TPTP Problem File: SLH0581^1.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_08400_388016__15030366_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1424 ( 479 unt; 151 typ; 0 def)
% Number of atoms : 4165 (1953 equ; 0 cnn)
% Maximal formula atoms : 38 ( 3 avg)
% Number of connectives : 18615 ( 868 ~; 38 |; 587 &;15317 @)
% ( 0 <=>;1805 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 1058 (1058 >; 0 *; 0 +; 0 <<)
% Number of symbols : 147 ( 144 usr; 12 con; 0-4 aty)
% Number of variables : 3376 ( 110 ^;3076 !; 190 ?;3376 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:49:14.134
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
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% Explicit typings (144)
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con_li79537494059641180lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
con_li2822789971228234210lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > list_l5168835686021156086lambda > list_l5168835686021156086lambda > $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_Oide_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
ide_li2235071267601372570lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > list_l5168835686021156086lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
trg_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
trg_li817795487149935061lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Otrg_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
trg_li7687935573203771867lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > list_l5168835686021156086lambda > list_l5168835686021156086lambda ).
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_Ocoinitial_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
coinit255303819856200130lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > list_l5168835686021156086lambda > list_l5168835686021156086lambda > $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__LambdaCalculus__Olambda____calculus__Olambda,type,
join_o1386658102433174083lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_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_Ojoin__of_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
join_o5099642474212670159lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda > $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_Oseq_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
seq_li7066278307288228763lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > list_l5168835686021156086lambda > list_l5168835686021156086lambda > $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_ResiduatedTransitionSystem_Osimulation_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula5235733974575307523lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
simula1175017789036123529lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula9164532646813739645lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
simula3069223088535412739lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
simula3219528783544830217lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > ( list_lambda_lambda > list_l5168835686021156086lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
simula4114611666396501757lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_l5168835686021156086lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J,type,
simula1230958359699143427lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > ( list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_J,type,
simula2627686087595152009lambda: ( list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda ) > ( list_l4928420112573288188lambda > list_l4928420112573288188lambda > list_l4928420112573288188lambda ) > ( list_l5168835686021156086lambda > list_l4928420112573288188lambda ) > $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_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_Oinsert_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
insert_lambda_lambda: lambda_lambda > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
insert8465602035467394112lambda: list_lambda_lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda ).
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_v_M____,type,
m: lambda_lambda ).
thf(sy_v_N____,type,
n: lambda_lambda ).
thf(sy_v_Ua____,type,
ua: list_lambda_lambda ).
thf(sy_v_u____,type,
u: lambda_lambda ).
% Relevant facts (1263)
thf(fact_0_finite__developments,axiom,
! [T: lambda_lambda] : ( reduction_FD @ T ) ).
% finite_developments
thf(fact_1__092_060Lambda_062_Ocube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.cube
thf(fact_2__092_060Lambda_062_Oex__un__null,axiom,
? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( lambda_resid @ X @ T2 )
= X )
& ( ( lambda_resid @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( lambda_resid @ Y @ T3 )
= Y )
& ( ( lambda_resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% \<Lambda>.ex_un_null
thf(fact_3_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_4_U,axiom,
ua != nil_lambda_lambda ).
% U
thf(fact_5__C3_C,axiom,
lambda8135948770199036598uction @ u ).
% "3"
thf(fact_6__C1_C,axiom,
~ ( lambda_Ide @ ( lambda_App @ m @ n ) ) ).
% "1"
thf(fact_7__C2_C,axiom,
~ ( lambda8135948770199036598uction @ ( lambda_App @ m @ n ) ) ).
% "2"
thf(fact_8_Std,axiom,
reduction_Std @ ( cons_lambda_lambda @ u @ ua ) ).
% Std
thf(fact_9__C4_C,axiom,
~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) ) ).
% "4"
thf(fact_10_paths__in__confluent__rts__axioms,axiom,
paths_9040249923598999359lambda @ lambda_resid ).
% paths_in_confluent_rts_axioms
thf(fact_11_complete__development__cons,axiom,
! [T: lambda_lambda,U: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ ( cons_lambda_lambda @ U @ U2 ) )
=> ( reduct7282684938351565978opment @ ( lambda_resid @ T @ U ) @ U2 ) ) ).
% complete_development_cons
thf(fact_12__092_060Lambda_062_Opartial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% \<Lambda>.partial_magma_axioms
thf(fact_13__C_K_K_C,axiom,
seq_lambda_lambda @ lambda_resid @ ( lambda_App @ m @ n ) @ u ).
% "**"
thf(fact_14_MN,axiom,
( ( lambda_Arr @ m )
& ( lambda_Arr @ n ) ) ).
% MN
thf(fact_15_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_16_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_17__C_K_C,axiom,
~ ( ( lambda_is_Lam @ m )
& ( lambda_is_Beta @ u ) ) ).
% "*"
thf(fact_18_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_19_seq,axiom,
seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_App @ m @ n ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ u @ ua ) ).
% seq
thf(fact_20_u,axiom,
lambda_Arr @ u ).
% u
thf(fact_21_Resid1x_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) ) ).
% Resid1x.simps(3)
thf(fact_22_paths__in__weakly__extensional__rts__axioms,axiom,
paths_4652746547811739206lambda @ lambda_resid ).
% paths_in_weakly_extensional_rts_axioms
thf(fact_23_ex__un__null,axiom,
? [X: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X @ T2 )
= X )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ X )
= X ) )
& ! [Y: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ Y @ T3 )
= Y )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% ex_un_null
thf(fact_24_cube,axiom,
! [V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ).
% cube
thf(fact_25_Con__sym,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 )
!= nil_lambda_lambda ) ) ).
% Con_sym
thf(fact_26_Resid_Osimps_I1_J,axiom,
! [Uu: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ nil_lambda_lambda @ Uu )
= nil_lambda_lambda ) ).
% Resid.simps(1)
thf(fact_27_Cube_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U2 ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V2 ) )
!= nil_lambda_lambda ) ) ).
% Cube(1)
thf(fact_28_Cube_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V2 ) ) ) ) ).
% Cube(2)
thf(fact_29_complete__development__Ide__iff,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ U2 )
=> ( ( lambda_Ide @ T )
= ( U2 = nil_lambda_lambda ) ) ) ).
% complete_development_Ide_iff
thf(fact_30_Con__initial__right,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ).
% Con_initial_right
thf(fact_31_Con__initial__left,axiom,
! [T: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ).
% Con_initial_left
thf(fact_32_Resid__rec_I2_J,axiom,
! [T4: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ).
% Resid_rec(2)
thf(fact_33_Resid__rec_I3_J,axiom,
! [U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U2 ) ) ) ) ).
% Resid_rec(3)
thf(fact_34_Resid__cons_I2_J,axiom,
! [U2: list_lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 ) ) ) ) ).
% Resid_cons(2)
thf(fact_35_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_36_Con__cons_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_cons(1)
thf(fact_37_Con__cons_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_cons(2)
thf(fact_38_Resid1x_Osimps_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( lambda_resid @ T @ U ) ) ).
% Resid1x.simps(2)
thf(fact_39_Trg__resid__sym,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) ) ) ).
% Trg_resid_sym
thf(fact_40_Resid__cons_H,axiom,
! [T4: list_lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% Resid_cons'
thf(fact_41_Resid1x__as__Resid,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) ) ) ).
% Resid1x_as_Resid
thf(fact_42_Resid__rec_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) ) ).
% Resid_rec(1)
thf(fact_43_Con__consI_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_consI(1)
thf(fact_44_mem__Collect__eq,axiom,
! [A: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( member2667661554319814297lambda @ A @ ( collec8830145823237851355lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ A @ ( collec621444948832755541lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( collec8830145823237851355lambda
@ ^ [X2: list_lambda_lambda] : ( member2667661554319814297lambda @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X2: lambda_lambda] : ( member_lambda_lambda @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__cong,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ! [X: list_lambda_lambda] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec8830145823237851355lambda @ P )
= ( collec8830145823237851355lambda @ Q ) ) ) ).
% Collect_cong
thf(fact_49_Collect__cong,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ! [X: lambda_lambda] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec621444948832755541lambda @ P )
= ( collec621444948832755541lambda @ Q ) ) ) ).
% Collect_cong
thf(fact_50_Con__consI_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_consI(2)
thf(fact_51_complete__development__bottom__up__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( reduct7282684938351565978opment @ T @ ( reduct6613843825294780173opment @ T ) ) ) ).
% complete_development_bottom_up_development
thf(fact_52_is__confluent__rts,axiom,
conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% is_confluent_rts
thf(fact_53_reduction__paths_OStd_Osimps_I1_J,axiom,
reduction_Std @ nil_lambda_lambda ).
% reduction_paths.Std.simps(1)
thf(fact_54_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_55_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_56_transpose_Ocases,axiom,
! [X3: list_l4928420112573288188lambda] :
( ( X3 != nil_li6069905736445159420lambda )
=> ( ! [Xss: list_l4928420112573288188lambda] :
( X3
!= ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ Xss ) )
=> ~ ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Xss: list_l4928420112573288188lambda] :
( X3
!= ( cons_l4910729503725484972lambda @ ( cons_l4558873032622161958lambda @ X @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_57_transpose_Ocases,axiom,
! [X3: list_l5168835686021156086lambda] :
( ( X3 != nil_li8379291187143545974lambda )
=> ( ! [Xss: list_l5168835686021156086lambda] :
( X3
!= ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ Xss ) )
=> ~ ! [X: lambda_lambda,Xs: list_lambda_lambda,Xss: list_l5168835686021156086lambda] :
( X3
!= ( cons_l4558873032622161958lambda @ ( cons_lambda_lambda @ X @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_58_lambda__calculus_OIde__implies__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Ide_implies_Arr
thf(fact_59_lambda__calculus_Ocontains__head__reduction_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( ( lambda8135948770199036598uction @ ( lambda_App @ V @ Va ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(6)
thf(fact_60_list__nonempty__induct,axiom,
! [Xs2: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
=> ( ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ( P @ Xs )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_61_list__nonempty__induct,axiom,
! [Xs2: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( Xs2 != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ( ! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( P @ Xs )
=> ( P @ ( cons_lambda_lambda @ X @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_62_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_l5168835686021156086lambda > $o,Xs2: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X: lambda_lambda,Xs: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ Xs ) @ nil_li8379291187143545974lambda )
=> ( ! [Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_lambda_lambda @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) )
=> ( ! [X: lambda_lambda,Xs: list_lambda_lambda,Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs ) @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_63_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_lambda_lambda > $o,Xs2: list_l5168835686021156086lambda,Ys: list_lambda_lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ Xs ) @ nil_lambda_lambda )
=> ( ! [Y2: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_lambda_lambda @ Y2 @ Ys2 ) )
=> ( ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Y2: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs ) @ ( cons_lambda_lambda @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_64_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o,Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ Xs ) @ nil_li8379291187143545974lambda )
=> ( ! [Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) )
=> ( ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Y2: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs ) @ ( cons_l4558873032622161958lambda @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_65_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_lambda_lambda > $o,Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ Xs ) @ nil_lambda_lambda )
=> ( ! [Y2: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_lambda_lambda @ ( cons_lambda_lambda @ Y2 @ Ys2 ) )
=> ( ! [X: lambda_lambda,Xs: list_lambda_lambda,Y2: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs ) @ ( cons_lambda_lambda @ Y2 @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_66_neq__Nil__conv,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
= ( ? [Y3: list_lambda_lambda,Ys3: list_l5168835686021156086lambda] :
( Xs2
= ( cons_l4558873032622161958lambda @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_67_neq__Nil__conv,axiom,
! [Xs2: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
= ( ? [Y3: lambda_lambda,Ys3: list_lambda_lambda] :
( Xs2
= ( cons_lambda_lambda @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_68_remdups__adj_Ocases,axiom,
! [X3: list_l5168835686021156086lambda] :
( ( X3 != nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda] :
( X3
!= ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
=> ~ ! [X: list_lambda_lambda,Y2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( X3
!= ( cons_l4558873032622161958lambda @ X @ ( cons_l4558873032622161958lambda @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_69_remdups__adj_Ocases,axiom,
! [X3: list_lambda_lambda] :
( ( X3 != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] :
( X3
!= ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ~ ! [X: lambda_lambda,Y2: lambda_lambda,Xs: list_lambda_lambda] :
( X3
!= ( cons_lambda_lambda @ X @ ( cons_lambda_lambda @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_70_list_Oexhaust,axiom,
! [Y4: list_l5168835686021156086lambda] :
( ( Y4 != nil_li8379291187143545974lambda )
=> ~ ! [X212: list_lambda_lambda,X222: list_l5168835686021156086lambda] :
( Y4
!= ( cons_l4558873032622161958lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_71_list_Oexhaust,axiom,
! [Y4: list_lambda_lambda] :
( ( Y4 != nil_lambda_lambda )
=> ~ ! [X212: lambda_lambda,X222: list_lambda_lambda] :
( Y4
!= ( cons_lambda_lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_72_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_73_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_74_list_Odistinct_I1_J,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( nil_li8379291187143545974lambda
!= ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_75_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_76_lambda__calculus_OArr_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_App @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(4)
thf(fact_77_lambda__calculus_OIde_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Ide @ ( lambda_App @ T @ U ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.Ide.simps(4)
thf(fact_78_lambda__calculus_OArr__head__strategy,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_head_strategy @ T ) ) ) ).
% lambda_calculus.Arr_head_strategy
thf(fact_79_reduction__paths_Ostandardize_Ocases,axiom,
! [X3: list_lambda_lambda] :
( ( X3 != nil_lambda_lambda )
=> ~ ! [V3: lambda_lambda,Va2: list_lambda_lambda] :
( X3
!= ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ).
% reduction_paths.standardize.cases
thf(fact_80_reduction__paths_OStd_Ocases,axiom,
! [X3: list_lambda_lambda] :
( ( X3 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( X3
!= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( X3
!= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ).
% reduction_paths.Std.cases
thf(fact_81_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_82_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_83_not__Cons__self2,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( cons_l4558873032622161958lambda @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_84_not__Cons__self2,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( cons_lambda_lambda @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_85_partial__magma__axioms,axiom,
partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% partial_magma_axioms
thf(fact_86_stdz__insert__Beta,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_is_Beta @ T )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( lambda_is_Beta @ ( hd_lambda_lambda @ ( reduct5390047737487997688insert @ T @ U2 ) ) ) ) ) ).
% stdz_insert_Beta
thf(fact_87_rts__with__composites__axioms,axiom,
rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% rts_with_composites_axioms
thf(fact_88_paths__in__weakly__extensional__rts_OTrg__resid__sym,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( ( paths_3312059784463760222lambda @ Resid @ T4 @ U2 )
!= nil_li8379291187143545974lambda )
=> ( ( paths_4823109512173946681lambda @ Resid @ ( paths_3312059784463760222lambda @ Resid @ T4 @ U2 ) )
= ( paths_4823109512173946681lambda @ Resid @ ( paths_3312059784463760222lambda @ Resid @ U2 @ T4 ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg_resid_sym
thf(fact_89_paths__in__weakly__extensional__rts_OTrg__resid__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ U2 @ T4 ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg_resid_sym
thf(fact_90_complete__development__cong,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ U2 )
=> ( ~ ( lambda_Ide @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).
% complete_development_cong
thf(fact_91_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_92_composable__imp__seq,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% composable_imp_seq
thf(fact_93_has__composites,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% has_composites
thf(fact_94_paths__in__weakly__extensional__rts_OTrg_Osimps_I3_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda,Va: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_4823109512173946681lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ ( cons_l4558873032622161958lambda @ V @ Va ) ) )
= ( paths_4823109512173946681lambda @ Resid @ ( cons_l4558873032622161958lambda @ V @ Va ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.simps(3)
thf(fact_95_paths__in__weakly__extensional__rts_OTrg_Osimps_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_2695323870737867955lambda @ Resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( paths_2695323870737867955lambda @ Resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.simps(3)
thf(fact_96_paths__in__weakly__extensional__rts_OTrg_Ocases,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,X3: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( X3 != nil_li8379291187143545974lambda )
=> ( ! [T3: list_lambda_lambda] :
( X3
!= ( cons_l4558873032622161958lambda @ T3 @ nil_li8379291187143545974lambda ) )
=> ~ ! [T3: list_lambda_lambda,V3: list_lambda_lambda,Va2: list_l5168835686021156086lambda] :
( X3
!= ( cons_l4558873032622161958lambda @ T3 @ ( cons_l4558873032622161958lambda @ V3 @ Va2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.cases
thf(fact_97_paths__in__weakly__extensional__rts_OTrg_Ocases,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,X3: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( X3 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( X3
!= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( X3
!= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.cases
thf(fact_98_cong__symmetric,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ).
% cong_symmetric
thf(fact_99_cong__transitive,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) )
=> ( ( 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_100_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_101_prfx__transitive,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) ) ) ) ).
% prfx_transitive
thf(fact_102_cong__respects__seq_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ) ) ).
% cong_respects_seq\<^sub>P
thf(fact_103_cong__respects__seq,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,T6: list_lambda_lambda,U4: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U4 ) ) ) ) ).
% cong_respects_seq
thf(fact_104_complete__developments__cong,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ~ ( lambda_Ide @ T )
=> ( ( reduct7282684938351565978opment @ T @ U2 )
=> ( ( reduct7282684938351565978opment @ T @ V2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U2 ) ) ) ) ) ) ).
% complete_developments_cong
thf(fact_105_rts__with__composites_Ohas__composites,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( seq_lambda_lambda @ Resid @ T @ U )
=> ( compos7127019122090555255lambda @ Resid @ T @ U ) ) ) ).
% rts_with_composites.has_composites
thf(fact_106_rts__with__composites_Ohas__composites,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ Resid @ T @ U )
=> ( compos3359172601056258045lambda @ Resid @ T @ U ) ) ) ).
% rts_with_composites.has_composites
thf(fact_107_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_108_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_109_residuation_Oide_Ocong,axiom,
ide_li6773656263178250516lambda = ide_li6773656263178250516lambda ).
% residuation.ide.cong
thf(fact_110_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_111_rts_Ocomposable_Ocong,axiom,
compos3359172601056258045lambda = compos3359172601056258045lambda ).
% rts.composable.cong
thf(fact_112_rts_Ocomposable_Ocong,axiom,
compos7127019122090555255lambda = compos7127019122090555255lambda ).
% rts.composable.cong
thf(fact_113_list_Osel_I1_J,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( hd_lis2183642875561147995lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_114_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_115_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_116_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_117_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_118_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_119_paths__in__rts_OResid_Ocong,axiom,
paths_3312059784463760222lambda = paths_3312059784463760222lambda ).
% paths_in_rts.Resid.cong
thf(fact_120_paths__in__rts_OResid_Ocong,axiom,
paths_8607357831768207192lambda = paths_8607357831768207192lambda ).
% paths_in_rts.Resid.cong
thf(fact_121_rts_Oseq_Ocong,axiom,
seq_lambda_lambda = seq_lambda_lambda ).
% rts.seq.cong
thf(fact_122_rts_Oseq_Ocong,axiom,
seq_li8093550137657879445lambda = seq_li8093550137657879445lambda ).
% rts.seq.cong
thf(fact_123_paths__in__rts_OResid1x_Ocong,axiom,
paths_3319247187298047121lambda = paths_3319247187298047121lambda ).
% paths_in_rts.Resid1x.cong
thf(fact_124_paths__in__weakly__extensional__rts_OTrg_Ocong,axiom,
paths_4823109512173946681lambda = paths_4823109512173946681lambda ).
% paths_in_weakly_extensional_rts.Trg.cong
thf(fact_125_paths__in__weakly__extensional__rts_OTrg_Ocong,axiom,
paths_2695323870737867955lambda = paths_2695323870737867955lambda ).
% paths_in_weakly_extensional_rts.Trg.cong
thf(fact_126_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X2: lambda_lambda] :
( ! [T7: lambda_lambda] :
( ( ( OP @ X2 @ T7 )
= X2 )
& ( ( OP @ T7 @ X2 )
= X2 ) )
& ! [Y3: lambda_lambda] :
( ! [T7: lambda_lambda] :
( ( ( OP @ Y3 @ T7 )
= Y3 )
& ( ( OP @ T7 @ Y3 )
= Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_127_partial__magma__def,axiom,
( partia748546660539056525lambda
= ( ^ [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
? [X2: list_lambda_lambda] :
( ! [T7: list_lambda_lambda] :
( ( ( OP @ X2 @ T7 )
= X2 )
& ( ( OP @ T7 @ X2 )
= X2 ) )
& ! [Y3: list_lambda_lambda] :
( ! [T7: list_lambda_lambda] :
( ( ( OP @ Y3 @ T7 )
= Y3 )
& ( ( OP @ T7 @ Y3 )
= Y3 ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% partial_magma_def
thf(fact_128_partial__magma_Oex__un__null,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP2 @ X @ T2 )
= X )
& ( ( OP2 @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP2 @ Y @ T3 )
= Y )
& ( ( OP2 @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_129_partial__magma_Oex__un__null,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ? [X: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP2 @ X @ T2 )
= X )
& ( ( OP2 @ T2 @ X )
= X ) )
& ! [Y: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP2 @ Y @ T3 )
= Y )
& ( ( OP2 @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_130_partial__magma_Ointro,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ? [X4: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP2 @ X4 @ T3 )
= X4 )
& ( ( OP2 @ T3 @ X4 )
= X4 ) )
& ! [Y2: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP2 @ Y2 @ T2 )
= Y2 )
& ( ( OP2 @ T2 @ Y2 )
= Y2 ) )
=> ( Y2 = X4 ) ) )
=> ( partia3924075026459072775lambda @ OP2 ) ) ).
% partial_magma.intro
thf(fact_131_partial__magma_Ointro,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ? [X4: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP2 @ X4 @ T3 )
= X4 )
& ( ( OP2 @ T3 @ X4 )
= X4 ) )
& ! [Y2: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP2 @ Y2 @ T2 )
= Y2 )
& ( ( OP2 @ T2 @ Y2 )
= Y2 ) )
=> ( Y2 = X4 ) ) )
=> ( partia748546660539056525lambda @ OP2 ) ) ).
% partial_magma.intro
thf(fact_132_complete__development__def,axiom,
( reduct7282684938351565978opment
= ( ^ [T7: lambda_lambda,U5: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T7 @ U5 )
& ( ( lambda_Ide @ T7 )
| ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T7 @ nil_lambda_lambda ) @ U5 ) ) ) ) ) ) ).
% complete_development_def
thf(fact_133_B,axiom,
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ nil_lambda_lambda ) ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ m @ n ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ u @ nil_lambda_lambda ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ m @ n ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ u @ nil_lambda_lambda ) ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ nil_lambda_lambda ) ) ) ) ) ).
% B
thf(fact_134_cong__implies__coterminal,axiom,
! [U: list_lambda_lambda,U4: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U4 ) ) ).
% cong_implies_coterminal
thf(fact_135_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_136_stdz__insert__Ide__Std,axiom,
! [U: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ U )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ U2 )
=> ( ( reduction_Std @ U2 )
=> ( ( reduct5390047737487997688insert @ U @ U2 )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ U2 ) @ ( tl_lambda_lambda @ U2 ) ) ) ) ) ) ).
% stdz_insert_Ide_Std
thf(fact_137_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_138_prfx__decomp,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ 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 @ U @ T ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ).
% prfx_decomp
thf(fact_139_Std__consE,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ U2 ) )
=> ~ ( ( lambda_Arr @ T )
=> ( ( ( U2 != nil_lambda_lambda )
=> ( lambda_sseq @ T @ ( hd_lambda_lambda @ U2 ) ) )
=> ~ ( reduction_Std @ U2 ) ) ) ) ).
% Std_consE
thf(fact_140_Resid1x__as__Resid_H,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 )
= ( hd_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
= nil_lambda_lambda )
=> ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ).
% Resid1x_as_Resid'
thf(fact_141_cong__map__App1,axiom,
! [X3: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X3 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X3 ) @ T4 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X3 ) @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X3 ) @ U2 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X3 ) @ T4 ) ) ) ) ) ) ).
% cong_map_App1
thf(fact_142__092_060Lambda_062_Oweak__extensionality,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( T = U ) ) ) ) ).
% \<Lambda>.weak_extensionality
thf(fact_143__092_060Lambda_062_Oprfx__transitive,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) ) ) ) ).
% \<Lambda>.prfx_transitive
thf(fact_144__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_145__092_060Lambda_062_Ocong__transitive,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) ) ) ) ) ).
% \<Lambda>.cong_transitive
thf(fact_146__092_060Lambda_062_Ocong__symmetric,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ) ).
% \<Lambda>.cong_symmetric
thf(fact_147__092_060Lambda_062_Ocong__respects__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T6 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T6 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T6 @ U4 ) ) ) ) ).
% \<Lambda>.cong_respects_seq
thf(fact_148__092_060Lambda_062_Ocomposable__imp__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.composable_imp_seq
thf(fact_149__092_060Lambda_062_Onull__eqI,axiom,
! [N: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( lambda_resid @ N @ T3 )
= N )
& ( ( lambda_resid @ T3 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.null_eqI
thf(fact_150__092_060Lambda_062_Ocube__ax,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ).
% \<Lambda>.cube_ax
thf(fact_151__092_060Lambda_062_Ocon__sym__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ U @ T )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.con_sym_ax
thf(fact_152__092_060Lambda_062_Ocon__imp__arr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.con_imp_arr_resid
thf(fact_153_development__imp__Arr,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U2 )
=> ( lambda_Arr @ T ) ) ).
% development_imp_Arr
thf(fact_154_Std__standard__development,axiom,
! [T: lambda_lambda] : ( reduction_Std @ ( reduct1603535164923377302opment @ T ) ) ).
% Std_standard_development
thf(fact_155_development__Ide,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ T )
=> ( ( reduct9208763521642324371opment @ T @ U2 )
= ( U2 = nil_lambda_lambda ) ) ) ).
% development_Ide
thf(fact_156_Resid1x__ide,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_3319247187298047121lambda @ lambda_resid @ A @ T4 )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( paths_3319247187298047121lambda @ lambda_resid @ A @ T4 ) ) ) ) ).
% Resid1x_ide
thf(fact_157_Resid1x__null,axiom,
! [T4: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T4 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x_null
thf(fact_158_development__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( reduct9208763521642324371opment @ T @ ( reduct1603535164923377302opment @ T ) ) ) ).
% development_standard_development
thf(fact_159_Con__append_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V2 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(2)
thf(fact_160_Con__append_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T4 ) )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(1)
thf(fact_161_Resid__append__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V2 != nil_lambda_lambda )
=> ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V2 @ T4 ) @ U2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V2 ) )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V2 @ U2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 ) @ U2 )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V2 @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V2 @ T4 ) @ U2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V2 ) ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V2 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V2 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 ) @ U2 ) ) ) ) ) ) ) ).
% Resid_append_ind
thf(fact_162_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_163_Resid1x_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x.simps(1)
thf(fact_164_Trg_Osimps_I1_J,axiom,
( ( paths_2695323870737867955lambda @ lambda_resid @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Trg.simps(1)
thf(fact_165_development__append,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U2 )
=> ( ( reduct9208763521642324371opment @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 ) @ V2 )
=> ( reduct9208763521642324371opment @ T @ ( append_lambda_lambda @ U2 @ V2 ) ) ) ) ).
% development_append
thf(fact_166_same__append__eq,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs2 @ Ys )
= ( append1017360404025248779lambda @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_167_same__append__eq,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs2 @ Ys )
= ( append_lambda_lambda @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_168_append__same__eq,axiom,
! [Ys: list_l5168835686021156086lambda,Xs2: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Xs2 )
= ( append1017360404025248779lambda @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_169_append__same__eq,axiom,
! [Ys: list_lambda_lambda,Xs2: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Xs2 )
= ( append_lambda_lambda @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_170_append__assoc,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) @ Zs )
= ( append1017360404025248779lambda @ Xs2 @ ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_171_append__assoc,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) @ Zs )
= ( append_lambda_lambda @ Xs2 @ ( append_lambda_lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_172_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_173_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_174_Resid__rec_I4_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) ) ) ) ) ) ) ).
% Resid_rec(4)
thf(fact_175_Resid__cons_I1_J,axiom,
! [U2: list_lambda_lambda,T: lambda_lambda,T4: list_lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% Resid_cons(1)
thf(fact_176_Resid1x_Oelims,axiom,
! [X3: lambda_lambda,Xa: list_lambda_lambda,Y4: lambda_lambda] :
( ( ( paths_3319247187298047121lambda @ lambda_resid @ X3 @ Xa )
= Y4 )
=> ( ( ( Xa = nil_lambda_lambda )
=> ( Y4
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [U6: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U6 @ nil_lambda_lambda ) )
=> ( Y4
!= ( lambda_resid @ X3 @ U6 ) ) )
=> ~ ! [U6: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U6 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y4
!= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ X3 @ U6 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Resid1x.elims
thf(fact_177_lambda__calculus_Oprfx__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( lambda_Ide @ ( lambda_resid @ T @ U ) ) ) ).
% lambda_calculus.prfx_char
thf(fact_178_lambda__calculus_Oide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.ide_char
thf(fact_179_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_180_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_181_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_182_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_183_Nil__is__map__conv,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( nil_lambda_lambda
= ( map_li5201084558375397438lambda @ F @ Xs2 ) )
= ( Xs2 = nil_li8379291187143545974lambda ) ) ).
% Nil_is_map_conv
thf(fact_184_Nil__is__map__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs2: list_lambda_lambda] :
( ( nil_li8379291187143545974lambda
= ( map_la6434941737452557130lambda @ F @ Xs2 ) )
= ( Xs2 = nil_lambda_lambda ) ) ).
% Nil_is_map_conv
thf(fact_185_Nil__is__map__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( nil_li8379291187143545974lambda
= ( map_li385728805257398596lambda @ F @ Xs2 ) )
= ( Xs2 = nil_li8379291187143545974lambda ) ) ).
