TPTP Problem File: SLH0443^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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_03286_137268__14440540_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1446 ( 358 unt; 165 typ; 0 def)
% Number of atoms : 4367 (1835 equ; 0 cnn)
% Maximal formula atoms : 38 ( 3 avg)
% Number of connectives : 18193 ( 763 ~; 24 |; 532 &;14801 @)
% ( 0 <=>;2073 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 1505 (1505 >; 0 *; 0 +; 0 <<)
% Number of symbols : 159 ( 156 usr; 12 con; 0-5 aty)
% Number of variables : 3610 ( 105 ^;3373 !; 132 ?;3610 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:48.648
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
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thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
arr_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Oarr_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
arr_li3691443969012319837lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
con_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_Ocon_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
con_li79537494059641180lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oresiduation_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_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_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_Ocomposite__of_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
compos9183582412961930448lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ocomposite__of_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
compos7114946393191943382lambda: ( list_lambda_lambda > 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_Ojoinable_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
joinab2519788767067900224lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Ojoinable_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
joinab677607494065406278lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
seq_lambda_lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Oseq_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
seq_li8093550137657879445lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
source5627170679743844886lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > set_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Osources_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
source6119337877621565212lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
target5490931411331770124lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > set_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
target4026644918049360146lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_ax7298520571313827089lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
rts_ax2283037410143324311lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > $o ).
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__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula826895837143331878lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osimulation__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
simula6289889787749269804lambda: ( 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__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
simula5056032608672110112lambda: ( 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__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
simula6307967448222309926lambda: ( 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_Osub__rts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
sub_rt3575465509250962105lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > $o ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osub__rts__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
sub_rt3099298114314975807lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > $o ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
transf356316610583137064lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
transf4653517080217399086lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
transf3419659901140239394lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
transf8838427370478101288lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > 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_Oweakly__extensional__rts_Osrc_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
weakly4554751870655413501lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
weakly2957279434024022659lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda ).
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_Oimage_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
image_3692823078739859851lambda: ( lambda_lambda > list_lambda_lambda ) > set_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
image_2458965899662700159lambda: ( list_lambda_lambda > lambda_lambda ) > set_li7878742357367608272lambda > set_lambda_lambda ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
image_5886195720093285125lambda: ( list_lambda_lambda > list_lambda_lambda ) > set_li7878742357367608272lambda > set_li7878742357367608272lambda ).
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_Set_Othe__elem_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
the_el5431214018478934239lambda: set_lambda_lambda > lambda_lambda ).
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_Ua____,type,
ua: list_lambda_lambda ).
thf(sy_v_ta____,type,
ta: lambda_lambda ).
thf(sy_v_u____,type,
u: lambda_lambda ).
% Relevant facts (1275)
thf(fact_0__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_1__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_2_U,axiom,
ua != nil_lambda_lambda ).
% U
thf(fact_3_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_4_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_5_Trgs_Ocases,axiom,
! [X2: list_lambda_lambda] :
( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ).
% Trgs.cases
thf(fact_6_Cube_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) ) ) ) ).
% Cube(2)
thf(fact_7_Cube_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) ).
% Cube(1)
thf(fact_8_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_9_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_10_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_11_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_12_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_13_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_14_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_15_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_16_Resid_Osimps_I2_J,axiom,
! [V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% Resid.simps(2)
thf(fact_17_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_18_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_19_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_20_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_21_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_22_Con__append_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(2)
thf(fact_23_Con__append_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(1)
thf(fact_24_Resid__append__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ U2 )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ U2 ) ) ) ) ) ) ) ).
% Resid_append_ind
thf(fact_25_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_26_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_27_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_28_uU,axiom,
reduct9208763521642324371opment @ ta @ ( cons_lambda_lambda @ u @ ua ) ).
% uU
thf(fact_29_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_30_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_31_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_32_Con__appendI_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(2)
thf(fact_33_Con__appendI_I1_J,axiom,
! [T4: list_lambda_lambda,V3: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(1)
thf(fact_34_Resid__append_I2_J,axiom,
! [U2: list_lambda_lambda,V3: list_lambda_lambda,T4: list_lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V3 ) ) ) ) ) ).
% Resid_append(2)
thf(fact_35_Resid__append_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) ) ) ) ) ) ).
% Resid_append(1)
thf(fact_36_Resid__append2,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda,W: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( W != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V3 @ W ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V3 @ W ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ W ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) ) ) ) ) ) ) ) ) ) ).
% Resid_append2
thf(fact_37_paths__in__confluent__rts__axioms,axiom,
paths_9040249923598999359lambda @ lambda_resid ).
% paths_in_confluent_rts_axioms
thf(fact_38__092_060open_062U_A_061_A_091_093_A_092_060Longrightarrow_062_Aelementary__reduction__path_A_Iu_A_D_AU_J_A_092_060and_062_A_Iu_A_D_AU_A_092_060noteq_062_A_091_093_A_092_060longrightarrow_062_Au_A_D_AU_A_092_060_094sup_062_K_092_060lesssim_062_092_060_094sup_062_K_A_091t_093_J_092_060close_062,axiom,
( ( ua = nil_lambda_lambda )
=> ( ( reduct8314398545252454212n_path @ ( cons_lambda_lambda @ u @ ua ) )
& ( ( ( cons_lambda_lambda @ u @ ua )
!= nil_lambda_lambda )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ u @ ua ) @ ( cons_lambda_lambda @ ta @ nil_lambda_lambda ) ) ) ) ) ) ).
% \<open>U = [] \<Longrightarrow> elementary_reduction_path (u # U) \<and> (u # U \<noteq> [] \<longrightarrow> u # U \<^sup>*\<lesssim>\<^sup>* [t])\<close>
thf(fact_39_is__confluent__rts,axiom,
conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% is_confluent_rts
thf(fact_40_partial__magma__axioms,axiom,
partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% partial_magma_axioms
thf(fact_41_mem__Collect__eq,axiom,
! [A: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( member2667661554319814297lambda @ A @ ( collec8830145823237851355lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ A @ ( collec621444948832755541lambda @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_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_46_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_47__092_060Lambda_062_Opartial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% \<Lambda>.partial_magma_axioms
thf(fact_48_cong__implies__coterminal,axiom,
! [U: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U3 ) ) ).
% cong_implies_coterminal
thf(fact_49_append1__eq__conv,axiom,
! [Xs: list_s345339197461576016lambda,X2: set_lambda_lambda,Ys: list_s345339197461576016lambda,Y2: set_lambda_lambda] :
( ( ( append2117223597850273125lambda @ Xs @ ( cons_s3132251510596564736lambda @ X2 @ nil_se6914803783601386320lambda ) )
= ( append2117223597850273125lambda @ Ys @ ( cons_s3132251510596564736lambda @ Y2 @ nil_se6914803783601386320lambda ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_50_append1__eq__conv,axiom,
! [Xs: list_l5168835686021156086lambda,X2: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Y2: list_lambda_lambda] :
( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) )
= ( append1017360404025248779lambda @ Ys @ ( cons_l4558873032622161958lambda @ Y2 @ nil_li8379291187143545974lambda ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_51_append1__eq__conv,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda,Ys: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) )
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ Y2 @ nil_lambda_lambda ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_52_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_53_append_Oright__neutral,axiom,
! [A: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ A @ nil_li8379291187143545974lambda )
= A ) ).
% append.right_neutral
thf(fact_54_append_Oright__neutral,axiom,
! [A: list_s345339197461576016lambda] :
( ( append2117223597850273125lambda @ A @ nil_se6914803783601386320lambda )
= A ) ).
% append.right_neutral
thf(fact_55_append_Oright__neutral,axiom,
! [A: list_lambda_lambda] :
( ( append_lambda_lambda @ A @ nil_lambda_lambda )
= A ) ).
% append.right_neutral
thf(fact_56_append__Nil2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ Xs @ nil_li8379291187143545974lambda )
= Xs ) ).
% append_Nil2
thf(fact_57_append__Nil2,axiom,
! [Xs: list_s345339197461576016lambda] :
( ( append2117223597850273125lambda @ Xs @ nil_se6914803783601386320lambda )
= Xs ) ).
% append_Nil2
thf(fact_58_append__Nil2,axiom,
! [Xs: list_lambda_lambda] :
( ( append_lambda_lambda @ Xs @ nil_lambda_lambda )
= Xs ) ).
% append_Nil2
thf(fact_59_append__self__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= Xs )
= ( Ys = nil_li8379291187143545974lambda ) ) ).
% append_self_conv
thf(fact_60_append__self__conv,axiom,
! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
( ( ( append2117223597850273125lambda @ Xs @ Ys )
= Xs )
= ( Ys = nil_se6914803783601386320lambda ) ) ).
% append_self_conv
thf(fact_61_append__self__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= Xs )
= ( Ys = nil_lambda_lambda ) ) ).
% append_self_conv
thf(fact_62_self__append__conv,axiom,
! [Y2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Y2
= ( append1017360404025248779lambda @ Y2 @ Ys ) )
= ( Ys = nil_li8379291187143545974lambda ) ) ).
% self_append_conv
thf(fact_63_self__append__conv,axiom,
! [Y2: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
( ( Y2
= ( append2117223597850273125lambda @ Y2 @ Ys ) )
= ( Ys = nil_se6914803783601386320lambda ) ) ).
% self_append_conv
thf(fact_64_self__append__conv,axiom,
! [Y2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Y2
= ( append_lambda_lambda @ Y2 @ Ys ) )
= ( Ys = nil_lambda_lambda ) ) ).
% self_append_conv
thf(fact_65_append__self__conv2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= Ys )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% append_self_conv2
thf(fact_66_append__self__conv2,axiom,
! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
( ( ( append2117223597850273125lambda @ Xs @ Ys )
= Ys )
= ( Xs = nil_se6914803783601386320lambda ) ) ).
% append_self_conv2
thf(fact_67_append__self__conv2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= Ys )
= ( Xs = nil_lambda_lambda ) ) ).
% append_self_conv2
thf(fact_68_self__append__conv2,axiom,
! [Y2: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda] :
( ( Y2
= ( append1017360404025248779lambda @ Xs @ Y2 ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% self_append_conv2
thf(fact_69_self__append__conv2,axiom,
! [Y2: list_s345339197461576016lambda,Xs: list_s345339197461576016lambda] :
( ( Y2
= ( append2117223597850273125lambda @ Xs @ Y2 ) )
= ( Xs = nil_se6914803783601386320lambda ) ) ).
% self_append_conv2
thf(fact_70_self__append__conv2,axiom,
! [Y2: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Y2
= ( append_lambda_lambda @ Xs @ Y2 ) )
= ( Xs = nil_lambda_lambda ) ) ).
% self_append_conv2
thf(fact_71_Nil__is__append__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( nil_li8379291187143545974lambda
= ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( ( Xs = nil_li8379291187143545974lambda )
& ( Ys = nil_li8379291187143545974lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_72_Nil__is__append__conv,axiom,
! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
( ( nil_se6914803783601386320lambda
= ( append2117223597850273125lambda @ Xs @ Ys ) )
= ( ( Xs = nil_se6914803783601386320lambda )
& ( Ys = nil_se6914803783601386320lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_73_Nil__is__append__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( append_lambda_lambda @ Xs @ Ys ) )
= ( ( Xs = nil_lambda_lambda )
& ( Ys = nil_lambda_lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_74_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_75_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_76_same__append__eq,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_77_same__append__eq,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_78_append__same__eq,axiom,
! [Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Xs )
= ( append1017360404025248779lambda @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_79_append__same__eq,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Xs )
= ( append_lambda_lambda @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_80_append__assoc,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) @ Zs )
= ( append1017360404025248779lambda @ Xs @ ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_81_append__assoc,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) @ Zs )
= ( append_lambda_lambda @ Xs @ ( append_lambda_lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_82_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_83_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_84_append__is__Nil__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= nil_li8379291187143545974lambda )
= ( ( Xs = nil_li8379291187143545974lambda )
& ( Ys = nil_li8379291187143545974lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_85_append__is__Nil__conv,axiom,
! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
( ( ( append2117223597850273125lambda @ Xs @ Ys )
= nil_se6914803783601386320lambda )
= ( ( Xs = nil_se6914803783601386320lambda )
& ( Ys = nil_se6914803783601386320lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_86_append__is__Nil__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= nil_lambda_lambda )
= ( ( Xs = nil_lambda_lambda )
& ( Ys = nil_lambda_lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_87_ind,axiom,
! [T: lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ ua )
=> ( ( reduct8314398545252454212n_path @ ua )
& ( ( ua != nil_lambda_lambda )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ua @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).
