
!= ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.distinct(9)
thf(fact_1254_lambda__calculus_Olambda_Odistinct_I13_J,axiom,
! [X23: nat,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Var @ X23 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(13)
thf(fact_1255_lambda__calculus_Olambda_Osize_I7_J,axiom,
! [X23: nat] :
( ( size_s1768714712973771222lambda @ ( lambda_Var @ X23 ) )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size(7)
thf(fact_1256_lambda__calculus_Olambda_Odistinct_I11_J,axiom,
! [X23: nat,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Var @ X23 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(11)
thf(fact_1257_list_Omap__ident,axiom,
! [T: list_lambda_lambda] :
( ( map_la336867333220232516lambda
@ ^ [X2: lambda_lambda] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_1258_lambda__calculus_Obottom__up__redex_Osimps_I2_J,axiom,
! [X: nat] :
( ( lambda5654762366400029632_redex @ ( lambda_Var @ X ) )
= ( lambda_Var @ X ) ) ).
% lambda_calculus.bottom_up_redex.simps(2)
thf(fact_1259_lambda__calculus_Ohead__redex_Osimps_I2_J,axiom,
! [X: nat] :
( ( lambda_head_redex @ ( lambda_Var @ X ) )
= ( lambda_Var @ X ) ) ).
% lambda_calculus.head_redex.simps(2)
thf(fact_1260_reduction__paths_Ostandard__development_Osimps_I2_J,axiom,
! [Uu2: nat] :
( ( reduct1603535164923377302opment @ ( lambda_Var @ Uu2 ) )
= nil_lambda_lambda ) ).
% reduction_paths.standard_development.simps(2)
thf(fact_1261_lambda__calculus_Ohgt_Osimps_I2_J,axiom,
! [Uu2: nat] :
( ( lambda_hgt @ ( lambda_Var @ Uu2 ) )
= zero_zero_nat ) ).
% lambda_calculus.hgt.simps(2)
thf(fact_1262_lambda__calculus_OApp__Var__contains__no__head__reduction,axiom,
! [X: nat,U2: lambda_lambda] :
~ ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Var @ X ) @ U2 ) ) ).
% lambda_calculus.App_Var_contains_no_head_reduction
thf(fact_1263_lambda__calculus_Osseq_Osimps_I14_J,axiom,
! [Va: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_Lam @ Va ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(14)
thf(fact_1264_lambda__calculus_Osseq_Osimps_I20_J,axiom,
! [Va: nat,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Var @ Va ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(20)
thf(fact_1265_lambda__calculus_Osseq_Osimps_I29_J,axiom,
! [Vb2: nat,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Var @ Vb2 ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(29)
thf(fact_1266_lambda__calculus_Osseq_Osimps_I16_J,axiom,
! [Vc2: nat,Vb2: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Var @ Vc2 ) @ Vb2 ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(16)
thf(fact_1267_lambda__calculus_Osseq_Osimps_I18_J,axiom,
! [Vc2: lambda_lambda,Vd: lambda_lambda,Vb2: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vc2 @ Vd ) @ Vb2 ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(18)
thf(fact_1268_lambda__calculus_Osseq_Osimps_I26_J,axiom,
! [Vb2: nat,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Var @ Vb2 ) @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(26)
thf(fact_1269_lambda__calculus_Oprfx__Var__iff,axiom,
! [U2: lambda_lambda,I: nat] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ ( lambda_Var @ I ) ) )
= ( U2
= ( lambda_Var @ I ) ) ) ).
% lambda_calculus.prfx_Var_iff
thf(fact_1270_lambda__calculus_Olambda_Odisc_I7_J,axiom,
! [X23: nat] :
~ ( lambda_is_Lam @ ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.disc(7)
thf(fact_1271_lambda__calculus_Olambda_Odisc__eq__case_I3_J,axiom,
( lambda_is_Lam
= ( lambda_case_lambda_o @ $false
@ ^ [Uu3: nat] : $false
@ ^ [Uu3: lambda_lambda] : $true
@ ^ [Uu3: lambda_lambda,Uv2: lambda_lambda] : $false
@ ^ [Uu3: lambda_lambda,Uv2: lambda_lambda] : $false ) ) ).
% lambda_calculus.lambda.disc_eq_case(3)
thf(fact_1272_set__n__lists,axiom,
! [N: nat,Xs: list_lambda_lambda] :
( ( set_li3846194476672328965lambda @ ( n_list4163717777911954552lambda @ N @ Xs ) )
= ( collec8830145823237851355lambda
@ ^ [Ys3: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Ys3 )
= N )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Ys3 ) @ ( set_lambda_lambda2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1273_set__n__lists,axiom,
! [N: nat,Xs: list_l5168835686021156086lambda] :
( ( set_li6257665199064048907lambda @ ( n_list7041979136695072894lambda @ N @ Xs ) )
= ( collec4999033578922223073lambda
@ ^ [Ys3: list_l5168835686021156086lambda] :
( ( ( size_s6298197441921507170lambda @ Ys3 )
= N )
& ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Ys3 ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1274_lambda__calculus_OIde_Oelims_I2_J,axiom,
! [X: lambda_lambda] :
( ( lambda_Ide @ X )
=> ( ! [Uu: nat] :
( X
!= ( lambda_Var @ Uu ) )
=> ( ! [T4: lambda_lambda] :
( ( X
= ( lambda_Lam @ T4 ) )
=> ~ ( lambda_Ide @ T4 ) )
=> ~ ! [T4: lambda_lambda,U5: lambda_lambda] :
( ( X
= ( lambda_App @ T4 @ U5 ) )
=> ~ ( ( lambda_Ide @ T4 )
& ( lambda_Ide @ U5 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(2)
% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ~ ( lambda_Arr @ ( lambda_Beta @ t1 @ t2 ) )
| ( reduct9208763521642324371opment @ ( lambda_Beta @ t1 @ t2 ) @ ( reduct1603535164923377302opment @ ( lambda_Beta @ t1 @ t2 ) ) ) ) ).
%------------------------------------------------------------------------------