% Nil_is_map_conv
thf(fact_186_Nil__is__map__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( map_la336867333220232516lambda @ F @ Xs2 ) )
= ( Xs2 = nil_lambda_lambda ) ) ).
% Nil_is_map_conv
thf(fact_187_map__is__Nil__conv,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( ( map_li5201084558375397438lambda @ F @ Xs2 )
= nil_lambda_lambda )
= ( Xs2 = nil_li8379291187143545974lambda ) ) ).
% map_is_Nil_conv
thf(fact_188_map__is__Nil__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( map_la6434941737452557130lambda @ F @ Xs2 )
= nil_li8379291187143545974lambda )
= ( Xs2 = nil_lambda_lambda ) ) ).
% map_is_Nil_conv
thf(fact_189_map__is__Nil__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs2 )
= nil_li8379291187143545974lambda )
= ( Xs2 = nil_li8379291187143545974lambda ) ) ).
% map_is_Nil_conv
thf(fact_190_map__is__Nil__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs2 )
= nil_lambda_lambda )
= ( Xs2 = nil_lambda_lambda ) ) ).
% map_is_Nil_conv
thf(fact_191_append_Oright__neutral,axiom,
! [A: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ A @ nil_li8379291187143545974lambda )
= A ) ).
% append.right_neutral
thf(fact_192_append_Oright__neutral,axiom,
! [A: list_lambda_lambda] :
( ( append_lambda_lambda @ A @ nil_lambda_lambda )
= A ) ).
% append.right_neutral
thf(fact_193_append__Nil2,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ Xs2 @ nil_li8379291187143545974lambda )
= Xs2 ) ).
% append_Nil2
thf(fact_194_append__Nil2,axiom,
! [Xs2: list_lambda_lambda] :
( ( append_lambda_lambda @ Xs2 @ nil_lambda_lambda )
= Xs2 ) ).
% append_Nil2
thf(fact_195_append__self__conv,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_li8379291187143545974lambda ) ) ).
% append_self_conv
thf(fact_196_append__self__conv,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_lambda_lambda ) ) ).
% append_self_conv
thf(fact_197_self__append__conv,axiom,
! [Y4: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Y4
= ( append1017360404025248779lambda @ Y4 @ Ys ) )
= ( Ys = nil_li8379291187143545974lambda ) ) ).
% self_append_conv
thf(fact_198_self__append__conv,axiom,
! [Y4: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Y4
= ( append_lambda_lambda @ Y4 @ Ys ) )
= ( Ys = nil_lambda_lambda ) ) ).
% self_append_conv
thf(fact_199_append__self__conv2,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_li8379291187143545974lambda ) ) ).
% append_self_conv2
thf(fact_200_append__self__conv2,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_lambda_lambda ) ) ).
% append_self_conv2
thf(fact_201_self__append__conv2,axiom,
! [Y4: list_l5168835686021156086lambda,Xs2: list_l5168835686021156086lambda] :
( ( Y4
= ( append1017360404025248779lambda @ Xs2 @ Y4 ) )
= ( Xs2 = nil_li8379291187143545974lambda ) ) ).
% self_append_conv2
thf(fact_202_self__append__conv2,axiom,
! [Y4: list_lambda_lambda,Xs2: list_lambda_lambda] :
( ( Y4
= ( append_lambda_lambda @ Xs2 @ Y4 ) )
= ( Xs2 = nil_lambda_lambda ) ) ).
% self_append_conv2
thf(fact_203_Nil__is__append__conv,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( nil_li8379291187143545974lambda
= ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_li8379291187143545974lambda )
& ( Ys = nil_li8379291187143545974lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_204_Nil__is__append__conv,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_lambda_lambda )
& ( Ys = nil_lambda_lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_205_append__is__Nil__conv,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs2 @ Ys )
= nil_li8379291187143545974lambda )
= ( ( Xs2 = nil_li8379291187143545974lambda )
& ( Ys = nil_li8379291187143545974lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_206_append__is__Nil__conv,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs2 @ Ys )
= nil_lambda_lambda )
= ( ( Xs2 = nil_lambda_lambda )
& ( Ys = nil_lambda_lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_207_map__append,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( map_li5201084558375397438lambda @ F @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( append_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ Xs2 ) @ ( map_li5201084558375397438lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_208_map__append,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( map_la6434941737452557130lambda @ F @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( append1017360404025248779lambda @ ( map_la6434941737452557130lambda @ F @ Xs2 ) @ ( map_la6434941737452557130lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_209_map__append,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( map_li385728805257398596lambda @ F @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( append1017360404025248779lambda @ ( map_li385728805257398596lambda @ F @ Xs2 ) @ ( map_li385728805257398596lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_210_map__append,axiom,
! [F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( append_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs2 ) @ ( map_la336867333220232516lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_211__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_212__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_213_development__implies,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U2 )
=> ( ( reduct8314398545252454212n_path @ U2 )
& ( ( U2 != nil_lambda_lambda )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).
% development_implies
thf(fact_214_append1__eq__conv,axiom,
! [Xs2: list_l5168835686021156086lambda,X3: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Y4: list_lambda_lambda] :
( ( ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) )
= ( append1017360404025248779lambda @ Ys @ ( cons_l4558873032622161958lambda @ Y4 @ nil_li8379291187143545974lambda ) ) )
= ( ( Xs2 = Ys )
& ( X3 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_215_append1__eq__conv,axiom,
! [Xs2: list_lambda_lambda,X3: lambda_lambda,Ys: list_lambda_lambda,Y4: lambda_lambda] :
( ( ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) )
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ Y4 @ nil_lambda_lambda ) ) )
= ( ( Xs2 = Ys )
& ( X3 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_216_hd__append2,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Xs2 ) ) ) ).
% hd_append2
thf(fact_217_hd__append2,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( hd_lambda_lambda @ Xs2 ) ) ) ).
% hd_append2
thf(fact_218_tl__append2,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( tl_lis5905685823335986783lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( append1017360404025248779lambda @ ( tl_lis5905685823335986783lambda @ Xs2 ) @ Ys ) ) ) ).
% tl_append2
thf(fact_219_tl__append2,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( append_lambda_lambda @ ( tl_lambda_lambda @ Xs2 ) @ Ys ) ) ) ).
% tl_append2
thf(fact_220_Resid__single__ide_I2_J,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
= T4 ) ) ) ).
% Resid_single_ide(2)
thf(fact_221_list_Ocollapse,axiom,
! [List: list_l5168835686021156086lambda] :
( ( List != nil_li8379291187143545974lambda )
=> ( ( cons_l4558873032622161958lambda @ ( hd_lis2183642875561147995lambda @ List ) @ ( tl_lis5905685823335986783lambda @ List ) )
= List ) ) ).
% list.collapse
thf(fact_222_list_Ocollapse,axiom,
! [List: list_lambda_lambda] :
( ( List != nil_lambda_lambda )
=> ( ( cons_lambda_lambda @ ( hd_lambda_lambda @ List ) @ ( tl_lambda_lambda @ List ) )
= List ) ) ).
% list.collapse
thf(fact_223_hd__Cons__tl,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( cons_l4558873032622161958lambda @ ( hd_lis2183642875561147995lambda @ Xs2 ) @ ( tl_lis5905685823335986783lambda @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_224_hd__Cons__tl,axiom,
! [Xs2: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( cons_lambda_lambda @ ( hd_lambda_lambda @ Xs2 ) @ ( tl_lambda_lambda @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_225_Con__appendI_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V2 ) )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(2)
thf(fact_226_Con__appendI_I1_J,axiom,
! [T4: list_lambda_lambda,V2: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T4 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V2 )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(1)
thf(fact_227_Resid__append_I2_J,axiom,
! [U2: list_lambda_lambda,V2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( V2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V2 ) ) ) ) ) ).
% Resid_append(2)
thf(fact_228_Resid__append_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T4 ) ) ) ) ) ) ).
% Resid_append(1)
thf(fact_229_Resid__append2,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V2: list_lambda_lambda,W: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V2 != nil_lambda_lambda )
=> ( ( W != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V2 @ W ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V2 @ W ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 ) @ W ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ T4 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V2 ) ) ) ) ) ) ) ) ) ) ).
% Resid_append2
thf(fact_230_cong__append__ideI_I4_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ T4 @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ T4 ) ) ) ) ) ).
% cong_append_ideI(4)
thf(fact_231_cong__append__ideI_I3_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( append_lambda_lambda @ T4 @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ U2 ) ) ) ) ) ).
% cong_append_ideI(3)
thf(fact_232_cong__append__ideI_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ T4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ).
% cong_append_ideI(2)
thf(fact_233_cong__append__ideI_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ).
% cong_append_ideI(1)
thf(fact_234_cong__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ T5 @ U3 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T5 @ U3 ) @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ) ).
% cong_append
thf(fact_235_Trg__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ U2 ) ) ) ).
% Trg_append
thf(fact_236_cong__cons__ideI_I2_J,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ U2 ) ) ) ) ) ).
% cong_cons_ideI(2)
thf(fact_237_cong__cons__ideI_I1_J,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ).
% cong_cons_ideI(1)
thf(fact_238_cong__cons,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,T6: lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T6 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T6 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ ( cons_lambda_lambda @ T6 @ U3 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T6 @ U3 ) @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ) ).
% cong_cons
thf(fact_239_partial__magma_Onull_Ocong,axiom,
partia1461432607296028304lambda = partia1461432607296028304lambda ).
% partial_magma.null.cong
thf(fact_240_partial__magma_Onull_Ocong,axiom,
partia6873046715171720854lambda = partia6873046715171720854lambda ).
% partial_magma.null.cong
thf(fact_241_rts_Ocoterminal_Ocong,axiom,
coterm81126318419254786lambda = coterm81126318419254786lambda ).
% rts.coterminal.cong
thf(fact_242_rts_Ocoterminal_Ocong,axiom,
coterm4142657152830665724lambda = coterm4142657152830665724lambda ).
% rts.coterminal.cong
thf(fact_243_list_Omap__sel_I2_J,axiom,
! [A: list_l5168835686021156086lambda,F: list_lambda_lambda > lambda_lambda] :
( ( A != nil_li8379291187143545974lambda )
=> ( ( tl_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ A ) )
= ( map_li5201084558375397438lambda @ F @ ( tl_lis5905685823335986783lambda @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_244_list_Omap__sel_I2_J,axiom,
! [A: list_lambda_lambda,F: lambda_lambda > list_lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( ( tl_lis5905685823335986783lambda @ ( map_la6434941737452557130lambda @ F @ A ) )
= ( map_la6434941737452557130lambda @ F @ ( tl_lambda_lambda @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_245_list_Omap__sel_I2_J,axiom,
! [A: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( A != nil_li8379291187143545974lambda )
=> ( ( tl_lis5905685823335986783lambda @ ( map_li385728805257398596lambda @ F @ A ) )
= ( map_li385728805257398596lambda @ F @ ( tl_lis5905685823335986783lambda @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_246_list_Omap__sel_I2_J,axiom,
! [A: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( map_la336867333220232516lambda @ F @ A ) )
= ( map_la336867333220232516lambda @ F @ ( tl_lambda_lambda @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_247_tl__append__if,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( Xs2 = nil_li8379291187143545974lambda )
=> ( ( tl_lis5905685823335986783lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( tl_lis5905685823335986783lambda @ Ys ) ) )
& ( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( tl_lis5905685823335986783lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( append1017360404025248779lambda @ ( tl_lis5905685823335986783lambda @ Xs2 ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_248_tl__append__if,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( Xs2 = nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( tl_lambda_lambda @ Ys ) ) )
& ( ( Xs2 != nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( append_lambda_lambda @ ( tl_lambda_lambda @ Xs2 ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_249_append__eq__append__conv2,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ts: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs2 @ Ys )
= ( append1017360404025248779lambda @ Zs @ Ts ) )
= ( ? [Us: list_l5168835686021156086lambda] :
( ( ( Xs2
= ( append1017360404025248779lambda @ Zs @ Us ) )
& ( ( append1017360404025248779lambda @ Us @ Ys )
= Ts ) )
| ( ( ( append1017360404025248779lambda @ Xs2 @ Us )
= Zs )
& ( Ys
= ( append1017360404025248779lambda @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_250_append__eq__append__conv2,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ts: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs2 @ Ys )
= ( append_lambda_lambda @ Zs @ Ts ) )
= ( ? [Us: list_lambda_lambda] :
( ( ( Xs2
= ( append_lambda_lambda @ Zs @ Us ) )
& ( ( append_lambda_lambda @ Us @ Ys )
= Ts ) )
| ( ( ( append_lambda_lambda @ Xs2 @ Us )
= Zs )
& ( Ys
= ( append_lambda_lambda @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_251_map__eq__append__conv,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs2: list_l5168835686021156086lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( map_li5201084558375397438lambda @ F @ Xs2 )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ? [Us: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Us @ Vs ) )
& ( Ys
= ( map_li5201084558375397438lambda @ F @ Us ) )
& ( Zs
= ( map_li5201084558375397438lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_252_map__eq__append__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs2: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( map_la6434941737452557130lambda @ F @ Xs2 )
= ( append1017360404025248779lambda @ Ys @ Zs ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Us @ Vs ) )
& ( Ys
= ( map_la6434941737452557130lambda @ F @ Us ) )
& ( Zs
= ( map_la6434941737452557130lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_253_map__eq__append__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs2 )
= ( append1017360404025248779lambda @ Ys @ Zs ) )
= ( ? [Us: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Us @ Vs ) )
& ( Ys
= ( map_li385728805257398596lambda @ F @ Us ) )
& ( Zs
= ( map_li385728805257398596lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_254_map__eq__append__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs2 )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Us @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_255_append__eq__map__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,F: list_lambda_lambda > lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( map_li5201084558375397438lambda @ F @ Xs2 ) )
= ( ? [Us: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Us @ Vs ) )
& ( Ys
= ( map_li5201084558375397438lambda @ F @ Us ) )
& ( Zs
= ( map_li5201084558375397438lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_256_append__eq__map__conv,axiom,
! [Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,F: lambda_lambda > list_lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Zs )
= ( map_la6434941737452557130lambda @ F @ Xs2 ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Us @ Vs ) )
& ( Ys
= ( map_la6434941737452557130lambda @ F @ Us ) )
& ( Zs
= ( map_la6434941737452557130lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_257_append__eq__map__conv,axiom,
! [Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Zs )
= ( map_li385728805257398596lambda @ F @ Xs2 ) )
= ( ? [Us: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Us @ Vs ) )
& ( Ys
= ( map_li385728805257398596lambda @ F @ Us ) )
& ( Zs
= ( map_li385728805257398596lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_258_append__eq__map__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( map_la336867333220232516lambda @ F @ Xs2 ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Us @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_259_append__eq__appendI,axiom,
! [Xs2: list_l5168835686021156086lambda,Xs1: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Us2: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append1017360404025248779lambda @ Xs1 @ Us2 ) )
=> ( ( append1017360404025248779lambda @ Xs2 @ Ys )
= ( append1017360404025248779lambda @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_260_append__eq__appendI,axiom,
! [Xs2: list_lambda_lambda,Xs1: list_lambda_lambda,Zs: list_lambda_lambda,Ys: list_lambda_lambda,Us2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_lambda_lambda @ Xs1 @ Us2 ) )
=> ( ( append_lambda_lambda @ Xs2 @ Ys )
= ( append_lambda_lambda @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_261_map__tl,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( map_li5201084558375397438lambda @ F @ ( tl_lis5905685823335986783lambda @ Xs2 ) )
= ( tl_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_262_map__tl,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs2: list_lambda_lambda] :
( ( map_la6434941737452557130lambda @ F @ ( tl_lambda_lambda @ Xs2 ) )
= ( tl_lis5905685823335986783lambda @ ( map_la6434941737452557130lambda @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_263_map__tl,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( map_li385728805257398596lambda @ F @ ( tl_lis5905685823335986783lambda @ Xs2 ) )
= ( tl_lis5905685823335986783lambda @ ( map_li385728805257398596lambda @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_264_map__tl,axiom,
! [F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ ( tl_lambda_lambda @ Xs2 ) )
= ( tl_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_265_map__eq__Cons__conv,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs2: list_l5168835686021156086lambda,Y4: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_li5201084558375397438lambda @ F @ Xs2 )
= ( cons_lambda_lambda @ Y4 @ Ys ) )
= ( ? [Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs2
= ( cons_l4558873032622161958lambda @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y4 )
& ( ( map_li5201084558375397438lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_266_map__eq__Cons__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs2: list_lambda_lambda,Y4: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( map_la6434941737452557130lambda @ F @ Xs2 )
= ( cons_l4558873032622161958lambda @ Y4 @ Ys ) )
= ( ? [Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs2
= ( cons_lambda_lambda @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y4 )
& ( ( map_la6434941737452557130lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_267_map__eq__Cons__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y4: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs2 )
= ( cons_l4558873032622161958lambda @ Y4 @ Ys ) )
= ( ? [Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs2
= ( cons_l4558873032622161958lambda @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y4 )
& ( ( map_li385728805257398596lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_268_map__eq__Cons__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda,Y4: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs2 )
= ( cons_lambda_lambda @ Y4 @ Ys ) )
= ( ? [Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs2
= ( cons_lambda_lambda @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y4 )
& ( ( map_la336867333220232516lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_269_Cons__eq__map__conv,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,F: list_lambda_lambda > lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( cons_lambda_lambda @ X3 @ Xs2 )
= ( map_li5201084558375397438lambda @ F @ Ys ) )
= ( ? [Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ Z @ Zs2 ) )
& ( X3
= ( F @ Z ) )
& ( Xs2
= ( map_li5201084558375397438lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_270_Cons__eq__map__conv,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,F: lambda_lambda > list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_l4558873032622161958lambda @ X3 @ Xs2 )
= ( map_la6434941737452557130lambda @ F @ Ys ) )
= ( ? [Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z @ Zs2 ) )
& ( X3
= ( F @ Z ) )
& ( Xs2
= ( map_la6434941737452557130lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_271_Cons__eq__map__conv,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X3 @ Xs2 )
= ( map_li385728805257398596lambda @ F @ Ys ) )
= ( ? [Z: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ Z @ Zs2 ) )
& ( X3
= ( F @ Z ) )
& ( Xs2
= ( map_li385728805257398596lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_272_Cons__eq__map__conv,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X3 @ Xs2 )
= ( map_la336867333220232516lambda @ F @ Ys ) )
= ( ? [Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z @ Zs2 ) )
& ( X3
= ( F @ Z ) )
& ( Xs2
= ( map_la336867333220232516lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_273_map__eq__Cons__D,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs2: list_l5168835686021156086lambda,Y4: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_li5201084558375397438lambda @ F @ Xs2 )
= ( cons_lambda_lambda @ Y4 @ Ys ) )
=> ? [Z2: list_lambda_lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs2
= ( cons_l4558873032622161958lambda @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y4 )
& ( ( map_li5201084558375397438lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_274_map__eq__Cons__D,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs2: list_lambda_lambda,Y4: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( map_la6434941737452557130lambda @ F @ Xs2 )
= ( cons_l4558873032622161958lambda @ Y4 @ Ys ) )
=> ? [Z2: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs2
= ( cons_lambda_lambda @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y4 )
& ( ( map_la6434941737452557130lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_275_map__eq__Cons__D,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y4: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs2 )
= ( cons_l4558873032622161958lambda @ Y4 @ Ys ) )
=> ? [Z2: list_lambda_lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs2
= ( cons_l4558873032622161958lambda @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y4 )
& ( ( map_li385728805257398596lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_276_map__eq__Cons__D,axiom,
! [F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda,Y4: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs2 )
= ( cons_lambda_lambda @ Y4 @ Ys ) )
=> ? [Z2: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs2
= ( cons_lambda_lambda @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y4 )
& ( ( map_la336867333220232516lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_277_Cons__eq__map__D,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,F: list_lambda_lambda > lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( cons_lambda_lambda @ X3 @ Xs2 )
= ( map_li5201084558375397438lambda @ F @ Ys ) )
=> ? [Z2: list_lambda_lambda,Zs3: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ Z2 @ Zs3 ) )
& ( X3
= ( F @ Z2 ) )
& ( Xs2
= ( map_li5201084558375397438lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_278_Cons__eq__map__D,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,F: lambda_lambda > list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_l4558873032622161958lambda @ X3 @ Xs2 )
= ( map_la6434941737452557130lambda @ F @ Ys ) )
=> ? [Z2: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z2 @ Zs3 ) )
& ( X3
= ( F @ Z2 ) )
& ( Xs2
= ( map_la6434941737452557130lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_279_Cons__eq__map__D,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X3 @ Xs2 )
= ( map_li385728805257398596lambda @ F @ Ys ) )
=> ? [Z2: list_lambda_lambda,Zs3: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ Z2 @ Zs3 ) )
& ( X3
= ( F @ Z2 ) )
& ( Xs2
= ( map_li385728805257398596lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_280_Cons__eq__map__D,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X3 @ Xs2 )
= ( map_la336867333220232516lambda @ F @ Ys ) )
=> ? [Z2: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z2 @ Zs3 ) )
& ( X3
= ( F @ Z2 ) )
& ( Xs2
= ( map_la336867333220232516lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_281_list_Osimps_I9_J,axiom,
! [F: lambda_lambda > list_lambda_lambda,X21: lambda_lambda,X22: list_lambda_lambda] :
( ( map_la6434941737452557130lambda @ F @ ( cons_lambda_lambda @ X21 @ X22 ) )
= ( cons_l4558873032622161958lambda @ ( F @ X21 ) @ ( map_la6434941737452557130lambda @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_282_list_Osimps_I9_J,axiom,
! [F: list_lambda_lambda > lambda_lambda,X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( map_li5201084558375397438lambda @ F @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
= ( cons_lambda_lambda @ ( F @ X21 ) @ ( map_li5201084558375397438lambda @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_283_list_Osimps_I9_J,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( map_li385728805257398596lambda @ F @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
= ( cons_l4558873032622161958lambda @ ( F @ X21 ) @ ( map_li385728805257398596lambda @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_284_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_285_list_Osimps_I8_J,axiom,
! [F: list_lambda_lambda > lambda_lambda] :
( ( map_li5201084558375397438lambda @ F @ nil_li8379291187143545974lambda )
= nil_lambda_lambda ) ).
% list.simps(8)
thf(fact_286_list_Osimps_I8_J,axiom,
! [F: lambda_lambda > list_lambda_lambda] :
( ( map_la6434941737452557130lambda @ F @ nil_lambda_lambda )
= nil_li8379291187143545974lambda ) ).
% list.simps(8)
thf(fact_287_list_Osimps_I8_J,axiom,
! [F: list_lambda_lambda > list_lambda_lambda] :
( ( map_li385728805257398596lambda @ F @ nil_li8379291187143545974lambda )
= nil_li8379291187143545974lambda ) ).
% list.simps(8)
thf(fact_288_list_Osimps_I8_J,axiom,
! [F: lambda_lambda > lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.simps(8)
thf(fact_289_Cons__eq__appendI,axiom,
! [X3: list_lambda_lambda,Xs1: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs2: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X3 @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append1017360404025248779lambda @ Xs1 @ Zs ) )
=> ( ( cons_l4558873032622161958lambda @ X3 @ Xs2 )
= ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_290_Cons__eq__appendI,axiom,
! [X3: lambda_lambda,Xs1: list_lambda_lambda,Ys: list_lambda_lambda,Xs2: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X3 @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_lambda_lambda @ Xs1 @ Zs ) )
=> ( ( cons_lambda_lambda @ X3 @ Xs2 )
= ( append_lambda_lambda @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_291_append__Cons,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) @ Ys )
= ( cons_l4558873032622161958lambda @ X3 @ ( append1017360404025248779lambda @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_292_append__Cons,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ ( cons_lambda_lambda @ X3 @ Xs2 ) @ Ys )
= ( cons_lambda_lambda @ X3 @ ( append_lambda_lambda @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_293_append__Nil,axiom,
! [Ys: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_294_append__Nil,axiom,
! [Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_295_append_Oleft__neutral,axiom,
! [A: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ A )
= A ) ).
% append.left_neutral
thf(fact_296_append_Oleft__neutral,axiom,
! [A: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ A )
= A ) ).
% append.left_neutral
thf(fact_297_eq__Nil__appendI,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_298_eq__Nil__appendI,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_lambda_lambda @ nil_lambda_lambda @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_299_list_Osel_I3_J,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( tl_lis5905685823335986783lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_300_list_Osel_I3_J,axiom,
! [X21: lambda_lambda,X22: list_lambda_lambda] :
( ( tl_lambda_lambda @ ( cons_lambda_lambda @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_301_list_Osel_I2_J,axiom,
( ( tl_lis5905685823335986783lambda @ nil_li8379291187143545974lambda )
= nil_li8379291187143545974lambda ) ).
% list.sel(2)
thf(fact_302_list_Osel_I2_J,axiom,
( ( tl_lambda_lambda @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.sel(2)
thf(fact_303_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_304_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_305_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_306_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_307_partial__magma_Onull__eqI,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda,N: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ( ! [T3: lambda_lambda] :
( ( ( OP2 @ N @ T3 )
= N )
& ( ( OP2 @ T3 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_308_partial__magma_Onull__eqI,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,N: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ( ! [T3: list_lambda_lambda] :
( ( ( OP2 @ N @ T3 )
= N )
& ( ( OP2 @ T3 @ N )
= N ) )
=> ( N
= ( partia6873046715171720854lambda @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_309_list_Omap__sel_I1_J,axiom,
! [A: list_l5168835686021156086lambda,F: list_lambda_lambda > lambda_lambda] :
( ( A != nil_li8379291187143545974lambda )
=> ( ( hd_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ A ) )
= ( F @ ( hd_lis2183642875561147995lambda @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_310_list_Omap__sel_I1_J,axiom,
! [A: list_lambda_lambda,F: lambda_lambda > list_lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( map_la6434941737452557130lambda @ F @ A ) )
= ( F @ ( hd_lambda_lambda @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_311_list_Omap__sel_I1_J,axiom,
! [A: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( A != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( map_li385728805257398596lambda @ F @ A ) )
= ( F @ ( hd_lis2183642875561147995lambda @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_312_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_313_hd__map,axiom,
! [Xs2: list_l5168835686021156086lambda,F: list_lambda_lambda > lambda_lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( hd_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ Xs2 ) )
= ( F @ ( hd_lis2183642875561147995lambda @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_314_hd__map,axiom,
! [Xs2: list_lambda_lambda,F: lambda_lambda > list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( map_la6434941737452557130lambda @ F @ Xs2 ) )
= ( F @ ( hd_lambda_lambda @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_315_hd__map,axiom,
! [Xs2: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( map_li385728805257398596lambda @ F @ Xs2 ) )
= ( F @ ( hd_lis2183642875561147995lambda @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_316_hd__map,axiom,
! [Xs2: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs2 ) )
= ( F @ ( hd_lambda_lambda @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_317_lambda__calculus_Oeq__Ide__are__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( T = U )
=> ( ( lambda_Ide @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ) ).
% lambda_calculus.eq_Ide_are_cong
thf(fact_318_lambda__calculus_Ocong__Ide__are__eq,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( ( lambda_Ide @ T )
=> ( ( lambda_Ide @ U )
=> ( T = U ) ) ) ) ).
% lambda_calculus.cong_Ide_are_eq
thf(fact_319_rev__induct,axiom,
! [P: list_l5168835686021156086lambda > $o,Xs2: list_l5168835686021156086lambda] :
( ( P @ nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( P @ Xs )
=> ( P @ ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_320_rev__induct,axiom,
! [P: list_lambda_lambda > $o,Xs2: list_lambda_lambda] :
( ( P @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( P @ Xs )
=> ( P @ ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_321_rev__exhaust,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ~ ! [Ys2: list_l5168835686021156086lambda,Y2: list_lambda_lambda] :
( Xs2
!= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ Y2 @ nil_li8379291187143545974lambda ) ) ) ) ).
% rev_exhaust
thf(fact_322_rev__exhaust,axiom,
! [Xs2: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ~ ! [Ys2: list_lambda_lambda,Y2: lambda_lambda] :
( Xs2
!= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ Y2 @ nil_lambda_lambda ) ) ) ) ).
% rev_exhaust
thf(fact_323_Cons__eq__append__conv,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X3 @ Xs2 )
= ( append1017360404025248779lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_li8379291187143545974lambda )
& ( ( cons_l4558873032622161958lambda @ X3 @ Xs2 )
= Zs ) )
| ? [Ys4: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X3 @ Ys4 )
= Ys )
& ( Xs2
= ( append1017360404025248779lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_324_Cons__eq__append__conv,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X3 @ Xs2 )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( ( cons_lambda_lambda @ X3 @ Xs2 )
= Zs ) )
| ? [Ys4: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X3 @ Ys4 )
= Ys )
& ( Xs2
= ( append_lambda_lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_325_append__eq__Cons__conv,axiom,
! [Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Zs )
= ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) )
= ( ( ( Ys = nil_li8379291187143545974lambda )
& ( Zs
= ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) ) )
| ? [Ys4: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ X3 @ Ys4 ) )
& ( ( append1017360404025248779lambda @ Ys4 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_326_append__eq__Cons__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( cons_lambda_lambda @ X3 @ Xs2 ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( Zs
= ( cons_lambda_lambda @ X3 @ Xs2 ) ) )
| ? [Ys4: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ X3 @ Ys4 ) )
& ( ( append_lambda_lambda @ Ys4 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_327_rev__nonempty__induct,axiom,
! [Xs2: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
=> ( ! [X: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ( P @ Xs )
=> ( P @ ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_328_rev__nonempty__induct,axiom,
! [Xs2: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( Xs2 != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ( ! [X: lambda_lambda,Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( P @ Xs )
=> ( P @ ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_329_lambda__calculus_Osseq__imp__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_sseq @ T @ U )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% lambda_calculus.sseq_imp_seq
thf(fact_330_Nil__tl,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( nil_li8379291187143545974lambda
= ( tl_lis5905685823335986783lambda @ Xs2 ) )
= ( ( Xs2 = nil_li8379291187143545974lambda )
| ? [X2: list_lambda_lambda] :
( Xs2
= ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) ) ) ) ).