% ind
thf(fact_88_not__Cons__self2,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( cons_l4558873032622161958lambda @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_89_not__Cons__self2,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( cons_lambda_lambda @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_90_append__eq__append__conv2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ts: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Zs @ Ts ) )
= ( ? [Us: list_l5168835686021156086lambda] :
( ( ( Xs
= ( append1017360404025248779lambda @ Zs @ Us ) )
& ( ( append1017360404025248779lambda @ Us @ Ys )
= Ts ) )
| ( ( ( append1017360404025248779lambda @ Xs @ Us )
= Zs )
& ( Ys
= ( append1017360404025248779lambda @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_91_append__eq__append__conv2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ts: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs @ Ts ) )
= ( ? [Us: list_lambda_lambda] :
( ( ( Xs
= ( append_lambda_lambda @ Zs @ Us ) )
& ( ( append_lambda_lambda @ Us @ Ys )
= Ts ) )
| ( ( ( append_lambda_lambda @ Xs @ Us )
= Zs )
& ( Ys
= ( append_lambda_lambda @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_92_append__eq__appendI,axiom,
! [Xs: list_l5168835686021156086lambda,Xs1: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Us2: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append1017360404025248779lambda @ Xs1 @ Us2 ) )
=> ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_93_append__eq__appendI,axiom,
! [Xs: list_lambda_lambda,Xs1: list_lambda_lambda,Zs: list_lambda_lambda,Ys: list_lambda_lambda,Us2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_lambda_lambda @ Xs1 @ Us2 ) )
=> ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_94_list__nonempty__induct,axiom,
! [Xs: list_s345339197461576016lambda,P: list_s345339197461576016lambda > $o] :
( ( Xs != nil_se6914803783601386320lambda )
=> ( ! [X: set_lambda_lambda] : ( P @ ( cons_s3132251510596564736lambda @ X @ nil_se6914803783601386320lambda ) )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
( ( Xs2 != nil_se6914803783601386320lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_s3132251510596564736lambda @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_95_list__nonempty__induct,axiom,
! [Xs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_96_list__nonempty__induct,axiom,
! [Xs: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( Xs != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_97_list__induct2_H,axiom,
! [P: list_s345339197461576016lambda > list_s345339197461576016lambda > $o,Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
( ( P @ nil_se6914803783601386320lambda @ nil_se6914803783601386320lambda )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda] : ( P @ ( cons_s3132251510596564736lambda @ X @ Xs2 ) @ nil_se6914803783601386320lambda )
=> ( ! [Y3: set_lambda_lambda,Ys2: list_s345339197461576016lambda] : ( P @ nil_se6914803783601386320lambda @ ( cons_s3132251510596564736lambda @ Y3 @ Ys2 ) )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda,Y3: set_lambda_lambda,Ys2: list_s345339197461576016lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_s3132251510596564736lambda @ X @ Xs2 ) @ ( cons_s3132251510596564736lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_98_list__induct2_H,axiom,
! [P: list_s345339197461576016lambda > list_lambda_lambda > $o,Xs: list_s345339197461576016lambda,Ys: list_lambda_lambda] :
( ( P @ nil_se6914803783601386320lambda @ nil_lambda_lambda )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda] : ( P @ ( cons_s3132251510596564736lambda @ X @ Xs2 ) @ nil_lambda_lambda )
=> ( ! [Y3: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_se6914803783601386320lambda @ ( cons_lambda_lambda @ Y3 @ Ys2 ) )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda,Y3: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_s3132251510596564736lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_99_list__induct2_H,axiom,
! [P: list_s345339197461576016lambda > list_l5168835686021156086lambda > $o,Xs: list_s345339197461576016lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_se6914803783601386320lambda @ nil_li8379291187143545974lambda )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda] : ( P @ ( cons_s3132251510596564736lambda @ X @ Xs2 ) @ nil_li8379291187143545974lambda )
=> ( ! [Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_se6914803783601386320lambda @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda,Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_s3132251510596564736lambda @ X @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_100_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_s345339197461576016lambda > $o,Xs: list_lambda_lambda,Ys: list_s345339197461576016lambda] :
( ( P @ nil_lambda_lambda @ nil_se6914803783601386320lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ nil_se6914803783601386320lambda )
=> ( ! [Y3: set_lambda_lambda,Ys2: list_s345339197461576016lambda] : ( P @ nil_lambda_lambda @ ( cons_s3132251510596564736lambda @ Y3 @ Ys2 ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: set_lambda_lambda,Ys2: list_s345339197461576016lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_s3132251510596564736lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_101_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_l5168835686021156086lambda > $o,Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ nil_li8379291187143545974lambda )
=> ( ! [Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_lambda_lambda @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_102_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_s345339197461576016lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_s345339197461576016lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_se6914803783601386320lambda )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ nil_se6914803783601386320lambda )
=> ( ! [Y3: set_lambda_lambda,Ys2: list_s345339197461576016lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_s3132251510596564736lambda @ Y3 @ Ys2 ) )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y3: set_lambda_lambda,Ys2: list_s345339197461576016lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ ( cons_s3132251510596564736lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_103_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_lambda_lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ nil_lambda_lambda )
=> ( ! [Y3: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_lambda_lambda @ Y3 @ Ys2 ) )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y3: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_104_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ nil_li8379291187143545974lambda )
=> ( ! [Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_105_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_lambda_lambda > $o,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ nil_lambda_lambda )
=> ( ! [Y3: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_lambda_lambda @ ( cons_lambda_lambda @ Y3 @ Ys2 ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_106_neq__Nil__conv,axiom,
! [Xs: list_s345339197461576016lambda] :
( ( Xs != nil_se6914803783601386320lambda )
= ( ? [Y4: set_lambda_lambda,Ys3: list_s345339197461576016lambda] :
( Xs
= ( cons_s3132251510596564736lambda @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_107_neq__Nil__conv,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
= ( ? [Y4: list_lambda_lambda,Ys3: list_l5168835686021156086lambda] :
( Xs
= ( cons_l4558873032622161958lambda @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_108_neq__Nil__conv,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
= ( ? [Y4: lambda_lambda,Ys3: list_lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_109_remdups__adj_Ocases,axiom,
! [X2: list_s345339197461576016lambda] :
( ( X2 != nil_se6914803783601386320lambda )
=> ( ! [X: set_lambda_lambda] :
( X2
!= ( cons_s3132251510596564736lambda @ X @ nil_se6914803783601386320lambda ) )
=> ~ ! [X: set_lambda_lambda,Y3: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
( X2
!= ( cons_s3132251510596564736lambda @ X @ ( cons_s3132251510596564736lambda @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_110_remdups__adj_Ocases,axiom,
! [X2: list_l5168835686021156086lambda] :
( ( X2 != nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda] :
( X2
!= ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
=> ~ ! [X: list_lambda_lambda,Y3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( X2
!= ( cons_l4558873032622161958lambda @ X @ ( cons_l4558873032622161958lambda @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_111_remdups__adj_Ocases,axiom,
! [X2: list_lambda_lambda] :
( ( X2 != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ~ ! [X: lambda_lambda,Y3: lambda_lambda,Xs2: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ X @ ( cons_lambda_lambda @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_112_transpose_Ocases,axiom,
! [X2: list_l7543285377636199382lambda] :
( ( X2 != nil_li3187382615321999190lambda )
=> ( ! [Xss: list_l7543285377636199382lambda] :
( X2
!= ( cons_l203434155165082374lambda @ nil_se6914803783601386320lambda @ Xss ) )
=> ~ ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda,Xss: list_l7543285377636199382lambda] :
( X2
!= ( cons_l203434155165082374lambda @ ( cons_s3132251510596564736lambda @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_113_transpose_Ocases,axiom,
! [X2: list_l4928420112573288188lambda] :
( ( X2 != nil_li6069905736445159420lambda )
=> ( ! [Xss: list_l4928420112573288188lambda] :
( X2
!= ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ Xss ) )
=> ~ ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Xss: list_l4928420112573288188lambda] :
( X2
!= ( cons_l4910729503725484972lambda @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_114_transpose_Ocases,axiom,
! [X2: list_l5168835686021156086lambda] :
( ( X2 != nil_li8379291187143545974lambda )
=> ( ! [Xss: list_l5168835686021156086lambda] :
( X2
!= ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ Xss ) )
=> ~ ! [X: lambda_lambda,Xs2: list_lambda_lambda,Xss: list_l5168835686021156086lambda] :
( X2
!= ( cons_l4558873032622161958lambda @ ( cons_lambda_lambda @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_115_min__list_Ocases,axiom,
! [X2: list_s345339197461576016lambda] :
( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
( X2
!= ( cons_s3132251510596564736lambda @ X @ Xs2 ) )
=> ( X2 = nil_se6914803783601386320lambda ) ) ).
% min_list.cases
thf(fact_116_list_Oexhaust,axiom,
! [Y2: list_s345339197461576016lambda] :
( ( Y2 != nil_se6914803783601386320lambda )
=> ~ ! [X212: set_lambda_lambda,X222: list_s345339197461576016lambda] :
( Y2
!= ( cons_s3132251510596564736lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_117_list_Oexhaust,axiom,
! [Y2: list_l5168835686021156086lambda] :
( ( Y2 != nil_li8379291187143545974lambda )
=> ~ ! [X212: list_lambda_lambda,X222: list_l5168835686021156086lambda] :
( Y2
!= ( cons_l4558873032622161958lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_118_list_Oexhaust,axiom,
! [Y2: list_lambda_lambda] :
( ( Y2 != nil_lambda_lambda )
=> ~ ! [X212: lambda_lambda,X222: list_lambda_lambda] :
( Y2
!= ( cons_lambda_lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_119_list_OdiscI,axiom,
! [List: list_s345339197461576016lambda,X21: set_lambda_lambda,X22: list_s345339197461576016lambda] :
( ( List
= ( cons_s3132251510596564736lambda @ X21 @ X22 ) )
=> ( List != nil_se6914803783601386320lambda ) ) ).
% list.discI
thf(fact_120_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_121_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_122_list_Odistinct_I1_J,axiom,
! [X21: set_lambda_lambda,X22: list_s345339197461576016lambda] :
( nil_se6914803783601386320lambda
!= ( cons_s3132251510596564736lambda @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_123_list_Odistinct_I1_J,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( nil_li8379291187143545974lambda
!= ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_124_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_125_Cons__eq__appendI,axiom,
! [X2: list_lambda_lambda,Xs1: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append1017360404025248779lambda @ Xs1 @ Zs ) )
=> ( ( cons_l4558873032622161958lambda @ X2 @ Xs )
= ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_126_Cons__eq__appendI,axiom,
! [X2: lambda_lambda,Xs1: list_lambda_lambda,Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_lambda_lambda @ Xs1 @ Zs ) )
=> ( ( cons_lambda_lambda @ X2 @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_127_append__Cons,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( cons_l4558873032622161958lambda @ X2 @ Xs ) @ Ys )
= ( cons_l4558873032622161958lambda @ X2 @ ( append1017360404025248779lambda @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_128_append__Cons,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) @ Ys )
= ( cons_lambda_lambda @ X2 @ ( append_lambda_lambda @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_129_eq__Nil__appendI,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs = Ys )
=> ( Xs
= ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_130_eq__Nil__appendI,axiom,
! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
( ( Xs = Ys )
=> ( Xs
= ( append2117223597850273125lambda @ nil_se6914803783601386320lambda @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_131_eq__Nil__appendI,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs = Ys )
=> ( Xs
= ( append_lambda_lambda @ nil_lambda_lambda @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_132_append_Oleft__neutral,axiom,
! [A: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ A )
= A ) ).
% append.left_neutral
thf(fact_133_append_Oleft__neutral,axiom,
! [A: list_s345339197461576016lambda] :
( ( append2117223597850273125lambda @ nil_se6914803783601386320lambda @ A )
= A ) ).
% append.left_neutral
thf(fact_134_append_Oleft__neutral,axiom,
! [A: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ A )
= A ) ).
% append.left_neutral
thf(fact_135_append__Nil,axiom,
! [Ys: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ nil_li8379291187143545974lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_136_append__Nil,axiom,
! [Ys: list_s345339197461576016lambda] :
( ( append2117223597850273125lambda @ nil_se6914803783601386320lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_137_append__Nil,axiom,
! [Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_138_rev__nonempty__induct,axiom,
! [Xs: list_s345339197461576016lambda,P: list_s345339197461576016lambda > $o] :
( ( Xs != nil_se6914803783601386320lambda )
=> ( ! [X: set_lambda_lambda] : ( P @ ( cons_s3132251510596564736lambda @ X @ nil_se6914803783601386320lambda ) )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
( ( Xs2 != nil_se6914803783601386320lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( append2117223597850273125lambda @ Xs2 @ ( cons_s3132251510596564736lambda @ X @ nil_se6914803783601386320lambda ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_139_rev__nonempty__induct,axiom,
! [Xs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_140_rev__nonempty__induct,axiom,
! [Xs: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( Xs != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] :
( ( Xs2 != nil_lambda_lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_141_append__eq__Cons__conv,axiom,
! [Ys: list_s345339197461576016lambda,Zs: list_s345339197461576016lambda,X2: set_lambda_lambda,Xs: list_s345339197461576016lambda] :
( ( ( append2117223597850273125lambda @ Ys @ Zs )
= ( cons_s3132251510596564736lambda @ X2 @ Xs ) )
= ( ( ( Ys = nil_se6914803783601386320lambda )
& ( Zs
= ( cons_s3132251510596564736lambda @ X2 @ Xs ) ) )
| ? [Ys4: list_s345339197461576016lambda] :
( ( Ys
= ( cons_s3132251510596564736lambda @ X2 @ Ys4 ) )
& ( ( append2117223597850273125lambda @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_142_append__eq__Cons__conv,axiom,
! [Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Zs )
= ( cons_l4558873032622161958lambda @ X2 @ Xs ) )
= ( ( ( Ys = nil_li8379291187143545974lambda )
& ( Zs
= ( cons_l4558873032622161958lambda @ X2 @ Xs ) ) )
| ? [Ys4: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ X2 @ Ys4 ) )
& ( ( append1017360404025248779lambda @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_143_append__eq__Cons__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( cons_lambda_lambda @ X2 @ Xs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( Zs
= ( cons_lambda_lambda @ X2 @ Xs ) ) )
| ? [Ys4: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ X2 @ Ys4 ) )
& ( ( append_lambda_lambda @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_144_Cons__eq__append__conv,axiom,
! [X2: set_lambda_lambda,Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda,Zs: list_s345339197461576016lambda] :
( ( ( cons_s3132251510596564736lambda @ X2 @ Xs )
= ( append2117223597850273125lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_se6914803783601386320lambda )
& ( ( cons_s3132251510596564736lambda @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_s345339197461576016lambda] :
( ( ( cons_s3132251510596564736lambda @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append2117223597850273125lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_145_Cons__eq__append__conv,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X2 @ Xs )
= ( append1017360404025248779lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_li8379291187143545974lambda )
& ( ( cons_l4558873032622161958lambda @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append1017360404025248779lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_146_Cons__eq__append__conv,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( ( cons_lambda_lambda @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_lambda_lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_147_rev__exhaust,axiom,
! [Xs: list_s345339197461576016lambda] :
( ( Xs != nil_se6914803783601386320lambda )
=> ~ ! [Ys2: list_s345339197461576016lambda,Y3: set_lambda_lambda] :
( Xs
!= ( append2117223597850273125lambda @ Ys2 @ ( cons_s3132251510596564736lambda @ Y3 @ nil_se6914803783601386320lambda ) ) ) ) ).
% rev_exhaust
thf(fact_148_rev__exhaust,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ~ ! [Ys2: list_l5168835686021156086lambda,Y3: list_lambda_lambda] :
( Xs
!= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ Y3 @ nil_li8379291187143545974lambda ) ) ) ) ).
% rev_exhaust
thf(fact_149_rev__exhaust,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ~ ! [Ys2: list_lambda_lambda,Y3: lambda_lambda] :
( Xs
!= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ Y3 @ nil_lambda_lambda ) ) ) ) ).
% rev_exhaust
thf(fact_150_rev__induct,axiom,
! [P: list_s345339197461576016lambda > $o,Xs: list_s345339197461576016lambda] :
( ( P @ nil_se6914803783601386320lambda )
=> ( ! [X: set_lambda_lambda,Xs2: list_s345339197461576016lambda] :
( ( P @ Xs2 )
=> ( P @ ( append2117223597850273125lambda @ Xs2 @ ( cons_s3132251510596564736lambda @ X @ nil_se6914803783601386320lambda ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_151_rev__induct,axiom,
! [P: list_l5168835686021156086lambda > $o,Xs: list_l5168835686021156086lambda] :
( ( P @ nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( P @ Xs2 )
=> ( P @ ( append1017360404025248779lambda @ Xs2 @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_152_rev__induct,axiom,
! [P: list_lambda_lambda > $o,Xs: list_lambda_lambda] :
( ( P @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] :
( ( P @ Xs2 )
=> ( P @ ( append_lambda_lambda @ Xs2 @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_153_rts__with__composites__axioms,axiom,
rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% rts_with_composites_axioms
thf(fact_154_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_155_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_156_cong__implies__coinitial,axiom,
! [U: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U3 ) ) ).
% cong_implies_coinitial
thf(fact_157_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_158_Resid__parallel,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T ) ) )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ U @ T5 ) ) ) ) ).
% Resid_parallel
thf(fact_159_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_160_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_161__C1_C,axiom,
( ( lambda2670285437103676873uction @ u )
& ( reduct8314398545252454212n_path @ ua )
& ( lambda_subs @ u @ ta )
& ( ( ua != nil_lambda_lambda )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ua @ ( cons_lambda_lambda @ ( lambda_resid @ ta @ u ) @ nil_lambda_lambda ) ) ) ) ) ).
% "1"
thf(fact_162_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_163_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_164_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_165_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_166__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_167__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_168_Trg_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ).
% Trg.simps(3)
thf(fact_169_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_170_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_171__092_060open_062_092_060Lambda_062_Oelementary__reduction_Au_A_092_060and_062_Au_A_092_060sqsubseteq_062_At_A_092_060and_062_Adevelopment_A_It_A_092_Au_J_AU_092_060close_062,axiom,
( ( lambda2670285437103676873uction @ u )
& ( lambda_subs @ u @ ta )
& ( reduct9208763521642324371opment @ ( lambda_resid @ ta @ u ) @ ua ) ) ).
% \<open>\<Lambda>.elementary_reduction u \<and> u \<sqsubseteq> t \<and> development (t \ u) U\<close>
thf(fact_172_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_173_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_174_paths__in__weakly__extensional__rts_OTrg_Ocong,axiom,
paths_2695323870737867955lambda = paths_2695323870737867955lambda ).
% paths_in_weakly_extensional_rts.Trg.cong
thf(fact_175_residuation_Ocon_Ocong,axiom,
con_lambda_lambda = con_lambda_lambda ).
% residuation.con.cong
thf(fact_176_residuation_Ocon_Ocong,axiom,
con_li79537494059641180lambda = con_li79537494059641180lambda ).
% residuation.con.cong
thf(fact_177_rts_Ocoinitial_Ocong,axiom,
coinit1063259437155584316lambda = coinit1063259437155584316lambda ).
% rts.coinitial.cong
thf(fact_178_rts_Ocoinitial_Ocong,axiom,
coinit1392077014771647542lambda = coinit1392077014771647542lambda ).
% rts.coinitial.cong
thf(fact_179_reduction__paths_Odevelopment_Osimps_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ ( cons_lambda_lambda @ U @ U2 ) )
= ( ( lambda2670285437103676873uction @ U )
& ( lambda_subs @ U @ T )
& ( reduct9208763521642324371opment @ ( lambda_resid @ T @ U ) @ U2 ) ) ) ).
% reduction_paths.development.simps(2)
thf(fact_180_lambda__calculus_Ois__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% lambda_calculus.is_confluent_rts
thf(fact_181_paths__in__rts_OResid_Ocong,axiom,
paths_3312059784463760222lambda = paths_3312059784463760222lambda ).
% paths_in_rts.Resid.cong
thf(fact_182_paths__in__rts_OResid_Ocong,axiom,
paths_8607357831768207192lambda = paths_8607357831768207192lambda ).
% paths_in_rts.Resid.cong
thf(fact_183_residuation_Oide_Ocong,axiom,
ide_li6773656263178250516lambda = ide_li6773656263178250516lambda ).
% residuation.ide.cong
thf(fact_184_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_185_rts_Ocoterminal_Ocong,axiom,
coterm81126318419254786lambda = coterm81126318419254786lambda ).
% rts.coterminal.cong
thf(fact_186_rts_Ocoterminal_Ocong,axiom,
coterm4142657152830665724lambda = coterm4142657152830665724lambda ).