% Nil_tl
thf(fact_331_Nil__tl,axiom,
! [Xs2: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( tl_lambda_lambda @ Xs2 ) )
= ( ( Xs2 = nil_lambda_lambda )
| ? [X2: lambda_lambda] :
( Xs2
= ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) ) ) ) ).
% Nil_tl
thf(fact_332_tl__Nil,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( ( tl_lis5905685823335986783lambda @ Xs2 )
= nil_li8379291187143545974lambda )
= ( ( Xs2 = nil_li8379291187143545974lambda )
| ? [X2: list_lambda_lambda] :
( Xs2
= ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) ) ) ) ).
% tl_Nil
thf(fact_333_tl__Nil,axiom,
! [Xs2: list_lambda_lambda] :
( ( ( tl_lambda_lambda @ Xs2 )
= nil_lambda_lambda )
= ( ( Xs2 = nil_lambda_lambda )
| ? [X2: lambda_lambda] :
( Xs2
= ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) ) ) ) ).
% tl_Nil
thf(fact_334_hd__append,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( Xs2 = nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Ys ) ) )
& ( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_335_hd__append,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( Xs2 = nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( hd_lambda_lambda @ Ys ) ) )
& ( ( Xs2 != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( hd_lambda_lambda @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_336_longest__common__prefix,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
? [Ps: list_l5168835686021156086lambda,Xs3: list_l5168835686021156086lambda,Ys5: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Ps @ Xs3 ) )
& ( Ys
= ( append1017360404025248779lambda @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_li8379291187143545974lambda )
| ( Ys5 = nil_li8379291187143545974lambda )
| ( ( hd_lis2183642875561147995lambda @ Xs3 )
!= ( hd_lis2183642875561147995lambda @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_337_longest__common__prefix,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ps: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs2
= ( 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_338_lambda__calculus_Ois__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% lambda_calculus.is_confluent_rts
thf(fact_339_list_Oexpand,axiom,
! [List: list_l5168835686021156086lambda,List2: list_l5168835686021156086lambda] :
( ( ( List = nil_li8379291187143545974lambda )
= ( List2 = nil_li8379291187143545974lambda ) )
=> ( ( ( List != nil_li8379291187143545974lambda )
=> ( ( List2 != nil_li8379291187143545974lambda )
=> ( ( ( hd_lis2183642875561147995lambda @ List )
= ( hd_lis2183642875561147995lambda @ List2 ) )
& ( ( tl_lis5905685823335986783lambda @ List )
= ( tl_lis5905685823335986783lambda @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_340_list_Oexpand,axiom,
! [List: list_lambda_lambda,List2: list_lambda_lambda] :
( ( ( List = nil_lambda_lambda )
= ( List2 = nil_lambda_lambda ) )
=> ( ( ( List != nil_lambda_lambda )
=> ( ( List2 != nil_lambda_lambda )
=> ( ( ( hd_lambda_lambda @ List )
= ( hd_lambda_lambda @ List2 ) )
& ( ( tl_lambda_lambda @ List )
= ( tl_lambda_lambda @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_341_reduction__paths_Ostdz__insert_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( reduct5390047737487997688insert @ T @ nil_lambda_lambda )
= ( reduct1603535164923377302opment @ T ) ) ).
% reduction_paths.stdz_insert.simps(1)
thf(fact_342_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_343_paths__in__weakly__extensional__rts_OTrg_Osimps_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_4823109512173946681lambda @ Resid @ nil_li8379291187143545974lambda )
= ( partia6873046715171720854lambda @ Resid ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.simps(1)
thf(fact_344_paths__in__weakly__extensional__rts_OTrg_Osimps_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_2695323870737867955lambda @ Resid @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ Resid ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.simps(1)
thf(fact_345_list_Oexhaust__sel,axiom,
! [List: list_l5168835686021156086lambda] :
( ( List != nil_li8379291187143545974lambda )
=> ( List
= ( cons_l4558873032622161958lambda @ ( hd_lis2183642875561147995lambda @ List ) @ ( tl_lis5905685823335986783lambda @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_346_list_Oexhaust__sel,axiom,
! [List: list_lambda_lambda] :
( ( List != nil_lambda_lambda )
=> ( List
= ( cons_lambda_lambda @ ( hd_lambda_lambda @ List ) @ ( tl_lambda_lambda @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_347_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_348_paths__in__weakly__extensional__rts_OTrg__append,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( seq_li7066278307288228763lambda @ ( paths_3312059784463760222lambda @ Resid ) @ T4 @ U2 )
=> ( ( paths_4823109512173946681lambda @ Resid @ ( append1017360404025248779lambda @ T4 @ U2 ) )
= ( paths_4823109512173946681lambda @ Resid @ U2 ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg_append
thf(fact_349_paths__in__weakly__extensional__rts_OTrg__append,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( ( paths_2695323870737867955lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ Resid @ U2 ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg_append
thf(fact_350_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_351_map__App1__Resid,axiom,
! [X3: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X3 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda @ ( lambda_App @ X3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X3 ) @ T4 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X3 ) @ U2 ) ) ) ) ) ).
% map_App1_Resid
thf(fact_352_Resid_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(4)
thf(fact_353_Std__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( ( reduction_Std @ U2 )
=> ( ( ( T4 = nil_lambda_lambda )
| ( U2 = nil_lambda_lambda )
| ( lambda_sseq @ ( last_lambda_lambda @ T4 ) @ ( hd_lambda_lambda @ U2 ) ) )
=> ( reduction_Std @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Std_append
thf(fact_354_Std__imp__sseq__last__hd,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( reduction_Std @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( lambda_sseq @ ( last_lambda_lambda @ T4 ) @ ( hd_lambda_lambda @ U2 ) ) ) ) ) ).
% Std_imp_sseq_last_hd
thf(fact_355_Std__append__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( lambda_sseq @ ( last_lambda_lambda @ T4 ) @ U )
=> ( reduction_Std @ ( append_lambda_lambda @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ) ) ).
% Std_append_single
thf(fact_356_cong__map__Lam,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) @ ( map_la336867333220232516lambda @ lambda_Lam @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ U2 ) @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) ) ) ) ) ).
% cong_map_Lam
thf(fact_357_arr__append__imp__seq,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% arr_append_imp_seq
thf(fact_358_reduction__paths_Ostandardize_Oelims,axiom,
! [X3: list_lambda_lambda,Y4: list_lambda_lambda] :
( ( ( reduct6308743955541908015ardize @ X3 )
= Y4 )
=> ( ( ( X3 = nil_lambda_lambda )
=> ( Y4 != nil_lambda_lambda ) )
=> ~ ! [V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ V3 @ Va2 ) )
=> ( Y4
!= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V3 @ Va2 ) ) @ ( reduct6308743955541908015ardize @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).
% reduction_paths.standardize.elims
thf(fact_359_con__imp__arr__resid,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% con_imp_arr_resid
thf(fact_360_con__sym__ax,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% con_sym_ax
thf(fact_361_cube__ax,axiom,
! [V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ).
% cube_ax
thf(fact_362_null__eqI,axiom,
! [N: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ N @ T3 )
= N )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ N )
= N ) )
=> ( N
= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% null_eqI
thf(fact_363__092_060Lambda_062_Ocong__implies__coterminal,axiom,
! [U: lambda_lambda,U4: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ U @ U4 ) ) ).
% \<Lambda>.cong_implies_coterminal
thf(fact_364__092_060Lambda_062_Ocon__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ U @ T ) ) ).
% \<Lambda>.con_sym
thf(fact_365__092_060Lambda_062_Oresid__reflects__con,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ U @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% \<Lambda>.resid_reflects_con
thf(fact_366_null__char,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= nil_lambda_lambda ) ).
% null_char
thf(fact_367__092_060Lambda_062_Ocong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T6 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T6 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T6 @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T6 @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_left(2)
thf(fact_368__092_060Lambda_062_Ocong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T6 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T6 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T6 @ U ) ) ) ).
% \<Lambda>.cong_subst_left(1)
thf(fact_369__092_060Lambda_062_Ocong__subst__right_I2_J,axiom,
! [U: lambda_lambda,U4: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U4 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U4 ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_right(2)
thf(fact_370__092_060Lambda_062_Ocong__subst__right_I1_J,axiom,
! [U: lambda_lambda,U4: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U4 ) ) ) ).
% \<Lambda>.cong_subst_right(1)
thf(fact_371__092_060Lambda_062_Ocon__ide__are__eq,axiom,
! [A: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A3 )
=> ( A = A3 ) ) ) ) ).
% \<Lambda>.con_ide_are_eq
thf(fact_372__092_060Lambda_062_Ocon__imp__coinitial__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ? [A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A4 )
& ( con_lambda_lambda @ lambda_resid @ A4 @ T )
& ( con_lambda_lambda @ lambda_resid @ A4 @ U ) ) ) ).
% \<Lambda>.con_imp_coinitial_ax
thf(fact_373__092_060Lambda_062_Ocon__target,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
=> ( ( con_lambda_lambda @ lambda_resid @ U @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ V @ U ) ) ) ) ).
% \<Lambda>.con_target
thf(fact_374__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_375__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_376__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_377__092_060Lambda_062_Oide__imp__con__iff__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ) ) ).
% \<Lambda>.ide_imp_con_iff_cong
thf(fact_378__092_060Lambda_062_Oprfx__implies__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.prfx_implies_con
thf(fact_379__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_380__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_381_not__arr__null,axiom,
~ ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% not_arr_null
thf(fact_382_Std__map__Lam,axiom,
! [T4: list_lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( reduction_Std @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) ) ) ).
% Std_map_Lam
thf(fact_383__092_060Lambda_062_OconE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.conE
thf(fact_384__092_060Lambda_062_Ocon__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.con_def
thf(fact_385_development__map__Lam,axiom,
! [T: lambda_lambda,T4: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ T4 )
=> ( reduct9208763521642324371opment @ ( lambda_Lam @ T ) @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) ) ) ).
% development_map_Lam
thf(fact_386_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_387_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_388_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_389_cong__implies__coinitial,axiom,
! [U: list_lambda_lambda,U4: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U4 ) ) ).
% cong_implies_coinitial
thf(fact_390_coinitial__ide__are__cong,axiom,
! [A: list_lambda_lambda,A3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_391_Resid__parallel,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T ) ) )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) ) ) ).
% Resid_parallel
thf(fact_392_composableD_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% composableD(1)
thf(fact_393_composableD_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ).
% composableD(2)
thf(fact_394_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_395_Con__rec_I4_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_rec(4)
thf(fact_396_Con__rec_I3_J,axiom,
! [U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_rec(3)
thf(fact_397_Con__rec_I2_J,axiom,
! [T4: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_rec(2)
thf(fact_398_Con__rec_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
= ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% Con_rec(1)
thf(fact_399_Resid_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(3)
thf(fact_400_map__Lam__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda @ lambda_Lam @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) @ ( map_la336867333220232516lambda @ lambda_Lam @ U2 ) ) ) ) ).
% map_Lam_Resid
thf(fact_401_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_402_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_403_last__appendR,axiom,
! [Ys: list_l5168835686021156086lambda,Xs2: list_l5168835686021156086lambda] :
( ( Ys != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( last_l1874196399739661071lambda @ Ys ) ) ) ).
% last_appendR
thf(fact_404_last__appendR,axiom,
! [Ys: list_lambda_lambda,Xs2: list_lambda_lambda] :
( ( Ys != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( last_lambda_lambda @ Ys ) ) ) ).
% last_appendR
thf(fact_405_last__appendL,axiom,
! [Ys: list_l5168835686021156086lambda,Xs2: list_l5168835686021156086lambda] :
( ( Ys = nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( last_l1874196399739661071lambda @ Xs2 ) ) ) ).
% last_appendL
thf(fact_406_last__appendL,axiom,
! [Ys: list_lambda_lambda,Xs2: list_lambda_lambda] :
( ( Ys = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( last_lambda_lambda @ Xs2 ) ) ) ).
% last_appendL
thf(fact_407__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_408__092_060Lambda_062_OconI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.conI
thf(fact_409_last__snoc,axiom,
! [Xs2: list_l5168835686021156086lambda,X3: list_lambda_lambda] :
( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X3 @ nil_li8379291187143545974lambda ) ) )
= X3 ) ).
% last_snoc
thf(fact_410_last__snoc,axiom,
! [Xs2: list_lambda_lambda,X3: lambda_lambda] :
( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) ) )
= X3 ) ).
% last_snoc
thf(fact_411_arr__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ).
% arr_append
thf(fact_412_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_413_rts_Ocoinitial_Ocong,axiom,
coinit1063259437155584316lambda = coinit1063259437155584316lambda ).
% rts.coinitial.cong
thf(fact_414_rts_Ocoinitial_Ocong,axiom,
coinit1392077014771647542lambda = coinit1392077014771647542lambda ).
% rts.coinitial.cong
thf(fact_415_residuation_Oarr_Ocong,axiom,
arr_li3691443969012319837lambda = arr_li3691443969012319837lambda ).
% residuation.arr.cong
thf(fact_416_residuation_Oarr_Ocong,axiom,
arr_lambda_lambda = arr_lambda_lambda ).
% residuation.arr.cong
thf(fact_417_residuation_Ocon_Ocong,axiom,
con_lambda_lambda = con_lambda_lambda ).
% residuation.con.cong
thf(fact_418_residuation_Ocon_Ocong,axiom,
con_li79537494059641180lambda = con_li79537494059641180lambda ).
% residuation.con.cong
thf(fact_419_confluent__rts_Oconfluence,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( conflu4927472611300549594lambda @ Resid )
=> ( ( coinit1392077014771647542lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_420_confluent__rts_Oconfluence,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( conflu4036547884742428384lambda @ Resid )
=> ( ( coinit1063259437155584316lambda @ Resid @ T @ U )
=> ( con_li79537494059641180lambda @ Resid @ T @ U ) ) ) ).
% confluent_rts.confluence
thf(fact_421_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_422_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_423_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_424_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_425_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_426_lambda__calculus_Ohead__strategy_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_head_strategy @ T ) ) ) ).
% lambda_calculus.head_strategy.simps(2)
thf(fact_427_lambda__calculus_Osseq_Osimps_I3_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T6 ) )
= ( lambda_sseq @ T @ T6 ) ) ).
% lambda_calculus.sseq.simps(3)
thf(fact_428_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_429_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_430_lambda__calculus_Olambda_Odisc_I8_J,axiom,
! [X32: lambda_lambda] : ( lambda_is_Lam @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(8)
thf(fact_431_lambda__calculus_Olambda_Odisc_I18_J,axiom,
! [X32: lambda_lambda] :
~ ( lambda_is_Beta @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(18)
thf(fact_432_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_433_lambda__calculus_Osseq_Osimps_I23_J,axiom,
! [Vc: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Lam @ Vc ) @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(23)
thf(fact_434_lambda__calculus_Osseq_Osimps_I24_J,axiom,
! [Vc: lambda_lambda,Vd: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vc @ Vd ) @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(24)
thf(fact_435_lambda__calculus_Osseq_Osimps_I27_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(27)
thf(fact_436_last__ConsR,axiom,
! [Xs2: list_l5168835686021156086lambda,X3: list_lambda_lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) )
= ( last_l1874196399739661071lambda @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_437_last__ConsR,axiom,
! [Xs2: list_lambda_lambda,X3: lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X3 @ Xs2 ) )
= ( last_lambda_lambda @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_438_last__ConsL,axiom,
! [Xs2: list_l5168835686021156086lambda,X3: list_lambda_lambda] :
( ( Xs2 = nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) )
= X3 ) ) ).
% last_ConsL
thf(fact_439_last__ConsL,axiom,
! [Xs2: list_lambda_lambda,X3: lambda_lambda] :
( ( Xs2 = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X3 @ Xs2 ) )
= X3 ) ) ).
% last_ConsL
thf(fact_440_last_Osimps,axiom,
! [Xs2: list_l5168835686021156086lambda,X3: list_lambda_lambda] :
( ( ( Xs2 = nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) )
= X3 ) )
& ( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) )
= ( last_l1874196399739661071lambda @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_441_last_Osimps,axiom,
! [Xs2: list_lambda_lambda,X3: lambda_lambda] :
( ( ( Xs2 = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X3 @ Xs2 ) )
= X3 ) )
& ( ( Xs2 != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X3 @ Xs2 ) )
= ( last_lambda_lambda @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_442_last__map,axiom,
! [Xs2: list_l5168835686021156086lambda,F: list_lambda_lambda > lambda_lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( last_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ Xs2 ) )
= ( F @ ( last_l1874196399739661071lambda @ Xs2 ) ) ) ) ).
% last_map
thf(fact_443_last__map,axiom,
! [Xs2: list_lambda_lambda,F: lambda_lambda > list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( last_l1874196399739661071lambda @ ( map_la6434941737452557130lambda @ F @ Xs2 ) )
= ( F @ ( last_lambda_lambda @ Xs2 ) ) ) ) ).
% last_map
thf(fact_444_last__map,axiom,
! [Xs2: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( map_li385728805257398596lambda @ F @ Xs2 ) )
= ( F @ ( last_l1874196399739661071lambda @ Xs2 ) ) ) ) ).
% last_map
thf(fact_445_last__map,axiom,
! [Xs2: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs2 ) )
= ( F @ ( last_lambda_lambda @ Xs2 ) ) ) ) ).
% last_map
thf(fact_446_last__append,axiom,
! [Ys: list_l5168835686021156086lambda,Xs2: list_l5168835686021156086lambda] :
( ( ( Ys = nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( last_l1874196399739661071lambda @ Xs2 ) ) )
& ( ( Ys != nil_li8379291187143545974lambda )
=> ( ( last_l1874196399739661071lambda @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( last_l1874196399739661071lambda @ Ys ) ) ) ) ).
% last_append
thf(fact_447_last__append,axiom,
! [Ys: list_lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( Ys = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( last_lambda_lambda @ Xs2 ) ) )
& ( ( Ys != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( last_lambda_lambda @ Ys ) ) ) ) ).
% last_append
thf(fact_448_longest__common__suffix,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
? [Ss: list_l5168835686021156086lambda,Xs3: list_l5168835686021156086lambda,Ys5: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Xs3 @ Ss ) )
& ( Ys
= ( append1017360404025248779lambda @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_li8379291187143545974lambda )
| ( Ys5 = nil_li8379291187143545974lambda )
| ( ( last_l1874196399739661071lambda @ Xs3 )
!= ( last_l1874196399739661071lambda @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_449_longest__common__suffix,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ss: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs2
= ( 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_450_hd__Nil__eq__last,axiom,
( ( hd_lis2183642875561147995lambda @ nil_li8379291187143545974lambda )
= ( last_l1874196399739661071lambda @ nil_li8379291187143545974lambda ) ) ).
% hd_Nil_eq_last
thf(fact_451_hd__Nil__eq__last,axiom,
( ( hd_lambda_lambda @ nil_lambda_lambda )
= ( last_lambda_lambda @ nil_lambda_lambda ) ) ).
% hd_Nil_eq_last
thf(fact_452_last__tl,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( ( Xs2 = nil_li8379291187143545974lambda )
| ( ( tl_lis5905685823335986783lambda @ Xs2 )
!= nil_li8379291187143545974lambda ) )
=> ( ( last_l1874196399739661071lambda @ ( tl_lis5905685823335986783lambda @ Xs2 ) )
= ( last_l1874196399739661071lambda @ Xs2 ) ) ) ).
% last_tl
thf(fact_453_last__tl,axiom,
! [Xs2: list_lambda_lambda] :
( ( ( Xs2 = nil_lambda_lambda )
| ( ( tl_lambda_lambda @ Xs2 )
!= nil_lambda_lambda ) )
=> ( ( last_lambda_lambda @ ( tl_lambda_lambda @ Xs2 ) )
= ( last_lambda_lambda @ Xs2 ) ) ) ).
% last_tl
thf(fact_454_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_455_reduction__paths_Ostandardize_Osimps_I1_J,axiom,
( ( reduct6308743955541908015ardize @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% reduction_paths.standardize.simps(1)
thf(fact_456_lambda__calculus_Ocontains__head__reduction__iff,axiom,
( lambda8135948770199036598uction
= ( ^ [T7: lambda_lambda] :
( ( lambda_Arr @ T7 )
& ~ ( lambda_Ide @ ( lambda_head_redex @ T7 ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction_iff
thf(fact_457_reduction__paths_Ostandardize_Osimps_I2_J,axiom,
! [V: lambda_lambda,Va: list_lambda_lambda] :
( ( reduct6308743955541908015ardize @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( reduct6308743955541908015ardize @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ).
% reduction_paths.standardize.simps(2)
thf(fact_458_seq__map__un__Lam,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Lam @ T ) @ nil_lambda_lambda ) @ U2 )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( map_la336867333220232516lambda @ lambda_un_Lam @ U2 ) ) ) ).
% seq_map_un_Lam
thf(fact_459_joinable__implies__coinitial,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% joinable_implies_coinitial
thf(fact_460__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_461_Resid_Oelims,axiom,
! [X3: list_lambda_lambda,Xa: list_lambda_lambda,Y4: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X3 @ Xa )
= Y4 )
=> ( ( ( X3 = nil_lambda_lambda )
=> ( Y4 != nil_lambda_lambda ) )
=> ( ( ? [V3: lambda_lambda,Va2: list_lambda_lambda] :
( X3
= ( cons_lambda_lambda @ V3 @ Va2 ) )
=> ( ( Xa = nil_lambda_lambda )
=> ( Y4 != nil_lambda_lambda ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U6: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U6 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U6 )
=> ( Y4
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U6 ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ U6 )
=> ( Y4 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U6: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U6 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U6 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U6 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y4
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U6 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U6 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U6 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y4 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ! [U6: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U6 @ nil_lambda_lambda ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U6 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U6 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y4
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U6 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U6 @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U6 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U6 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y4 = nil_lambda_lambda ) ) ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ! [U6: lambda_lambda,Vb2: lambda_lambda,Vc2: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U6 @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U6 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U6 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U6 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U6 ) ) )
!= nil_lambda_lambda ) )
=> ( Y4
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U6 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U6 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U6 ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U6 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U6 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U6 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U6 ) ) )
!= nil_lambda_lambda ) )
=> ( Y4 = nil_lambda_lambda ) ) ) ) ) ) ) ) ) ) ) ).
% Resid.elims
thf(fact_462_Resid_Osimps_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb: lambda_lambda,Vc: list_lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(6)
thf(fact_463_Resid_Osimps_I7_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda,Vb: lambda_lambda,Vc: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(7)
thf(fact_464_resid__reflects__con,axiom,
! [T: list_lambda_lambda,V: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ V )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ V )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ) ) ).
% resid_reflects_con
thf(fact_465_con__sym,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T ) ) ).
% con_sym
thf(fact_466__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_467__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_468__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_469__092_060Lambda_062_Ocong__implies__coinitial,axiom,
! [U: lambda_lambda,U4: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ U @ U4 ) ) ).
% \<Lambda>.cong_implies_coinitial
thf(fact_470__092_060Lambda_062_Ocoinitial__ide__are__eq,axiom,
! [A: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A3 )
=> ( A = A3 ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_eq
thf(fact_471__092_060Lambda_062_Ocoinitial__ide__are__cong,axiom,
! [A: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A @ A3 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A ) ) ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_cong
thf(fact_472__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_473__092_060Lambda_062_Oarr__resid__iff__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.arr_resid_iff_con
thf(fact_474__092_060Lambda_062_Oarr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.arr_resid
thf(fact_475__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_476__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_477__092_060Lambda_062_Ocon__implies__arr_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.con_implies_arr(1)
thf(fact_478__092_060Lambda_062_Ocon__implies__arr_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ U ) ) ).
% \<Lambda>.con_implies_arr(2)
thf(fact_479__092_060Lambda_062_Oconfluence,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.confluence
thf(fact_480__092_060Lambda_062_Ocon__imp__coinitial,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.con_imp_coinitial
thf(fact_481__092_060Lambda_062_Onot__arr__null,axiom,
~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.not_arr_null
thf(fact_482__092_060Lambda_062_OcomposableD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.composableD(1)
thf(fact_483__092_060Lambda_062_OcomposableD_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
=> ( arr_lambda_lambda @ lambda_resid @ U ) ) ).
% \<Lambda>.composableD(2)
thf(fact_484_con__char,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda ) ) ).
% con_char
thf(fact_485_cong__subst__left_I2_J,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_486_cong__subst__left_I1_J,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_487_cong__subst__right_I2_J,axiom,
! [U: list_lambda_lambda,U4: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( 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 @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_488_cong__subst__right_I1_J,axiom,
! [U: list_lambda_lambda,U4: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U4 ) ) ) ).
% cong_subst_right(1)
thf(fact_489_con__imp__coinitial__ax,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ? [A4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 @ T )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 @ U ) ) ) ).
% con_imp_coinitial_ax
thf(fact_490_con__target,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) ) ) ).
% con_target
thf(fact_491_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_492_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_493_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_494_ide__imp__con__iff__cong,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) ) ) ) ) ).
% ide_imp_con_iff_cong
thf(fact_495_prfx__implies__con,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% prfx_implies_con
thf(fact_496_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_497_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_498_Residx1_Osimps_I1_J,axiom,
! [U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ nil_lambda_lambda @ U )
= nil_lambda_lambda ) ).
% Residx1.simps(1)
thf(fact_499_con__implies__arr_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ).
% con_implies_arr(2)
thf(fact_500_con__implies__arr_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% con_implies_arr(1)
thf(fact_501_arrE,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).
% arrE
thf(fact_502_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_503_arr__resid,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ).
% arr_resid
thf(fact_504_arr__resid__iff__con,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% arr_resid_iff_con
thf(fact_505_conE,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% conE
thf(fact_506_con__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% con_def
thf(fact_507_con__imp__coinitial,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% con_imp_coinitial
thf(fact_508_confluence,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% confluence
thf(fact_509_confluence_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ).
% confluence\<^sub>P
thf(fact_510_joinable__implies__con,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% joinable_implies_con
thf(fact_511_Residx1__as__Resid,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ T4 @ U )
= ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ).
% Residx1_as_Resid
thf(fact_512_Residx1_Oelims,axiom,
! [X3: list_lambda_lambda,Xa: lambda_lambda,Y4: list_lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ X3 @ Xa )
= Y4 )
=> ( ( ( X3 = nil_lambda_lambda )
=> ( Y4 != nil_lambda_lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
=> ( Y4
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
=> ( Y4 = nil_lambda_lambda ) ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y4
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y4 = nil_lambda_lambda ) ) ) ) ) ) ) ).
% Residx1.elims
thf(fact_513_Residx1_Osimps_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
= nil_lambda_lambda ) ) ) ).
% Residx1.simps(2)
thf(fact_514_Residx1_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ U )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ U )
= nil_lambda_lambda ) ) ) ).
% Residx1.simps(3)
thf(fact_515_Con__sym1,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ T4 @ U )
!= nil_lambda_lambda )
= ( ( paths_3319247187298047121lambda @ lambda_resid @ U @ T4 )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% Con_sym1
thf(fact_516_Resid_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(5)
thf(fact_517_reflects__con,axiom,
! [T: lambda_lambda,U: 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 @ U ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) )
!= nil_lambda_lambda )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% reflects_con
thf(fact_518_lambda__calculus_Oarr__char,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.arr_char
thf(fact_519__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_520_conI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ).
% conI\<^sub>P
thf(fact_521_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_522_arrI,axiom,
! [T: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% arrI
thf(fact_523_conI,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ U )
!= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ).
% conI
thf(fact_524_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_525_paths__in__rts_OResidx1_Ocong,axiom,
paths_2112734093361750239lambda = paths_2112734093361750239lambda ).
% paths_in_rts.Residx1.cong
thf(fact_526_rts_Ojoinable_Ocong,axiom,
joinab677607494065406278lambda = joinab677607494065406278lambda ).
% rts.joinable.cong
thf(fact_527_rts_Ojoinable_Ocong,axiom,
joinab2519788767067900224lambda = joinab2519788767067900224lambda ).
% rts.joinable.cong
thf(fact_528_lambda__calculus_Olambda_Osel_I2_J,axiom,
! [X32: lambda_lambda] :
( ( lambda_un_Lam @ ( lambda_Lam @ X32 ) )
= X32 ) ).
% lambda_calculus.lambda.sel(2)
thf(fact_529_paths__in__confluent__rts_Oconfluence_092_060_094sub_062P,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( ( coinit255303819856200130lambda @ ( paths_3312059784463760222lambda @ Resid ) @ T4 @ U2 )
=> ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ Resid ) @ T4 @ U2 ) ) ) ).
% paths_in_confluent_rts.confluence\<^sub>P
thf(fact_530_paths__in__confluent__rts_Oconfluence_092_060_094sub_062P,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 ) ) ) ).
% paths_in_confluent_rts.confluence\<^sub>P
thf(fact_531_lambda__calculus_Oprfx__Lam__iff,axiom,
! [U: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Lam @ T ) ) )
= ( ( lambda_is_Lam @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_Lam @ U ) @ T ) ) ) ) ).
% lambda_calculus.prfx_Lam_iff
thf(fact_532_reduction__paths_Ostdz__insert_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ T )
=> ( ( reduct5390047737487997688insert @ ( lambda_Lam @ T ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ T )
=> ( ( reduct5390047737487997688insert @ ( lambda_Lam @ T ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( map_la336867333220232516lambda @ lambda_Lam @ ( reduct5390047737487997688insert @ T @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(3)
thf(fact_533__092_060Lambda_062_Ojoinable__implies__con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.joinable_implies_con
thf(fact_534__092_060Lambda_062_Ohas__joins,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( joinab2519788767067900224lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.has_joins
thf(fact_535__092_060open_062_092_060Lambda_062_Ois__App_Au_A_092_060or_062_A_092_060Lambda_062_Ois__Beta_Au_092_060close_062,axiom,
( ( lambda_is_App @ u )
| ( lambda_is_Beta @ u ) ) ).
% \<open>\<Lambda>.is_App u \<or> \<Lambda>.is_Beta u\<close>
thf(fact_536_Trg_Oelims,axiom,
! [X3: list_lambda_lambda,Y4: lambda_lambda] :
( ( ( paths_2695323870737867955lambda @ lambda_resid @ X3 )
= Y4 )
=> ( ( ( X3 = nil_lambda_lambda )
=> ( Y4
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y4
!= ( trg_lambda_lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y4
!= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Trg.elims
thf(fact_537_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_538_joinable__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ? [X5: list_lambda_lambda] : ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ X5 ) ) ) ).
% joinable_def
thf(fact_539__092_060Lambda_062_Oapex__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ).
% \<Lambda>.apex_sym
thf(fact_540__092_060Lambda_062_Otrg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_resid @ T @ T ) ) ).
% \<Lambda>.trg_def
thf(fact_541__092_060Lambda_062_Otrg__resid__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).
% \<Lambda>.trg_resid_sym
thf(fact_542_join__of__symmetric,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V ) ) ).
% join_of_symmetric
thf(fact_543__092_060Lambda_062_Ojoinable__implies__coinitial,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ).
% \<Lambda>.joinable_implies_coinitial
thf(fact_544_join__of__un__upto__cong,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ 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_545__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_546__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_547_join__of__resid,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V @ W2 )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ V ) ) ) ) ).
% join_of_resid
thf(fact_548_con__with__join__of__iff_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ V )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_549_con__with__join__of__iff_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ 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 @ U @ T ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_550_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_551__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_552__092_060Lambda_062_Ocoterminal__iff__con__trg,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
= ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.coterminal_iff_con_trg
thf(fact_553__092_060Lambda_062_Ocoterminal__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ) ).
% \<Lambda>.coterminal_iff\<^sub>W\<^sub>E
thf(fact_554__092_060Lambda_062_OcoterminalE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
!= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ) ) ).
% \<Lambda>.coterminalE\<^sub>W\<^sub>E
thf(fact_555_Src__hd__eqI,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( lambda_Src @ ( hd_lambda_lambda @ T4 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) ) ).
% Src_hd_eqI
thf(fact_556_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_557_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_558__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_559_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_560__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_561__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_562_lambda__calculus_Ocoinitial__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) ) ) ).
% lambda_calculus.coinitial_char
thf(fact_563__092_060Lambda_062_OcoterminalI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( trg_lambda_lambda @ lambda_resid @ U ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.coterminalI\<^sub>W\<^sub>E
thf(fact_564_Trg__simp,axiom,
! [T4: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( trg_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T4 ) ) ) ) ).
% Trg_simp
thf(fact_565_conI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( ( lambda_Src @ ( hd_lambda_lambda @ T4 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% conI\<^sub>\<Lambda>\<^sub>P
thf(fact_566_rts_Ojoin__of_Ocong,axiom,
join_o6956785509312998601lambda = join_o6956785509312998601lambda ).
% rts.join_of.cong
thf(fact_567_rts_Ojoin__of_Ocong,axiom,
join_o1386658102433174083lambda = join_o1386658102433174083lambda ).
% rts.join_of.cong
thf(fact_568_residuation_Otrg_Ocong,axiom,
trg_lambda_lambda = trg_lambda_lambda ).
% residuation.trg.cong
thf(fact_569_residuation_Otrg_Ocong,axiom,
trg_li817795487149935061lambda = trg_li817795487149935061lambda ).
% residuation.trg.cong
thf(fact_570_lambda__calculus_OSrc_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_App @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.Src.simps(4)
thf(fact_571_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_Src @ ( lambda_App @ T6 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T6 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_572_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_573_lambda__calculus_OArr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Arr_Src
thf(fact_574_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_575_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_576_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_577_lambda__calculus_Olambda_Odisc_I13_J,axiom,
! [X32: lambda_lambda] :
~ ( lambda_is_App @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(13)
thf(fact_578_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_579_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_580_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_581_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_582_lambda__calculus_Ohead__strategy_Osimps_I6_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_App @ V @ Va ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(6)
thf(fact_583_lambda__calculus_Ohead__redex_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_App @ V @ Va ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(7)
thf(fact_584_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_585_lambda__calculus_OIde__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Ide_Src
thf(fact_586_lambda__calculus_OSrc__head__strategy,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_head_strategy @ T ) )
= ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Src_head_strategy
thf(fact_587_lambda__calculus_Ohead__strategy__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_head_strategy @ ( lambda_Src @ T ) )
= ( lambda_head_strategy @ T ) ) ) ).
% lambda_calculus.head_strategy_Src
thf(fact_588_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_589_lambda__calculus_Osseq__preserves__App__and__no__head__reduction,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_sseq @ T @ U )
=> ( ( ( lambda_is_App @ T )
& ~ ( lambda8135948770199036598uction @ T ) )
=> ( ( lambda_is_App @ U )
& ~ ( lambda8135948770199036598uction @ U ) ) ) ) ).
% lambda_calculus.sseq_preserves_App_and_no_head_reduction
thf(fact_590_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_591_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_592_lambda__calculus_Ohead__redex_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(4)
thf(fact_593_paths__in__weakly__extensional__rts_OTrg_Osimps_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_4823109512173946681lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) )
= ( trg_li817795487149935061lambda @ Resid @ T ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.simps(2)
thf(fact_594_paths__in__weakly__extensional__rts_OTrg_Osimps_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_2695323870737867955lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( trg_lambda_lambda @ Resid @ T ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.simps(2)
thf(fact_595_paths__in__weakly__extensional__rts_OTrg__simp,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( T4 != nil_li8379291187143545974lambda )
=> ( ( paths_4823109512173946681lambda @ Resid @ T4 )
= ( trg_li817795487149935061lambda @ Resid @ ( last_l1874196399739661071lambda @ T4 ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg_simp
thf(fact_596_paths__in__weakly__extensional__rts_OTrg__simp,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ Resid @ T4 )
= ( trg_lambda_lambda @ Resid @ ( last_lambda_lambda @ T4 ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg_simp
thf(fact_597_paths__in__weakly__extensional__rts_OTrg_Oelims,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,X3: list_l5168835686021156086lambda,Y4: list_lambda_lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( ( paths_4823109512173946681lambda @ Resid @ X3 )
= Y4 )
=> ( ( ( X3 = nil_li8379291187143545974lambda )
=> ( Y4
!= ( partia6873046715171720854lambda @ Resid ) ) )
=> ( ! [T3: list_lambda_lambda] :
( ( X3
= ( cons_l4558873032622161958lambda @ T3 @ nil_li8379291187143545974lambda ) )
=> ( Y4
!= ( trg_li817795487149935061lambda @ Resid @ T3 ) ) )
=> ~ ! [T3: list_lambda_lambda,V3: list_lambda_lambda,Va2: list_l5168835686021156086lambda] :
( ( X3
= ( cons_l4558873032622161958lambda @ T3 @ ( cons_l4558873032622161958lambda @ V3 @ Va2 ) ) )
=> ( Y4
!= ( paths_4823109512173946681lambda @ Resid @ ( cons_l4558873032622161958lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.elims
thf(fact_598_paths__in__weakly__extensional__rts_OTrg_Oelims,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,X3: list_lambda_lambda,Y4: lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( ( paths_2695323870737867955lambda @ Resid @ X3 )
= Y4 )
=> ( ( ( X3 = nil_lambda_lambda )
=> ( Y4
!= ( partia1461432607296028304lambda @ Resid ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y4
!= ( trg_lambda_lambda @ Resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y4
!= ( paths_2695323870737867955lambda @ Resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Trg.elims
thf(fact_599_seq__imp__cong__Trg__last__Src__hd,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) ) ) ) ).
% seq_imp_cong_Trg_last_Src_hd
thf(fact_600_Trg__last__Src__hd__eqI,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ T4 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) ) ).
% Trg_last_Src_hd_eqI
thf(fact_601_ind2,axiom,
( ~ ( lambda_Ide @ ( lambda_App @ m @ n ) )
=> ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ m @ n ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_App @ m @ n ) )
=> ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ nil_lambda_lambda ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
& ( reduction_Std @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ind2
thf(fact_602_ind4,axiom,
( ~ ( lambda_Ide @ ( lambda_App @ m @ n ) )
=> ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ m @ n ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ m @ n ) )
=> ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ nil_lambda_lambda ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
& ( reduction_Std @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) @ ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) @ ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ind4
thf(fact_603_ind5,axiom,
( ~ ( lambda_Ide @ ( lambda_App @ m @ n ) )
=> ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ m @ n ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ m @ n ) )
=> ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ nil_lambda_lambda ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
& ( reduction_Std @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ind5
thf(fact_604__092_060Lambda_062_Ojoin__of__symmetric,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ lambda_resid @ U @ T @ V ) ) ).
% \<Lambda>.join_of_symmetric
thf(fact_605_trg__def,axiom,
! [T: list_lambda_lambda] :
( ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ).
% trg_def
thf(fact_606__092_060Lambda_062_Ojoin__of__un__upto__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V4: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V4 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ V4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V4 @ V ) ) ) ) ) ).
% \<Lambda>.join_of_un_upto_cong
thf(fact_607__092_060Lambda_062_Ojoin__of__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ V @ W2 )
=> ( join_o1386658102433174083lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) @ ( lambda_resid @ W2 @ V ) ) ) ) ).
% \<Lambda>.join_of_resid
thf(fact_608__092_060Lambda_062_Ocon__with__join__of__iff_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( ( con_lambda_lambda @ lambda_resid @ U @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) )
=> ( con_lambda_lambda @ lambda_resid @ W2 @ V ) ) ) ).
% \<Lambda>.con_with_join_of_iff(1)
thf(fact_609__092_060Lambda_062_Ocon__with__join__of__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) ) ) ) ) ).
% \<Lambda>.con_with_join_of_iff(2)
thf(fact_610__092_060Lambda_062_Ojoin__of__arr__self,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ T @ T ) ) ).
% \<Lambda>.join_of_arr_self
thf(fact_611__092_060Lambda_062_Otrg__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
= ( trg_lambda_lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.trg_join_of(2)
thf(fact_612__092_060Lambda_062_Otrg__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.trg_join_of(1)
thf(fact_613__092_060Lambda_062_Ojoinable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
= ( ? [X5: lambda_lambda] : ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ X5 ) ) ) ).
% \<Lambda>.joinable_def
thf(fact_614_Ide_Osimps_I1_J,axiom,
~ ( paths_7717490669901197737lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Ide.simps(1)
thf(fact_615_Resid__Ide_I1_J,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A2 )
= T4 ) ) ) ).
% Resid_Ide(1)
thf(fact_616_Resid__Arr__Ide__ind,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A2 )
= T4 ) ) ) ).
% Resid_Arr_Ide_ind
thf(fact_617_Resid__Ide__Arr__ind,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 ) ) ) ) ).
% Resid_Ide_Arr_ind
thf(fact_618_ide__char,axiom,
! [T4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ).
% ide_char
thf(fact_619_Ide__imp__Ide__tl,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( ( tl_lambda_lambda @ T4 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( tl_lambda_lambda @ T4 ) ) ) ) ).
% Ide_imp_Ide_tl
thf(fact_620_Ide__imp__Ide__hd,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) ) ) ).
% Ide_imp_Ide_hd
thf(fact_621_ide__trg,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% ide_trg
thf(fact_622_Ide__imp__Ide__last,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ide_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T4 ) ) ) ).
% Ide_imp_Ide_last
thf(fact_623_Std__map__App1,axiom,
! [B: lambda_lambda,T4: list_lambda_lambda] :
( ( lambda_Ide @ B )
=> ( ( reduction_Std @ T4 )
=> ( reduction_Std
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ B )
@ T4 ) ) ) ) ).
% Std_map_App1
thf(fact_624_Std__map__App2,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( reduction_Std @ T4 )
=> ( reduction_Std @ ( map_la336867333220232516lambda @ ( lambda_App @ A ) @ T4 ) ) ) ) ).
% Std_map_App2
thf(fact_625_coterminal__iff__con__trg,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).
% coterminal_iff_con_trg
thf(fact_626_stdz__insert__Ide,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) )
=> ( ( reduct5390047737487997688insert @ T @ U2 )
= nil_lambda_lambda ) ) ).
% stdz_insert_Ide
thf(fact_627_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_628_Trg__last__eqI,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ T4 ) )
= ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) ) ) ).
% Trg_last_eqI
thf(fact_629_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_630_development__map__App__2,axiom,
! [T: lambda_lambda,U: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( reduct9208763521642324371opment @ U @ U2 )
=> ( reduct9208763521642324371opment @ ( lambda_App @ T @ U ) @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ T ) ) @ U2 ) ) ) ) ).
% development_map_App_2
thf(fact_631_development__map__App__1,axiom,
! [T: lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ T4 )
=> ( ( lambda_Arr @ U )
=> ( reduct9208763521642324371opment @ ( lambda_App @ T @ U )
@ ( map_la336867333220232516lambda
@ ^ [X2: lambda_lambda] : ( lambda_App @ X2 @ ( lambda_Src @ U ) )
@ T4 ) ) ) ) ).
% development_map_App_1
thf(fact_632_map__ident,axiom,
( ( map_li385728805257398596lambda
@ ^ [X2: list_lambda_lambda] : X2 )
= ( ^ [Xs4: list_l5168835686021156086lambda] : Xs4 ) ) ).
% map_ident
thf(fact_633_map__ident,axiom,
( ( map_la336867333220232516lambda
@ ^ [X2: lambda_lambda] : X2 )
= ( ^ [Xs4: list_lambda_lambda] : Xs4 ) ) ).
% map_ident
thf(fact_634_cong__map__App2,axiom,
! [X3: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X3 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X3 )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X3 )
@ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X3 )
@ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X3 )
@ T4 ) ) ) ) ) ) ).
% cong_map_App2
thf(fact_635_map__App2__Resid,axiom,
! [X3: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X3 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda
@ ^ [T7: lambda_lambda] : ( lambda_App @ T7 @ X3 )
@ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T7: lambda_lambda] : ( lambda_App @ T7 @ X3 )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [T7: lambda_lambda] : ( lambda_App @ T7 @ X3 )
@ U2 ) ) ) ) ) ).
% map_App2_Resid
thf(fact_636_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_637__092_060open_062_092_060And_062t_O_Aseq_A_091t_093_A_091_093_A_092_060and_062_AStd_A_091_093_A_092_060longrightarrow_062_AStd_A_Istdz__insert_At_A_091_093_J_A_092_060and_062_A_I_092_060not_062_AIde_A_091t_093_A_092_060longrightarrow_062_Astdz__insert_At_A_091_093_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_091t_093_J_092_060close_062,axiom,
! [T: lambda_lambda] :
( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ nil_lambda_lambda )
& ( reduction_Std @ nil_lambda_lambda ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ T @ nil_lambda_lambda ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ 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 ) @ ( reduct5390047737487997688insert @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% \<open>\<And>t. seq [t] [] \<and> Std [] \<longrightarrow> Std (stdz_insert t []) \<and> (\<not> Ide [t] \<longrightarrow> stdz_insert t [] \<^sup>*\<sim>\<^sup>* [t])\<close>
thf(fact_638_ind1,axiom,
( ( lambda_Ide @ ( lambda_App @ m @ n ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ nil_lambda_lambda ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
& ( reduction_Std @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ) ) ) ).
% ind1
thf(fact_639_ind3,axiom,
( ~ ( lambda_Ide @ ( lambda_App @ m @ n ) )
=> ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ m @ n ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_App @ m @ n ) )
=> ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ u @ ua ) )
& ( reduction_Std @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) @ ( cons_lambda_lambda @ u @ ua ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) @ ( cons_lambda_lambda @ u @ ua ) ) @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) @ ( cons_lambda_lambda @ u @ ua ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) @ ( cons_lambda_lambda @ u @ ua ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_redex @ ( lambda_App @ m @ n ) ) ) @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ) ) ) ) ) ).
% ind3
thf(fact_640_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_641_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_642_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_643__092_060open_062_092_060And_062u_AU_AM_O_A_092_060lbrakk_062_092_060Lambda_062_OIde_AM_A_092_060Longrightarrow_062_Aseq_A_091hd_A_Iu_A_D_AU_J_093_A_Itl_A_Iu_A_D_AU_J_J_A_092_060and_062_AStd_A_Itl_A_Iu_A_D_AU_J_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_Ihd_A_Iu_A_D_AU_J_J_A_Itl_A_Iu_A_D_AU_J_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_Ihd_A_Iu_A_D_AU_J_A_D_Atl_A_Iu_A_D_AU_J_J_A_092_060longrightarrow_062_Astdz__insert_A_Ihd_A_Iu_A_D_AU_J_J_A_Itl_A_Iu_A_D_AU_J_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_Ahd_A_Iu_A_D_AU_J_A_D_Atl_A_Iu_A_D_AU_J_J_059_A_092_060not_062_A_092_060Lambda_062_OIde_AM_A_092_060Longrightarrow_062_Aseq_A_091M_093_A_Imap_A_092_060Lambda_062_Oun__Lam_A_Iu_A_D_AU_J_J_A_092_060and_062_AStd_A_Imap_A_092_060Lambda_062_Oun__Lam_A_Iu_A_D_AU_J_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_AM_A_Imap_A_092_060Lambda_062_Oun__Lam_A_Iu_A_D_AU_J_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_IM_A_D_Amap_A_092_060Lambda_062_Oun__Lam_A_Iu_A_D_AU_J_J_A_092_060longrightarrow_062_Astdz__insert_AM_A_Imap_A_092_060Lambda_062_Oun__Lam_A_Iu_A_D_AU_J_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_AM_A_D_Amap_A_092_060Lambda_062_Oun__Lam_A_Iu_A_D_AU_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Aseq_A_091_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_093_A_Iu_A_D_AU_J_A_092_060and_062_AStd_A_Iu_A_D_AU_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_Iu_A_D_AU_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_D_Au_A_D_AU_J_A_092_060longrightarrow_062_Astdz__insert_A_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_Iu_A_D_AU_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_D_Au_A_D_AU_J_092_060close_062,axiom,
! [M: lambda_lambda,U: lambda_lambda,U2: list_lambda_lambda] :
( ( ( lambda_Ide @ M )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ nil_lambda_lambda ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) )
& ( reduction_Std @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( lambda_Ide @ M )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ M @ nil_lambda_lambda ) @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ U @ U2 ) ) )
& ( reduction_Std @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ M @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ M @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ M @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ U @ U2 ) ) ) @ ( cons_lambda_lambda @ M @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ M @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ U @ U2 ) ) ) @ ( reduct5390047737487997688insert @ M @ ( map_la336867333220232516lambda @ lambda_un_Lam @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) ) ) ) ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Lam @ M ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
& ( reduction_Std @ ( cons_lambda_lambda @ U @ U2 ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_Lam @ M ) @ ( cons_lambda_lambda @ U @ U2 ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Lam @ M ) @ ( cons_lambda_lambda @ U @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_Lam @ M ) @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( cons_lambda_lambda @ ( lambda_Lam @ M ) @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Lam @ M ) @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( reduct5390047737487997688insert @ ( lambda_Lam @ M ) @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) ) ) ) ) ) ) ).
% \<open>\<And>u U M. \<lbrakk>\<Lambda>.Ide M \<Longrightarrow> seq [hd (u # U)] (tl (u # U)) \<and> Std (tl (u # U)) \<longrightarrow> Std (stdz_insert (hd (u # U)) (tl (u # U))) \<and> (\<not> Ide (hd (u # U) # tl (u # U)) \<longrightarrow> stdz_insert (hd (u # U)) (tl (u # U)) \<^sup>*\<sim>\<^sup>* hd (u # U) # tl (u # U)); \<not> \<Lambda>.Ide M \<Longrightarrow> seq [M] (map \<Lambda>.un_Lam (u # U)) \<and> Std (map \<Lambda>.un_Lam (u # U)) \<longrightarrow> Std (stdz_insert M (map \<Lambda>.un_Lam (u # U))) \<and> (\<not> Ide (M # map \<Lambda>.un_Lam (u # U)) \<longrightarrow> stdz_insert M (map \<Lambda>.un_Lam (u # U)) \<^sup>*\<sim>\<^sup>* M # map \<Lambda>.un_Lam (u # U))\<rbrakk> \<Longrightarrow> seq [\<^bold>\<lambda>\<^bold>[M\<^bold>]] (u # U) \<and> Std (u # U) \<longrightarrow> Std (stdz_insert \<^bold>\<lambda>\<^bold>[M\<^bold>] (u # U)) \<and> (\<not> Ide (\<^bold>\<lambda>\<^bold>[M\<^bold>] # u # U) \<longrightarrow> stdz_insert \<^bold>\<lambda>\<^bold>[M\<^bold>] (u # U) \<^sup>*\<sim>\<^sup>* \<^bold>\<lambda>\<^bold>[M\<^bold>] # u # U)\<close>
thf(fact_644_lambda__calculus_OseqI_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Trg @ T )
= ( lambda_Src @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% lambda_calculus.seqI\<^sub>\<Lambda>
thf(fact_645_lambda__calculus_OTrg_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Trg @ ( lambda_App @ T @ U ) )
= ( lambda_App @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) ) ).
% lambda_calculus.Trg.simps(3)
thf(fact_646_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_647_lambda__calculus_OArr__Trg,axiom,
! [X6: lambda_lambda] :
( ( lambda_Arr @ X6 )
=> ( lambda_Arr @ ( lambda_Trg @ X6 ) ) ) ).
% lambda_calculus.Arr_Trg
thf(fact_648_list_Omap__ident,axiom,
! [T: list_l5168835686021156086lambda] :
( ( map_li385728805257398596lambda
@ ^ [X2: list_lambda_lambda] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_649_list_Omap__ident,axiom,
! [T: list_lambda_lambda] :
( ( map_la336867333220232516lambda
@ ^ [X2: lambda_lambda] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_650_paths__in__rts_OIde_Ocong,axiom,
paths_5097954201535619375lambda = paths_5097954201535619375lambda ).
% paths_in_rts.Ide.cong
thf(fact_651_paths__in__rts_OIde_Ocong,axiom,
paths_7717490669901197737lambda = paths_7717490669901197737lambda ).
% paths_in_rts.Ide.cong
thf(fact_652_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_653_lambda__calculus_OIde__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Ide_Trg
thf(fact_654_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_655_reduction__paths_Ostandard__development_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) )
=> ( ( reduct1603535164923377302opment @ ( lambda_App @ T @ U ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [V5: lambda_lambda] : ( lambda_App @ V5 @ ( lambda_Src @ U ) )
@ ( reduct1603535164923377302opment @ T ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ T ) ) @ ( reduct1603535164923377302opment @ U ) ) ) ) )
& ( ~ ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) )
=> ( ( reduct1603535164923377302opment @ ( lambda_App @ T @ U ) )
= nil_lambda_lambda ) ) ) ).
% reduction_paths.standard_development.simps(4)
thf(fact_656_lambda__calculus_Oseq__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Trg @ T )
= ( lambda_Src @ U ) ) ) ) ).
% lambda_calculus.seq_char
thf(fact_657_lambda__calculus_OseqE_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ~ ( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( lambda_Trg @ T )
!= ( lambda_Src @ U ) ) ) ) ) ).
% lambda_calculus.seqE\<^sub>\<Lambda>
thf(fact_658_reduction__paths_Oorthogonal__App__single__single_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ U ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ U ) ) @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ T ) @ U ) @ nil_lambda_lambda ) ) ) ) ).
% reduction_paths.orthogonal_App_single_single(1)
thf(fact_659_reduction__paths_Oorthogonal__App__single__single_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ U ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ U ) @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Trg @ U ) ) @ nil_lambda_lambda ) ) ) ) ).
% reduction_paths.orthogonal_App_single_single(2)
thf(fact_660__092_060open_062_092_060And_062U_AN_AM_AB_AA_O_Aseq_A_091_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AN_093_AU_A_092_060and_062_AStd_AU_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AN_J_AU_J_A_092_060and_062_A_I_092_060not_062_AIde_A_I_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AN_J_A_D_AU_J_A_092_060longrightarrow_062_Astdz__insert_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AN_J_AU_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AN_J_A_D_AU_J_A_092_060Longrightarrow_062_Aseq_A_091_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060circ_062_AN_093_A_I_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091A_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AB_J_A_D_AU_J_A_092_060and_062_AStd_A_I_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091A_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AB_J_A_D_AU_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060circ_062_AN_J_A_I_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091A_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AB_J_A_D_AU_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_I_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060circ_062_AN_J_A_D_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091A_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AB_J_A_D_AU_J_A_092_060longrightarrow_062_Astdz__insert_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060circ_062_AN_J_A_I_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091A_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AB_J_A_D_AU_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091M_092_060_094bold_062_093_A_092_060_094bold_062_092_060circ_062_AN_J_A_D_A_I_092_060_094bold_062_092_060lambda_062_092_060_094bold_062_091A_092_060_094bold_062_093_A_092_060_094bold_062_092_060Zspot_062_AB_J_A_D_AU_J_092_060close_062,axiom,
! [M: lambda_lambda,N2: lambda_lambda,U2: list_lambda_lambda,A2: lambda_lambda,B2: lambda_lambda] :
( ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Beta @ M @ N2 ) @ nil_lambda_lambda ) @ U2 )
& ( reduction_Std @ U2 ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_Beta @ M @ N2 ) @ U2 ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ M @ N2 ) @ U2 ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_Beta @ M @ N2 ) @ U2 ) @ ( cons_lambda_lambda @ ( lambda_Beta @ M @ N2 ) @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ M @ N2 ) @ U2 ) @ ( reduct5390047737487997688insert @ ( lambda_Beta @ M @ N2 ) @ U2 ) ) ) ) ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ M ) @ N2 ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Beta @ A2 @ B2 ) @ U2 ) )
& ( reduction_Std @ ( cons_lambda_lambda @ ( lambda_Beta @ A2 @ B2 ) @ U2 ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Lam @ M ) @ N2 ) @ ( cons_lambda_lambda @ ( lambda_Beta @ A2 @ B2 ) @ U2 ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ M ) @ N2 ) @ ( cons_lambda_lambda @ ( lambda_Beta @ A2 @ B2 ) @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Lam @ M ) @ N2 ) @ ( cons_lambda_lambda @ ( lambda_Beta @ A2 @ B2 ) @ U2 ) ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ M ) @ N2 ) @ ( cons_lambda_lambda @ ( lambda_Beta @ A2 @ B2 ) @ U2 ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ M ) @ N2 ) @ ( cons_lambda_lambda @ ( lambda_Beta @ A2 @ B2 ) @ U2 ) ) @ ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Lam @ M ) @ N2 ) @ ( cons_lambda_lambda @ ( lambda_Beta @ A2 @ B2 ) @ U2 ) ) ) ) ) ) ) ) ) ).
% \<open>\<And>U N M B A. seq [\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<Zspot> N] U \<and> Std U \<longrightarrow> Std (stdz_insert (\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<Zspot> N) U) \<and> (\<not> Ide ((\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<Zspot> N) # U) \<longrightarrow> stdz_insert (\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<Zspot> N) U \<^sup>*\<sim>\<^sup>* (\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<Zspot> N) # U) \<Longrightarrow> seq [\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<circ> N] ((\<^bold>\<lambda>\<^bold>[A\<^bold>] \<^bold>\<Zspot> B) # U) \<and> Std ((\<^bold>\<lambda>\<^bold>[A\<^bold>] \<^bold>\<Zspot> B) # U) \<longrightarrow> Std (stdz_insert (\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<circ> N) ((\<^bold>\<lambda>\<^bold>[A\<^bold>] \<^bold>\<Zspot> B) # U)) \<and> (\<not> Ide ((\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<circ> N) # (\<^bold>\<lambda>\<^bold>[A\<^bold>] \<^bold>\<Zspot> B) # U) \<longrightarrow> stdz_insert (\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<circ> N) ((\<^bold>\<lambda>\<^bold>[A\<^bold>] \<^bold>\<Zspot> B) # U) \<^sup>*\<sim>\<^sup>* (\<^bold>\<lambda>\<^bold>[M\<^bold>] \<^bold>\<circ> N) # (\<^bold>\<lambda>\<^bold>[A\<^bold>] \<^bold>\<Zspot> B) # U)\<close>
thf(fact_661__092_060open_062_092_060And_062x_Au_AU_O_Aseq_A_091hd_A_Iu_A_D_AU_J_093_A_Itl_A_Iu_A_D_AU_J_J_A_092_060and_062_AStd_A_Itl_A_Iu_A_D_AU_J_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_Ihd_A_Iu_A_D_AU_J_J_A_Itl_A_Iu_A_D_AU_J_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_Ihd_A_Iu_A_D_AU_J_A_D_Atl_A_Iu_A_D_AU_J_J_A_092_060longrightarrow_062_Astdz__insert_A_Ihd_A_Iu_A_D_AU_J_J_A_Itl_A_Iu_A_D_AU_J_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_Ahd_A_Iu_A_D_AU_J_A_D_Atl_A_Iu_A_D_AU_J_J_A_092_060Longrightarrow_062_Aseq_A_091_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_093_A_Iu_A_D_AU_J_A_092_060and_062_AStd_A_Iu_A_D_AU_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_A_Iu_A_D_AU_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_I_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_A_D_Au_A_D_AU_J_A_092_060longrightarrow_062_Astdz__insert_A_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_A_Iu_A_D_AU_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_092_060_094bold_062_092_060guillemotleft_062x_092_060_094bold_062_092_060guillemotright_062_A_D_Au_A_D_AU_J_092_060close_062,axiom,
! [U: lambda_lambda,U2: list_lambda_lambda,X3: nat] :
( ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ nil_lambda_lambda ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) )
& ( reduction_Std @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) ) ) ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Var @ X3 ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
& ( reduction_Std @ ( cons_lambda_lambda @ U @ U2 ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ ( lambda_Var @ X3 ) @ ( cons_lambda_lambda @ U @ U2 ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Var @ X3 ) @ ( cons_lambda_lambda @ U @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ ( lambda_Var @ X3 ) @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( cons_lambda_lambda @ ( lambda_Var @ X3 ) @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Var @ X3 ) @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( reduct5390047737487997688insert @ ( lambda_Var @ X3 ) @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) ) ) ) ) ) ).