% rts.coterminal.cong
thf(fact_187_partial__magma__def,axiom,
( partia748546660539056525lambda
= ( ^ [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
? [X3: list_lambda_lambda] :
( ! [T6: list_lambda_lambda] :
( ( ( OP @ X3 @ T6 )
= X3 )
& ( ( OP @ T6 @ X3 )
= X3 ) )
& ! [Y4: list_lambda_lambda] :
( ! [T6: list_lambda_lambda] :
( ( ( OP @ Y4 @ T6 )
= Y4 )
& ( ( OP @ T6 @ Y4 )
= Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_188_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X3: lambda_lambda] :
( ! [T6: lambda_lambda] :
( ( ( OP @ X3 @ T6 )
= X3 )
& ( ( OP @ T6 @ X3 )
= X3 ) )
& ! [Y4: lambda_lambda] :
( ! [T6: lambda_lambda] :
( ( ( OP @ Y4 @ T6 )
= Y4 )
& ( ( OP @ T6 @ Y4 )
= Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_189_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_190_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_191_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 ) )
& ! [Y3: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP2 @ Y3 @ T2 )
= Y3 )
& ( ( OP2 @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X4 ) ) )
=> ( partia748546660539056525lambda @ OP2 ) ) ).
% partial_magma.intro
thf(fact_192_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 ) )
& ! [Y3: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP2 @ Y3 @ T2 )
= Y3 )
& ( ( OP2 @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X4 ) ) )
=> ( partia3924075026459072775lambda @ OP2 ) ) ).
% partial_magma.intro
thf(fact_193_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_194__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_195__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_196_Resid_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(5)
thf(fact_197_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_198_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_199_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_200_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_201_residuation__axioms,axiom,
residu7698045915734027664lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% residuation_axioms
thf(fact_202__092_060Lambda_062_Ocong__implies__coinitial,axiom,
! [U: lambda_lambda,U3: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ U @ U3 ) ) ).
% \<Lambda>.cong_implies_coinitial
thf(fact_203__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_204__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_205__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_206__092_060Lambda_062_Ocong__implies__coterminal,axiom,
! [U: lambda_lambda,U3: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ U @ U3 ) ) ).
% \<Lambda>.cong_implies_coterminal
thf(fact_207__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_208__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_209__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_210__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_211__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_212_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_213_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_214__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_215__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_216__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_217__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_218__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_219__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_220__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_221__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_222__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_223__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_224__092_060Lambda_062_Ocong__subst__right_I1_J,axiom,
! [U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U3 ) ) ) ).
% \<Lambda>.cong_subst_right(1)
thf(fact_225__092_060Lambda_062_Ocong__subst__right_I2_J,axiom,
! [U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U3 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U3 ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_right(2)
thf(fact_226__092_060Lambda_062_Ocong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T5 @ U ) ) ) ).
% \<Lambda>.cong_subst_left(1)
thf(fact_227__092_060Lambda_062_Ocong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T5 @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T5 @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_left(2)
thf(fact_228__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_229__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_230__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_231__092_060Lambda_062_Otrg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_resid @ T @ T ) ) ).
% \<Lambda>.trg_def
thf(fact_232__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_233__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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_cong__subst__right_I1_J,axiom,
! [U: list_lambda_lambda,U3: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U3 ) ) ) ).
% cong_subst_right(1)
thf(fact_245_cong__subst__right_I2_J,axiom,
! [U: list_lambda_lambda,U3: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ 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 @ U3 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_246_cong__subst__left_I1_J,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_247_cong__subst__left_I2_J,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ 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 @ T5 @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_248_Arr_Osimps_I1_J,axiom,
~ ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Arr.simps(1)
thf(fact_249_Residx1_Osimps_I1_J,axiom,
! [U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ nil_lambda_lambda @ U )
= nil_lambda_lambda ) ).
% Residx1.simps(1)
thf(fact_250_Resid1x_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ).
% Resid1x.simps(3)
thf(fact_251__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_252__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_253_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_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_Residx1_Oelims,axiom,
! [X2: list_lambda_lambda,Xa: lambda_lambda,Y2: list_lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
=> ( Y2 = nil_lambda_lambda ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ).
% Residx1.elims
thf(fact_265_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_266_Residx1_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ U )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ U )
= nil_lambda_lambda ) ) ) ).
% Residx1.simps(3)
thf(fact_267__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_268__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_269__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_270_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_271_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_272_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_273_residuation_Oresid__arr__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_274_residuation_Oresid__arr__self,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_li817795487149935061lambda @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_275_paths__in__rts_OResidx1_Ocong,axiom,
paths_7088367479088905573lambda = paths_7088367479088905573lambda ).
% paths_in_rts.Residx1.cong
thf(fact_276_paths__in__rts_OResidx1_Ocong,axiom,
paths_2112734093361750239lambda = paths_2112734093361750239lambda ).
% paths_in_rts.Residx1.cong
thf(fact_277_paths__in__rts_OResid1x_Ocong,axiom,
paths_4460507232930246679lambda = paths_4460507232930246679lambda ).
% paths_in_rts.Resid1x.cong
thf(fact_278_paths__in__rts_OResid1x_Ocong,axiom,
paths_3319247187298047121lambda = paths_3319247187298047121lambda ).
% paths_in_rts.Resid1x.cong
thf(fact_279_paths__in__rts_OArr_Ocong,axiom,
paths_2015741907369688696lambda = paths_2015741907369688696lambda ).
% paths_in_rts.Arr.cong
thf(fact_280_paths__in__rts_OArr_Ocong,axiom,
paths_366302053746739058lambda = paths_366302053746739058lambda ).
% paths_in_rts.Arr.cong
thf(fact_281_residuation_Otrg_Ocong,axiom,
trg_lambda_lambda = trg_lambda_lambda ).
% residuation.trg.cong
thf(fact_282_residuation_Otrg_Ocong,axiom,
trg_li817795487149935061lambda = trg_li817795487149935061lambda ).
% residuation.trg.cong
thf(fact_283_residuation_Otrg__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( trg_lambda_lambda @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_284_residuation_Otrg__def,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( trg_li817795487149935061lambda @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_285_rts_Ojoinable_Ocong,axiom,
joinab677607494065406278lambda = joinab677607494065406278lambda ).
% rts.joinable.cong
thf(fact_286_rts_Ojoinable_Ocong,axiom,
joinab2519788767067900224lambda = joinab2519788767067900224lambda ).
% rts.joinable.cong
thf(fact_287_residuation_Ocube,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_288_residuation_Ocube,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_289_residuation_Ocon__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_290_residuation_Ocon__sym,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( con_li79537494059641180lambda @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_291_residuation_Oaxioms_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( partia748546660539056525lambda @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_292_residuation_Oaxioms_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( partia3924075026459072775lambda @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_293_lambda__calculus_Osubs__implies__prfx,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subs @ T @ U )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ).
% lambda_calculus.subs_implies_prfx
thf(fact_294_lambda__calculus_Ocong__elementary__reductions__are__equal,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ( ( lambda2670285437103676873uction @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( T = U ) ) ) ) ).
% lambda_calculus.cong_elementary_reductions_are_equal
thf(fact_295_lambda__calculus_Oelementary__reduction__not__ide,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T ) ) ).
% lambda_calculus.elementary_reduction_not_ide
thf(fact_296_residuation_OideE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A )
=> ~ ( ( con_li79537494059641180lambda @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_297_residuation_OideE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ~ ( ( con_lambda_lambda @ Resid @ A @ A )
=> ( ( Resid @ A @ A )
!= A ) ) ) ) ).
% residuation.ideE
thf(fact_298_residuation_OideI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_li6773656263178250516lambda @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_299_residuation_OideI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ A @ A )
=> ( ( ( Resid @ A @ A )
= A )
=> ( ide_lambda_lambda @ Resid @ A ) ) ) ) ).
% residuation.ideI
thf(fact_300_residuation_Oide__def,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A )
= ( ( con_li79537494059641180lambda @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_301_residuation_Oide__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
= ( ( con_lambda_lambda @ Resid @ A @ A )
& ( ( Resid @ A @ A )
= A ) ) ) ) ).
% residuation.ide_def
thf(fact_302_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_303_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_304_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_305_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_306_Resid_Oelims,axiom,
! [X2: list_lambda_lambda,Xa: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) )
=> ( ( ? [V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
= ( cons_lambda_lambda @ V2 @ Va ) )
=> ( ( Xa = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U4: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U4 ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U4: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y2
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ! [U4: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ nil_lambda_lambda ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U4 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ! [U4: lambda_lambda,Vb: lambda_lambda,Vc: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ ( cons_lambda_lambda @ Vb @ Vc ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T3 @ U4 ) ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T3 @ U4 ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U4 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U4 ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U4 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T3 @ U4 ) ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ) ) ) ) ).
% Resid.elims
thf(fact_307_Resid_Osimps_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb2: lambda_lambda,Vc2: list_lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(6)
thf(fact_308_Resid_Osimps_I7_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda,Vb2: lambda_lambda,Vc2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(7)
thf(fact_309_Resid_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(4)
thf(fact_310_Trg_Oelims,axiom,
! [X2: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_2695323870737867955lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
!= ( trg_lambda_lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Trg.elims
thf(fact_311_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_312_trg__def,axiom,
! [T: list_lambda_lambda] :
( ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ).
% trg_def
thf(fact_313__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_314__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_315__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_316__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_317__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_318__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_319_Resid1x__null,axiom,
! [T4: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T4 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x_null
thf(fact_320_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_321_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_322_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_323_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_324_Resid1x_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x.simps(1)
thf(fact_325_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_326_Trg_Osimps_I1_J,axiom,
( ( paths_2695323870737867955lambda @ lambda_resid @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Trg.simps(1)
thf(fact_327_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_328_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_329_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_330_Resid1x_Oelims,axiom,
! [X2: lambda_lambda,Xa: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_3319247187298047121lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_lambda_lambda )
=> ( Y2
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [U4: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ nil_lambda_lambda ) )
=> ( Y2
!= ( lambda_resid @ X2 @ U4 ) ) )
=> ~ ! [U4: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ X2 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Resid1x.elims
thf(fact_331_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_332__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_333__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_334__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_335_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_336_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_337_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_338_rts_Osources_Ocong,axiom,
source6119337877621565212lambda = source6119337877621565212lambda ).
% rts.sources.cong
thf(fact_339_rts_Osources_Ocong,axiom,
source5627170679743844886lambda = source5627170679743844886lambda ).
% rts.sources.cong
thf(fact_340_partial__magma_Onull_Ocong,axiom,
partia1461432607296028304lambda = partia1461432607296028304lambda ).
% partial_magma.null.cong
thf(fact_341_partial__magma_Onull_Ocong,axiom,
partia6873046715171720854lambda = partia6873046715171720854lambda ).
% partial_magma.null.cong
thf(fact_342_paths__in__weakly__extensional__rts_OSrc_Ocong,axiom,
paths_1651982994781619320lambda = paths_1651982994781619320lambda ).
% paths_in_weakly_extensional_rts.Src.cong
thf(fact_343_residuation_Ocube__ax,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_344_residuation_Ocube__ax,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_345_residuation_Ocon__sym__ax,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_346_residuation_Ocon__sym__ax,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_347_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_348_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_349_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_350_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_351_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_352_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_353_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_354_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_355_lambda__calculus_Oresid__is__residuation,axiom,
residu6197316955523523210lambda @ lambda_resid ).
% lambda_calculus.resid_is_residuation
thf(fact_356_residuation_OconE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_357_residuation_OconE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_358_residuation_OconI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_359_residuation_OconI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( con_li79537494059641180lambda @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_360_residuation_Ocon__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_361_residuation_Ocon__def,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_362_red__iff,axiom,
( lambda_red
= ( ^ [A5: lambda_lambda,B2: lambda_lambda] :
? [T7: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T7 )
& ( ( paths_1651982994781619320lambda @ lambda_resid @ T7 )
= A5 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T7 )
= B2 ) ) ) ) ).
% red_iff
thf(fact_363__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_364_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_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373__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_374__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_375__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_376__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_377__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_378_null__char,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= nil_lambda_lambda ) ).
% null_char
thf(fact_379_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_380_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_381_Srcs_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( source5627170679743844886lambda @ lambda_resid @ T ) ) ).
% Srcs.simps(3)
thf(fact_382__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_383__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_384__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_385__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_386__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_387_Ide_Osimps_I1_J,axiom,
~ ( paths_7717490669901197737lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Ide.simps(1)
thf(fact_388__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_389__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_390__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_391__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_392__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_393__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_394_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_395_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_396_Ide__implies__Arr,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% Ide_implies_Arr
thf(fact_397__092_060Lambda_062_Onot__arr__null,axiom,
~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.not_arr_null
thf(fact_398__092_060Lambda_062_Ocoinitial__iff,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T5 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T5 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T5 ) ) ) ) ).
% \<Lambda>.coinitial_iff
thf(fact_399__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_400_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_401_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_402_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_403_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_404_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_405_ide__char,axiom,
! [T4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ).
% ide_char
thf(fact_406_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_407_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_408__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_409__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_410_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_411_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_412_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_413_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_414__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_415__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_416_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_417_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_418_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_419_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_420_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_421_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_422_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_423_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_424_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_425_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_426_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_427_confluence__single,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T3 @ U4 )
=> ( con_lambda_lambda @ lambda_resid @ T3 @ U4 ) )
=> ( ( 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_428_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_429_hd__append2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Xs ) ) ) ).
% hd_append2
thf(fact_430_hd__append2,axiom,
! [Xs: list_s345339197461576016lambda,Ys: list_s345339197461576016lambda] :
( ( Xs != nil_se6914803783601386320lambda )
=> ( ( hd_set_lambda_lambda @ ( append2117223597850273125lambda @ Xs @ Ys ) )
= ( hd_set_lambda_lambda @ Xs ) ) ) ).
% hd_append2
thf(fact_431_hd__append2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Xs ) ) ) ).
% hd_append2
thf(fact_432__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_433__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_434__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_435__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_436_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_437__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_438_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_439_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_440_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_441_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_442_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_443_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_444_paths__in__rts_OSrcs_Ocong,axiom,
paths_2897053694880971696lambda = paths_2897053694880971696lambda ).
% paths_in_rts.Srcs.cong
thf(fact_445_paths__in__rts_OIde_Ocong,axiom,
paths_7717490669901197737lambda = paths_7717490669901197737lambda ).
% paths_in_rts.Ide.cong
thf(fact_446_residuation_Oarr_Ocong,axiom,
arr_lambda_lambda = arr_lambda_lambda ).
% residuation.arr.cong
thf(fact_447_residuation_Oarr_Ocong,axiom,
arr_li3691443969012319837lambda = arr_li3691443969012319837lambda ).
% residuation.arr.cong
thf(fact_448_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_449_lambda__calculus_Oweak__diamond,axiom,
! [A: lambda_lambda,B: lambda_lambda,B3: lambda_lambda] :
( ( lambda_red @ A @ B )
=> ( ( lambda_red @ A @ B3 )
=> ~ ! [C2: lambda_lambda] :
( ( lambda_red @ B @ C2 )
=> ~ ( lambda_red @ B3 @ C2 ) ) ) ) ).
% lambda_calculus.weak_diamond
thf(fact_450_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_451_lambda__calculus_Oelementary__reduction__is__arr,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% lambda_calculus.elementary_reduction_is_arr
thf(fact_452_residuation_Oide__implies__arr,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A )
=> ( arr_li3691443969012319837lambda @ Resid @ A ) ) ) ).
% residuation.ide_implies_arr
thf(fact_453_residuation_Oide__implies__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( arr_lambda_lambda @ Resid @ A ) ) ) ).
% residuation.ide_implies_arr
thf(fact_454_residuation_Oarr__resid__iff__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
= ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_455_residuation_Oarr__resid__iff__con,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( arr_li3691443969012319837lambda @ Resid @ ( Resid @ T @ U ) )
= ( con_li79537494059641180lambda @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_456_residuation_Oarr__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_457_residuation_Oarr__resid,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( arr_li3691443969012319837lambda @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_458_residuation_Oarr__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
= ( con_lambda_lambda @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_459_residuation_Oarr__def,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
= ( con_li79537494059641180lambda @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_460_residuation_OarrI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ T )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_461_residuation_OarrI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ T )
=> ( arr_li3691443969012319837lambda @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_462_residuation_OarrE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( con_lambda_lambda @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_463_residuation_OarrE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( con_li79537494059641180lambda @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_464_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_465_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( arr_li3691443969012319837lambda @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_466_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_467_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( arr_li3691443969012319837lambda @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_468_longest__common__prefix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ps: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ps @ Xs3 ) )
& ( Ys
= ( append_lambda_lambda @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_lambda_lambda )
| ( Ys5 = nil_lambda_lambda )
| ( ( hd_lambda_lambda @ Xs3 )
!= ( hd_lambda_lambda @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_469_hd__append,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Ys ) ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Xs ) ) ) ) ).
% hd_append
thf(fact_470_residuation_Onot__arr__null,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ~ ( arr_lambda_lambda @ Resid @ ( partia1461432607296028304lambda @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_471_residuation_Onot__arr__null,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ~ ( arr_li3691443969012319837lambda @ Resid @ ( partia6873046715171720854lambda @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_472_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,U4: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ Resid @ T3 @ U4 )
=> ( con_li79537494059641180lambda @ Resid @ T3 @ U4 ) )
=> ( ( 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_473_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,U4: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid @ T3 @ U4 )
=> ( con_lambda_lambda @ Resid @ T3 @ U4 ) )
=> ( ( 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_474_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_475_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_476_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_477_Srcs_Oelims,axiom,
! [X2: list_lambda_lambda,Y2: set_lambda_lambda] :
( ( ( paths_2897053694880971696lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != bot_bo1271416176687173302lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
!= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda] :
( ? [V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) ) ) ) ) ).
% Srcs.elims
thf(fact_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_arr__char,axiom,
! [T4: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% arr_char
thf(fact_486_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_487_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_488_arrE,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).
% arrE
thf(fact_489_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_490_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_491_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_492__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_493__092_060Lambda_062_Otargets__are__cong,axiom,
! [B: lambda_lambda,T: lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ B ) ) ) ) ) ).
% \<Lambda>.targets_are_cong
thf(fact_494__092_060Lambda_062_Otargets__cong__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ B ) ) )
=> ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.targets_cong_closed
thf(fact_495__092_060Lambda_062_Otargets__are__con,axiom,
! [B: lambda_lambda,T: lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ B @ B3 ) ) ) ).
% \<Lambda>.targets_are_con
thf(fact_496__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_497__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_498_not__arr__null,axiom,
~ ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% not_arr_null
thf(fact_499__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_500_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_501_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_502_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_503_targets__are__cong,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ B3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_504_targets__cong__closed,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ B3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ B ) ) )
=> ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% targets_cong_closed
thf(fact_505_targets__are__con,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ B3 ) ) ) ).
% targets_are_con
thf(fact_506_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_507__092_060Lambda_062_Otargets__con__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B3 )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ B3 )
=> ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.targets_con_closed
thf(fact_508_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_509_Srcs_Osimps_I1_J,axiom,
( ( paths_2897053694880971696lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Srcs.simps(1)
thf(fact_510_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_511__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_512__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_513_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_514_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_515_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_516__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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_coinitial__iff,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T5 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
& ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ) ).
% coinitial_iff
thf(fact_525_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_526_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_527__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_528__092_060Lambda_062_Ocoterminal__iff,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T5 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T5 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T5 ) ) ) ) ).
% \<Lambda>.coterminal_iff
thf(fact_529_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_530_coterminal__iff,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T5 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ) ).
% coterminal_iff
thf(fact_531_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_532_targets__con__closed,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ B3 )
=> ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% targets_con_closed
thf(fact_533_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_534__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_535_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_536_arrI,axiom,
! [T: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% arrI
thf(fact_537__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_538_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_539__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_540_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_541_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_542_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_543__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_544_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_545_rts_Ojoin__of_Ocong,axiom,
join_o6956785509312998601lambda = join_o6956785509312998601lambda ).
% rts.join_of.cong
thf(fact_546_rts_Ojoin__of_Ocong,axiom,
join_o1386658102433174083lambda = join_o1386658102433174083lambda ).
% rts.join_of.cong
thf(fact_547_rts_Otargets_Ocong,axiom,
target4026644918049360146lambda = target4026644918049360146lambda ).
% rts.targets.cong
thf(fact_548_rts_Otargets_Ocong,axiom,
target5490931411331770124lambda = target5490931411331770124lambda ).
% rts.targets.cong
thf(fact_549_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,T5: 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 @ T5 @ V4 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T5 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T5 @ W3 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T5 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_550_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,T5: 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 @ T5 @ V4 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T5 @ W3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T5 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_551_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,U3: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( member2667661554319814297lambda @ U @ NN )
=> ( ( member2667661554319814297lambda @ U3 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ U )
= ( source6119337877621565212lambda @ Resid @ U3 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ U )
= ( target4026644918049360146lambda @ Resid @ U3 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( source6119337877621565212lambda @ Resid @ U ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U3 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_552_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U3 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U )
= ( source5627170679743844886lambda @ Resid @ U3 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U )
= ( target5490931411331770124lambda @ Resid @ U3 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_553_lambda__calculus_Osubs__Ide,axiom,
! [U: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ U )
=> ( ( ( lambda_Src @ T )
= ( lambda_Src @ U ) )
=> ( lambda_subs @ U @ T ) ) ) ).
% lambda_calculus.subs_Ide
thf(fact_554_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_555_lambda__calculus_ONF__unique,axiom,
! [T: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( lambda_red @ T @ U )
=> ( ( lambda_red @ T @ U3 )
=> ( ( lambda_NF @ U )
=> ( ( lambda_NF @ U3 )
=> ( U = U3 ) ) ) ) ) ).
% lambda_calculus.NF_unique
thf(fact_556_Ide_Oelims_I1_J,axiom,
! [X2: list_lambda_lambda,Y2: $o] :
( ( ( paths_7717490669901197737lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> Y2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
= ( ~ ( ide_lambda_lambda @ lambda_resid @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Ide.elims(1)
thf(fact_557_Ide_Oelims_I2_J,axiom,
! [X2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ X2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Ide.elims(2)
thf(fact_558_Ide_Oelims_I3_J,axiom,
! [X2: list_lambda_lambda] :
( ~ ( paths_7717490669901197737lambda @ lambda_resid @ X2 )
=> ( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( ide_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ).
% Ide.elims(3)
thf(fact_559_Arr_Oelims_I1_J,axiom,
! [X2: list_lambda_lambda,Y2: $o] :
( ( ( paths_366302053746739058lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> Y2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
= ( ~ ( arr_lambda_lambda @ lambda_resid @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Arr.elims(1)
thf(fact_560_Arr_Oelims_I2_J,axiom,
! [X2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ X2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( arr_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Arr.elims(2)
thf(fact_561__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_562__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_563__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_564__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_565__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_566__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_567__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_568__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_569__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_570__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_571__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_572__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_573__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_574__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_575__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_576_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_577_Arr_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ) ) ).
% Arr.simps(3)
thf(fact_578_Ide_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( ( ide_lambda_lambda @ lambda_resid @ T )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ) ) ).
% Ide.simps(3)
thf(fact_579_Arr_Oelims_I3_J,axiom,
! [X2: list_lambda_lambda] :
( ~ ( paths_366302053746739058lambda @ lambda_resid @ X2 )
=> ( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( arr_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ).
% Arr.elims(3)
thf(fact_580_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_581__092_060open_062u_A_D_AU_A_061_A_091_093_A_092_060or_062_AArr_A_Iu_A_D_AU_J_A_092_060and_062_Aset_A_Iu_A_D_AU_J_A_092_060subseteq_062_ACollect_A_092_060Lambda_062_Oelementary__reduction_092_060close_062,axiom,
( ( ( cons_lambda_lambda @ u @ ua )
= nil_lambda_lambda )
| ( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ u @ ua ) )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ u @ ua ) ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ) ).
% \<open>u # U = [] \<or> Arr (u # U) \<and> set (u # U) \<subseteq> Collect \<Lambda>.elementary_reduction\<close>
thf(fact_582_elementary__reduction__path__def,axiom,
( reduct8314398545252454212n_path
= ( ^ [T7: list_lambda_lambda] :
( ( T7 = nil_lambda_lambda )
| ( ( paths_366302053746739058lambda @ lambda_resid @ T7 )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T7 ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ) ) ) ).
% elementary_reduction_path_def
thf(fact_583_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_584_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_585_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_586_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_587_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_588_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_589_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_590_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_591_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_592_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_593_set__empty,axiom,
! [Xs: list_lambda_lambda] :
( ( ( set_lambda_lambda2 @ Xs )
= bot_bo1271416176687173302lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty
thf(fact_594_set__empty,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( ( set_li3846194476672328965lambda @ Xs )
= bot_bo1397037886704484924lambda )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty
thf(fact_595_set__empty2,axiom,
! [Xs: list_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty2
thf(fact_596_set__empty2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ Xs ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty2
thf(fact_597_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_598_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_599_subset__code_I1_J,axiom,
! [Xs: list_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ B4 )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
=> ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_600_subset__code_I1_J,axiom,
! [Xs: list_l5168835686021156086lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ B4 )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( member2667661554319814297lambda @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_601_rts_Ocomposable_Ocong,axiom,
compos3359172601056258045lambda = compos3359172601056258045lambda ).
% rts.composable.cong
thf(fact_602_rts_Ocomposable_Ocong,axiom,
compos7127019122090555255lambda = compos7127019122090555255lambda ).
% rts.composable.cong
thf(fact_603_set__ConsD,axiom,
! [Y2: list_lambda_lambda,X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_604_set__ConsD,axiom,
! [Y2: lambda_lambda,X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_605_list_Oset__cases,axiom,
! [E: list_lambda_lambda,A: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ A ) )
=> ( ! [Z2: list_l5168835686021156086lambda] :
( A
!= ( cons_l4558873032622161958lambda @ E @ Z2 ) )
=> ~ ! [Z1: list_lambda_lambda,Z2: list_l5168835686021156086lambda] :
( ( A
= ( cons_l4558873032622161958lambda @ Z1 @ Z2 ) )
=> ~ ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_606_list_Oset__cases,axiom,
! [E: lambda_lambda,A: list_lambda_lambda] :
( ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ A ) )
=> ( ! [Z2: list_lambda_lambda] :
( A
!= ( cons_lambda_lambda @ E @ Z2 ) )
=> ~ ! [Z1: lambda_lambda,Z2: list_lambda_lambda] :
( ( A
= ( cons_lambda_lambda @ Z1 @ Z2 ) )
=> ~ ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_607_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_608_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_609_list_Oset__intros_I2_J,axiom,
! [Y2: list_lambda_lambda,X22: list_l5168835686021156086lambda,X21: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ X22 ) )
=> ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_610_list_Oset__intros_I2_J,axiom,
! [Y2: lambda_lambda,X22: list_lambda_lambda,X21: lambda_lambda] :
( ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ X22 ) )
=> ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_611_empty__set,axiom,
( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ nil_lambda_lambda ) ) ).
% empty_set
thf(fact_612_empty__set,axiom,
( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ nil_li8379291187143545974lambda ) ) ).
% empty_set
thf(fact_613_set__subset__Cons,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] : ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_614_set__subset__Cons,axiom,
! [Xs: list_l5168835686021156086lambda,X2: list_lambda_lambda] : ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_615_split__list,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_616_split__list,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_617_split__list__last,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_618_split__list__last,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_619_split__list__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_620_split__list__first,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_621_split__list__first,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_622_split__list__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_623_append__Cons__eq__iff,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs4: list_l5168835686021156086lambda,Ys6: list_l5168835686021156086lambda] :
( ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys ) )
=> ( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X2 @ Ys ) )
= ( append1017360404025248779lambda @ Xs4 @ ( cons_l4558873032622161958lambda @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_624_append__Cons__eq__iff,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Xs4: list_lambda_lambda,Ys6: list_lambda_lambda] :
( ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X2 @ Ys ) )
= ( append_lambda_lambda @ Xs4 @ ( cons_lambda_lambda @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_625_in__set__conv__decomp,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_626_in__set__conv__decomp,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_627_split__list__last__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_628_split__list__first__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_629_split__list__last__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_630_split__list__first__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_631_in__set__conv__decomp__last,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_632_in__set__conv__decomp__last,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_633_in__set__conv__decomp__first,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_634_in__set__conv__decomp__first,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_635_split__list__last__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_lambda_lambda,X3: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_636_split__list__first__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_lambda_lambda,X3: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_637_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_638_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_639_hd__in__set,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ Xs ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ).
% hd_in_set
thf(fact_640_hd__in__set,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( member_lambda_lambda @ ( hd_lambda_lambda @ Xs ) @ ( set_lambda_lambda2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_641__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_642_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_643_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_644__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_645__092_060Lambda_062_Ocong__respects__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T5 @ U3 ) ) ) ) ).
% \<Lambda>.cong_respects_seq
thf(fact_646__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_647__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_648__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_649__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_650__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_651_rts_Oseq_Ocong,axiom,
seq_lambda_lambda = seq_lambda_lambda ).
% rts.seq.cong
thf(fact_652_rts_Oseq_Ocong,axiom,
seq_li8093550137657879445lambda = seq_li8093550137657879445lambda ).
% rts.seq.cong
thf(fact_653_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_654_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_655_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_656_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_657_coterminal__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
!= bot_bo1397037886704484924lambda ) ) ).
% coterminal_def
thf(fact_658_coinitial__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
!= bot_bo1397037886704484924lambda ) ) ).
% coinitial_def
thf(fact_659_con__imp__common__source,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) )
!= bot_bo1397037886704484924lambda ) ) ).
% con_imp_common_source
thf(fact_660_Ide__append__iff_092_060_094sub_062P,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 )
& ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P
thf(fact_661_cong__respects__seq,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 ) ) ) ) ).
% cong_respects_seq
thf(fact_662_cong__respects__seq_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T8: list_lambda_lambda,U5: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T8 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U5 @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T8 @ U5 ) ) ) ) ).
% cong_respects_seq\<^sub>P
thf(fact_663_Trgs_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ).
% Trgs.simps(3)
thf(fact_664_ex__un__Trg,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( Y = X ) ) ) ) ).
% ex_un_Trg
thf(fact_665_Trgs__are__con,axiom,
! [B: lambda_lambda,T4: list_lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( ( member_lambda_lambda @ B3 @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( con_lambda_lambda @ lambda_resid @ B @ B3 ) ) ) ).
% Trgs_are_con
thf(fact_666_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_667_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_668_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_669_Trgs__con__closed,axiom,
! [B: lambda_lambda,T4: list_lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B3 )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ B3 )
=> ( member_lambda_lambda @ B3 @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Trgs_con_closed
thf(fact_670_Trgs_Osimps_I1_J,axiom,
( ( paths_7594662086398683701lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Trgs.simps(1)
thf(fact_671_Trgs__are__ide,axiom,
! [T4: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).
% Trgs_are_ide
thf(fact_672_Arr__has__Trg,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
!= bot_bo1271416176687173302lambda ) ) ).
% Arr_has_Trg
thf(fact_673_Trgs__Resid__sym__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_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T4 ) ) ) ) ).
% Trgs_Resid_sym_Arr_single
thf(fact_674_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_675_Trgs_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( target5490931411331770124lambda @ lambda_resid @ T ) ) ).
% Trgs.simps(2)
thf(fact_676_seq__char,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ U2 )
& ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ).
% seq_char
thf(fact_677_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_678_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_679_seq__implies__Trgs__eq__Srcs,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ) ).
% seq_implies_Trgs_eq_Srcs
thf(fact_680_sources__eqI,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ).
% sources_eqI
thf(fact_681_targets__eqI,axiom,
! [T: list_lambda_lambda,T5: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 ) ) ) ).
% targets_eqI
thf(fact_682_Srcs__Resid__single__Arr,axiom,
! [U: lambda_lambda,T4: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T4 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ).
% Srcs_Resid_single_Arr
thf(fact_683_Trgs_Oelims,axiom,
! [X2: list_lambda_lambda,Y2: set_lambda_lambda] :
( ( ( paths_7594662086398683701lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != bot_bo1271416176687173302lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
!= ( target5490931411331770124lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Trgs.elims
thf(fact_684_Arr__appendE_092_060_094sub_062P,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_7594662086398683701lambda @ lambda_resid @ T4 )
!= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P
thf(fact_685_Arr__append__iff_092_060_094sub_062P,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 )
& ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P
thf(fact_686_Trgs__Resid__sym,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) ) ) ).
% Trgs_Resid_sym
thf(fact_687_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_688_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_689_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_690_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_691_cong__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T8: list_lambda_lambda,U5: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T8 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U5 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ T8 @ U5 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T8 @ U5 ) @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ) ).
% cong_append
thf(fact_692_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_693_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_694_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_695_Trgs__append,axiom,
! [U2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ U2 ) ) ) ).
% Trgs_append
thf(fact_696_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_697_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_698_Srcs__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ U2 ) ) ) ).
% Srcs_Resid
thf(fact_699_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_700_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_701_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_702_cong__cons,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,T5: lambda_lambda,U5: 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 @ T5 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T5 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U5 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ ( cons_lambda_lambda @ T5 @ U5 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T5 @ U5 ) @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ) ).
% cong_cons
thf(fact_703_Arr__appendI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Arr_appendI\<^sub>P
thf(fact_704_Ide__appendI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Ide_appendI\<^sub>P
thf(fact_705_Trgs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( insert_lambda_lambda @ ( paths_2695323870737867955lambda @ lambda_resid @ T4 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Trgs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_706_paths__in__rts_OTrgs_Ocong,axiom,
paths_7594662086398683701lambda = paths_7594662086398683701lambda ).