% \<open>\<And>x u U. seq [hd (u # U)] (tl (u # U)) \<and> Std (tl (u # U)) \<longrightarrow> Std (stdz_insert (hd (u # U)) (tl (u # U))) \<and> (\<not> Ide (hd (u # U) # tl (u # U)) \<longrightarrow> stdz_insert (hd (u # U)) (tl (u # U)) \<^sup>*\<sim>\<^sup>* hd (u # U) # tl (u # U)) \<Longrightarrow> seq [\<^bold>\<guillemotleft>x\<^bold>\<guillemotright>] (u # U) \<and> Std (u # U) \<longrightarrow> Std (stdz_insert \<^bold>\<guillemotleft>x\<^bold>\<guillemotright> (u # U)) \<and> (\<not> Ide (\<^bold>\<guillemotleft>x\<^bold>\<guillemotright> # u # U) \<longrightarrow> stdz_insert \<^bold>\<guillemotleft>x\<^bold>\<guillemotright> (u # U) \<^sup>*\<sim>\<^sup>* \<^bold>\<guillemotleft>x\<^bold>\<guillemotright> # u # U)\<close>
thf(fact_662__092_060open_062_092_060And_062u_AU_O_Aseq_A_091_092_060_094bold_062_092_060sharp_062_093_A_Iu_A_D_AU_J_A_092_060and_062_AStd_A_Iu_A_D_AU_J_A_092_060longrightarrow_062_AStd_A_Istdz__insert_A_092_060_094bold_062_092_060sharp_062_A_Iu_A_D_AU_J_J_A_092_060and_062_A_I_092_060not_062_AIde_A_I_092_060_094bold_062_092_060sharp_062_A_D_Au_A_D_AU_J_A_092_060longrightarrow_062_Astdz__insert_A_092_060_094bold_062_092_060sharp_062_A_Iu_A_D_AU_J_A_092_060_094sup_062_K_092_060sim_062_092_060_094sup_062_K_A_092_060_094bold_062_092_060sharp_062_A_D_Au_A_D_AU_J_092_060close_062,axiom,
! [U: lambda_lambda,U2: list_lambda_lambda] :
( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ lambda_Nil @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
& ( reduction_Std @ ( cons_lambda_lambda @ U @ U2 ) ) )
=> ( ( reduction_Std @ ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ U @ U2 ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ lambda_Nil @ ( cons_lambda_lambda @ U @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( cons_lambda_lambda @ lambda_Nil @ ( cons_lambda_lambda @ U @ U2 ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ lambda_Nil @ ( cons_lambda_lambda @ U @ U2 ) ) @ ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ U @ U2 ) ) ) ) ) ) ) ) ).
% \<open>\<And>u U. seq [\<^bold>\<sharp>] (u # U) \<and> Std (u # U) \<longrightarrow> Std (stdz_insert \<^bold>\<sharp> (u # U)) \<and> (\<not> Ide (\<^bold>\<sharp> # u # U) \<longrightarrow> stdz_insert \<^bold>\<sharp> (u # U) \<^sup>*\<sim>\<^sup>* \<^bold>\<sharp> # u # U)\<close>
thf(fact_663_ind8,axiom,
( ~ ( lambda_Ide @ ( lambda_App @ m @ n ) )
=> ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ m @ n ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ m @ n ) )
=> ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ n @ nil_lambda_lambda )
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) )
& ( reduction_Std
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
=> ( ( reduction_Std
@ ( reduct5390047737487997688insert @ n
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid
@ ( cons_lambda_lambda @ n
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( reduct5390047737487997688insert @ n
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) )
@ ( cons_lambda_lambda @ n
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( cons_lambda_lambda @ n
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) )
@ ( reduct5390047737487997688insert @ n
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ind8
thf(fact_664_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_665_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_666_filter__notIde__Ide,axiom,
! [U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ U2 )
= nil_lambda_lambda ) ) ).
% filter_notIde_Ide
thf(fact_667_filter__filter,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o,Xs2: list_l5168835686021156086lambda] :
( ( filter1131015166755930349lambda @ P @ ( filter1131015166755930349lambda @ Q @ Xs2 ) )
= ( filter1131015166755930349lambda
@ ^ [X2: list_lambda_lambda] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_668_filter__filter,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o,Xs2: list_lambda_lambda] :
( ( filter_lambda_lambda @ P @ ( filter_lambda_lambda @ Q @ Xs2 ) )
= ( filter_lambda_lambda
@ ^ [X2: lambda_lambda] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_669_filter__append,axiom,
! [P: list_lambda_lambda > $o,Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( filter1131015166755930349lambda @ P @ ( append1017360404025248779lambda @ Xs2 @ Ys ) )
= ( append1017360404025248779lambda @ ( filter1131015166755930349lambda @ P @ Xs2 ) @ ( filter1131015166755930349lambda @ P @ Ys ) ) ) ).
% filter_append
thf(fact_670_filter__append,axiom,
! [P: lambda_lambda > $o,Xs2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( filter_lambda_lambda @ P @ ( append_lambda_lambda @ Xs2 @ Ys ) )
= ( append_lambda_lambda @ ( filter_lambda_lambda @ P @ Xs2 ) @ ( filter_lambda_lambda @ P @ Ys ) ) ) ).
% filter_append
thf(fact_671_lambda__calculus_Ocon__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ).
% lambda_calculus.con_char
thf(fact_672_lambda__calculus_Onull__char,axiom,
( ( partia1461432607296028304lambda @ lambda_resid )
= lambda_Nil ) ).
% lambda_calculus.null_char
thf(fact_673_lambda__calculus_OConI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Src @ T )
= ( lambda_Src @ U ) )
=> ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ) ) ).
% lambda_calculus.ConI
thf(fact_674_lambda__calculus_Otrg__char,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_Trg @ T ) ) )
& ( ~ ( lambda_Arr @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= lambda_Nil ) ) ) ).
% lambda_calculus.trg_char
thf(fact_675_ind7,axiom,
( ~ ( lambda_Ide @ ( lambda_App @ m @ n ) )
=> ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ m @ n ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ m @ n ) )
=> ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ u @ ua ) ) )
=> ( ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ m @ nil_lambda_lambda )
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) ) )
& ( reduction_Std
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
=> ( ( reduction_Std
@ ( reduct5390047737487997688insert @ m
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
& ( ~ ( paths_7717490669901197737lambda @ lambda_resid
@ ( cons_lambda_lambda @ m
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( reduct5390047737487997688insert @ m
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) ) )
@ ( cons_lambda_lambda @ m
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( cons_lambda_lambda @ m
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) ) )
@ ( reduct5390047737487997688insert @ m
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ u @ ua ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ind7
thf(fact_676_lambda__calculus_OTrg_Osimps_I5_J,axiom,
( ( lambda_Trg @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Trg.simps(5)
thf(fact_677_lambda__calculus_OTrg_Osimps_I1_J,axiom,
! [I: nat] :
( ( lambda_Trg @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Trg.simps(1)
thf(fact_678_lambda__calculus_OCon__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
= ( ( lambda_resid @ U @ T )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_sym
thf(fact_679_lambda__calculus_OCube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ).
% lambda_calculus.Cube
thf(fact_680_lambda__calculus_Oresid_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(7)
thf(fact_681_lambda__calculus_Oresid_Osimps_I26_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_resid @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(26)
thf(fact_682_lambda__calculus_OArr_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Arr @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Arr.simps(2)
thf(fact_683_lambda__calculus_Olambda_Odistinct_I5_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( lambda_Nil
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(5)
thf(fact_684_lambda__calculus_OSrc_Osimps_I2_J,axiom,
! [I: nat] :
( ( lambda_Src @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Src.simps(2)
thf(fact_685_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_686_lambda__calculus_OArr_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Beta @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(5)
thf(fact_687_lambda__calculus_Ocontains__head__reduction_Osimps_I1_J,axiom,
! [Uu: nat] :
~ ( lambda8135948770199036598uction @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.contains_head_reduction.simps(1)
thf(fact_688_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T6 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T6 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(3)
thf(fact_689_lambda__calculus_Ohead__strategy_Osimps_I1_J,axiom,
! [I: nat] :
( ( lambda_head_strategy @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.head_strategy.simps(1)
thf(fact_690_lambda__calculus_Ohead__redex_Osimps_I2_J,axiom,
! [X3: nat] :
( ( lambda_head_redex @ ( lambda_Var @ X3 ) )
= ( lambda_Var @ X3 ) ) ).
% lambda_calculus.head_redex.simps(2)
thf(fact_691_lambda__calculus_Olambda_Odistinct_I3_J,axiom,
! [X32: lambda_lambda] :
( lambda_Nil
!= ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.distinct(3)
thf(fact_692_lambda__calculus_OIde_Osimps_I1_J,axiom,
~ ( lambda_Ide @ lambda_Nil ) ).
% lambda_calculus.Ide.simps(1)
thf(fact_693_lambda__calculus_OArr_Osimps_I1_J,axiom,
~ ( lambda_Arr @ lambda_Nil ) ).
% lambda_calculus.Arr.simps(1)
thf(fact_694_lambda__calculus_OArr__not__Nil,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( T != lambda_Nil ) ) ).
% lambda_calculus.Arr_not_Nil
thf(fact_695_lambda__calculus_OCoinitial__cases,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ T6 )
=> ( ( ( lambda_Src @ T )
= ( lambda_Src @ T6 ) )
=> ( ( ( T = lambda_Nil )
& ( T6 = lambda_Nil ) )
| ? [X: nat] :
( ( T
= ( lambda_Var @ X ) )
& ( T6
= ( lambda_Var @ X ) ) )
| ? [U6: lambda_lambda,U8: lambda_lambda] :
( ( T
= ( lambda_Lam @ U6 ) )
& ( T6
= ( lambda_Lam @ U8 ) ) )
| ? [U6: lambda_lambda,V3: lambda_lambda,U8: lambda_lambda,V6: lambda_lambda] :
( ( T
= ( lambda_App @ U6 @ V3 ) )
& ( T6
= ( lambda_App @ U8 @ V6 ) ) )
| ? [U6: lambda_lambda,V3: lambda_lambda,U8: lambda_lambda,V6: lambda_lambda] :
( ( T
= ( lambda_Beta @ U6 @ V3 ) )
& ( T6
= ( lambda_Beta @ U8 @ V6 ) ) )
| ? [U6: lambda_lambda,V3: lambda_lambda,U8: lambda_lambda,V6: lambda_lambda] :
( ( T
= ( lambda_App @ ( lambda_Lam @ U6 ) @ V3 ) )
& ( T6
= ( lambda_Beta @ U8 @ V6 ) ) )
| ? [U6: lambda_lambda,V3: lambda_lambda,U8: lambda_lambda,V6: lambda_lambda] :
( ( T
= ( lambda_Beta @ U6 @ V3 ) )
& ( T6
= ( lambda_App @ ( lambda_Lam @ U8 ) @ V6 ) ) ) ) ) ) ) ).
% lambda_calculus.Coinitial_cases
thf(fact_696_lambda__calculus_Ocontains__head__reduction_Oelims_I3_J,axiom,
! [X3: lambda_lambda] :
( ~ ( lambda8135948770199036598uction @ X3 )
=> ( ! [Uu2: nat] :
( X3
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( lambda8135948770199036598uction @ T3 ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ( ! [U6: lambda_lambda] :
( ( X3
= ( lambda_App @ lambda_Nil @ U6 ) )
=> ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U6 ) ) )
=> ( ! [V3: nat,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Var @ V3 ) @ U6 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U6 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U6 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U6 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U6 ) )
=> ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U6 ) ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) )
=> ( X3 = lambda_Nil ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.elims(3)
thf(fact_697_lambda__calculus_Ocontains__head__reduction_Oelims_I2_J,axiom,
! [X3: lambda_lambda] :
( ( lambda8135948770199036598uction @ X3 )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda8135948770199036598uction @ T3 ) )
=> ( ! [U6: lambda_lambda] :
( ( X3
= ( lambda_App @ lambda_Nil @ U6 ) )
=> ~ ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U6 ) ) )
=> ( ! [V3: nat,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Var @ V3 ) @ U6 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U6 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U6 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U6 ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U6 ) )
=> ~ ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U6 ) ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.elims(2)
thf(fact_698_lambda__calculus_Ocontains__head__reduction_Oelims_I1_J,axiom,
! [X3: lambda_lambda,Y4: $o] :
( ( ( lambda8135948770199036598uction @ X3 )
= Y4 )
=> ( ( ? [Uu2: nat] :
( X3
= ( lambda_Var @ Uu2 ) )
=> Y4 )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( Y4
= ( ~ ( lambda8135948770199036598uction @ T3 ) ) ) )
=> ( ( ? [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X3
= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> Y4 )
=> ( ! [U6: lambda_lambda] :
( ( X3
= ( lambda_App @ lambda_Nil @ U6 ) )
=> ( Y4
= ( ~ ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U6 ) ) ) ) )
=> ( ! [V3: nat,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Var @ V3 ) @ U6 ) )
=> ( Y4
= ( ~ ( ( lambda8135948770199036598uction @ ( lambda_Var @ V3 ) )
& ( lambda_Arr @ U6 ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U6 ) )
=> ( Y4
= ( ~ ( ( lambda8135948770199036598uction @ ( lambda_App @ V3 @ Va2 ) )
& ( lambda_Arr @ U6 ) ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U6 ) )
=> ( Y4
= ( ~ ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V3 @ Va2 ) )
& ( lambda_Arr @ U6 ) ) ) ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ( Y4
= ( ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) ) ) )
=> ~ ( ( X3 = lambda_Nil )
=> Y4 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.elims(1)
thf(fact_699_lambda__calculus_OSrc_Osimps_I1_J,axiom,
( ( lambda_Src @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Src.simps(1)
thf(fact_700_lambda__calculus_Ohead__strategy_Oelims,axiom,
! [X3: lambda_lambda,Y4: lambda_lambda] :
( ( ( lambda_head_strategy @ X3 )
= Y4 )
=> ( ! [I3: nat] :
( ( X3
= ( lambda_Var @ I3 ) )
=> ( Y4
!= ( lambda_Var @ I3 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( Y4
!= ( lambda_Lam @ ( lambda_head_strategy @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Lam @ T3 ) @ U6 ) )
=> ( Y4
!= ( lambda_Beta @ ( lambda_Src @ T3 ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [U6: lambda_lambda] :
( ( X3
= ( lambda_App @ lambda_Nil @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_head_strategy @ lambda_Nil ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [V3: nat,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Var @ V3 ) @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Var @ V3 ) ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_head_strategy @ ( lambda_App @ V3 @ Va2 ) ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Beta @ V3 @ Va2 ) ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ( Y4
!= ( lambda_Beta @ ( lambda_Src @ T3 ) @ ( lambda_Src @ U6 ) ) ) )
=> ~ ( ( X3 = lambda_Nil )
=> ( Y4 != lambda_Nil ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.head_strategy.elims
thf(fact_701_lambda__calculus_Ocontains__head__reduction_Osimps_I9_J,axiom,
~ ( lambda8135948770199036598uction @ lambda_Nil ) ).
% lambda_calculus.contains_head_reduction.simps(9)
thf(fact_702_lambda__calculus_Ohead__redex_Oelims,axiom,
! [X3: lambda_lambda,Y4: lambda_lambda] :
( ( ( lambda_head_redex @ X3 )
= Y4 )
=> ( ( ( X3 = lambda_Nil )
=> ( Y4 != lambda_Nil ) )
=> ( ! [X: nat] :
( ( X3
= ( lambda_Var @ X ) )
=> ( Y4
!= ( lambda_Var @ X ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( Y4
!= ( lambda_Lam @ ( lambda_head_redex @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Lam @ T3 ) @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T3 ) ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [U6: lambda_lambda] :
( ( X3
= ( lambda_App @ lambda_Nil @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_head_redex @ lambda_Nil ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [V3: nat,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Var @ V3 ) @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_head_redex @ ( lambda_Var @ V3 ) ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_head_redex @ ( lambda_App @ V3 @ Va2 ) ) @ ( lambda_Src @ U6 ) ) ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_head_redex @ ( lambda_Beta @ V3 @ Va2 ) ) @ ( lambda_Src @ U6 ) ) ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ( Y4
!= ( lambda_Beta @ ( lambda_Src @ T3 ) @ ( lambda_Src @ U6 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.head_redex.elims
thf(fact_703_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_704_lambda__calculus_Ohead__strategy_Osimps_I9_J,axiom,
( ( lambda_head_strategy @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.head_strategy.simps(9)
thf(fact_705_lambda__calculus_Ohead__redex_Osimps_I1_J,axiom,
( ( lambda_head_redex @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.head_redex.simps(1)
thf(fact_706_lambda__calculus_Olambda_Odisc_I12_J,axiom,
! [X23: nat] :
~ ( lambda_is_App @ ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.disc(12)
thf(fact_707_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_708_lambda__calculus_OIde_Oelims_I1_J,axiom,
! [X3: lambda_lambda,Y4: $o] :
( ( ( lambda_Ide @ X3 )
= Y4 )
=> ( ( ( X3 = lambda_Nil )
=> Y4 )
=> ( ( ? [Uu2: nat] :
( X3
= ( lambda_Var @ Uu2 ) )
=> ~ Y4 )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( Y4
= ( ~ ( lambda_Ide @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ T3 @ U6 ) )
=> ( Y4
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U6 ) ) ) ) )
=> ~ ( ? [T3: lambda_lambda,U6: lambda_lambda] :
( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> Y4 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(1)
thf(fact_709_lambda__calculus_OArr_Oelims_I1_J,axiom,
! [X3: lambda_lambda,Y4: $o] :
( ( ( lambda_Arr @ X3 )
= Y4 )
=> ( ( ( X3 = lambda_Nil )
=> Y4 )
=> ( ( ? [Uu2: nat] :
( X3
= ( lambda_Var @ Uu2 ) )
=> ~ Y4 )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( Y4
= ( ~ ( lambda_Arr @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ T3 @ U6 ) )
=> ( Y4
= ( ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) ) ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ( Y4
= ( ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(1)
thf(fact_710_filter_Osimps_I2_J,axiom,
! [P: list_lambda_lambda > $o,X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( ( P @ X3 )
=> ( ( filter1131015166755930349lambda @ P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) )
= ( cons_l4558873032622161958lambda @ X3 @ ( filter1131015166755930349lambda @ P @ Xs2 ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( filter1131015166755930349lambda @ P @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) )
= ( filter1131015166755930349lambda @ P @ Xs2 ) ) ) ) ).
% filter.simps(2)
thf(fact_711_filter_Osimps_I2_J,axiom,
! [P: lambda_lambda > $o,X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( P @ X3 )
=> ( ( filter_lambda_lambda @ P @ ( cons_lambda_lambda @ X3 @ Xs2 ) )
= ( cons_lambda_lambda @ X3 @ ( filter_lambda_lambda @ P @ Xs2 ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( filter_lambda_lambda @ P @ ( cons_lambda_lambda @ X3 @ Xs2 ) )
= ( filter_lambda_lambda @ P @ Xs2 ) ) ) ) ).
% filter.simps(2)
thf(fact_712_filter_Osimps_I1_J,axiom,
! [P: list_lambda_lambda > $o] :
( ( filter1131015166755930349lambda @ P @ nil_li8379291187143545974lambda )
= nil_li8379291187143545974lambda ) ).
% filter.simps(1)
thf(fact_713_filter_Osimps_I1_J,axiom,
! [P: lambda_lambda > $o] :
( ( filter_lambda_lambda @ P @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% filter.simps(1)
thf(fact_714_lambda__calculus_Oresid_Osimps_I13_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(13)
thf(fact_715_lambda__calculus_Oresid_Osimps_I32_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(32)
thf(fact_716_lambda__calculus_Oresid_Osimps_I15_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(15)
thf(fact_717_lambda__calculus_Oresid_Osimps_I17_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(17)
thf(fact_718_lambda__calculus_Oresid_Osimps_I18_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(18)
thf(fact_719_lambda__calculus_Oresid_Osimps_I22_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ lambda_Nil @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(22)
thf(fact_720_lambda__calculus_Oresid_Osimps_I24_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(24)
thf(fact_721_lambda__calculus_Oresid_Osimps_I25_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(25)
thf(fact_722_lambda__calculus_Oresid_Osimps_I34_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ lambda_Nil @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(34)
thf(fact_723_lambda__calculus_Oresid_Osimps_I36_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(36)
thf(fact_724_lambda__calculus_Oresid_Osimps_I37_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(37)
thf(fact_725_lambda__calculus_Oresid_Osimps_I41_J,axiom,
! [Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(41)
thf(fact_726_lambda__calculus_Oresid_Osimps_I43_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(43)
thf(fact_727_lambda__calculus_Oresid_Osimps_I44_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(44)
thf(fact_728_lambda__calculus_Oresid_Osimps_I28_J,axiom,
! [Va: nat,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Va ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(28)
thf(fact_729_lambda__calculus_Oresid_Osimps_I9_J,axiom,
! [V: lambda_lambda,Va: nat] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Var @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(9)
thf(fact_730_lambda__calculus_OConD_I2_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,V4: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_Beta @ V4 @ U4 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Lam @ V4 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U4 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(2)
thf(fact_731_lambda__calculus_Oresid_Osimps_I40_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(40)
thf(fact_732_lambda__calculus_Oresid_Osimps_I30_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(30)
thf(fact_733_lambda__calculus_Oresid_Osimps_I21_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(21)
thf(fact_734_lambda__calculus_Oresid_Osimps_I11_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Beta @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(11)
thf(fact_735_lambda__calculus_Osseq_Osimps_I29_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(29)
thf(fact_736_lambda__calculus_Osseq_Osimps_I28_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ lambda_Nil @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(28)
thf(fact_737_lambda__calculus_Osseq_Osimps_I13_J,axiom,
! [V: nat] :
~ ( lambda_sseq @ lambda_Nil @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(13)
thf(fact_738_lambda__calculus_Olambda_Odistinct_I1_J,axiom,
! [X23: nat] :
( lambda_Nil
!= ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.distinct(1)
thf(fact_739_lambda__calculus_Olambda_Odistinct_I7_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( lambda_Nil
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(7)
thf(fact_740_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_741_lambda__calculus_Oresid_Osimps_I20_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(20)
thf(fact_742_lambda__calculus_Oresid_Osimps_I39_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(39)
thf(fact_743_lambda__calculus_Oresid_Osimps_I19_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(19)
thf(fact_744_lambda__calculus_Oresid_Osimps_I38_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(38)
thf(fact_745_lambda__calculus_Oresid_Osimps_I1_J,axiom,
! [I: nat,I2: nat] :
( ( ( I = I2 )
=> ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I2 ) )
= ( lambda_Var @ I ) ) )
& ( ( I != I2 )
=> ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I2 ) )
= lambda_Nil ) ) ) ).
% lambda_calculus.resid.simps(1)
thf(fact_746_lambda__calculus_Oresid_Osimps_I16_J,axiom,
! [Vd: nat,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vd ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(16)
thf(fact_747_lambda__calculus_Oresid_Osimps_I23_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: nat,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(23)
thf(fact_748_lambda__calculus_Oresid_Osimps_I35_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: nat,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_Var @ Vd ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(35)
thf(fact_749_lambda__calculus_Oresid_Osimps_I42_J,axiom,
! [Vd: nat,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(42)
thf(fact_750_lambda__calculus_OTrg_Ocases,axiom,
! [X3: lambda_lambda] :
( ! [I3: nat] :
( X3
!= ( lambda_Var @ I3 ) )
=> ( ! [T3: lambda_lambda] :
( X3
!= ( lambda_Lam @ T3 ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_App @ T3 @ U6 ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_Beta @ T3 @ U6 ) )
=> ( X3 = lambda_Nil ) ) ) ) ) ).
% lambda_calculus.Trg.cases
thf(fact_751_lambda__calculus_Olambda_Oexhaust,axiom,
! [Y4: lambda_lambda] :
( ( Y4 != lambda_Nil )
=> ( ! [X24: nat] :
( Y4
!= ( lambda_Var @ X24 ) )
=> ( ! [X34: lambda_lambda] :
( Y4
!= ( lambda_Lam @ X34 ) )
=> ( ! [X413: lambda_lambda,X423: lambda_lambda] :
( Y4
!= ( lambda_App @ X413 @ X423 ) )
=> ~ ! [X512: lambda_lambda,X522: lambda_lambda] :
( Y4
!= ( lambda_Beta @ X512 @ X522 ) ) ) ) ) ) ).
% lambda_calculus.lambda.exhaust
thf(fact_752_lambda__calculus_Ohead__redex_Ocases,axiom,
! [X3: lambda_lambda] :
( ( X3 != lambda_Nil )
=> ( ! [X: nat] :
( X3
!= ( lambda_Var @ X ) )
=> ( ! [T3: lambda_lambda] :
( X3
!= ( lambda_Lam @ T3 ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_Lam @ T3 ) @ U6 ) )
=> ( ! [U6: lambda_lambda] :
( X3
!= ( lambda_App @ lambda_Nil @ U6 ) )
=> ( ! [V3: nat,U6: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_Var @ V3 ) @ U6 ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U6 ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U6 ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_Beta @ T3 @ U6 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.head_redex.cases
thf(fact_753_reduction__paths_Ostandard__development_Ocases,axiom,
! [X3: lambda_lambda] :
( ( X3 != lambda_Nil )
=> ( ! [Uu2: nat] :
( X3
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( X3
!= ( lambda_Lam @ T3 ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_App @ T3 @ U6 ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_Beta @ T3 @ U6 ) ) ) ) ) ) ).
% reduction_paths.standard_development.cases
thf(fact_754_lambda__calculus_Ocontains__head__reduction_Ocases,axiom,
! [X3: lambda_lambda] :
( ! [Uu2: nat] :
( X3
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( X3
!= ( lambda_Lam @ T3 ) )
=> ( ! [Uv2: lambda_lambda,Uw2: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_Lam @ Uv2 ) @ Uw2 ) )
=> ( ! [U6: lambda_lambda] :
( X3
!= ( lambda_App @ lambda_Nil @ U6 ) )
=> ( ! [V3: nat,U6: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_Var @ V3 ) @ U6 ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_App @ V3 @ Va2 ) @ U6 ) )
=> ( ! [V3: lambda_lambda,Va2: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_App @ ( lambda_Beta @ V3 @ Va2 ) @ U6 ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_Beta @ T3 @ U6 ) )
=> ( X3 = lambda_Nil ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.contains_head_reduction.cases
thf(fact_755_lambda__calculus_OIde_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
~ ( lambda_Ide @ ( lambda_Beta @ T @ U ) ) ).
% lambda_calculus.Ide.simps(5)
thf(fact_756_lambda__calculus_OIde_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Ide @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Ide.simps(2)
thf(fact_757_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_758_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_759_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_760_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_761_lambda__calculus_Osseq_Osimps_I31_J,axiom,
! [Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ lambda_Nil @ Vc ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(31)
thf(fact_762_lambda__calculus_Osseq_Osimps_I15_J,axiom,
! [Vb: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ lambda_Nil @ Vb ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(15)
thf(fact_763_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_764_lambda__calculus_Osseq_Osimps_I32_J,axiom,
! [Vd: nat,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(32)
thf(fact_765_lambda__calculus_OSrc_Oelims,axiom,
! [X3: lambda_lambda,Y4: lambda_lambda] :
( ( ( lambda_Src @ X3 )
= Y4 )
=> ( ( ( X3 = lambda_Nil )
=> ( Y4 != lambda_Nil ) )
=> ( ! [I3: nat] :
( ( X3
= ( lambda_Var @ I3 ) )
=> ( Y4
!= ( lambda_Var @ I3 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( Y4
!= ( lambda_Lam @ ( lambda_Src @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ T3 @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_Src @ T3 ) @ ( lambda_Src @ U6 ) ) ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ( Y4
!= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T3 ) ) @ ( lambda_Src @ U6 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Src.elims
thf(fact_766_lambda__calculus_Olambda_Odisc_I7_J,axiom,
! [X23: nat] :
~ ( lambda_is_Lam @ ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.disc(7)
thf(fact_767_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_768_lambda__calculus_Osseq_Osimps_I1_J,axiom,
! [Uu: lambda_lambda] :
~ ( lambda_sseq @ Uu @ lambda_Nil ) ).
% lambda_calculus.sseq.simps(1)
thf(fact_769_lambda__calculus_Olambda_Odisc_I17_J,axiom,
! [X23: nat] :
~ ( lambda_is_Beta @ ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.disc(17)
thf(fact_770_lambda__calculus_Ois__Beta__def,axiom,
( lambda_is_Beta
= ( ^ [Lambda2: lambda_lambda] :
? [X513: lambda_lambda,X523: lambda_lambda] :
( Lambda2
= ( lambda_Beta @ X513 @ X523 ) ) ) ) ).
% lambda_calculus.is_Beta_def
thf(fact_771_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_772_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_773_lambda__calculus_Olambda_Odistinct__disc_I5_J,axiom,
! [Lambda: lambda_lambda] :
( ( Lambda = lambda_Nil )
=> ~ ( lambda_is_App @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(5)
thf(fact_774_lambda__calculus_Olambda_Odistinct__disc_I6_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_App @ Lambda )
=> ( Lambda != lambda_Nil ) ) ).
% lambda_calculus.lambda.distinct_disc(6)
thf(fact_775_lambda__calculus_Olambda_Odisc_I11_J,axiom,
~ ( lambda_is_App @ lambda_Nil ) ).