% paths_in_rts.Trgs.cong
thf(fact_707_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_708_seq__char_H,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ U2 )
& ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
!= bot_bo1271416176687173302lambda ) ) ) ).
% seq_char'
thf(fact_709__092_060Lambda_062_Ocoterminal__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ U )
= ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>.coterminal_def
thf(fact_710__092_060Lambda_062_Ocoinitial__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
= ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>.coinitial_def
thf(fact_711__092_060Lambda_062_Otargets__eqI,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T5 ) ) ) ).
% \<Lambda>.targets_eqI
thf(fact_712_Srcs__eqI,axiom,
! [T4: list_lambda_lambda,T8: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ T8 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T8 ) ) ) ).
% Srcs_eqI
thf(fact_713__092_060Lambda_062_Osources__eqI,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T5 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T5 ) ) ) ).
% \<Lambda>.sources_eqI
thf(fact_714_Trgs__eqI,axiom,
! [T4: list_lambda_lambda,T8: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_7594662086398683701lambda @ lambda_resid @ T8 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T8 ) ) ) ).
% Trgs_eqI
thf(fact_715__092_060Lambda_062_Ocon__imp__common__source,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>.con_imp_common_source
thf(fact_716_seqI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
!= bot_bo1271416176687173302lambda )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% seqI\<^sub>P
thf(fact_717_Trgs__simp_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( target5490931411331770124lambda @ lambda_resid @ ( last_lambda_lambda @ T4 ) ) ) ) ).
% Trgs_simp\<^sub>P
thf(fact_718_rts__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( arr_lambda_lambda @ Resid @ T3 )
=> ( ide_lambda_lambda @ Resid @ ( trg_lambda_lambda @ Resid @ T3 ) ) )
=> ( ! [A4: lambda_lambda,T3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A4 )
=> ( ( con_lambda_lambda @ Resid @ T3 @ A4 )
=> ( ( Resid @ T3 @ A4 )
= T3 ) ) )
=> ( ! [A4: lambda_lambda,T3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A4 )
=> ( ( con_lambda_lambda @ Resid @ A4 @ T3 )
=> ( ide_lambda_lambda @ Resid @ ( Resid @ A4 @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ Resid @ T3 @ U4 )
=> ? [A6: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A6 )
& ( con_lambda_lambda @ Resid @ A6 @ T3 )
& ( con_lambda_lambda @ Resid @ A6 @ U4 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda,V2: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ ( Resid @ T3 @ U4 ) )
=> ( ( con_lambda_lambda @ Resid @ U4 @ V2 )
=> ( con_lambda_lambda @ Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ V2 @ U4 ) ) ) )
=> ( rts_ax7298520571313827089lambda @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_719_rts__axioms_Ointro,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ Resid @ T3 )
=> ( ide_li6773656263178250516lambda @ Resid @ ( trg_li817795487149935061lambda @ Resid @ T3 ) ) )
=> ( ! [A4: list_lambda_lambda,T3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A4 )
=> ( ( con_li79537494059641180lambda @ Resid @ T3 @ A4 )
=> ( ( Resid @ T3 @ A4 )
= T3 ) ) )
=> ( ! [A4: list_lambda_lambda,T3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A4 )
=> ( ( con_li79537494059641180lambda @ Resid @ A4 @ T3 )
=> ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ A4 @ T3 ) ) ) )
=> ( ! [T3: list_lambda_lambda,U4: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ Resid @ T3 @ U4 )
=> ? [A6: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A6 )
& ( con_li79537494059641180lambda @ Resid @ A6 @ T3 )
& ( con_li79537494059641180lambda @ Resid @ A6 @ U4 ) ) )
=> ( ! [T3: list_lambda_lambda,U4: list_lambda_lambda,V2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ T3 @ U4 ) )
=> ( ( con_li79537494059641180lambda @ Resid @ U4 @ V2 )
=> ( con_li79537494059641180lambda @ Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ V2 @ U4 ) ) ) )
=> ( rts_ax2283037410143324311lambda @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_720_rts__axioms__def,axiom,
( rts_ax7298520571313827089lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T6: lambda_lambda] :
( ( arr_lambda_lambda @ Resid2 @ T6 )
=> ( ide_lambda_lambda @ Resid2 @ ( trg_lambda_lambda @ Resid2 @ T6 ) ) )
& ! [A5: lambda_lambda,T6: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A5 )
=> ( ( con_lambda_lambda @ Resid2 @ T6 @ A5 )
=> ( ( Resid2 @ T6 @ A5 )
= T6 ) ) )
& ! [A5: lambda_lambda,T6: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A5 )
=> ( ( con_lambda_lambda @ Resid2 @ A5 @ T6 )
=> ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ A5 @ T6 ) ) ) )
& ! [T6: lambda_lambda,U6: lambda_lambda] :
( ( con_lambda_lambda @ Resid2 @ T6 @ U6 )
=> ? [A5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A5 )
& ( con_lambda_lambda @ Resid2 @ A5 @ T6 )
& ( con_lambda_lambda @ Resid2 @ A5 @ U6 ) ) )
& ! [T6: lambda_lambda,U6: lambda_lambda,V5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ T6 @ U6 ) )
=> ( ( con_lambda_lambda @ Resid2 @ U6 @ V5 )
=> ( con_lambda_lambda @ Resid2 @ ( Resid2 @ T6 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_721_rts__axioms__def,axiom,
( rts_ax2283037410143324311lambda
= ( ^ [Resid2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T6: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ Resid2 @ T6 )
=> ( ide_li6773656263178250516lambda @ Resid2 @ ( trg_li817795487149935061lambda @ Resid2 @ T6 ) ) )
& ! [A5: list_lambda_lambda,T6: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A5 )
=> ( ( con_li79537494059641180lambda @ Resid2 @ T6 @ A5 )
=> ( ( Resid2 @ T6 @ A5 )
= T6 ) ) )
& ! [A5: list_lambda_lambda,T6: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A5 )
=> ( ( con_li79537494059641180lambda @ Resid2 @ A5 @ T6 )
=> ( ide_li6773656263178250516lambda @ Resid2 @ ( Resid2 @ A5 @ T6 ) ) ) )
& ! [T6: list_lambda_lambda,U6: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ Resid2 @ T6 @ U6 )
=> ? [A5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A5 )
& ( con_li79537494059641180lambda @ Resid2 @ A5 @ T6 )
& ( con_li79537494059641180lambda @ Resid2 @ A5 @ U6 ) ) )
& ! [T6: list_lambda_lambda,U6: list_lambda_lambda,V5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ ( Resid2 @ T6 @ U6 ) )
=> ( ( con_li79537494059641180lambda @ Resid2 @ U6 @ V5 )
=> ( con_li79537494059641180lambda @ Resid2 @ ( Resid2 @ T6 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_722_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_723_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_724_last__appendL,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Ys = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_appendL
thf(fact_725_last__appendR,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Ys != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Ys ) ) ) ).
% last_appendR
thf(fact_726_last__snoc,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] :
( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) ) )
= X2 ) ).
% last_snoc
thf(fact_727_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_728_last_Osimps,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ) ).
% last.simps
thf(fact_729_last__ConsL,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] :
( ( Xs = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_730_last__ConsR,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_ConsR
thf(fact_731_last__in__set,axiom,
! [As: list_l5168835686021156086lambda] :
( ( As != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( last_l1874196399739661071lambda @ As ) @ ( set_li3846194476672328965lambda @ As ) ) ) ).
% last_in_set
thf(fact_732_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_733_last__append,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( ( Ys = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Xs ) ) )
& ( ( Ys != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Ys ) ) ) ) ).
% last_append
thf(fact_734_longest__common__suffix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ss: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Xs3 @ Ss ) )
& ( Ys
= ( append_lambda_lambda @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_lambda_lambda )
| ( Ys5 = nil_lambda_lambda )
| ( ( last_lambda_lambda @ Xs3 )
!= ( last_lambda_lambda @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_735_hd__Nil__eq__last,axiom,
( ( hd_lambda_lambda @ nil_lambda_lambda )
= ( last_lambda_lambda @ nil_lambda_lambda ) ) ).
% hd_Nil_eq_last
thf(fact_736_in__targets__iff,axiom,
! [T4: list_lambda_lambda,B4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B4 @ ( 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 ) @ B4 ) ) ) ) ) ).
% in_targets_iff
thf(fact_737_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_738_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_739_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_740_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_741_Trgs__simp_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T ) ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Trgs_simp\<^sub>\<Lambda>\<^sub>P
thf(fact_742_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_743_sub__rts__axioms__def,axiom,
( sub_rt3575465509250962105lambda
= ( ^ [R: lambda_lambda > lambda_lambda > lambda_lambda,Arr: lambda_lambda > $o] :
( ! [T6: lambda_lambda] :
( ( Arr @ T6 )
=> ( arr_lambda_lambda @ R @ T6 ) )
& ! [T6: lambda_lambda] :
( ( Arr @ T6 )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R @ T6 ) @ ( collec621444948832755541lambda @ Arr ) ) )
& ! [T6: lambda_lambda,U6: lambda_lambda] :
( ( Arr @ T6 )
=> ( ( Arr @ U6 )
=> ( ( con_lambda_lambda @ R @ T6 @ U6 )
=> ( Arr @ ( R @ T6 @ U6 ) ) ) ) ) ) ) ) ).
% sub_rts_axioms_def
thf(fact_744_sub__rts__axioms__def,axiom,
( sub_rt3099298114314975807lambda
= ( ^ [R: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Arr: list_lambda_lambda > $o] :
( ! [T6: list_lambda_lambda] :
( ( Arr @ T6 )
=> ( arr_li3691443969012319837lambda @ R @ T6 ) )
& ! [T6: list_lambda_lambda] :
( ( Arr @ T6 )
=> ( ord_le3064231726521830256lambda @ ( source6119337877621565212lambda @ R @ T6 ) @ ( collec8830145823237851355lambda @ Arr ) ) )
& ! [T6: list_lambda_lambda,U6: list_lambda_lambda] :
( ( Arr @ T6 )
=> ( ( Arr @ U6 )
=> ( ( con_li79537494059641180lambda @ R @ T6 @ U6 )
=> ( Arr @ ( R @ T6 @ U6 ) ) ) ) ) ) ) ) ).
% sub_rts_axioms_def
thf(fact_745_sub__rts__axioms_Ointro,axiom,
! [Arr2: lambda_lambda > $o,R2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( arr_lambda_lambda @ R2 @ T3 ) )
=> ( ! [T3: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R2 @ T3 ) @ ( collec621444948832755541lambda @ Arr2 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ( Arr2 @ U4 )
=> ( ( con_lambda_lambda @ R2 @ T3 @ U4 )
=> ( Arr2 @ ( R2 @ T3 @ U4 ) ) ) ) )
=> ( sub_rt3575465509250962105lambda @ R2 @ Arr2 ) ) ) ) ).
% sub_rts_axioms.intro
thf(fact_746_sub__rts__axioms_Ointro,axiom,
! [Arr2: list_lambda_lambda > $o,R2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( Arr2 @ T3 )
=> ( arr_li3691443969012319837lambda @ R2 @ T3 ) )
=> ( ! [T3: list_lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ord_le3064231726521830256lambda @ ( source6119337877621565212lambda @ R2 @ T3 ) @ ( collec8830145823237851355lambda @ Arr2 ) ) )
=> ( ! [T3: list_lambda_lambda,U4: list_lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ( Arr2 @ U4 )
=> ( ( con_li79537494059641180lambda @ R2 @ T3 @ U4 )
=> ( Arr2 @ ( R2 @ T3 @ U4 ) ) ) ) )
=> ( sub_rt3099298114314975807lambda @ R2 @ Arr2 ) ) ) ) ).
% sub_rts_axioms.intro
thf(fact_747__092_060Lambda_062_OseqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
!= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ) ).
% \<Lambda>.seqE\<^sub>W\<^sub>E
thf(fact_748__092_060Lambda_062_Ocon__imp__eq__src,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.con_imp_eq_src
thf(fact_749__092_060Lambda_062_Osrc__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ U )
= ( weakly4554751870655413501lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.src_join_of(2)
thf(fact_750__092_060Lambda_062_Osrc__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.src_join_of(1)
thf(fact_751__092_060Lambda_062_Osrc__eqI,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( con_lambda_lambda @ lambda_resid @ A @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= A ) ) ) ).
% \<Lambda>.src_eqI
thf(fact_752__092_060Lambda_062_Oide__src,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.ide_src
thf(fact_753__092_060Lambda_062_Oide__iff__src__self,axiom,
! [A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ A )
=> ( ( ide_lambda_lambda @ lambda_resid @ A )
= ( ( weakly4554751870655413501lambda @ lambda_resid @ A )
= A ) ) ) ).
% \<Lambda>.ide_iff_src_self
thf(fact_754__092_060Lambda_062_Osrc__in__sources,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( member_lambda_lambda @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.src_in_sources
thf(fact_755__092_060Lambda_062_OcoinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
!= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ) ).
% \<Lambda>.coinitialE\<^sub>W\<^sub>E
thf(fact_756__092_060Lambda_062_Ocoinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ).
% \<Lambda>.coinitial_iff\<^sub>W\<^sub>E
thf(fact_757_Src_Osimps,axiom,
! [T4: list_lambda_lambda] :
( ( paths_1651982994781619320lambda @ lambda_resid @ T4 )
= ( weakly4554751870655413501lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) ) ) ).
% Src.simps
thf(fact_758_Src_Oelims,axiom,
! [X2: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_1651982994781619320lambda @ lambda_resid @ X2 )
= Y2 )
=> ( Y2
= ( weakly4554751870655413501lambda @ lambda_resid @ ( hd_lambda_lambda @ X2 ) ) ) ) ).
% Src.elims
thf(fact_759__092_060Lambda_062_Osrc__src,axiom,
! [T: lambda_lambda] :
( ( weakly4554751870655413501lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.src_src
thf(fact_760__092_060Lambda_062_Osrc__ide,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ A )
= A ) ) ).
% \<Lambda>.src_ide
thf(fact_761__092_060Lambda_062_Oarr__src__iff__arr,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.arr_src_iff_arr
thf(fact_762__092_060Lambda_062_Otrg__src,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.trg_src
thf(fact_763__092_060Lambda_062_Osrc__trg,axiom,
! [T: lambda_lambda] :
( ( weakly4554751870655413501lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.src_trg
thf(fact_764__092_060Lambda_062_Osrc__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.src_resid
thf(fact_765__092_060Lambda_062_OcoinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.coinitialI\<^sub>W\<^sub>E
thf(fact_766__092_060Lambda_062_OseqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [U: lambda_lambda,T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% \<Lambda>.seqI\<^sub>W\<^sub>E
thf(fact_767_weakly__extensional__rts_Osrc_Ocong,axiom,
weakly4554751870655413501lambda = weakly4554751870655413501lambda ).