% lambda_calculus.lambda.disc(11)
thf(fact_776_lambda__calculus_Olambda_Odistinct__disc_I3_J,axiom,
! [Lambda: lambda_lambda] :
( ( Lambda = lambda_Nil )
=> ~ ( lambda_is_Lam @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(3)
thf(fact_777_lambda__calculus_Olambda_Odistinct__disc_I4_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ( Lambda != lambda_Nil ) ) ).
% lambda_calculus.lambda.distinct_disc(4)
thf(fact_778_lambda__calculus_Olambda_Odisc_I6_J,axiom,
~ ( lambda_is_Lam @ lambda_Nil ) ).
% lambda_calculus.lambda.disc(6)
thf(fact_779_lambda__calculus_Olambda_Odistinct__disc_I7_J,axiom,
! [Lambda: lambda_lambda] :
( ( Lambda = lambda_Nil )
=> ~ ( lambda_is_Beta @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(7)
thf(fact_780_lambda__calculus_Olambda_Odistinct__disc_I8_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Beta @ Lambda )
=> ( Lambda != lambda_Nil ) ) ).
% lambda_calculus.lambda.distinct_disc(8)
thf(fact_781_lambda__calculus_Olambda_Odisc_I16_J,axiom,
~ ( lambda_is_Beta @ lambda_Nil ) ).
% lambda_calculus.lambda.disc(16)
thf(fact_782_lambda__calculus_OArrE,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ! [I3: nat] :
( T
!= ( lambda_Var @ I3 ) )
=> ( ! [U6: lambda_lambda] :
( T
!= ( lambda_Lam @ U6 ) )
=> ( ! [U6: lambda_lambda,V3: lambda_lambda] :
( T
!= ( lambda_App @ U6 @ V3 ) )
=> ~ ! [U6: lambda_lambda,V3: lambda_lambda] :
( T
!= ( lambda_Beta @ U6 @ V3 ) ) ) ) ) ) ).
% lambda_calculus.ArrE
thf(fact_783_lambda__calculus_OArr_Oelims_I2_J,axiom,
! [X3: lambda_lambda] :
( ( lambda_Arr @ X3 )
=> ( ! [Uu2: nat] :
( X3
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ T3 @ U6 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(2)
thf(fact_784_lambda__calculus_OConD_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V4: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_Beta @ V4 @ U4 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ ( lambda_Lam @ V4 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U4 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(4)
thf(fact_785_lambda__calculus_OConD_I3_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_App @ T6 @ U4 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ T6 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U4 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(3)
thf(fact_786_lambda__calculus_Oresid_Osimps_I6_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda,U: lambda_lambda,U4: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U4 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T6 ) @ U4 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U4 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T6 ) @ U4 ) )
= ( lambda_Beta @ ( lambda_resid @ T @ T6 ) @ ( lambda_resid @ U @ U4 ) ) ) ) ) ).
% lambda_calculus.resid.simps(6)
thf(fact_787_lambda__calculus_OIde_Oelims_I3_J,axiom,
! [X3: lambda_lambda] :
( ~ ( lambda_Ide @ X3 )
=> ( ( X3 != lambda_Nil )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( lambda_Ide @ T3 ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ T3 @ U6 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U6 ) ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( X3
!= ( lambda_Beta @ T3 @ U6 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(3)
thf(fact_788_lambda__calculus_OArr_Oelims_I3_J,axiom,
! [X3: lambda_lambda] :
( ~ ( lambda_Arr @ X3 )
=> ( ( X3 != lambda_Nil )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ( lambda_Arr @ T3 ) )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ T3 @ U6 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_Beta @ T3 @ U6 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U6 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(3)
thf(fact_789_lambda__calculus_OApp__Var__contains__no__head__reduction,axiom,
! [X3: nat,U: lambda_lambda] :
~ ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Var @ X3 ) @ U ) ) ).
% lambda_calculus.App_Var_contains_no_head_reduction
thf(fact_790_lambda__calculus_Oprfx__Var__iff,axiom,
! [U: lambda_lambda,I: nat] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Var @ I ) ) )
= ( U
= ( lambda_Var @ I ) ) ) ).
% lambda_calculus.prfx_Var_iff
thf(fact_791_lambda__calculus_OConD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T6 @ U4 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ T6 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U4 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(1)
thf(fact_792_lambda__calculus_Oresid_Osimps_I31_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(31)
thf(fact_793_lambda__calculus_Oresid_Osimps_I12_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(12)
thf(fact_794_lambda__calculus_Oresid_Osimps_I3_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda,U: lambda_lambda,U4: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U4 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T6 @ U4 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U4 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T6 @ U4 ) )
= ( lambda_App @ ( lambda_resid @ T @ T6 ) @ ( lambda_resid @ U @ U4 ) ) ) ) ) ).
% lambda_calculus.resid.simps(3)
thf(fact_795_lambda__calculus_Oresid_Osimps_I2_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( ( ( lambda_resid @ T @ T6 )
= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T6 ) )
= lambda_Nil ) )
& ( ( ( lambda_resid @ T @ T6 )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T6 ) )
= ( lambda_Lam @ ( lambda_resid @ T @ T6 ) ) ) ) ) ).
% lambda_calculus.resid.simps(2)
thf(fact_796_lambda__calculus_Oresid_Osimps_I8_J,axiom,
! [V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(8)
thf(fact_797_lambda__calculus_Oresid_Osimps_I27_J,axiom,
! [V: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(27)
thf(fact_798_lambda__calculus_Osseq_Osimps_I16_J,axiom,
! [Vc: nat,Vb: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Var @ Vc ) @ Vb ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(16)
thf(fact_799_lambda__calculus_Osseq_Osimps_I18_J,axiom,
! [Vc: lambda_lambda,Vd: lambda_lambda,Vb: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vc @ Vd ) @ Vb ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(18)
thf(fact_800_lambda__calculus_Osseq_Osimps_I26_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Var @ Vb ) @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(26)
thf(fact_801_lambda__calculus_Ocontains__head__reduction_Osimps_I8_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_Beta @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(8)
thf(fact_802_lambda__calculus_Osseq_Osimps_I33_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(33)
thf(fact_803_lambda__calculus_Ohead__strategy_Osimps_I8_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_Beta @ T @ U ) )
= ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(8)
thf(fact_804_lambda__calculus_Ohead__redex_Osimps_I9_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_Beta @ T @ U ) )
= ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(9)
thf(fact_805_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_806_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_807_lambda__calculus_OArr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ T @ U ) ) ) ).
% lambda_calculus.Arr_resid
thf(fact_808_lambda__calculus_OCon__implies__Arr1,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Con_implies_Arr1
thf(fact_809_lambda__calculus_OCon__implies__Arr2,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ U ) ) ).
% lambda_calculus.Con_implies_Arr2
thf(fact_810_lambda__calculus_Osseq_Osimps_I30_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(30)
thf(fact_811_reduction__paths_Ostandard__development_Osimps_I2_J,axiom,
! [Uu: nat] :
( ( reduct1603535164923377302opment @ ( lambda_Var @ Uu ) )
= nil_lambda_lambda ) ).
% reduction_paths.standard_development.simps(2)
thf(fact_812_lambda__calculus_Osseq_Osimps_I25_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
~ ( lambda_sseq @ lambda_Nil @ ( lambda_App @ V @ Va ) ) ).
% lambda_calculus.sseq.simps(25)
thf(fact_813_lambda__calculus_Osseq_Osimps_I19_J,axiom,
! [V: lambda_lambda] :
~ ( lambda_sseq @ lambda_Nil @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(19)
thf(fact_814_lambda__calculus_OCon__implies__is__Lam__iff__is__Lam,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_is_Lam @ T )
= ( lambda_is_Lam @ U ) ) ) ).
% lambda_calculus.Con_implies_is_Lam_iff_is_Lam
thf(fact_815_reduction__paths_Ostandard__development_Osimps_I1_J,axiom,
( ( reduct1603535164923377302opment @ lambda_Nil )
= nil_lambda_lambda ) ).
% reduction_paths.standard_development.simps(1)
thf(fact_816_lambda__calculus_OIde_Oelims_I2_J,axiom,
! [X3: lambda_lambda] :
( ( lambda_Ide @ X3 )
=> ( ! [Uu2: nat] :
( X3
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Ide @ T3 ) )
=> ~ ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( X3
= ( lambda_App @ T3 @ U6 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U6 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(2)
thf(fact_817_lambda__calculus_OSrc_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.Src.simps(5)
thf(fact_818_lambda__calculus_OSrc__eq__iff_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T6 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T6 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(4)
thf(fact_819_lambda__calculus_Ocontains__head__reduction_Osimps_I5_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( ( lambda8135948770199036598uction @ ( lambda_Var @ V ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(5)
thf(fact_820_lambda__calculus_Ohead__strategy_Osimps_I5_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Var @ V ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(5)
thf(fact_821_lambda__calculus_Ocontains__head__reduction_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( ( lambda8135948770199036598uction @ ( lambda_Beta @ V @ Va ) )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(7)
thf(fact_822_lambda__calculus_Ohead__redex_Osimps_I6_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Var @ V ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(6)
thf(fact_823_lambda__calculus_Ohead__strategy_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_head_strategy @ ( lambda_Beta @ V @ Va ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(7)
thf(fact_824_lambda__calculus_Ohead__redex_Osimps_I8_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ V @ Va ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Beta @ V @ Va ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(8)
thf(fact_825_lambda__calculus_Osseq_Osimps_I17_J,axiom,
! [Vc: lambda_lambda,Vb: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Lam @ Vc ) @ Vb ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(17)
thf(fact_826_lambda__calculus_Osseq_Osimps_I22_J,axiom,
! [Vc: nat,Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Var @ Vc ) @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(22)
thf(fact_827_lambda__calculus_Oresid_Osimps_I33_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(33)
thf(fact_828_lambda__calculus_Oresid_Osimps_I29_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(29)
thf(fact_829_lambda__calculus_Oresid_Osimps_I14_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(14)
thf(fact_830_lambda__calculus_Oresid_Osimps_I10_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_App @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(10)
thf(fact_831_lambda__calculus_Osseq_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( lambda_Beta @ T6 @ U4 ) ) ).
% lambda_calculus.sseq.simps(5)
thf(fact_832_reduction__paths_Ostdz__insert_Osimps_I14_J,axiom,
! [V: lambda_lambda,Va: list_lambda_lambda] :
( ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ V @ Va ) )
= nil_lambda_lambda ) ).
% reduction_paths.stdz_insert.simps(14)
thf(fact_833_lambda__calculus_OCon__Arr__Src_I2_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ ( lambda_Src @ T ) @ T )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_Src(2)
thf(fact_834_lambda__calculus_OCon__Arr__Src_I1_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ ( lambda_Src @ T ) )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_Src(1)
thf(fact_835_lambda__calculus_OCon__implies__Coinitial3_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ U )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ U )
= ( lambda_Src @ T ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(3)
thf(fact_836_lambda__calculus_OCon__implies__Coinitial3_I2_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ V )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ V )
= ( lambda_Src @ T ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(2)
thf(fact_837_lambda__calculus_OCon__implies__Coinitial3_I1_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) )
!= lambda_Nil )
=> ( ( lambda_Arr @ V )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ V )
= ( lambda_Src @ U ) ) ) ) ).
% lambda_calculus.Con_implies_Coinitial3(1)
thf(fact_838_lambda__calculus_OCoinitial__iff__Con,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) )
= ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ).
% lambda_calculus.Coinitial_iff_Con
thf(fact_839_lambda__calculus_OCoinitial__resid__resid,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ V )
!= lambda_Nil )
=> ( ( ( lambda_resid @ U @ V )
!= lambda_Nil )
=> ( ( lambda_Arr @ ( lambda_resid @ T @ V ) )
& ( lambda_Arr @ ( lambda_resid @ U @ V ) )
& ( ( lambda_Src @ ( lambda_resid @ T @ V ) )
= ( lambda_Src @ ( lambda_resid @ U @ V ) ) ) ) ) ) ).
% lambda_calculus.Coinitial_resid_resid
thf(fact_840_lambda__calculus_Ocontains__head__reduction_Osimps_I4_J,axiom,
! [U: lambda_lambda] :
( ( lambda8135948770199036598uction @ ( lambda_App @ lambda_Nil @ U ) )
= ( ( lambda8135948770199036598uction @ lambda_Nil )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.contains_head_reduction.simps(4)
thf(fact_841_lambda__calculus_OCon__head__strategy,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ ( lambda_head_strategy @ T ) )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_head_strategy
thf(fact_842_lambda__calculus_OCon__Arr__head__redex,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ ( lambda_head_redex @ T ) )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_Arr_head_redex
thf(fact_843_lambda__calculus_Ohead__strategy_Osimps_I4_J,axiom,
! [U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ lambda_Nil @ U ) )
= ( lambda_App @ ( lambda_head_strategy @ lambda_Nil ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(4)
thf(fact_844_lambda__calculus_Ohead__redex_Osimps_I5_J,axiom,
! [U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ lambda_Nil @ U ) )
= ( lambda_App @ ( lambda_head_redex @ lambda_Nil ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(5)
thf(fact_845_lambda__calculus_OSrc__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_Src @ ( lambda_resid @ T @ U ) )
= ( lambda_Trg @ U ) ) ) ).
% lambda_calculus.Src_resid
thf(fact_846_lambda__calculus_Osseq_Osimps_I21_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ lambda_Nil @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(21)
thf(fact_847_reduction__paths_Ostdz__insert_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Uv: lambda_lambda,Uw: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( cons_lambda_lambda @ ( lambda_Beta @ Uv @ Uw ) @ U2 ) )
= ( reduct5390047737487997688insert @ ( lambda_Beta @ T @ U ) @ U2 ) ) ).
% reduction_paths.stdz_insert.simps(4)
thf(fact_848_lambda__calculus_Ohead__strategy_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_strategy @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_strategy.simps(3)
thf(fact_849_reduction__paths_Ostdz__insert_Osimps_I2_J,axiom,
! [Uu: nat,V: lambda_lambda,Va: list_lambda_lambda] :
( ( reduct5390047737487997688insert @ ( lambda_Var @ Uu ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ).
% reduction_paths.stdz_insert.simps(2)
thf(fact_850_reduction__paths_OBeta__decomp_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U ) @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% reduction_paths.Beta_decomp(2)
thf(fact_851_orthogonal__App__cong,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) )
@ ( append_lambda_lambda @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) ) ) ) ) ) ) ).
% orthogonal_App_cong
thf(fact_852_orthogonal__App__single__Arr,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ T ) ) @ U2 ) @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) @ nil_lambda_lambda ) )
= ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ T ) ) @ U2 ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) @ nil_lambda_lambda ) @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ T ) ) @ U2 ) )
= ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) ) @ nil_lambda_lambda ) ) ) ) ) ).
% orthogonal_App_single_Arr
thf(fact_853_reduction__paths_Ostdz__insert_Osimps_I11_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb: lambda_lambda,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ ( lambda_App @ T @ U ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_App @ T @ U ) )
=> ( ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ T @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) )
=> ( ( ( lambda8135948770199036598uction @ ( lambda_App @ T @ U ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ T @ U ) )
=> ( ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) )
= ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [A5: lambda_lambda] : ( lambda_App @ A5 @ ( lambda_Src @ U ) )
@ ( reduct5390047737487997688insert @ T
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) ) ) )
@ ( reduct5390047737487997688insert @ U
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ T @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Lam @ Vb ) @ Va ) )
= nil_lambda_lambda ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(11)
thf(fact_854_reduction__paths_Ostdz__insert_Osimps_I6_J,axiom,
! [Vb: nat,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) )
=> ( ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) ) @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) )
=> ( ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [A5: lambda_lambda] : ( lambda_App @ A5 @ ( lambda_Src @ U ) )
@ ( reduct5390047737487997688insert @ ( lambda_Var @ Vb )
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
@ ( reduct5390047737487997688insert @ U
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Var @ Vb ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= nil_lambda_lambda ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(6)
thf(fact_855_Arr_Osimps_I1_J,axiom,
~ ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Arr.simps(1)
thf(fact_856_Ide__implies__Arr,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% Ide_implies_Arr
thf(fact_857_Con__implies__Arr_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ U2 ) ) ).
% Con_implies_Arr(2)
thf(fact_858_Con__implies__Arr_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% Con_implies_Arr(1)
thf(fact_859_Arr__iff__Con__self,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T4 )
!= nil_lambda_lambda ) ) ).
% Arr_iff_Con_self
thf(fact_860_Con__Arr__self,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T4 )
!= nil_lambda_lambda ) ) ).
% Con_Arr_self
thf(fact_861_Con__imp__Arr__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) ) ) ).
% Con_imp_Arr_Resid
thf(fact_862_Arr__map__Lam,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) ) ) ).
% Arr_map_Lam
thf(fact_863_Arr__imp__Arr__tl,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ( tl_lambda_lambda @ T4 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( tl_lambda_lambda @ T4 ) ) ) ) ).
% Arr_imp_Arr_tl
thf(fact_864_Resid__Arr__self,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T4 ) ) ) ).
% Resid_Arr_self
thf(fact_865_Arr__imp__arr__hd,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( arr_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) ) ) ).
% Arr_imp_arr_hd
thf(fact_866_Std__imp__Arr,axiom,
! [T4: list_lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( ( T4 != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ) ).
% Std_imp_Arr
thf(fact_867_arr__char,axiom,
! [T4: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% arr_char
thf(fact_868_Arr__imp__arr__last,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( arr_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T4 ) ) ) ).
% Arr_imp_arr_last
thf(fact_869__092_060open_062U_A_092_060noteq_062_A_091_093_A_092_060Longrightarrow_062_AArr_AU_092_060close_062,axiom,
( ( ua != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ua ) ) ).
% \<open>U \<noteq> [] \<Longrightarrow> Arr U\<close>
thf(fact_870_Arr__Resid__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ).
% Arr_Resid_single
thf(fact_871_Arr__map__App2,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( lambda_Ide @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ A ) @ T4 ) ) ) ) ).
% Arr_map_App2
thf(fact_872_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_873_Arr__map__App1,axiom,
! [B: lambda_lambda,T4: list_lambda_lambda] :
( ( lambda_Ide @ B )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T7: lambda_lambda] : ( lambda_App @ T7 @ B )
@ T4 ) ) ) ) ).
% Arr_map_App1
thf(fact_874_cong__filter__notIde,axiom,
! [U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ~ ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ U2 )
@ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid @ U2
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ U2 ) ) ) ) ) ) ).
% cong_filter_notIde
thf(fact_875_orthogonal__App__Arr__Arr,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 ) )
= ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 ) )
= ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) ) ) ) ) ).
% orthogonal_App_Arr_Arr
thf(fact_876_arrI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ).
% arrI\<^sub>P
thf(fact_877_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_878_seqI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( lambda_Trg @ ( last_lambda_lambda @ T4 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% seqI\<^sub>\<Lambda>\<^sub>P
thf(fact_879_paths__in__rts_OArr_Ocong,axiom,
paths_2015741907369688696lambda = paths_2015741907369688696lambda ).
% paths_in_rts.Arr.cong
thf(fact_880_paths__in__rts_OArr_Ocong,axiom,
paths_366302053746739058lambda = paths_366302053746739058lambda ).
% paths_in_rts.Arr.cong
thf(fact_881_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_882_lambda__calculus_Oprfx__App__iff,axiom,
! [U: lambda_lambda,T1: lambda_lambda,T22: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_App @ T1 @ T22 ) ) )
= ( ( lambda_is_App @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_App1 @ U ) @ T1 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_App2 @ U ) @ T22 ) ) ) ) ).
% lambda_calculus.prfx_App_iff
thf(fact_883_reduction__paths_Ostdz__insert_Osimps_I12_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ ( lambda_App @ T @ U ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_App @ T @ U ) )
=> ( ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ T @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) )
=> ( ( ( lambda8135948770199036598uction @ ( lambda_App @ T @ U ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ T @ U ) )
=> ( ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [A5: lambda_lambda] : ( lambda_App @ A5 @ ( lambda_Src @ U ) )
@ ( reduct5390047737487997688insert @ T
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) ) ) )
@ ( reduct5390047737487997688insert @ U
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ T @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_App @ Vb @ Vc ) @ Va ) )
= nil_lambda_lambda ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(12)
thf(fact_884_reduction__paths_Ostdz__insert_Osimps_I7_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) )
=> ( ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) ) @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) )
=> ( ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [A5: lambda_lambda] : ( lambda_App @ A5 @ ( lambda_Src @ U ) )
@ ( reduct5390047737487997688insert @ ( lambda_App @ Vb @ Vc )
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
@ ( reduct5390047737487997688insert @ U
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_App @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= nil_lambda_lambda ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(7)
thf(fact_885_reduction__paths_Ostdz__insert_Osimps_I8_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) )
=> ( ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) ) @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) )
=> ( ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( lambda_head_strategy @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [A5: lambda_lambda] : ( lambda_App @ A5 @ ( lambda_Src @ U ) )
@ ( reduct5390047737487997688insert @ ( lambda_Beta @ Vb @ Vc )
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
@ ( reduct5390047737487997688insert @ U
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Beta @ Vb @ Vc ) @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= nil_lambda_lambda ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(8)
thf(fact_886_reduction__paths_Ostdz__insert_Osimps_I9_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ ( lambda_App @ T @ U ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ lambda_Nil @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_App @ T @ U ) )
=> ( ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ T @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) )
=> ( ( ( lambda8135948770199036598uction @ ( lambda_App @ T @ U ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ lambda_Nil @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ lambda_Nil @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ T @ U ) )
=> ( ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ lambda_Nil @ Va ) )
= ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ lambda_Nil @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ lambda_Nil @ Va ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [A5: lambda_lambda] : ( lambda_App @ A5 @ ( lambda_Src @ U ) )
@ ( reduct5390047737487997688insert @ T
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) ) ) )
@ ( reduct5390047737487997688insert @ U
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ T @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ lambda_Nil @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ lambda_Nil @ Va ) )
= nil_lambda_lambda ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(9)
thf(fact_887_reduction__paths_Ostdz__insert_Osimps_I5_J,axiom,
! [U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ ( lambda_App @ lambda_Nil @ U ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ lambda_Nil @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_App @ lambda_Nil @ U ) )
=> ( ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ lambda_Nil @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( ( lambda8135948770199036598uction @ ( lambda_App @ lambda_Nil @ U ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ lambda_Nil @ U ) @ ( lambda_head_redex @ ( lambda_App @ lambda_Nil @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ lambda_Nil @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ lambda_Nil @ U ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ lambda_Nil @ U ) @ ( lambda_head_redex @ ( lambda_App @ lambda_Nil @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ lambda_Nil @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ lambda_Nil @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ lambda_Nil @ U ) @ ( lambda_head_redex @ ( lambda_App @ lambda_Nil @ U ) ) ) @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ lambda_Nil @ U ) )
=> ( ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ lambda_Nil @ U ) @ ( lambda_head_strategy @ ( lambda_App @ lambda_Nil @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ lambda_Nil @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ lambda_Nil @ U ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ lambda_Nil @ U ) @ ( lambda_head_strategy @ ( lambda_App @ lambda_Nil @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ lambda_Nil @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ lambda_Nil @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ lambda_Nil @ U ) @ ( lambda_head_strategy @ ( lambda_App @ lambda_Nil @ U ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ lambda_Nil @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [A5: lambda_lambda] : ( lambda_App @ A5 @ ( lambda_Src @ U ) )
@ ( reduct5390047737487997688insert @ lambda_Nil
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) ) ) )
@ ( reduct5390047737487997688insert @ U
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ lambda_Nil @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ lambda_Nil @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
= nil_lambda_lambda ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(5)
thf(fact_888_reduction__paths_Ostdz__insert_Osimps_I10_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb: nat,Va: list_lambda_lambda] :
( ( ( lambda_Ide @ ( lambda_App @ T @ U ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) )
= ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_App @ T @ U ) )
=> ( ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ T @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) )
=> ( ( ( lambda8135948770199036598uction @ ( lambda_App @ T @ U ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_redex @ ( lambda_App @ T @ U ) ) ) @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( lambda_App @ T @ U ) )
=> ( ( ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) )
=> ( ( ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) )
= ( reduct5390047737487997688insert @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) ) ) )
& ( ~ ( lambda_Ide @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_head_strategy @ ( lambda_App @ T @ U ) ) ) @ ( tl_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) ) ) ) ) ) )
& ( ~ ( lambda8135948770199036598uction @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) )
= ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [A5: lambda_lambda] : ( lambda_App @ A5 @ ( lambda_Src @ U ) )
@ ( reduct5390047737487997688insert @ T
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) ) ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) ) ) )
@ ( reduct5390047737487997688insert @ U
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) ) ) ) ) ) ) ) ) ) )
& ( ~ ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ T @ U ) @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) ) )
=> ( ( reduct5390047737487997688insert @ ( lambda_App @ T @ U ) @ ( cons_lambda_lambda @ ( lambda_Var @ Vb ) @ Va ) )
= nil_lambda_lambda ) ) ) ) ) ).
% reduction_paths.stdz_insert.simps(10)
thf(fact_889_Arr__consE,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( U2 != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ U2 ) )
=> ~ ( ( U2 != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Arr_consE
thf(fact_890_Ide__consE,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ( U2 != nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U2 ) )
=> ~ ( ( U2 != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Ide_consE
thf(fact_891_sources__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [A6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A6 @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ A6 ) ) ) ) ) ).
% sources_char\<^sub>\<Lambda>\<^sub>P
thf(fact_892_sources__cong__closed,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A ) ) )
=> ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% sources_cong_closed
thf(fact_893_sources__are__cong,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A ) ) ) ) ) ).
% sources_are_cong
thf(fact_894_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_895_sources__are__con,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A3 ) ) ) ).
% sources_are_con
thf(fact_896_sources__join__of_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).
% sources_join_of(1)
thf(fact_897_sources__join__of_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).
% sources_join_of(2)
thf(fact_898_Ide__imp__Src__eq__Trg,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ T4 )
= ( paths_2695323870737867955lambda @ lambda_resid @ T4 ) ) ) ).
% Ide_imp_Src_eq_Trg
thf(fact_899_sources__con__closed,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A3 )
=> ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% sources_con_closed
thf(fact_900_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_901_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_902_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_903_coinitial__iff,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T6 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
& ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ) ).
% coinitial_iff
thf(fact_904_coinitialE,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
!= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ) ) ).
% coinitialE
thf(fact_905_sources__def,axiom,
! [T: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [A5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A5 ) ) ) ) ).
% sources_def
thf(fact_906_sources__cons,axiom,
! [T: lambda_lambda,T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ T4 ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% sources_cons
thf(fact_907_Arr__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ U2 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_908_Arr__appendE_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
!= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P\<^sub>W\<^sub>E
thf(fact_909_Ide__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
& ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_910_Ide__appendE,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ~ ( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
!= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ) ).
% Ide_appendE
thf(fact_911_single__Src__hd__in__sources,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% single_Src_hd_in_sources
thf(fact_912_in__sources__iff,axiom,
! [T4: list_lambda_lambda,A2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ A2 ) ) ) ) ) ).
% in_sources_iff
thf(fact_913_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_914_sources__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% sources_append
thf(fact_915_Src__resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ U2 ) ) ) ).
% Src_resid
thf(fact_916_Src__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_1651982994781619320lambda @ lambda_resid @ T4 ) ) ) ).
% Src_append
thf(fact_917_coinitialI,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ) ).
% coinitialI
thf(fact_918_Arr__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Arr_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_919_Ide__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Ide_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_920_Ide__consI,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ).
% Ide_consI
thf(fact_921_Arr__consI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ).
% Arr_consI\<^sub>P\<^sub>W\<^sub>E
thf(fact_922_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_923_rts_Osources_Ocong,axiom,
source6119337877621565212lambda = source6119337877621565212lambda ).
% rts.sources.cong
thf(fact_924_rts_Osources_Ocong,axiom,
source5627170679743844886lambda = source5627170679743844886lambda ).
% rts.sources.cong
thf(fact_925_paths__in__weakly__extensional__rts_OSrc_Ocong,axiom,
paths_1982998465148223614lambda = paths_1982998465148223614lambda ).
% paths_in_weakly_extensional_rts.Src.cong
thf(fact_926_paths__in__weakly__extensional__rts_OSrc_Ocong,axiom,
paths_1651982994781619320lambda = paths_1651982994781619320lambda ).
% paths_in_weakly_extensional_rts.Src.cong
thf(fact_927_paths__in__weakly__extensional__rts_OIde__imp__Src__eq__Trg,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_5097954201535619375lambda @ Resid @ T4 )
=> ( ( paths_1982998465148223614lambda @ Resid @ T4 )
= ( paths_4823109512173946681lambda @ Resid @ T4 ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_imp_Src_eq_Trg
thf(fact_928_paths__in__weakly__extensional__rts_OIde__imp__Src__eq__Trg,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ T4 )
=> ( ( paths_1651982994781619320lambda @ Resid @ T4 )
= ( paths_2695323870737867955lambda @ Resid @ T4 ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_imp_Src_eq_Trg
thf(fact_929_paths__in__weakly__extensional__rts_OSrc__append,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( seq_li7066278307288228763lambda @ ( paths_3312059784463760222lambda @ Resid ) @ T4 @ U2 )
=> ( ( paths_1982998465148223614lambda @ Resid @ ( append1017360404025248779lambda @ T4 @ U2 ) )
= ( paths_1982998465148223614lambda @ Resid @ T4 ) ) ) ) ).
% paths_in_weakly_extensional_rts.Src_append
thf(fact_930_paths__in__weakly__extensional__rts_OSrc__append,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( ( paths_1651982994781619320lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_1651982994781619320lambda @ Resid @ T4 ) ) ) ) ).