% weakly_extensional_rts.src.cong
thf(fact_768_transformation__axioms__def,axiom,
( transf8838427370478101288lambda
= ( ^ [A7: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,G2: list_lambda_lambda > list_lambda_lambda,Tau: list_lambda_lambda > list_lambda_lambda] :
( ! [F3: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A7 @ F3 )
=> ( ( Tau @ F3 )
= ( partia6873046715171720854lambda @ B5 ) ) )
& ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A7 @ F3 )
=> ( ( weakly2957279434024022659lambda @ B5 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly2957279434024022659lambda @ A7 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A7 @ F3 )
=> ( ( trg_li817795487149935061lambda @ B5 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_li817795487149935061lambda @ A7 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A7 @ F3 )
=> ( ( B5 @ ( Tau @ ( weakly2957279434024022659lambda @ A7 @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_li817795487149935061lambda @ A7 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A7 @ F3 )
=> ( ( B5 @ ( F2 @ F3 ) @ ( Tau @ ( weakly2957279434024022659lambda @ A7 @ F3 ) ) )
= ( G2 @ F3 ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A7 @ F3 )
=> ( join_o6956785509312998601lambda @ B5 @ ( Tau @ ( weakly2957279434024022659lambda @ A7 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_769_transformation__axioms__def,axiom,
( transf4653517080217399086lambda
= ( ^ [A7: lambda_lambda > lambda_lambda > lambda_lambda,B5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,G2: lambda_lambda > list_lambda_lambda,Tau: lambda_lambda > list_lambda_lambda] :
( ! [F3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A7 @ F3 )
=> ( ( Tau @ F3 )
= ( partia6873046715171720854lambda @ B5 ) ) )
& ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A7 @ F3 )
=> ( ( weakly2957279434024022659lambda @ B5 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly4554751870655413501lambda @ A7 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A7 @ F3 )
=> ( ( trg_li817795487149935061lambda @ B5 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_lambda_lambda @ A7 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F3 )
=> ( ( B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A7 @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_lambda_lambda @ A7 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F3 )
=> ( ( B5 @ ( F2 @ F3 ) @ ( Tau @ ( weakly4554751870655413501lambda @ A7 @ F3 ) ) )
= ( G2 @ F3 ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F3 )
=> ( join_o6956785509312998601lambda @ B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A7 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_770_transformation__axioms__def,axiom,
( transf3419659901140239394lambda
= ( ^ [A7: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,G2: list_lambda_lambda > lambda_lambda,Tau: list_lambda_lambda > lambda_lambda] :
( ! [F3: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A7 @ F3 )
=> ( ( Tau @ F3 )
= ( partia1461432607296028304lambda @ B5 ) ) )
& ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A7 @ F3 )
=> ( ( weakly4554751870655413501lambda @ B5 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly2957279434024022659lambda @ A7 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A7 @ F3 )
=> ( ( trg_lambda_lambda @ B5 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_li817795487149935061lambda @ A7 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A7 @ F3 )
=> ( ( B5 @ ( Tau @ ( weakly2957279434024022659lambda @ A7 @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_li817795487149935061lambda @ A7 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A7 @ F3 )
=> ( ( B5 @ ( F2 @ F3 ) @ ( Tau @ ( weakly2957279434024022659lambda @ A7 @ F3 ) ) )
= ( G2 @ F3 ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A7 @ F3 )
=> ( join_o1386658102433174083lambda @ B5 @ ( Tau @ ( weakly2957279434024022659lambda @ A7 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_771_transformation__axioms__def,axiom,
( transf356316610583137064lambda
= ( ^ [A7: lambda_lambda > lambda_lambda > lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,G2: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda] :
( ! [F3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A7 @ F3 )
=> ( ( Tau @ F3 )
= ( partia1461432607296028304lambda @ B5 ) ) )
& ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A7 @ F3 )
=> ( ( weakly4554751870655413501lambda @ B5 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly4554751870655413501lambda @ A7 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A7 @ F3 )
=> ( ( trg_lambda_lambda @ B5 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_lambda_lambda @ A7 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F3 )
=> ( ( B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A7 @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_lambda_lambda @ A7 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F3 )
=> ( ( B5 @ ( F2 @ F3 ) @ ( Tau @ ( weakly4554751870655413501lambda @ A7 @ F3 ) ) )
= ( G2 @ F3 ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A7 @ F3 )
=> ( join_o1386658102433174083lambda @ B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A7 @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_772_transformation__axioms_Ointro,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Tau2: list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,G3: list_lambda_lambda > list_lambda_lambda] :
( ! [F5: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A2 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia6873046715171720854lambda @ B4 ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A2 @ F5 )
=> ( ( weakly2957279434024022659lambda @ B4 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly2957279434024022659lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A2 @ F5 )
=> ( ( trg_li817795487149935061lambda @ B4 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_li817795487149935061lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A2 @ F5 )
=> ( ( B4 @ ( Tau2 @ ( weakly2957279434024022659lambda @ A2 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_li817795487149935061lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A2 @ F5 )
=> ( ( B4 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly2957279434024022659lambda @ A2 @ F5 ) ) )
= ( G3 @ F5 ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A2 @ F5 )
=> ( join_o6956785509312998601lambda @ B4 @ ( Tau2 @ ( weakly2957279434024022659lambda @ A2 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) )
=> ( transf8838427370478101288lambda @ A2 @ B4 @ F4 @ G3 @ Tau2 ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_773_transformation__axioms_Ointro,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,Tau2: lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,G3: lambda_lambda > list_lambda_lambda] :
( ! [F5: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A2 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia6873046715171720854lambda @ B4 ) ) )
=> ( ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A2 @ F5 )
=> ( ( weakly2957279434024022659lambda @ B4 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly4554751870655413501lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A2 @ F5 )
=> ( ( trg_li817795487149935061lambda @ B4 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_lambda_lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A2 @ F5 )
=> ( ( B4 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A2 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_lambda_lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A2 @ F5 )
=> ( ( B4 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly4554751870655413501lambda @ A2 @ F5 ) ) )
= ( G3 @ F5 ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A2 @ F5 )
=> ( join_o6956785509312998601lambda @ B4 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A2 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) )
=> ( transf4653517080217399086lambda @ A2 @ B4 @ F4 @ G3 @ Tau2 ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_774_transformation__axioms_Ointro,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Tau2: list_lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,G3: list_lambda_lambda > lambda_lambda] :
( ! [F5: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A2 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia1461432607296028304lambda @ B4 ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A2 @ F5 )
=> ( ( weakly4554751870655413501lambda @ B4 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly2957279434024022659lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A2 @ F5 )
=> ( ( trg_lambda_lambda @ B4 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_li817795487149935061lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A2 @ F5 )
=> ( ( B4 @ ( Tau2 @ ( weakly2957279434024022659lambda @ A2 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_li817795487149935061lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A2 @ F5 )
=> ( ( B4 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly2957279434024022659lambda @ A2 @ F5 ) ) )
= ( G3 @ F5 ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A2 @ F5 )
=> ( join_o1386658102433174083lambda @ B4 @ ( Tau2 @ ( weakly2957279434024022659lambda @ A2 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) )
=> ( transf3419659901140239394lambda @ A2 @ B4 @ F4 @ G3 @ Tau2 ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_775_transformation__axioms_Ointro,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,Tau2: lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,G3: lambda_lambda > lambda_lambda] :
( ! [F5: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A2 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia1461432607296028304lambda @ B4 ) ) )
=> ( ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A2 @ F5 )
=> ( ( weakly4554751870655413501lambda @ B4 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly4554751870655413501lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A2 @ F5 )
=> ( ( trg_lambda_lambda @ B4 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_lambda_lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A2 @ F5 )
=> ( ( B4 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A2 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_lambda_lambda @ A2 @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A2 @ F5 )
=> ( ( B4 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly4554751870655413501lambda @ A2 @ F5 ) ) )
= ( G3 @ F5 ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A2 @ F5 )
=> ( join_o1386658102433174083lambda @ B4 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A2 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) )
=> ( transf356316610583137064lambda @ A2 @ B4 @ F4 @ G3 @ Tau2 ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_776_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_777_development__imp__Arr,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U2 )
=> ( lambda_Arr @ T ) ) ).
% development_imp_Arr
thf(fact_778_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_779_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_780_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_781_lambda__calculus_Oarr__char,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.arr_char
thf(fact_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_lambda__calculus_OArr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Arr_Src
thf(fact_793_lambda__calculus_OArr__Trg,axiom,
! [X6: lambda_lambda] :
( ( lambda_Arr @ X6 )
=> ( lambda_Arr @ ( lambda_Trg @ X6 ) ) ) ).
% lambda_calculus.Arr_Trg
thf(fact_794_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_795_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_796_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_797_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 ) ) )
=> ~ ! [B6: lambda_lambda] :
( ( lambda_red @ A1 @ B6 )
=> ~ ( lambda_red @ B6 @ A22 ) ) ) ) ).
% lambda_calculus.red.cases
thf(fact_798_lambda__calculus_Ored_Osimps,axiom,
( lambda_red
= ( ^ [A12: lambda_lambda,A23: lambda_lambda] :
( ? [T6: lambda_lambda] :
( ( A12
= ( lambda_Src @ T6 ) )
& ( A23
= ( lambda_Trg @ T6 ) )
& ( lambda_Arr @ T6 ) )
| ? [A5: lambda_lambda,B2: lambda_lambda,C3: lambda_lambda] :
( ( A12 = A5 )
& ( A23 = C3 )
& ( lambda_red @ A5 @ B2 )
& ( lambda_red @ B2 @ C3 ) ) ) ) ) ).
% lambda_calculus.red.simps
thf(fact_799_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_800_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_801_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_802_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_803_reduction__paths_Odevelopment_Oelims_I1_J,axiom,
! [X2: lambda_lambda,Xa: list_lambda_lambda,Y2: $o] :
( ( ( reduct9208763521642324371opment @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_lambda_lambda )
=> ( Y2
= ( ~ ( lambda_Arr @ X2 ) ) ) )
=> ~ ! [U4: lambda_lambda,U7: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ U7 ) )
=> ( Y2
= ( ~ ( ( lambda2670285437103676873uction @ U4 )
& ( lambda_subs @ U4 @ X2 )
& ( reduct9208763521642324371opment @ ( lambda_resid @ X2 @ U4 ) @ U7 ) ) ) ) ) ) ) ).
% reduction_paths.development.elims(1)
thf(fact_804_reduction__paths_Odevelopment_Oelims_I2_J,axiom,
! [X2: lambda_lambda,Xa: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ X2 @ Xa )
=> ( ( ( Xa = nil_lambda_lambda )
=> ~ ( lambda_Arr @ X2 ) )
=> ~ ! [U4: lambda_lambda,U7: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ U7 ) )
=> ~ ( ( lambda2670285437103676873uction @ U4 )
& ( lambda_subs @ U4 @ X2 )
& ( reduct9208763521642324371opment @ ( lambda_resid @ X2 @ U4 ) @ U7 ) ) ) ) ) ).
% reduction_paths.development.elims(2)
thf(fact_805_reduction__paths_Odevelopment_Oelims_I3_J,axiom,
! [X2: lambda_lambda,Xa: list_lambda_lambda] :
( ~ ( reduct9208763521642324371opment @ X2 @ Xa )
=> ( ( ( Xa = nil_lambda_lambda )
=> ( lambda_Arr @ X2 ) )
=> ~ ! [U4: lambda_lambda,U7: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ U7 ) )
=> ( ( lambda2670285437103676873uction @ U4 )
& ( lambda_subs @ U4 @ X2 )
& ( reduct9208763521642324371opment @ ( lambda_resid @ X2 @ U4 ) @ U7 ) ) ) ) ) ).
% reduction_paths.development.elims(3)
thf(fact_806_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_807_append__is__composite__of,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ).
% append_is_composite_of
thf(fact_808_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_809_composite__ofE,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) ) ) ) ) ).
% composite_ofE
thf(fact_810_composite__of__cancel__left,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,U3: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U3 @ V )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) ) ) ) ).
% composite_of_cancel_left
thf(fact_811_composite__of__def,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) ) ) ) ).
% composite_of_def
thf(fact_812_composite__of__ide__self,axiom,
! [A: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ A @ A ) ) ).
% composite_of_ide_self
thf(fact_813_composite__of__unq__upto__cong,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ 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 ) ) ) ) ) ).
% composite_of_unq_upto_cong
thf(fact_814_diamond__commutes__upto__cong,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ ( 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 ) ) ) ) ) ).
% diamond_commutes_upto_cong
thf(fact_815_con__prfx__composite__of_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ V ) ) ) ).
% con_prfx_composite_of(2)
thf(fact_816_con__prfx__composite__of_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ W2 ) ) ).
% con_prfx_composite_of(1)
thf(fact_817_resid__composite__of_I4_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ V ) ) ) ) ).
% resid_composite_of(4)
thf(fact_818_resid__composite__of_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ U ) ) ) ).
% resid_composite_of(2)
thf(fact_819_resid__composite__of_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ T ) ) ) ) ).
% resid_composite_of(1)
thf(fact_820_bounded__imp__con,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,T5: list_lambda_lambda,U3: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T5 @ U3 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T5 ) ) ) ).
% bounded_imp_con
thf(fact_821_con__composite__of__iff,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,W2: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ T ) @ U ) ) ) ).
% con_composite_of_iff
thf(fact_822_sources__composite__of,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).
% sources_composite_of
thf(fact_823_targets__composite__of,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% targets_composite_of
thf(fact_824_arr__composite__of,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ).
% arr_composite_of
thf(fact_825_obtains__composite__of,axiom,
! [G: list_lambda_lambda,F: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F )
=> ~ ! [H: list_lambda_lambda] :
~ ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F @ H ) ) ).
% obtains_composite_of
thf(fact_826_join__ofE,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ~ ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) @ V )
=> ~ ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ V ) ) ) ).
% join_ofE
thf(fact_827_join__of__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
= ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) @ V )
& ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ V ) ) ) ).
% join_of_def
thf(fact_828_composable__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ? [X5: list_lambda_lambda] : ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ X5 ) ) ) ).
% composable_def
thf(fact_829_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_830_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_831_resid__composite__of_I3_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ W2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ W2 ) ) ) ) ) ) ).
% resid_composite_of(3)
thf(fact_832_composite__of__arr__ide,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ B @ T )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) @ B ) ) ) ).
% composite_of_arr_ide
thf(fact_833_composite__of__ide__arr,axiom,
! [A: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ T @ T )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A ) ) ) ).
% composite_of_ide_arr
thf(fact_834_composite__of__source__arr,axiom,
! [T: list_lambda_lambda,A: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A @ T @ T ) ) ) ).
% composite_of_source_arr
thf(fact_835_composite__of__arr__target,axiom,
! [T: list_lambda_lambda,B: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ B @ T ) ) ) ).
% composite_of_arr_target
thf(fact_836_tl__append2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( append_lambda_lambda @ ( tl_lambda_lambda @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_837_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_838_hd__Cons__tl,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( cons_lambda_lambda @ ( hd_lambda_lambda @ Xs ) @ ( tl_lambda_lambda @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_839_composite__ofI,axiom,
! [U: list_lambda_lambda,V: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) ) )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V ) ) ) ).
% composite_ofI
thf(fact_840_join__ofI,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ V )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V ) ) ) ).
% join_ofI
thf(fact_841_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_842_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_843_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_844_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_845_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_846_list_Osel_I2_J,axiom,
( ( tl_lambda_lambda @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.sel(2)
thf(fact_847_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_848_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_Src @ ( lambda_App @ T5 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T5 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_849_lambda__calculus_Osubs_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( lambda_subs @ ( lambda_App @ T @ U ) @ ( lambda_App @ T5 @ U3 ) )
= ( ( lambda_subs @ T @ T5 )
& ( lambda_subs @ U @ U3 ) ) ) ).
% lambda_calculus.subs.simps(3)
thf(fact_850_rts_Ocomposite__of_Ocong,axiom,
compos7114946393191943382lambda = compos7114946393191943382lambda ).
% rts.composite_of.cong
thf(fact_851_rts_Ocomposite__of_Ocong,axiom,
compos9183582412961930448lambda = compos9183582412961930448lambda ).
% rts.composite_of.cong
thf(fact_852_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,T: list_lambda_lambda,U: list_lambda_lambda,T5: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( compos7114946393191943382lambda @ Resid @ T @ U @ T5 )
=> ( ( member2667661554319814297lambda @ U @ NN )
=> ( ( member2667661554319814297lambda @ ( Resid @ T5 @ T ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ T @ T5 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_853_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ T5 )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T5 @ T ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T @ T5 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_854_Nil__tl,axiom,
! [Xs: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( tl_lambda_lambda @ Xs ) )
= ( ( Xs = nil_lambda_lambda )
| ? [X3: lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) ) ) ) ).
% Nil_tl
thf(fact_855_tl__Nil,axiom,
! [Xs: list_lambda_lambda] :
( ( ( tl_lambda_lambda @ Xs )
= nil_lambda_lambda )
= ( ( Xs = nil_lambda_lambda )
| ? [X3: lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ X3 @ nil_lambda_lambda ) ) ) ) ).
% tl_Nil
thf(fact_856_list_Oset__sel_I2_J,axiom,
! [A: list_l5168835686021156086lambda,X2: list_lambda_lambda] :
( ( A != nil_li8379291187143545974lambda )
=> ( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ ( tl_lis5905685823335986783lambda @ A ) ) )
=> ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_857_list_Oset__sel_I2_J,axiom,
! [A: list_lambda_lambda,X2: lambda_lambda] :
( ( A != nil_lambda_lambda )
=> ( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ ( tl_lambda_lambda @ A ) ) )
=> ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_858_tl__append__if,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( tl_lambda_lambda @ Ys ) ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( append_lambda_lambda @ ( tl_lambda_lambda @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_859_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_860_last__tl,axiom,
! [Xs: list_lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
| ( ( tl_lambda_lambda @ Xs )
!= nil_lambda_lambda ) )
=> ( ( last_lambda_lambda @ ( tl_lambda_lambda @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_tl
thf(fact_861_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V4: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ ( Resid @ T @ U ) @ V4 )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ V @ V4 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ V4 @ V ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_862_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( compos7114946393191943382lambda @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( compos7114946393191943382lambda @ Resid @ U @ ( Resid @ T @ U ) @ V4 )
=> ( ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ V @ V4 ) )
& ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ V4 @ V ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_863_rts__with__composites_Oobtains__composite__of,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,G: lambda_lambda,F: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( seq_lambda_lambda @ Resid @ G @ F )
=> ~ ! [H: lambda_lambda] :
~ ( compos9183582412961930448lambda @ Resid @ G @ F @ H ) ) ) ).
% rts_with_composites.obtains_composite_of
thf(fact_864_rts__with__composites_Oobtains__composite__of,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda,F: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ Resid @ G @ F )
=> ~ ! [H: list_lambda_lambda] :
~ ( compos7114946393191943382lambda @ Resid @ G @ F @ H ) ) ) ).
% rts_with_composites.obtains_composite_of
thf(fact_865_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_866_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_867_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_868_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_869_normal__sub__rts__axioms__def,axiom,
( normal6810188024974762232lambda
= ( ^ [Resid2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN2: set_li7878742357367608272lambda] :
( ! [T6: list_lambda_lambda] :
( ( member2667661554319814297lambda @ T6 @ NN2 )
=> ( arr_li3691443969012319837lambda @ Resid2 @ T6 ) )
& ! [A5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A5 )
=> ( member2667661554319814297lambda @ A5 @ NN2 ) )
& ! [U6: list_lambda_lambda,T6: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( coinit1063259437155584316lambda @ Resid2 @ T6 @ U6 )
=> ( member2667661554319814297lambda @ ( Resid2 @ U6 @ T6 ) @ NN2 ) ) )
& ! [U6: list_lambda_lambda,T6: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( member2667661554319814297lambda @ ( Resid2 @ T6 @ U6 ) @ NN2 )
=> ( member2667661554319814297lambda @ T6 @ NN2 ) ) )
& ! [U6: list_lambda_lambda,T6: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( seq_li8093550137657879445lambda @ Resid2 @ U6 @ T6 )
=> ? [X5: list_lambda_lambda] : ( compos7114946393191943382lambda @ Resid2 @ U6 @ T6 @ X5 ) ) )
& ! [U6: list_lambda_lambda,T6: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( seq_li8093550137657879445lambda @ Resid2 @ T6 @ U6 )
=> ? [X5: list_lambda_lambda] : ( compos7114946393191943382lambda @ Resid2 @ T6 @ U6 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_870_normal__sub__rts__axioms__def,axiom,
( normal5162919049924913650lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda,NN2: set_lambda_lambda] :
( ! [T6: lambda_lambda] :
( ( member_lambda_lambda @ T6 @ NN2 )
=> ( arr_lambda_lambda @ Resid2 @ T6 ) )
& ! [A5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A5 )
=> ( member_lambda_lambda @ A5 @ NN2 ) )
& ! [U6: lambda_lambda,T6: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( coinit1392077014771647542lambda @ Resid2 @ T6 @ U6 )
=> ( member_lambda_lambda @ ( Resid2 @ U6 @ T6 ) @ NN2 ) ) )
& ! [U6: lambda_lambda,T6: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( member_lambda_lambda @ ( Resid2 @ T6 @ U6 ) @ NN2 )
=> ( member_lambda_lambda @ T6 @ NN2 ) ) )
& ! [U6: lambda_lambda,T6: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( seq_lambda_lambda @ Resid2 @ U6 @ T6 )
=> ? [X5: lambda_lambda] : ( compos9183582412961930448lambda @ Resid2 @ U6 @ T6 @ X5 ) ) )
& ! [U6: lambda_lambda,T6: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( seq_lambda_lambda @ Resid2 @ T6 @ U6 )
=> ? [X5: lambda_lambda] : ( compos9183582412961930448lambda @ Resid2 @ T6 @ U6 @ X5 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_871_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_li7878742357367608272lambda,Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ T3 @ NN )
=> ( arr_li3691443969012319837lambda @ Resid @ T3 ) )
=> ( ! [A4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A4 )
=> ( member2667661554319814297lambda @ A4 @ NN ) )
=> ( ! [U4: list_lambda_lambda,T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U4 @ NN )
=> ( ( coinit1063259437155584316lambda @ Resid @ T3 @ U4 )
=> ( member2667661554319814297lambda @ ( Resid @ U4 @ T3 ) @ NN ) ) )
=> ( ! [U4: list_lambda_lambda,T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U4 @ NN )
=> ( ( member2667661554319814297lambda @ ( Resid @ T3 @ U4 ) @ NN )
=> ( member2667661554319814297lambda @ T3 @ NN ) ) )
=> ( ! [U4: list_lambda_lambda,T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U4 @ NN )
=> ( ( seq_li8093550137657879445lambda @ Resid @ U4 @ T3 )
=> ? [X_1: list_lambda_lambda] : ( compos7114946393191943382lambda @ Resid @ U4 @ T3 @ X_1 ) ) )
=> ( ! [U4: list_lambda_lambda,T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U4 @ NN )
=> ( ( seq_li8093550137657879445lambda @ Resid @ T3 @ U4 )
=> ? [X_1: list_lambda_lambda] : ( compos7114946393191943382lambda @ Resid @ T3 @ U4 @ X_1 ) ) )
=> ( normal6810188024974762232lambda @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_872_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_lambda_lambda,Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( member_lambda_lambda @ T3 @ NN )
=> ( arr_lambda_lambda @ Resid @ T3 ) )
=> ( ! [A4: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A4 )
=> ( member_lambda_lambda @ A4 @ NN ) )
=> ( ! [U4: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ( ( coinit1392077014771647542lambda @ Resid @ T3 @ U4 )
=> ( member_lambda_lambda @ ( Resid @ U4 @ T3 ) @ NN ) ) )
=> ( ! [U4: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T3 @ U4 ) @ NN )
=> ( member_lambda_lambda @ T3 @ NN ) ) )
=> ( ! [U4: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ( ( seq_lambda_lambda @ Resid @ U4 @ T3 )
=> ? [X_1: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ U4 @ T3 @ X_1 ) ) )
=> ( ! [U4: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U4 @ NN )
=> ( ( seq_lambda_lambda @ Resid @ T3 @ U4 )
=> ? [X_1: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ T3 @ U4 @ X_1 ) ) )
=> ( normal5162919049924913650lambda @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_873__092_060Lambda_062_Ocomposite__of__unq__upto__cong,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda,V4: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V4 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ V4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V4 @ V ) ) ) ) ) ).
% \<Lambda>.composite_of_unq_upto_cong
thf(fact_874__092_060Lambda_062_Ocomposite__of__ide__self,axiom,
! [A: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( compos9183582412961930448lambda @ lambda_resid @ A @ A @ A ) ) ).
% \<Lambda>.composite_of_ide_self
thf(fact_875__092_060Lambda_062_Ocomposite__of__def,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) ) ) ).
% \<Lambda>.composite_of_def
thf(fact_876__092_060Lambda_062_Ocomposite__of__cancel__left,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,U3: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T @ U3 @ V )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) ) ) ) ).
% \<Lambda>.composite_of_cancel_left
thf(fact_877__092_060Lambda_062_Ocomposite__ofE,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) ) ) ) ).
% \<Lambda>.composite_ofE
thf(fact_878__092_060Lambda_062_Ocon__composite__of__iff,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,W2: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
= ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ W2 @ T ) @ U ) ) ) ).
% \<Lambda>.con_composite_of_iff
thf(fact_879__092_060Lambda_062_Obounded__imp__con,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T5 @ U3 @ V )
=> ( con_lambda_lambda @ lambda_resid @ T @ T5 ) ) ) ).
% \<Lambda>.bounded_imp_con
thf(fact_880__092_060Lambda_062_Oresid__composite__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ W2 @ T ) ) ) ) ).
% \<Lambda>.resid_composite_of(1)
thf(fact_881__092_060Lambda_062_Oresid__composite__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ U ) ) ) ).
% \<Lambda>.resid_composite_of(2)
thf(fact_882__092_060Lambda_062_Oresid__composite__of_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( compos9183582412961930448lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ ( lambda_resid @ V @ T ) ) @ ( lambda_resid @ W2 @ V ) ) ) ) ).
% \<Lambda>.resid_composite_of(4)
thf(fact_883__092_060Lambda_062_Ocon__prfx__composite__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( con_lambda_lambda @ lambda_resid @ T @ W2 ) ) ).
% \<Lambda>.con_prfx_composite_of(1)
thf(fact_884__092_060Lambda_062_Ocon__prfx__composite__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( con_lambda_lambda @ lambda_resid @ T @ V ) ) ) ).
% \<Lambda>.con_prfx_composite_of(2)
thf(fact_885__092_060Lambda_062_Oarr__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( arr_lambda_lambda @ lambda_resid @ V ) ) ).
% \<Lambda>.arr_composite_of
thf(fact_886__092_060Lambda_062_Otargets__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ V )
= ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.targets_composite_of
thf(fact_887__092_060Lambda_062_Osources__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ V )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.sources_composite_of
thf(fact_888__092_060Lambda_062_Otrg__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ V )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.trg_composite_of
thf(fact_889__092_060Lambda_062_Osrc__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ V )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.src_composite_of
thf(fact_890__092_060Lambda_062_Ojoin__of__def,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
= ( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
& ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V ) ) ) ).
% \<Lambda>.join_of_def
thf(fact_891__092_060Lambda_062_Ojoin__ofE,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ~ ( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
=> ~ ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V ) ) ) ).
% \<Lambda>.join_ofE
thf(fact_892__092_060Lambda_062_Ocomposable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
= ( ? [X5: lambda_lambda] : ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ X5 ) ) ) ).
% \<Lambda>.composable_def
thf(fact_893__092_060Lambda_062_Oresid__composite__of_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ W2 ) @ ( lambda_resid @ ( lambda_resid @ V @ T ) @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ T ) @ U ) @ ( lambda_resid @ V @ W2 ) ) ) ) ) ) ).
% \<Lambda>.resid_composite_of(3)
thf(fact_894__092_060Lambda_062_Ocomposite__of__arr__ide,axiom,
! [B: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T @ B @ T )
= ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) @ B ) ) ) ).
% \<Lambda>.composite_of_arr_ide
thf(fact_895__092_060Lambda_062_Ocomposite__of__ide__arr,axiom,
! [A: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ A @ T @ T )
= ( con_lambda_lambda @ lambda_resid @ T @ A ) ) ) ).
% \<Lambda>.composite_of_ide_arr
thf(fact_896__092_060Lambda_062_Ocomposite__of__arr__target,axiom,
! [T: lambda_lambda,B: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ T @ B @ T ) ) ) ).
% \<Lambda>.composite_of_arr_target
thf(fact_897__092_060Lambda_062_Ocomposite__of__source__arr,axiom,
! [T: lambda_lambda,A: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ A @ T @ T ) ) ) ).
% \<Lambda>.composite_of_source_arr
thf(fact_898_composite__of__single__single,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ).
% composite_of_single_single
thf(fact_899_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_900_Nil__is__map__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( map_la336867333220232516lambda @ F @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% Nil_is_map_conv
thf(fact_901_map__is__Nil__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= nil_lambda_lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% map_is_Nil_conv
thf(fact_902__092_060Lambda_062_Ocomposite__ofI,axiom,
! [U: lambda_lambda,V: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V ) ) ) ).
% \<Lambda>.composite_ofI
thf(fact_903_map__eq__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Xs ) )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_904_map__append,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( append_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) @ ( map_la336867333220232516lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_905__092_060Lambda_062_Ojoin__ofI,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V ) ) ) ).
% \<Lambda>.join_ofI
thf(fact_906_map__tl,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ ( tl_lambda_lambda @ Xs ) )
= ( tl_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) ) ) ).
% map_tl
thf(fact_907_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_908_Cons__eq__map__D,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( map_la336867333220232516lambda @ F @ Ys ) )
=> ? [Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z @ Zs2 ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_909_map__eq__Cons__D,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Y2: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( cons_lambda_lambda @ Y2 @ Ys ) )
=> ? [Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_910_Cons__eq__map__conv,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( map_la336867333220232516lambda @ F @ Ys ) )
= ( ? [Z3: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_911_map__eq__Cons__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Y2: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( cons_lambda_lambda @ Y2 @ Ys ) )
= ( ? [Z3: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_912_list_Osimps_I8_J,axiom,
! [F: lambda_lambda > lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.simps(8)
thf(fact_913_append__eq__map__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( map_la336867333220232516lambda @ F @ Xs ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_914_map__eq__append__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ? [Us: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_915_ex__map__conv,axiom,
! [Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( ? [Xs5: list_lambda_lambda] :
( Ys
= ( map_la336867333220232516lambda @ F @ Xs5 ) ) )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Ys ) )
=> ? [Y4: lambda_lambda] :
( X3
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_916_map__cong,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( Xs = Ys )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_917_map__idI,axiom,
! [Xs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_li385728805257398596lambda @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_918_map__idI,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_919_map__ext,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Xs ) ) ) ).
% map_ext
thf(fact_920_list_Omap__ident__strong,axiom,
! [T: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [Z: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z @ ( set_li3846194476672328965lambda @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li385728805257398596lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_921_list_Omap__ident__strong,axiom,
! [T: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [Z: lambda_lambda] :
( ( member_lambda_lambda @ Z @ ( set_lambda_lambda2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_la336867333220232516lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_922_list_Oinj__map__strong,axiom,
! [X2: list_lambda_lambda,Xa: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Fa: lambda_lambda > lambda_lambda] :
( ! [Z: lambda_lambda,Za: lambda_lambda] :
( ( member_lambda_lambda @ Z @ ( set_lambda_lambda2 @ X2 ) )
=> ( ( member_lambda_lambda @ Za @ ( set_lambda_lambda2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_923_list_Omap__cong0,axiom,
! [X2: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [Z: lambda_lambda] :
( ( member_lambda_lambda @ Z @ ( set_lambda_lambda2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_924_list_Omap__cong,axiom,
! [X2: list_lambda_lambda,Ya: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( X2 = Ya )
=> ( ! [Z: lambda_lambda] :
( ( member_lambda_lambda @ Z @ ( set_lambda_lambda2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_925_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_926_hd__map,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) )
= ( F @ ( hd_lambda_lambda @ Xs ) ) ) ) ).
% hd_map
thf(fact_927_last__map,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) )
= ( F @ ( last_lambda_lambda @ Xs ) ) ) ) ).
% last_map
thf(fact_928_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_929_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_930_map__App1__Resid,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ T4 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ U2 ) ) ) ) ) ).
% map_App1_Resid
thf(fact_931_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_932_development__Ide,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ T )
=> ( ( reduct9208763521642324371opment @ T @ U2 )
= ( U2 = nil_lambda_lambda ) ) ) ).
% development_Ide
thf(fact_933_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_934_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_935_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_936_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_937_cong__map__App1,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( ( 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 @ X2 ) @ T4 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ U2 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ T4 ) ) ) ) ) ) ).
% cong_map_App1
thf(fact_938_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_939_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_940_lambda__calculus_Oide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.ide_char
thf(fact_941_lambda__calculus_Olambda_Ocollapse_I2_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ( ( lambda_Lam @ ( lambda_un_Lam @ Lambda ) )
= Lambda ) ) ).
% lambda_calculus.lambda.collapse(2)
thf(fact_942_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_943_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_944_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_945_lambda__calculus_OIde__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Ide_Src
thf(fact_946_lambda__calculus_Oelementary__reduction_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_App @ T @ U ) )
= ( ( ( lambda2670285437103676873uction @ T )
& ( lambda_Ide @ U ) )
| ( ( lambda_Ide @ T )
& ( lambda2670285437103676873uction @ U ) ) ) ) ).
% lambda_calculus.elementary_reduction.simps(4)
thf(fact_947_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_948_lambda__calculus_OIde__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Ide_Trg
thf(fact_949_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_950_lambda__calculus_OIde__implies__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Ide_implies_Arr
thf(fact_951_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_952_lambda__calculus_Olambda_Odisc_I8_J,axiom,
! [X32: lambda_lambda] : ( lambda_is_Lam @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(8)
thf(fact_953_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_954_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_955_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_956_lambda__calculus_Osubs_Osimps_I2_J,axiom,
! [T: lambda_lambda,T5: lambda_lambda] :
( ( lambda_subs @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T5 ) )
= ( lambda_subs @ T @ T5 ) ) ).
% lambda_calculus.subs.simps(2)
thf(fact_957_lambda__calculus_Oelementary__reduction_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Lam @ T ) )
= ( lambda2670285437103676873uction @ T ) ) ).
% lambda_calculus.elementary_reduction.simps(3)
thf(fact_958_lambda__calculus_Olambda_Osel_I2_J,axiom,
! [X32: lambda_lambda] :
( ( lambda_un_Lam @ ( lambda_Lam @ X32 ) )
= X32 ) ).
% lambda_calculus.lambda.sel(2)
thf(fact_959_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_960_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_961_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_962_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_963_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_964_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_965_lambda__calculus_Osubs_Osimps_I29_J,axiom,
! [Vb2: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ Vb2 ) @ ( lambda_App @ V @ Va2 ) ) ).