% paths_in_weakly_extensional_rts.Src_append
thf(fact_931_paths__in__weakly__extensional__rts_OSrc__resid,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( ( paths_3312059784463760222lambda @ Resid @ T4 @ U2 )
!= nil_li8379291187143545974lambda )
=> ( ( paths_1982998465148223614lambda @ Resid @ ( paths_3312059784463760222lambda @ Resid @ T4 @ U2 ) )
= ( paths_4823109512173946681lambda @ Resid @ U2 ) ) ) ) ).
% paths_in_weakly_extensional_rts.Src_resid
thf(fact_932_paths__in__weakly__extensional__rts_OSrc__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_1651982994781619320lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ Resid @ U2 ) ) ) ) ).
% paths_in_weakly_extensional_rts.Src_resid
thf(fact_933_paths__in__weakly__extensional__rts_OArr__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( ( paths_4823109512173946681lambda @ Resid @ T4 )
= ( paths_1982998465148223614lambda @ Resid @ U2 ) )
=> ( paths_2015741907369688696lambda @ Resid @ ( append1017360404025248779lambda @ T4 @ U2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_934_paths__in__weakly__extensional__rts_OArr__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( ( paths_2695323870737867955lambda @ Resid @ T4 )
= ( paths_1651982994781619320lambda @ Resid @ U2 ) )
=> ( paths_366302053746739058lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_935_paths__in__weakly__extensional__rts_OIde__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_5097954201535619375lambda @ Resid @ T4 )
=> ( ( paths_5097954201535619375lambda @ Resid @ U2 )
=> ( ( ( paths_4823109512173946681lambda @ Resid @ T4 )
= ( paths_1982998465148223614lambda @ Resid @ U2 ) )
=> ( paths_5097954201535619375lambda @ Resid @ ( append1017360404025248779lambda @ T4 @ U2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_936_paths__in__weakly__extensional__rts_OIde__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ T4 )
=> ( ( paths_7717490669901197737lambda @ Resid @ U2 )
=> ( ( ( paths_2695323870737867955lambda @ Resid @ T4 )
= ( paths_1651982994781619320lambda @ Resid @ U2 ) )
=> ( paths_7717490669901197737lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_937_paths__in__weakly__extensional__rts_OIde__consI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ T )
=> ( ( paths_5097954201535619375lambda @ Resid @ U2 )
=> ( ( ( trg_li817795487149935061lambda @ Resid @ T )
= ( paths_1982998465148223614lambda @ Resid @ U2 ) )
=> ( paths_5097954201535619375lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ U2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_consI
thf(fact_938_paths__in__weakly__extensional__rts_OIde__consI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ T )
=> ( ( paths_7717490669901197737lambda @ Resid @ U2 )
=> ( ( ( trg_lambda_lambda @ Resid @ T )
= ( paths_1651982994781619320lambda @ Resid @ U2 ) )
=> ( paths_7717490669901197737lambda @ Resid @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_consI
thf(fact_939_paths__in__weakly__extensional__rts_OArr__consI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( ( trg_li817795487149935061lambda @ Resid @ T )
= ( paths_1982998465148223614lambda @ Resid @ U2 ) )
=> ( paths_2015741907369688696lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ U2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_consI\<^sub>P\<^sub>W\<^sub>E
thf(fact_940_paths__in__weakly__extensional__rts_OArr__consI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( ( trg_lambda_lambda @ Resid @ T )
= ( paths_1651982994781619320lambda @ Resid @ U2 ) )
=> ( paths_366302053746739058lambda @ Resid @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_consI\<^sub>P\<^sub>W\<^sub>E
thf(fact_941_paths__in__weakly__extensional__rts_OArr__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( T4 != nil_li8379291187143545974lambda )
=> ( ( U2 != nil_li8379291187143545974lambda )
=> ( ( paths_2015741907369688696lambda @ Resid @ ( append1017360404025248779lambda @ T4 @ U2 ) )
= ( ( paths_2015741907369688696lambda @ Resid @ T4 )
& ( paths_2015741907369688696lambda @ Resid @ U2 )
& ( ( paths_4823109512173946681lambda @ Resid @ T4 )
= ( paths_1982998465148223614lambda @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_942_paths__in__weakly__extensional__rts_OArr__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_366302053746739058lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( ( paths_366302053746739058lambda @ Resid @ T4 )
& ( paths_366302053746739058lambda @ Resid @ U2 )
& ( ( paths_2695323870737867955lambda @ Resid @ T4 )
= ( paths_1651982994781619320lambda @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_943_paths__in__weakly__extensional__rts_OArr__appendE_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ ( append1017360404025248779lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_li8379291187143545974lambda )
=> ( ( U2 != nil_li8379291187143545974lambda )
=> ~ ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( paths_4823109512173946681lambda @ Resid @ T4 )
!= ( paths_1982998465148223614lambda @ Resid @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_appendE\<^sub>P\<^sub>W\<^sub>E
thf(fact_944_paths__in__weakly__extensional__rts_OArr__appendE_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ~ ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( paths_2695323870737867955lambda @ Resid @ T4 )
!= ( paths_1651982994781619320lambda @ Resid @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_appendE\<^sub>P\<^sub>W\<^sub>E
thf(fact_945_paths__in__weakly__extensional__rts_OIde__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( T4 != nil_li8379291187143545974lambda )
=> ( ( U2 != nil_li8379291187143545974lambda )
=> ( ( paths_5097954201535619375lambda @ Resid @ ( append1017360404025248779lambda @ T4 @ U2 ) )
= ( ( paths_5097954201535619375lambda @ Resid @ T4 )
& ( paths_5097954201535619375lambda @ Resid @ U2 )
& ( ( paths_4823109512173946681lambda @ Resid @ T4 )
= ( paths_1982998465148223614lambda @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_946_paths__in__weakly__extensional__rts_OIde__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_7717490669901197737lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( ( paths_7717490669901197737lambda @ Resid @ T4 )
& ( paths_7717490669901197737lambda @ Resid @ U2 )
& ( ( paths_2695323870737867955lambda @ Resid @ T4 )
= ( paths_1651982994781619320lambda @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_947_paths__in__weakly__extensional__rts_OIde__appendE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_5097954201535619375lambda @ Resid @ ( append1017360404025248779lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_li8379291187143545974lambda )
=> ( ( U2 != nil_li8379291187143545974lambda )
=> ~ ( ( paths_5097954201535619375lambda @ Resid @ T4 )
=> ( ( paths_5097954201535619375lambda @ Resid @ U2 )
=> ( ( paths_4823109512173946681lambda @ Resid @ T4 )
!= ( paths_1982998465148223614lambda @ Resid @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_appendE
thf(fact_948_paths__in__weakly__extensional__rts_OIde__appendE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ~ ( ( paths_7717490669901197737lambda @ Resid @ T4 )
=> ( ( paths_7717490669901197737lambda @ Resid @ U2 )
=> ( ( paths_2695323870737867955lambda @ Resid @ T4 )
!= ( paths_1651982994781619320lambda @ Resid @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_appendE
thf(fact_949_paths__in__weakly__extensional__rts_OIde__consE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_5097954201535619375lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ U2 ) )
=> ~ ( ( ide_li6773656263178250516lambda @ Resid @ T )
=> ( ( ( U2 != nil_li8379291187143545974lambda )
=> ( paths_5097954201535619375lambda @ Resid @ U2 ) )
=> ~ ( ( U2 != nil_li8379291187143545974lambda )
=> ( ( trg_li817795487149935061lambda @ Resid @ T )
= ( paths_1982998465148223614lambda @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_consE
thf(fact_950_paths__in__weakly__extensional__rts_OIde__consE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ ( cons_lambda_lambda @ T @ U2 ) )
=> ~ ( ( ide_lambda_lambda @ Resid @ T )
=> ( ( ( U2 != nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ Resid @ U2 ) )
=> ~ ( ( U2 != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ Resid @ T )
= ( paths_1651982994781619320lambda @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Ide_consE
thf(fact_951_paths__in__weakly__extensional__rts_OArr__consE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ U2 ) )
=> ~ ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( ( U2 != nil_li8379291187143545974lambda )
=> ( paths_2015741907369688696lambda @ Resid @ U2 ) )
=> ~ ( ( U2 != nil_li8379291187143545974lambda )
=> ( ( trg_li817795487149935061lambda @ Resid @ T )
= ( paths_1982998465148223614lambda @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_consE
thf(fact_952_paths__in__weakly__extensional__rts_OArr__consE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ ( cons_lambda_lambda @ T @ U2 ) )
=> ~ ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( ( U2 != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ Resid @ U2 ) )
=> ~ ( ( U2 != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ Resid @ T )
= ( paths_1651982994781619320lambda @ Resid @ U2 ) ) ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.Arr_consE
thf(fact_953_path__from__NF__is__Ide,axiom,
! [A: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_NF @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_1651982994781619320lambda @ lambda_resid @ U2 )
= A )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U2 ) ) ) ) ).
% path_from_NF_is_Ide
thf(fact_954_red__iff,axiom,
( lambda_red
= ( ^ [A5: lambda_lambda,B3: lambda_lambda] :
? [T8: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T8 )
& ( ( paths_1651982994781619320lambda @ lambda_resid @ T8 )
= A5 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T8 )
= B3 ) ) ) ) ).
% red_iff
thf(fact_955__092_060Lambda_062_Osources__cong__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A ) ) )
=> ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.sources_cong_closed
thf(fact_956__092_060Lambda_062_Osources__are__cong,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A ) ) ) ) ) ).
% \<Lambda>.sources_are_cong
thf(fact_957__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_958__092_060Lambda_062_Osources__are__con,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ A @ A3 ) ) ) ).
% \<Lambda>.sources_are_con
thf(fact_959__092_060Lambda_062_Oarr__has__un__source,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( Y = X ) ) ) ) ).
% \<Lambda>.arr_has_un_source
thf(fact_960__092_060Lambda_062_Osources__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.sources_join_of(1)
thf(fact_961__092_060Lambda_062_Osources__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ U )
= ( source5627170679743844886lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.sources_join_of(2)
thf(fact_962__092_060Lambda_062_Osources__con__closed,axiom,
! [A: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A3 )
=> ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.sources_con_closed
thf(fact_963__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_964__092_060Lambda_062_Ojoin__of__arr__src_I2_J,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ A @ T ) ) ) ).
% \<Lambda>.join_of_arr_src(2)
thf(fact_965__092_060Lambda_062_Ojoin__of__arr__src_I1_J,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ A @ T @ T ) ) ) ).
% \<Lambda>.join_of_arr_src(1)
thf(fact_966__092_060Lambda_062_Ocoinitial__iff,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T6 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T6 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T6 ) ) ) ) ).
% \<Lambda>.coinitial_iff
thf(fact_967__092_060Lambda_062_OcoinitialE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
!= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ) ).
% \<Lambda>.coinitialE
thf(fact_968__092_060Lambda_062_Osources__def,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [A5: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A5 )
& ( con_lambda_lambda @ lambda_resid @ T @ A5 ) ) ) ) ).
% \<Lambda>.sources_def
thf(fact_969__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_970__092_060Lambda_062_OcoinitialI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.coinitialI
thf(fact_971_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_972_lambda__calculus_ONF__unique,axiom,
! [T: lambda_lambda,U: lambda_lambda,U4: lambda_lambda] :
( ( lambda_red @ T @ U )
=> ( ( lambda_red @ T @ U4 )
=> ( ( lambda_NF @ U )
=> ( ( lambda_NF @ U4 )
=> ( U = U4 ) ) ) ) ) ).
% lambda_calculus.NF_unique
thf(fact_973_lambda__calculus_Oweak__diamond,axiom,
! [A: lambda_lambda,B: lambda_lambda,B4: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( ( lambda_red @ A @ B4 )
=> ~ ! [C2: lambda_lambda] :
( ( lambda_red @ B @ C2 )
=> ~ ( lambda_red @ B4 @ C2 ) ) ) ) ).
% lambda_calculus.weak_diamond
thf(fact_974_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_975_lambda__calculus_ONF__Lam__iff,axiom,
! [T: lambda_lambda] :
( ( lambda_NF @ ( lambda_Lam @ T ) )
= ( lambda_NF @ T ) ) ).
% lambda_calculus.NF_Lam_iff
thf(fact_976_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_977_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_978_lambda__calculus_Ored_Ocases,axiom,
! [A1: lambda_lambda,A22: lambda_lambda] :
( ( lambda_red @ A1 @ A22 )
=> ( ! [T3: lambda_lambda] :
( ( A1
= ( lambda_Src @ T3 ) )
=> ( ( A22
= ( lambda_Trg @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) ) )
=> ~ ! [B5: lambda_lambda] :
( ( lambda_red @ A1 @ B5 )
=> ~ ( lambda_red @ B5 @ A22 ) ) ) ) ).
% lambda_calculus.red.cases
thf(fact_979_lambda__calculus_Ored_Osimps,axiom,
( lambda_red
= ( ^ [A12: lambda_lambda,A23: lambda_lambda] :
( ? [T7: lambda_lambda] :
( ( A12
= ( lambda_Src @ T7 ) )
& ( A23
= ( lambda_Trg @ T7 ) )
& ( lambda_Arr @ T7 ) )
| ? [A5: lambda_lambda,B3: lambda_lambda,C3: lambda_lambda] :
( ( A12 = A5 )
& ( A23 = C3 )
& ( lambda_red @ A5 @ B3 )
& ( lambda_red @ B3 @ C3 ) ) ) ) ) ).
% lambda_calculus.red.simps
thf(fact_980_lambda__calculus_ONF__def,axiom,
( lambda_NF
= ( ^ [A5: lambda_lambda] :
( ( lambda_Ide @ A5 )
& ! [T7: lambda_lambda] :
( ( ( lambda_Arr @ T7 )
& ( ( lambda_Src @ T7 )
= A5 ) )
=> ( lambda_Ide @ T7 ) ) ) ) ) ).
% lambda_calculus.NF_def
thf(fact_981_lambda__calculus_Onot__NF__elim,axiom,
! [T: lambda_lambda] :
( ~ ( lambda_NF @ T )
=> ( ( lambda_Ide @ T )
=> ~ ! [U6: lambda_lambda] :
~ ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U6 )
& ~ ( lambda_Ide @ U6 ) ) ) ) ).
% lambda_calculus.not_NF_elim
thf(fact_982_targets__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [B6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B6 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ B6 ) ) ) ) ) ).
% targets_char\<^sub>\<Lambda>\<^sub>P
thf(fact_983_confluence__single,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T3 @ U6 )
=> ( con_lambda_lambda @ lambda_resid @ T3 @ U6 ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% confluence_single
thf(fact_984_ex__un__Src,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( Y = X ) ) ) ) ).
% ex_un_Src
thf(fact_985_Srcs__are__con,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( con_lambda_lambda @ lambda_resid @ A @ A3 ) ) ) ).
% Srcs_are_con
thf(fact_986_Con__imp__eq__Srcs,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ).
% Con_imp_eq_Srcs
thf(fact_987_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_988_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_989_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_990_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_991_targets__resid__sym,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) ) ) ).
% targets_resid_sym
thf(fact_992_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_993_Srcs__con__closed,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ A3 )
=> ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_994_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_995_targets__join__of_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).
% targets_join_of(1)
thf(fact_996_targets__join__of_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ) ).
% targets_join_of(2)
thf(fact_997_confluence__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda ) ) ) ) ).
% confluence_ind
thf(fact_998_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_999_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_1000_Srcs__simp_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( source5627170679743844886lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) ) ) ) ).
% Srcs_simp\<^sub>P
thf(fact_1001_Ide__imp__sources__eq__targets,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% Ide_imp_sources_eq_targets
thf(fact_1002_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_1003_trg__in__targets,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( member2667661554319814297lambda @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% trg_in_targets
thf(fact_1004_composableD_I3_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).
% composableD(3)
thf(fact_1005_coterminal__iff,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T6 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ) ).
% coterminal_iff
thf(fact_1006_coterminalE,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
!= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ) ) ).
% coterminalE
thf(fact_1007_Resid__Arr__Src,axiom,
! [T4: list_lambda_lambda,A: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
= T4 ) ) ) ).
% Resid_Arr_Src
thf(fact_1008_Con__IdeI_I2_J,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A2 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(2)
thf(fact_1009_Con__IdeI_I1_J,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(1)
thf(fact_1010_Con__Ide__iff,axiom,
! [A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T4 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A2 ) ) ) ) ) ).
% Con_Ide_iff
thf(fact_1011_seq__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ) ).
% seq_def
thf(fact_1012_seqE,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ~ ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
!= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ) ) ).
% seqE
thf(fact_1013_in__targetsE,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B ) ) ) ).
% in_targetsE
thf(fact_1014_sources__char_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [A6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A6 )
& ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ A6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% sources_char\<^sub>P
thf(fact_1015_targets__def,axiom,
! [T: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [B3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B3 ) ) ) ) ).
% targets_def
thf(fact_1016_Con__single__ide__ind,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Con_single_ide_ind
thf(fact_1017_single__Trg__last__in__targets,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% single_Trg_last_in_targets
thf(fact_1018_in__targets__iff,axiom,
! [T4: list_lambda_lambda,B2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B2 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ B2 ) ) ) ) ) ).
% in_targets_iff
thf(fact_1019_Srcs__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ).
% Srcs_append
thf(fact_1020_targets__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ) ).
% targets_append
thf(fact_1021_sources__resid,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).
% sources_resid
thf(fact_1022_coterminalI,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
=> ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ) ).
% coterminalI
thf(fact_1023_seqI,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U )
=> ( ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U ) ) ) ) ).
% seqI
thf(fact_1024_in__targetsI,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B )
=> ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% in_targetsI
thf(fact_1025_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_1026_Con__single__ideI_I1_J,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(1)
thf(fact_1027_Con__single__ideI_I2_J,axiom,
! [A: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(2)
thf(fact_1028_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_1029_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_1030_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,V: list_lambda_lambda,V4: list_lambda_lambda,W2: list_lambda_lambda,W3: list_lambda_lambda,T: list_lambda_lambda,T6: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( member2667661554319814297lambda @ V @ NN )
=> ( ( member2667661554319814297lambda @ V4 @ NN )
=> ( ( member2667661554319814297lambda @ W2 @ NN )
=> ( ( member2667661554319814297lambda @ W3 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ V )
= ( source6119337877621565212lambda @ Resid @ W2 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ V4 )
= ( source6119337877621565212lambda @ Resid @ W3 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ W2 )
= ( target4026644918049360146lambda @ Resid @ W3 ) )
=> ( ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V4 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T6 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T6 @ W3 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T6 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_1031_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,V: lambda_lambda,V4: lambda_lambda,W2: lambda_lambda,W3: lambda_lambda,T: lambda_lambda,T6: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ V @ NN )
=> ( ( member_lambda_lambda @ V4 @ NN )
=> ( ( member_lambda_lambda @ W2 @ NN )
=> ( ( member_lambda_lambda @ W3 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ V )
= ( source5627170679743844886lambda @ Resid @ W2 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ V4 )
= ( source5627170679743844886lambda @ Resid @ W3 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ W2 )
= ( target5490931411331770124lambda @ Resid @ W3 ) )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V4 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T6 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T6 @ W3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T6 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_1032_rts_Otargets_Ocong,axiom,
target4026644918049360146lambda = target4026644918049360146lambda ).
% rts.targets.cong
thf(fact_1033_rts_Otargets_Ocong,axiom,
target5490931411331770124lambda = target5490931411331770124lambda ).
% rts.targets.cong
thf(fact_1034_paths__in__rts_OSrcs_Ocong,axiom,
paths_3172353640142224310lambda = paths_3172353640142224310lambda ).
% paths_in_rts.Srcs.cong
thf(fact_1035_paths__in__rts_OSrcs_Ocong,axiom,
paths_2897053694880971696lambda = paths_2897053694880971696lambda ).
% paths_in_rts.Srcs.cong
thf(fact_1036_paths__in__weakly__extensional__rts_Oex__un__Src,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ? [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ ( paths_3172353640142224310lambda @ Resid @ T4 ) )
& ! [Y: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y @ ( paths_3172353640142224310lambda @ Resid @ T4 ) )
=> ( Y = X ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.ex_un_Src
thf(fact_1037_paths__in__weakly__extensional__rts_Oex__un__Src,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( paths_2897053694880971696lambda @ Resid @ T4 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( paths_2897053694880971696lambda @ Resid @ T4 ) )
=> ( Y = X ) ) ) ) ) ).
% paths_in_weakly_extensional_rts.ex_un_Src
thf(fact_1038_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,T: list_lambda_lambda,U: list_lambda_lambda,U4: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( member2667661554319814297lambda @ U @ NN )
=> ( ( member2667661554319814297lambda @ U4 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ U )
= ( source6119337877621565212lambda @ Resid @ U4 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ U )
= ( target4026644918049360146lambda @ Resid @ U4 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( source6119337877621565212lambda @ Resid @ U ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U4 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_1039_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,U4: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U4 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U )
= ( source5627170679743844886lambda @ Resid @ U4 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U )
= ( target5490931411331770124lambda @ Resid @ U4 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U4 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_1040_paths__in__confluent__rts_Oconfluence__ind,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( ( paths_3172353640142224310lambda @ Resid @ T4 )
= ( paths_3172353640142224310lambda @ Resid @ U2 ) )
=> ( ( paths_3312059784463760222lambda @ Resid @ T4 @ U2 )
!= nil_li8379291187143545974lambda ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_ind
thf(fact_1041_paths__in__confluent__rts_Oconfluence__ind,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( ( paths_2897053694880971696lambda @ Resid @ T4 )
= ( paths_2897053694880971696lambda @ Resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_ind
thf(fact_1042_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_l5168835686021156086lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( ! [T3: list_lambda_lambda,U6: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ Resid @ T3 @ U6 )
=> ( con_li79537494059641180lambda @ Resid @ T3 @ U6 ) )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( paths_3172353640142224310lambda @ Resid @ U2 ) )
=> ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ U2 )
!= nil_li8379291187143545974lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_1043_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ! [T3: lambda_lambda,U6: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid @ T3 @ U6 )
=> ( con_lambda_lambda @ Resid @ T3 @ U6 ) )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( paths_2897053694880971696lambda @ Resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_1044_Srcs_Oelims,axiom,
! [X3: list_lambda_lambda,Y4: set_lambda_lambda] :
( ( ( paths_2897053694880971696lambda @ lambda_resid @ X3 )
= Y4 )
=> ( ( ( X3 = nil_lambda_lambda )
=> ( Y4 != bot_bo1271416176687173302lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y4
!= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda] :
( ? [V3: lambda_lambda,Va2: list_lambda_lambda] :
( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y4
!= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) ) ) ) ) ).
% Srcs.elims
thf(fact_1045_incl__is__simulation,axiom,
( simula1175017789036123529lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ^ [T7: lambda_lambda] : ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T7 ) @ ( cons_lambda_lambda @ T7 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% incl_is_simulation
thf(fact_1046__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_1047__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_1048__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_1049__092_060Lambda_062_Otargets__resid__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).
% \<Lambda>.targets_resid_sym
thf(fact_1050__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_1051__092_060Lambda_062_Oarr__has__un__target,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( Y = X ) ) ) ) ).
% \<Lambda>.arr_has_un_target
thf(fact_1052__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_1053__092_060Lambda_062_Otargets__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.targets_join_of(1)
thf(fact_1054__092_060Lambda_062_Otargets__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_resid @ U @ T ) )
= ( target5490931411331770124lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.targets_join_of(2)
thf(fact_1055__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_1056__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_1057__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_1058__092_060Lambda_062_OcoterminalE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
!= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ) ) ).
% \<Lambda>.coterminalE
thf(fact_1059__092_060Lambda_062_Ocoterminal__iff,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T6 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T6 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T6 ) ) ) ) ).
% \<Lambda>.coterminal_iff
thf(fact_1060__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_1061__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_1062__092_060Lambda_062_OcomposableD_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.composableD(3)
thf(fact_1063__092_060Lambda_062_Oseq__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ).
% \<Lambda>.seq_def
thf(fact_1064__092_060Lambda_062_OseqE,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
!= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ) ) ).
% \<Lambda>.seqE
thf(fact_1065_Srcs_Osimps_I1_J,axiom,
( ( paths_2897053694880971696lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Srcs.simps(1)
thf(fact_1066__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_1067_Arr__has__Src,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
!= bot_bo1271416176687173302lambda ) ) ).
% Arr_has_Src
thf(fact_1068__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_1069__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_1070_Srcs__Resid__Arr__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ).
% Srcs_Resid_Arr_single
thf(fact_1071_lambda__calculus_Otargets__simps_I1_J,axiom,
( ( target5490931411331770124lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.targets_simps(1)
thf(fact_1072__092_060Lambda_062_Osources__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.sources_resid
thf(fact_1073__092_060Lambda_062_OcoterminalI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ U ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.coterminalI
thf(fact_1074_lambda__calculus_Osources__simps_I1_J,axiom,
( ( source5627170679743844886lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.sources_simps(1)
thf(fact_1075__092_060Lambda_062_OseqI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% \<Lambda>.seqI
thf(fact_1076__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_1077_simulation_Opreserves__paths,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T4: list_l5168835686021156086lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( paths_2015741907369688696lambda @ A2 @ T4 )
=> ( paths_2836561004236663038lambda @ B2 @ ( map_li7955449278618656202lambda @ F2 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1078_simulation_Opreserves__paths,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( paths_2015741907369688696lambda @ A2 @ T4 )
=> ( paths_366302053746739058lambda @ B2 @ ( map_li5201084558375397438lambda @ F2 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1079_simulation_Opreserves__paths,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( paths_2015741907369688696lambda @ A2 @ T4 )
=> ( paths_2015741907369688696lambda @ B2 @ ( map_li385728805257398596lambda @ F2 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1080_simulation_Opreserves__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( paths_366302053746739058lambda @ A2 @ T4 )
=> ( paths_2015741907369688696lambda @ B2 @ ( map_la6434941737452557130lambda @ F2 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1081_simulation_Opreserves__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( paths_366302053746739058lambda @ A2 @ T4 )
=> ( paths_366302053746739058lambda @ B2 @ ( map_la336867333220232516lambda @ F2 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1082_simulation_Olifts__to__paths,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( simula2627686087595152009lambda @ ( paths_3312059784463760222lambda @ A2 ) @ ( paths_4946278009457343204lambda @ B2 )
@ ^ [T8: list_l5168835686021156086lambda] : ( if_lis8739384218277118914lambda @ ( paths_2015741907369688696lambda @ A2 @ T8 ) @ ( map_li7955449278618656202lambda @ F2 @ T8 ) @ nil_li6069905736445159420lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1083_simulation_Olifts__to__paths,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( simula4114611666396501757lambda @ ( paths_3312059784463760222lambda @ A2 ) @ ( paths_8607357831768207192lambda @ B2 )
@ ^ [T8: list_l5168835686021156086lambda] : ( if_lis7195930416592267574lambda @ ( paths_2015741907369688696lambda @ A2 @ T8 ) @ ( map_li5201084558375397438lambda @ F2 @ T8 ) @ nil_lambda_lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1084_simulation_Olifts__to__paths,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( simula1230958359699143427lambda @ ( paths_3312059784463760222lambda @ A2 ) @ ( paths_3312059784463760222lambda @ B2 )
@ ^ [T8: list_l5168835686021156086lambda] : ( if_lis4115203008988617916lambda @ ( paths_2015741907369688696lambda @ A2 @ T8 ) @ ( map_li385728805257398596lambda @ F2 @ T8 ) @ nil_li8379291187143545974lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1085_simulation_Olifts__to__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( simula3219528783544830217lambda @ ( paths_8607357831768207192lambda @ A2 ) @ ( paths_3312059784463760222lambda @ B2 )
@ ^ [T8: list_lambda_lambda] : ( if_lis4115203008988617916lambda @ ( paths_366302053746739058lambda @ A2 @ T8 ) @ ( map_la6434941737452557130lambda @ F2 @ T8 ) @ nil_li8379291187143545974lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1086_simulation_Olifts__to__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( simula3069223088535412739lambda @ ( paths_8607357831768207192lambda @ A2 ) @ ( paths_8607357831768207192lambda @ B2 )
@ ^ [T8: list_lambda_lambda] : ( if_lis7195930416592267574lambda @ ( paths_366302053746739058lambda @ A2 @ T8 ) @ ( map_la336867333220232516lambda @ F2 @ T8 ) @ nil_lambda_lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1087_simulation_Opreserves__joins,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( join_o6956785509312998601lambda @ A2 @ T @ U @ V )
=> ( join_o5099642474212670159lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1088_simulation_Opreserves__joins,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( join_o6956785509312998601lambda @ A2 @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1089_simulation_Opreserves__joins,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( join_o6956785509312998601lambda @ A2 @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1090_simulation_Opreserves__joins,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( join_o1386658102433174083lambda @ A2 @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1091_simulation_Opreserves__joins,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( join_o1386658102433174083lambda @ A2 @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) @ ( F2 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1092_simulation_Opreserves__prfx,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_li2235071267601372570lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1093_simulation_Opreserves__prfx,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_li6773656263178250516lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1094_simulation_Opreserves__prfx,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_lambda_lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1095_simulation_Opreserves__prfx,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_li6773656263178250516lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1096_simulation_Opreserves__prfx,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_lambda_lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1097_simulation_Opreserves__cong,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_li2235071267601372570lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_li2235071267601372570lambda @ B2 @ ( B2 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1098_simulation_Opreserves__cong,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_li6773656263178250516lambda @ B2 @ ( B2 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1099_simulation_Opreserves__cong,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_lambda_lambda @ B2 @ ( B2 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1100_simulation_Opreserves__cong,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_lambda_lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_li6773656263178250516lambda @ B2 @ ( B2 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1101_simulation_Opreserves__cong,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_lambda_lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B2 @ ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) )
& ( ide_lambda_lambda @ B2 @ ( B2 @ ( F2 @ U ) @ ( F2 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1102_simulation_Opreserves__ide,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,A: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ A )
=> ( ide_li2235071267601372570lambda @ B2 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1103_simulation_Opreserves__ide,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ A )
=> ( ide_li6773656263178250516lambda @ B2 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1104_simulation_Opreserves__ide,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,A: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ A )
=> ( ide_lambda_lambda @ B2 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1105_simulation_Opreserves__ide,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,A: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( ide_lambda_lambda @ A2 @ A )
=> ( ide_li6773656263178250516lambda @ B2 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1106_simulation_Opreserves__ide,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( ide_lambda_lambda @ A2 @ A )
=> ( ide_lambda_lambda @ B2 @ ( F2 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1107_simulation_Opreserves__con,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( con_li2822789971228234210lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1108_simulation_Opreserves__con,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( con_lambda_lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1109_simulation_Opreserves__con,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( con_li79537494059641180lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1110_simulation_Opreserves__con,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( con_lambda_lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1111_simulation_Opreserves__con,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( con_li79537494059641180lambda @ B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1112_simulation_Opreserves__resid,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( ( F2 @ ( A2 @ T @ U ) )
= ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1113_simulation_Opreserves__resid,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( ( F2 @ ( A2 @ T @ U ) )
= ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1114_simulation_Opreserves__resid,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( ( F2 @ ( A2 @ T @ U ) )
= ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1115_simulation_Opreserves__resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( ( F2 @ ( A2 @ T @ U ) )
= ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1116_simulation_Opreserves__resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( ( F2 @ ( A2 @ T @ U ) )
= ( B2 @ ( F2 @ T ) @ ( F2 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1117_simulation_Opreserves__reflects__arr,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( arr_li5380981774320532067lambda @ B2 @ ( F2 @ T ) )
= ( arr_li3691443969012319837lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1118_simulation_Opreserves__reflects__arr,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( arr_li3691443969012319837lambda @ B2 @ ( F2 @ T ) )
= ( arr_li3691443969012319837lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1119_simulation_Opreserves__reflects__arr,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( arr_li3691443969012319837lambda @ B2 @ ( F2 @ T ) )
= ( arr_lambda_lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1120_simulation_Opreserves__reflects__arr,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( arr_lambda_lambda @ B2 @ ( F2 @ T ) )
= ( arr_li3691443969012319837lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1121_simulation_Opreserves__reflects__arr,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( arr_lambda_lambda @ B2 @ ( F2 @ T ) )
= ( arr_lambda_lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1122_simulation_Opreserves__trg,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F2 @ ( trg_li817795487149935061lambda @ A2 @ T ) )
= ( trg_li7687935573203771867lambda @ B2 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1123_simulation_Opreserves__trg,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( arr_lambda_lambda @ A2 @ T )
=> ( ( F2 @ ( trg_lambda_lambda @ A2 @ T ) )
= ( trg_lambda_lambda @ B2 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1124_simulation_Opreserves__trg,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( arr_lambda_lambda @ A2 @ T )
=> ( ( F2 @ ( trg_lambda_lambda @ A2 @ T ) )
= ( trg_li817795487149935061lambda @ B2 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1125_simulation_Opreserves__trg,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F2 @ ( trg_li817795487149935061lambda @ A2 @ T ) )
= ( trg_lambda_lambda @ B2 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1126_simulation_Opreserves__trg,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F2 @ ( trg_li817795487149935061lambda @ A2 @ T ) )
= ( trg_li817795487149935061lambda @ B2 @ ( F2 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1127_simulation_Oextensional,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ~ ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia1037102945066357276lambda @ B2 ) ) ) ) ).
% simulation.extensional
thf(fact_1128_simulation_Oextensional,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ~ ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia1461432607296028304lambda @ B2 ) ) ) ) ).
% simulation.extensional
thf(fact_1129_simulation_Oextensional,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ~ ( arr_lambda_lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia1461432607296028304lambda @ B2 ) ) ) ) ).
% simulation.extensional
thf(fact_1130_simulation_Oextensional,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ~ ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia6873046715171720854lambda @ B2 ) ) ) ) ).
% simulation.extensional
thf(fact_1131_simulation_Oextensional,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ~ ( arr_lambda_lambda @ A2 @ T )
=> ( ( F2 @ T )
= ( partia6873046715171720854lambda @ B2 ) ) ) ) ).
% simulation.extensional
thf(fact_1132_simulation_Omap__Resid,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_li7955449278618656202lambda @ F2 @ ( paths_3312059784463760222lambda @ A2 @ T4 @ U2 ) )
= ( paths_4946278009457343204lambda @ B2 @ ( map_li7955449278618656202lambda @ F2 @ T4 ) @ ( map_li7955449278618656202lambda @ F2 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1133_simulation_Omap__Resid,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_li5201084558375397438lambda @ F2 @ ( paths_3312059784463760222lambda @ A2 @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ B2 @ ( map_li5201084558375397438lambda @ F2 @ T4 ) @ ( map_li5201084558375397438lambda @ F2 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1134_simulation_Omap__Resid,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_li385728805257398596lambda @ F2 @ ( paths_3312059784463760222lambda @ A2 @ T4 @ U2 ) )
= ( paths_3312059784463760222lambda @ B2 @ ( map_li385728805257398596lambda @ F2 @ T4 ) @ ( map_li385728805257398596lambda @ F2 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1135_simulation_Omap__Resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_la6434941737452557130lambda @ F2 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ U2 ) )
= ( paths_3312059784463760222lambda @ B2 @ ( map_la6434941737452557130lambda @ F2 @ T4 ) @ ( map_la6434941737452557130lambda @ F2 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1136_simulation_Omap__Resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda @ F2 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ B2 @ ( map_la336867333220232516lambda @ F2 @ T4 ) @ ( map_la336867333220232516lambda @ F2 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1137_simulation_Omap__Resid__single,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_l5168835686021156086lambda > list_l5168835686021156086lambda > list_l5168835686021156086lambda,F2: list_lambda_lambda > list_l5168835686021156086lambda,T4: list_l5168835686021156086lambda,U: list_lambda_lambda] :
( ( simula3219528783544830217lambda @ A2 @ B2 @ F2 )
=> ( ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ A2 ) @ T4 @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) )
=> ( ( map_li7955449278618656202lambda @ F2 @ ( paths_3312059784463760222lambda @ A2 @ T4 @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) ) )
= ( paths_4946278009457343204lambda @ B2 @ ( map_li7955449278618656202lambda @ F2 @ T4 ) @ ( cons_l4910729503725484972lambda @ ( F2 @ U ) @ nil_li6069905736445159420lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1138_simulation_Omap__Resid__single,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T4: list_l5168835686021156086lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B2 @ F2 )
=> ( ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ A2 ) @ T4 @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) )
=> ( ( map_li5201084558375397438lambda @ F2 @ ( paths_3312059784463760222lambda @ A2 @ T4 @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) ) )
= ( paths_8607357831768207192lambda @ B2 @ ( map_li5201084558375397438lambda @ F2 @ T4 ) @ ( cons_lambda_lambda @ ( F2 @ U ) @ nil_lambda_lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1139_simulation_Omap__Resid__single,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B2 @ F2 )
=> ( ( con_li2822789971228234210lambda @ ( paths_3312059784463760222lambda @ A2 ) @ T4 @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) )
=> ( ( map_li385728805257398596lambda @ F2 @ ( paths_3312059784463760222lambda @ A2 @ T4 @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) ) )
= ( paths_3312059784463760222lambda @ B2 @ ( map_li385728805257398596lambda @ F2 @ T4 ) @ ( cons_l4558873032622161958lambda @ ( F2 @ U ) @ nil_li8379291187143545974lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1140_simulation_Omap__Resid__single,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
=> ( ( map_la6434941737452557130lambda @ F2 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_3312059784463760222lambda @ B2 @ ( map_la6434941737452557130lambda @ F2 @ T4 ) @ ( cons_l4558873032622161958lambda @ ( F2 @ U ) @ nil_li8379291187143545974lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1141_simulation_Omap__Resid__single,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B2: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B2 @ F2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
=> ( ( map_la336867333220232516lambda @ F2 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_8607357831768207192lambda @ B2 @ ( map_la336867333220232516lambda @ F2 @ T4 ) @ ( cons_lambda_lambda @ ( F2 @ U ) @ nil_lambda_lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1142_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_1143_Arr_Oelims_I3_J,axiom,
! [X3: list_lambda_lambda] :
( ~ ( paths_366302053746739058lambda @ lambda_resid @ X3 )
=> ( ( X3 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( arr_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).
% Arr.elims(3)
thf(fact_1144_Srcs__are__ide,axiom,
! [T4: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).
% Srcs_are_ide
thf(fact_1145_Srcs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( insert_lambda_lambda @ ( paths_1651982994781619320lambda @ lambda_resid @ T4 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Srcs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_1146_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_1147_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_1148_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_1149_Ide_Oelims_I3_J,axiom,
! [X3: list_lambda_lambda] :
( ~ ( paths_7717490669901197737lambda @ lambda_resid @ X3 )
=> ( ( X3 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( ide_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).
% Ide.elims(3)
thf(fact_1150_Ide_Oelims_I2_J,axiom,
! [X3: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ X3 )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Ide.elims(2)
thf(fact_1151_Ide_Oelims_I1_J,axiom,
! [X3: list_lambda_lambda,Y4: $o] :
( ( ( paths_7717490669901197737lambda @ lambda_resid @ X3 )
= Y4 )
=> ( ( ( X3 = nil_lambda_lambda )
=> Y4 )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y4
= ( ~ ( ide_lambda_lambda @ lambda_resid @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y4
= ( ~ ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% Ide.elims(1)
thf(fact_1152_Arr_Oelims_I1_J,axiom,
! [X3: list_lambda_lambda,Y4: $o] :
( ( ( paths_366302053746739058lambda @ lambda_resid @ X3 )
= Y4 )
=> ( ( ( X3 = nil_lambda_lambda )
=> Y4 )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y4
= ( ~ ( arr_lambda_lambda @ lambda_resid @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y4
= ( ~ ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% Arr.elims(1)
thf(fact_1153_Arr_Oelims_I2_J,axiom,
! [X3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ X3 )
=> ( ! [T3: lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( arr_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X3
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Arr.elims(2)
thf(fact_1154_lambda__calculus_Otargets__simps_I2_J,axiom,
! [X3: nat] :
( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Var @ X3 ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X3 ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.targets_simps(2)
thf(fact_1155_lambda__calculus_Osources__simps_I2_J,axiom,
! [X3: nat] :
( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Var @ X3 ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X3 ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.sources_simps(2)
thf(fact_1156_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_1157_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_1158_Arr__consI_092_060_094sub_062P,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ).
% Arr_consI\<^sub>P
thf(fact_1159_lambda__calculus_Osources__simps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_App @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_simps(4)
thf(fact_1160_lambda__calculus_Otargets__simps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_App @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.targets_simps(4)
thf(fact_1161_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_1162_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_1163_lambda__calculus_Osources__simps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) )
= ( insert_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U ) ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% lambda_calculus.sources_simps(5)
thf(fact_1164_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_1165_lambda__calculus_OLam__is__simulation,axiom,
( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T7: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T7 ) @ ( lambda_Lam @ T7 ) @ lambda_Nil ) ) ).
% lambda_calculus.Lam_is_simulation
thf(fact_1166_lambda__calculus_OApp__is__simulation1,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T7: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T7 ) @ ( lambda_App @ T7 @ A ) @ lambda_Nil ) ) ) ).
% lambda_calculus.App_is_simulation1
thf(fact_1167_lambda__calculus_OApp__is__simulation2,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T7: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T7 ) @ ( lambda_App @ A @ T7 ) @ lambda_Nil ) ) ) ).
% lambda_calculus.App_is_simulation2
thf(fact_1168_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_1169_paths__in__weakly__extensional__rts_OSrcs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_7507126897227621196lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_3172353640142224310lambda @ Resid @ T4 )
= ( insert8465602035467394112lambda @ ( paths_1982998465148223614lambda @ Resid @ T4 ) @ bot_bo1397037886704484924lambda ) ) ) ) ).
% paths_in_weakly_extensional_rts.Srcs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_1170_paths__in__weakly__extensional__rts_OSrcs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_4652746547811739206lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ Resid @ T4 )
= ( insert_lambda_lambda @ ( paths_1651982994781619320lambda @ Resid @ T4 ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% paths_in_weakly_extensional_rts.Srcs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_1171_map__App__decomp,axiom,
! [U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ U2 ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U2 ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ U2 ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U2 ) ) )
@ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid @ U2
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ U2 ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U2 ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ U2 ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U2 ) ) ) ) ) ) ) ) ).
% map_App_decomp
thf(fact_1172_const__ide__is__Ide,axiom,
! [T4: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T4 ) @ bot_bo1271416176687173302lambda ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ) ) ).
% const_ide_is_Ide
thf(fact_1173_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_1174_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_1175_set__Ide__subset__ide,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ).
% set_Ide_subset_ide
thf(fact_1176_set__Arr__subset__arr,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ lambda_resid ) ) ) ) ).
% set_Arr_subset_arr
thf(fact_1177_Ide__char,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ) ).
% Ide_char
thf(fact_1178_seq__App__Std__implies,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ U2 ) )
=> ( ( ( lambda_is_App @ T )
& ~ ( lambda8135948770199036598uction @ T ) )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) ) ) ) ).
% seq_App_Std_implies
thf(fact_1179_Std__map__un__Lam,axiom,
! [T4: list_lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) )
=> ( reduction_Std @ ( map_la336867333220232516lambda @ lambda_un_Lam @ T4 ) ) ) ) ).
% Std_map_un_Lam
thf(fact_1180_Arr__map__un__App1,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_App1 @ T4 ) ) ) ) ).
% Arr_map_un_App1
thf(fact_1181_Arr__map__un__App2,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_App2 @ T4 ) ) ) ) ).
% Arr_map_un_App2
thf(fact_1182_set__Ide__subset__single__hd,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T4 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% set_Ide_subset_single_hd
thf(fact_1183_Arr__map__un__Lam,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_Lam @ T4 ) ) ) ) ).
% Arr_map_un_Lam
thf(fact_1184_map__eq__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs2: list_l5168835686021156086lambda,G: list_lambda_lambda > list_lambda_lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs2 )
= ( map_li385728805257398596lambda @ G @ Xs2 ) )
= ( ! [X2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1185_map__eq__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs2: list_lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs2 )
= ( map_la336867333220232516lambda @ G @ Xs2 ) )
= ( ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1186_filter__True,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs2 ) )
=> ( P @ X ) )
=> ( ( filter_lambda_lambda @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_1187_seq__Lam__Arr__implies,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( lambda_is_Lam @ T )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) ) ) ) ).
% seq_Lam_Arr_implies
thf(fact_1188_Std__filter__map__un__App1,axiom,
! [U2: list_lambda_lambda] :
( ( reduction_Std @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( reduction_Std
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U2 ) ) ) ) ) ).
% Std_filter_map_un_App1
thf(fact_1189_Std__filter__map__un__App2,axiom,
! [U2: list_lambda_lambda] :
( ( reduction_Std @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( reduction_Std
@ ( filter_lambda_lambda
@ ^ [U7: lambda_lambda] :
~ ( lambda_Ide @ U7 )
@ ( map_la336867333220232516lambda @ lambda_un_App2 @ U2 ) ) ) ) ) ).
% Std_filter_map_un_App2
thf(fact_1190_set__filter,axiom,
! [P: list_lambda_lambda > $o,Xs2: list_l5168835686021156086lambda] :
( ( set_li3846194476672328965lambda @ ( filter1131015166755930349lambda @ P @ Xs2 ) )
= ( collec8830145823237851355lambda
@ ^ [X2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs2 ) )
& ( P @ X2 ) ) ) ) ).
% set_filter
thf(fact_1191_set__filter,axiom,
! [P: lambda_lambda > $o,Xs2: list_lambda_lambda] :
( ( set_lambda_lambda2 @ ( filter_lambda_lambda @ P @ Xs2 ) )
= ( collec621444948832755541lambda
@ ^ [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X2 ) ) ) ) ).
% set_filter
thf(fact_1192_set__empty,axiom,
! [Xs2: list_lambda_lambda] :
( ( ( set_lambda_lambda2 @ Xs2 )
= bot_bo1271416176687173302lambda )
= ( Xs2 = nil_lambda_lambda ) ) ).
% set_empty
thf(fact_1193_set__empty,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( ( set_li3846194476672328965lambda @ Xs2 )
= bot_bo1397037886704484924lambda )
= ( Xs2 = nil_li8379291187143545974lambda ) ) ).
% set_empty
thf(fact_1194_set__empty2,axiom,
! [Xs2: list_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ Xs2 ) )
= ( Xs2 = nil_lambda_lambda ) ) ).
% set_empty2
thf(fact_1195_set__empty2,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ Xs2 ) )
= ( Xs2 = nil_li8379291187143545974lambda ) ) ).
% set_empty2
thf(fact_1196_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_1197_filter__False,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs2 ) )
=> ~ ( P @ X ) )
=> ( ( filter_lambda_lambda @ P @ Xs2 )
= nil_lambda_lambda ) ) ).
% filter_False
thf(fact_1198_IdeI,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ) ).
% IdeI
thf(fact_1199_filter__is__subset,axiom,
! [P: lambda_lambda > $o,Xs2: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ ( filter_lambda_lambda @ P @ Xs2 ) ) @ ( set_lambda_lambda2 @ Xs2 ) ) ).
% filter_is_subset
thf(fact_1200_subset__code_I1_J,axiom,
! [Xs2: list_l5168835686021156086lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs2 ) @ B2 )
= ( ! [X2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs2 ) )
=> ( member2667661554319814297lambda @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_1201_subset__code_I1_J,axiom,
! [Xs2: list_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs2 ) @ B2 )
= ( ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ( member_lambda_lambda @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_1202_set__subset__Cons,axiom,
! [Xs2: list_lambda_lambda,X3: lambda_lambda] : ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs2 ) @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_1203_Cons__eq__filterD,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,P: lambda_lambda > $o,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X3 @ Xs2 )
= ( filter_lambda_lambda @ P @ Ys ) )
=> ? [Us3: list_lambda_lambda,Vs2: list_lambda_lambda] :
( ( Ys
= ( append_lambda_lambda @ Us3 @ ( cons_lambda_lambda @ X3 @ Vs2 ) ) )
& ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Us3 ) )
=> ~ ( P @ X4 ) )
& ( P @ X3 )
& ( Xs2
= ( filter_lambda_lambda @ P @ Vs2 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_1204_filter__eq__ConsD,axiom,
! [P: lambda_lambda > $o,Ys: list_lambda_lambda,X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( filter_lambda_lambda @ P @ Ys )
= ( cons_lambda_lambda @ X3 @ Xs2 ) )
=> ? [Us3: list_lambda_lambda,Vs2: list_lambda_lambda] :
( ( Ys
= ( append_lambda_lambda @ Us3 @ ( cons_lambda_lambda @ X3 @ Vs2 ) ) )
& ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Us3 ) )
=> ~ ( P @ X4 ) )
& ( P @ X3 )
& ( Xs2
= ( filter_lambda_lambda @ P @ Vs2 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_1205_Cons__eq__filter__iff,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,P: lambda_lambda > $o,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X3 @ Xs2 )
= ( filter_lambda_lambda @ P @ Ys ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Ys
= ( append_lambda_lambda @ Us @ ( cons_lambda_lambda @ X3 @ Vs ) ) )
& ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Us ) )
=> ~ ( P @ X2 ) )
& ( P @ X3 )
& ( Xs2
= ( filter_lambda_lambda @ P @ Vs ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_1206_filter__eq__Cons__iff,axiom,
! [P: lambda_lambda > $o,Ys: list_lambda_lambda,X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( ( filter_lambda_lambda @ P @ Ys )
= ( cons_lambda_lambda @ X3 @ Xs2 ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Ys
= ( append_lambda_lambda @ Us @ ( cons_lambda_lambda @ X3 @ Vs ) ) )
& ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Us ) )
=> ~ ( P @ X2 ) )
& ( P @ X3 )
& ( Xs2
= ( filter_lambda_lambda @ P @ Vs ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_1207_list_Oset__intros_I2_J,axiom,
! [Y4: list_lambda_lambda,X22: list_l5168835686021156086lambda,X21: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y4 @ ( set_li3846194476672328965lambda @ X22 ) )
=> ( member2667661554319814297lambda @ Y4 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1208_list_Oset__intros_I2_J,axiom,
! [Y4: lambda_lambda,X22: list_lambda_lambda,X21: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ X22 ) )
=> ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1209_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_1210_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_1211_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_1212_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_1213_set__ConsD,axiom,
! [Y4: list_lambda_lambda,X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ Y4 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X3 @ Xs2 ) ) )
=> ( ( Y4 = X3 )
| ( member2667661554319814297lambda @ Y4 @ ( set_li3846194476672328965lambda @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_1214_set__ConsD,axiom,
! [Y4: lambda_lambda,X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X3 @ Xs2 ) ) )
=> ( ( Y4 = X3 )
| ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_1215_list_Omap__cong,axiom,
! [X3: list_lambda_lambda,Ya: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( X3 = Ya )
=> ( ! [Z2: lambda_lambda] :
( ( member_lambda_lambda @ Z2 @ ( set_lambda_lambda2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X3 )
= ( map_la336867333220232516lambda @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1216_list_Omap__cong0,axiom,
! [X3: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [Z2: lambda_lambda] :
( ( member_lambda_lambda @ Z2 @ ( set_lambda_lambda2 @ X3 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X3 )
= ( map_la336867333220232516lambda @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_1217_list_Oinj__map__strong,axiom,
! [X3: list_lambda_lambda,Xa: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Fa: lambda_lambda > lambda_lambda] :
( ! [Z2: lambda_lambda,Za: lambda_lambda] :
( ( member_lambda_lambda @ Z2 @ ( set_lambda_lambda2 @ X3 ) )
=> ( ( member_lambda_lambda @ Za @ ( set_lambda_lambda2 @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_la336867333220232516lambda @ F @ X3 )
= ( map_la336867333220232516lambda @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_1218_list_Omap__ident__strong,axiom,
! [T: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [Z2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z2 @ ( set_li3846194476672328965lambda @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_li385728805257398596lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1219_list_Omap__ident__strong,axiom,
! [T: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [Z2: lambda_lambda] :
( ( member_lambda_lambda @ Z2 @ ( set_lambda_lambda2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_la336867333220232516lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1220_map__ext,axiom,
! [Xs2: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs2 ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs2 )
= ( map_la336867333220232516lambda @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_1221_map__idI,axiom,
! [Xs2: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs2 ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_li385728805257398596lambda @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1222_map__idI,axiom,
! [Xs2: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs2 ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1223_map__cong,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( Xs2 = Ys )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs2 )
= ( map_la336867333220232516lambda @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_1224_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 ) )
=> ? [Y3: lambda_lambda] :
( X2
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1225_last__in__set,axiom,
! [As: list_l5168835686021156086lambda] :
( ( As != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( last_l1874196399739661071lambda @ As ) @ ( set_li3846194476672328965lambda @ As ) ) ) ).
% last_in_set
thf(fact_1226_last__in__set,axiom,
! [As: list_lambda_lambda] :
( ( As != nil_lambda_lambda )
=> ( member_lambda_lambda @ ( last_lambda_lambda @ As ) @ ( set_lambda_lambda2 @ As ) ) ) ).
% last_in_set
thf(fact_1227_empty__filter__conv,axiom,
! [P: lambda_lambda > $o,Xs2: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( filter_lambda_lambda @ P @ Xs2 ) )
= ( ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ~ ( P @ X2 ) ) ) ) ).
% empty_filter_conv
thf(fact_1228_filter__empty__conv,axiom,
! [P: lambda_lambda > $o,Xs2: list_lambda_lambda] :
( ( ( filter_lambda_lambda @ P @ Xs2 )
= nil_lambda_lambda )
= ( ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ~ ( P @ X2 ) ) ) ) ).
% filter_empty_conv
thf(fact_1229_list_Oset__sel_I2_J,axiom,
! [A: list_l5168835686021156086lambda,X3: list_lambda_lambda] :
( ( A != nil_li8379291187143545974lambda )
=> ( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ ( tl_lis5905685823335986783lambda @ A ) ) )
=> ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1230_list_Oset__sel_I2_J,axiom,
! [A: list_lambda_lambda,X3: lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ ( tl_lambda_lambda @ A ) ) )
=> ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_1231_hd__in__set,axiom,
! [Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ Xs2 ) @ ( set_li3846194476672328965lambda @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_1232_hd__in__set,axiom,
! [Xs2: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( member_lambda_lambda @ ( hd_lambda_lambda @ Xs2 ) @ ( set_lambda_lambda2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_1233_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_1234_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_1235_split__list,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs2 ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( Xs2
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X3 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_1236_split__list,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ? [Ys2: list_lambda_lambda,Zs3: list_lambda_lambda] :
( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_1237_split__list__last,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs2 ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X3 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_1238_split__list__last,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ? [Ys2: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_1239_split__list__prop,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_1240_split__list__first,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs2 ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X3 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1241_split__list__first,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ? [Ys2: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X3 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1242_split__list__propE,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_1243_append__Cons__eq__iff,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs5: list_l5168835686021156086lambda,Ys6: list_l5168835686021156086lambda] :
( ~ ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs2 ) )
=> ( ~ ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Ys ) )
=> ( ( ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X3 @ Ys ) )
= ( append1017360404025248779lambda @ Xs5 @ ( cons_l4558873032622161958lambda @ X3 @ Ys6 ) ) )
= ( ( Xs2 = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1244_append__Cons__eq__iff,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda,Ys: list_lambda_lambda,Xs5: list_lambda_lambda,Ys6: list_lambda_lambda] :
( ~ ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ( ~ ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X3 @ Ys ) )
= ( append_lambda_lambda @ Xs5 @ ( cons_lambda_lambda @ X3 @ Ys6 ) ) )
= ( ( Xs2 = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1245_in__set__conv__decomp,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs2 ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( Xs2
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X3 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1246_in__set__conv__decomp,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs2 ) )
= ( ? [Ys3: list_lambda_lambda,Zs2: list_lambda_lambda] :
( Xs2
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1247_split__list__last__prop,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_1248_split__list__first__prop,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_1249_split__list__last__propE,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1250_split__list__first__propE,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs2
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1251_in__set__conv__decomp__last,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs2 ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X3 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1252_in__set__conv__decomp__last,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs2 ) )
= ( ? [Ys3: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1253_in__set__conv__decomp__first,axiom,
! [X3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs2 ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs2
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X3 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1254_in__set__conv__decomp__first,axiom,
! [X3: lambda_lambda,Xs2: list_lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs2 ) )
= ( ? [Ys3: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1255_split__list__last__prop__iff,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list_lambda_lambda,X2: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs2
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y3: lambda_lambda] :
( ( member_lambda_lambda @ Y3 @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1256_split__list__first__prop__iff,axiom,
! [Xs2: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs2 ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list_lambda_lambda,X2: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs2
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y3: lambda_lambda] :
( ( member_lambda_lambda @ Y3 @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1257_filter__id__conv,axiom,
! [P: lambda_lambda > $o,Xs2: list_lambda_lambda] :
( ( ( filter_lambda_lambda @ P @ Xs2 )
= Xs2 )
= ( ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs2 ) )
=> ( P @ X2 ) ) ) ) ).
% filter_id_conv
thf(fact_1258_filter__cong,axiom,
! [Xs2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( Xs2 = Ys )
=> ( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( filter1131015166755930349lambda @ P @ Xs2 )
= ( filter1131015166755930349lambda @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1259_filter__cong,axiom,
! [Xs2: list_lambda_lambda,Ys: list_lambda_lambda,P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( Xs2 = Ys )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( filter_lambda_lambda @ P @ Xs2 )
= ( filter_lambda_lambda @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_1260_empty__set,axiom,
( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ nil_lambda_lambda ) ) ).
% empty_set
thf(fact_1261_empty__set,axiom,
( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ nil_li8379291187143545974lambda ) ) ).
% empty_set
thf(fact_1262_map__App__map__un__App1,axiom,
! [U2: list_lambda_lambda,B: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( lambda_Ide @ B )
=> ( ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ lambda_un_App2 @ ( set_lambda_lambda2 @ U2 ) ) @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
=> ( ( map_la336867333220232516lambda
@ ^ [T7: lambda_lambda] : ( lambda_App @ T7 @ B )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U2 ) )
= U2 ) ) ) ) ) ).
% map_App_map_un_App1
% Helper facts (9)
thf(help_If_2_1_If_001t__LambdaCalculus__Olambda____calculus__Olambda_T,axiom,
! [X3: lambda_lambda,Y4: lambda_lambda] :
( ( if_lambda_lambda @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__LambdaCalculus__Olambda____calculus__Olambda_T,axiom,
! [X3: lambda_lambda,Y4: lambda_lambda] :
( ( if_lambda_lambda @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X3: list_lambda_lambda,Y4: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X3: list_lambda_lambda,Y4: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_T,axiom,
! [X3: list_l5168835686021156086lambda,Y4: list_l5168835686021156086lambda] :
( ( if_lis4115203008988617916lambda @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_T,axiom,
! [X3: list_l5168835686021156086lambda,Y4: list_l5168835686021156086lambda] :
( ( if_lis4115203008988617916lambda @ $true @ X3 @ Y4 )
= X3 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_J_T,axiom,
! [X3: list_l4928420112573288188lambda,Y4: list_l4928420112573288188lambda] :
( ( if_lis8739384218277118914lambda @ $false @ X3 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_J_T,axiom,
! [X3: list_l4928420112573288188lambda,Y4: list_l4928420112573288188lambda] :
( ( if_lis8739384218277118914lambda @ $true @ X3 @ Y4 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( reduct5390047737487997688insert @ ( lambda_App @ m @ n ) @ ( cons_lambda_lambda @ u @ ua ) )
= ( cons_lambda_lambda @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) @ ( reduct5390047737487997688insert @ ( lambda_resid @ ( lambda_App @ m @ n ) @ ( lambda_head_strategy @ ( lambda_App @ m @ n ) ) ) @ ua ) ) ) ).
%------------------------------------------------------------------------------