% lambda_calculus.subs.simps(29)
thf(fact_966_lambda__calculus_Osubs_Osimps_I25_J,axiom,
! [Va2: lambda_lambda,Vb2: lambda_lambda,V: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ Va2 @ Vb2 ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.subs.simps(25)
thf(fact_967_lambda__calculus_Osubs_Osimps_I13_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ V @ Va2 ) @ ( lambda_Lam @ Vb2 ) ) ).
% lambda_calculus.subs.simps(13)
thf(fact_968_lambda__calculus_Osubs_Osimps_I9_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ V ) @ ( lambda_App @ Va2 @ Vb2 ) ) ).
% lambda_calculus.subs.simps(9)
thf(fact_969_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_970_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_971_lambda__calculus_ONF__def,axiom,
( lambda_NF
= ( ^ [A5: lambda_lambda] :
( ( lambda_Ide @ A5 )
& ! [T6: lambda_lambda] :
( ( ( lambda_Arr @ T6 )
& ( ( lambda_Src @ T6 )
= A5 ) )
=> ( lambda_Ide @ T6 ) ) ) ) ) ).
% lambda_calculus.NF_def
thf(fact_972_lambda__calculus_Oelementary__reduction__Lam__iff,axiom,
! [T: lambda_lambda] :
( ( lambda_is_Lam @ T )
=> ( ( lambda2670285437103676873uction @ T )
= ( lambda2670285437103676873uction @ ( lambda_un_Lam @ T ) ) ) ) ).
% lambda_calculus.elementary_reduction_Lam_iff
thf(fact_973_lambda__calculus_Onormalizable__def,axiom,
( lambda_normalizable
= ( ^ [A5: lambda_lambda] :
( ( lambda_Ide @ A5 )
& ? [B2: lambda_lambda] :
( ( lambda_red @ A5 @ B2 )
& ( lambda_NF @ B2 ) ) ) ) ) ).
% lambda_calculus.normalizable_def
thf(fact_974_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_975_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_976_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_977_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_978_lambda__calculus_Osubs_Osimps_I33_J,axiom,
! [Vb2: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ Vb2 ) @ ( lambda_Beta @ V @ Va2 ) ) ).
% lambda_calculus.subs.simps(33)
thf(fact_979_lambda__calculus_Osubs_Osimps_I26_J,axiom,
! [Va2: lambda_lambda,Vb2: lambda_lambda,V: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ Va2 @ Vb2 ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.subs.simps(26)
thf(fact_980_lambda__calculus_Osubs_Osimps_I20_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ V @ Va2 ) @ ( lambda_Lam @ Vb2 ) ) ).
% lambda_calculus.subs.simps(20)
thf(fact_981_lambda__calculus_Osubs_Osimps_I10_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Lam @ V ) @ ( lambda_Beta @ Va2 @ Vb2 ) ) ).
% lambda_calculus.subs.simps(10)
thf(fact_982_lambda__calculus_Oelementary__reduction_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Beta @ T @ U ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.elementary_reduction.simps(5)
thf(fact_983_lambda__calculus_Osubs_Osimps_I37_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc2: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc2 ) @ ( lambda_Beta @ V @ Va2 ) ) ).
% lambda_calculus.subs.simps(37)
thf(fact_984_lambda__calculus_Osubs_Osimps_I36_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc2: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc2 ) @ ( lambda_Beta @ V @ Va2 ) ) ).
% lambda_calculus.subs.simps(36)
thf(fact_985_lambda__calculus_Osubs_Osimps_I30_J,axiom,
! [Vb2: lambda_lambda,Vc2: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ Vb2 @ Vc2 ) @ ( lambda_App @ V @ Va2 ) ) ).
% lambda_calculus.subs.simps(30)
thf(fact_986_lambda__calculus_Osubs_Osimps_I21_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_Beta @ V @ Va2 ) @ ( lambda_App @ Vb2 @ Vc2 ) ) ).
% lambda_calculus.subs.simps(21)
thf(fact_987_lambda__calculus_Osubs_Osimps_I17_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) ).
% lambda_calculus.subs.simps(17)
thf(fact_988_lambda__calculus_Osubs_Osimps_I16_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va2: lambda_lambda,Vb2: lambda_lambda,Vc2: lambda_lambda] :
~ ( lambda_subs @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va2 ) @ ( lambda_Beta @ Vb2 @ Vc2 ) ) ).
% lambda_calculus.subs.simps(16)
thf(fact_989_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_990_lambda__calculus_Osubs_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( lambda_subs @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ T5 @ U3 ) )
= ( ( lambda_subs @ T @ T5 )
& ( lambda_subs @ U @ U3 ) ) ) ).
% lambda_calculus.subs.simps(5)
thf(fact_991_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T5 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T5 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(3)
thf(fact_992_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_993_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_994_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_995_lambda__calculus_OSrc__eq__iff_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T5 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T5 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(4)
thf(fact_996_lambda__calculus_Osubs_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T5: lambda_lambda,U3: lambda_lambda] :
( ( lambda_subs @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( lambda_Beta @ T5 @ U3 ) )
= ( ( lambda_subs @ T @ T5 )
& ( lambda_subs @ U @ U3 ) ) ) ).
% lambda_calculus.subs.simps(4)
thf(fact_997_lambda__calculus_Oelementary__reduction_Oelims_I2_J,axiom,
! [X2: lambda_lambda] :
( ( lambda2670285437103676873uction @ X2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda2670285437103676873uction @ T3 ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X2
= ( lambda_App @ T3 @ U4 ) )
=> ~ ( ( ( lambda2670285437103676873uction @ T3 )
& ( lambda_Ide @ U4 ) )
| ( ( lambda_Ide @ T3 )
& ( lambda2670285437103676873uction @ U4 ) ) ) )
=> ~ ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U4 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U4 ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.elims(2)
thf(fact_998_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_999_length__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( size_s4208335251162658524lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( size_s4208335251162658524lambda @ T4 ) ) ) ).
% length_Resid
thf(fact_1000_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda] :
( ! [T3: lambda_lambda,U4: lambda_lambda,U8: lambda_lambda] :
( ( arr_lambda_lambda @ Resid @ T3 )
=> ( ( member_lambda_lambda @ U4 @ NN )
=> ( ( member_lambda_lambda @ U8 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U4 )
= ( source5627170679743844886lambda @ Resid @ U8 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U4 )
= ( target5490931411331770124lambda @ Resid @ U8 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T3 )
= ( source5627170679743844886lambda @ Resid @ U4 ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ T3 @ U8 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T3 @ U8 ) @ ( Resid @ T3 @ U4 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere8844529169994681456lambda @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_1001_coherent__normal__sub__rts__axioms_Ointro,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda] :
( ! [T3: list_lambda_lambda,U4: list_lambda_lambda,U8: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ Resid @ T3 )
=> ( ( member2667661554319814297lambda @ U4 @ NN )
=> ( ( member2667661554319814297lambda @ U8 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ U4 )
= ( source6119337877621565212lambda @ Resid @ U8 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ U4 )
= ( target4026644918049360146lambda @ Resid @ U8 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ T3 )
= ( source6119337877621565212lambda @ Resid @ U4 ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T3 @ U4 ) @ ( Resid @ T3 @ U8 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T3 @ U8 ) @ ( Resid @ T3 @ U4 ) ) @ NN ) ) ) ) ) ) ) )
=> ( cohere6177641526713707638lambda @ Resid @ NN ) ) ).
% coherent_normal_sub_rts_axioms.intro
thf(fact_1002_append__eq__append__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Us2: list_lambda_lambda,Vs2: list_lambda_lambda] :
( ( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
| ( ( size_s4208335251162658524lambda @ Us2 )
= ( size_s4208335251162658524lambda @ Vs2 ) ) )
=> ( ( ( append_lambda_lambda @ Xs @ Us2 )
= ( append_lambda_lambda @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_1003_length__map,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( size_s4208335251162658524lambda @ ( map_la336867333220232516lambda @ F @ Xs ) )
= ( size_s4208335251162658524lambda @ Xs ) ) ).
% length_map
thf(fact_1004_list__induct4,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ws: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs )
= ( size_s4208335251162658524lambda @ Ws ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: lambda_lambda,Ys2: list_lambda_lambda,Z: lambda_lambda,Zs2: list_lambda_lambda,W4: lambda_lambda,Ws2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Zs2 )
= ( size_s4208335251162658524lambda @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) @ ( cons_lambda_lambda @ W4 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1005_list__induct3,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys )
= ( size_s4208335251162658524lambda @ Zs ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: lambda_lambda,Ys2: list_lambda_lambda,Z: lambda_lambda,Zs2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( ( size_s4208335251162658524lambda @ Ys2 )
= ( size_s4208335251162658524lambda @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys2 ) @ ( cons_lambda_lambda @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1006_list__induct2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,P: list_lambda_lambda > list_lambda_lambda > $o] :
( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: lambda_lambda,Ys2: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs2 )
= ( size_s4208335251162658524lambda @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1007_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_lambda_lambda] :
( ( size_s4208335251162658524lambda @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1008_neq__if__length__neq,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( size_s4208335251162658524lambda @ Xs )
!= ( size_s4208335251162658524lambda @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1009_map__eq__imp__length__eq,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,G: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Ys ) )
=> ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1010_same__length__different,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs != Ys )
=> ( ( ( size_s4208335251162658524lambda @ Xs )
= ( size_s4208335251162658524lambda @ Ys ) )
=> ? [Pre: list_lambda_lambda,X: lambda_lambda,Xs3: list_lambda_lambda,Y3: lambda_lambda,Ys5: list_lambda_lambda] :
( ( X != Y3 )
& ( Xs
= ( append_lambda_lambda @ Pre @ ( append_lambda_lambda @ ( cons_lambda_lambda @ X @ nil_lambda_lambda ) @ Xs3 ) ) )
& ( Ys
= ( append_lambda_lambda @ Pre @ ( append_lambda_lambda @ ( cons_lambda_lambda @ Y3 @ nil_lambda_lambda ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1011_coherent__normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( cohere6177641526713707638lambda @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(2)
thf(fact_1012_coherent__normal__sub__rts_Oaxioms_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( cohere8844529169994681456lambda @ Resid @ NN ) ) ).
% coherent_normal_sub_rts.axioms(2)
thf(fact_1013_coherent__normal__sub__rts__axioms__def,axiom,
( cohere8844529169994681456lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda,NN2: set_lambda_lambda] :
! [T6: lambda_lambda,U6: lambda_lambda,U9: lambda_lambda] :
( ( arr_lambda_lambda @ Resid2 @ T6 )
=> ( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( member_lambda_lambda @ U9 @ NN2 )
=> ( ( ( source5627170679743844886lambda @ Resid2 @ U6 )
= ( source5627170679743844886lambda @ Resid2 @ U9 ) )
=> ( ( ( target5490931411331770124lambda @ Resid2 @ U6 )
= ( target5490931411331770124lambda @ Resid2 @ U9 ) )
=> ( ( ( source5627170679743844886lambda @ Resid2 @ T6 )
= ( source5627170679743844886lambda @ Resid2 @ U6 ) )
=> ( ( member_lambda_lambda @ ( Resid2 @ ( Resid2 @ T6 @ U6 ) @ ( Resid2 @ T6 @ U9 ) ) @ NN2 )
& ( member_lambda_lambda @ ( Resid2 @ ( Resid2 @ T6 @ U9 ) @ ( Resid2 @ T6 @ U6 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_1014_coherent__normal__sub__rts__axioms__def,axiom,
( cohere6177641526713707638lambda
= ( ^ [Resid2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN2: set_li7878742357367608272lambda] :
! [T6: list_lambda_lambda,U6: list_lambda_lambda,U9: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ Resid2 @ T6 )
=> ( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( member2667661554319814297lambda @ U9 @ NN2 )
=> ( ( ( source6119337877621565212lambda @ Resid2 @ U6 )
= ( source6119337877621565212lambda @ Resid2 @ U9 ) )
=> ( ( ( target4026644918049360146lambda @ Resid2 @ U6 )
= ( target4026644918049360146lambda @ Resid2 @ U9 ) )
=> ( ( ( source6119337877621565212lambda @ Resid2 @ T6 )
= ( source6119337877621565212lambda @ Resid2 @ U6 ) )
=> ( ( member2667661554319814297lambda @ ( Resid2 @ ( Resid2 @ T6 @ U6 ) @ ( Resid2 @ T6 @ U9 ) ) @ NN2 )
& ( member2667661554319814297lambda @ ( Resid2 @ ( Resid2 @ T6 @ U9 ) @ ( Resid2 @ T6 @ U6 ) ) @ NN2 ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts_axioms_def
thf(fact_1015_length__Residx1,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] : ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ ( paths_2112734093361750239lambda @ lambda_resid @ T4 @ U ) ) @ ( size_s4208335251162658524lambda @ T4 ) ) ).
% length_Residx1
thf(fact_1016_simulation__axioms__def,axiom,
( simula826895837143331878lambda
= ( ^ [A7: lambda_lambda > lambda_lambda > lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda] :
( ! [T6: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A7 @ T6 )
=> ( ( F2 @ T6 )
= ( partia1461432607296028304lambda @ B5 ) ) )
& ! [T6: lambda_lambda,U6: lambda_lambda] :
( ( con_lambda_lambda @ A7 @ T6 @ U6 )
=> ( con_lambda_lambda @ B5 @ ( F2 @ T6 ) @ ( F2 @ U6 ) ) )
& ! [T6: lambda_lambda,U6: lambda_lambda] :
( ( con_lambda_lambda @ A7 @ T6 @ U6 )
=> ( ( F2 @ ( A7 @ T6 @ U6 ) )
= ( B5 @ ( F2 @ T6 ) @ ( F2 @ U6 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_1017_simulation__axioms__def,axiom,
( simula5056032608672110112lambda
= ( ^ [A7: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda] :
( ! [T6: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A7 @ T6 )
=> ( ( F2 @ T6 )
= ( partia1461432607296028304lambda @ B5 ) ) )
& ! [T6: list_lambda_lambda,U6: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ A7 @ T6 @ U6 )
=> ( con_lambda_lambda @ B5 @ ( F2 @ T6 ) @ ( F2 @ U6 ) ) )
& ! [T6: list_lambda_lambda,U6: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ A7 @ T6 @ U6 )
=> ( ( F2 @ ( A7 @ T6 @ U6 ) )
= ( B5 @ ( F2 @ T6 ) @ ( F2 @ U6 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_1018_simulation__axioms__def,axiom,
( simula6289889787749269804lambda
= ( ^ [A7: lambda_lambda > lambda_lambda > lambda_lambda,B5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda] :
( ! [T6: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A7 @ T6 )
=> ( ( F2 @ T6 )
= ( partia6873046715171720854lambda @ B5 ) ) )
& ! [T6: lambda_lambda,U6: lambda_lambda] :
( ( con_lambda_lambda @ A7 @ T6 @ U6 )
=> ( con_li79537494059641180lambda @ B5 @ ( F2 @ T6 ) @ ( F2 @ U6 ) ) )
& ! [T6: lambda_lambda,U6: lambda_lambda] :
( ( con_lambda_lambda @ A7 @ T6 @ U6 )
=> ( ( F2 @ ( A7 @ T6 @ U6 ) )
= ( B5 @ ( F2 @ T6 ) @ ( F2 @ U6 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_1019_simulation__axioms__def,axiom,
( simula6307967448222309926lambda
= ( ^ [A7: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda] :
( ! [T6: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A7 @ T6 )
=> ( ( F2 @ T6 )
= ( partia6873046715171720854lambda @ B5 ) ) )
& ! [T6: list_lambda_lambda,U6: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ A7 @ T6 @ U6 )
=> ( con_li79537494059641180lambda @ B5 @ ( F2 @ T6 ) @ ( F2 @ U6 ) ) )
& ! [T6: list_lambda_lambda,U6: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ A7 @ T6 @ U6 )
=> ( ( F2 @ ( A7 @ T6 @ U6 ) )
= ( B5 @ ( F2 @ T6 ) @ ( F2 @ U6 ) ) ) ) ) ) ) ).
% simulation_axioms_def
thf(fact_1020_impossible__Cons,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,X2: lambda_lambda] :
( ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ Xs ) @ ( size_s4208335251162658524lambda @ Ys ) )
=> ( Xs
!= ( cons_lambda_lambda @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1021_simulation__axioms_Ointro,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A2 @ T3 )
=> ( ( F4 @ T3 )
= ( partia1461432607296028304lambda @ B4 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ A2 @ T3 @ U4 )
=> ( con_lambda_lambda @ B4 @ ( F4 @ T3 ) @ ( F4 @ U4 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ A2 @ T3 @ U4 )
=> ( ( F4 @ ( A2 @ T3 @ U4 ) )
= ( B4 @ ( F4 @ T3 ) @ ( F4 @ U4 ) ) ) )
=> ( simula826895837143331878lambda @ A2 @ B4 @ F4 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_1022_simulation__axioms_Ointro,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A2 @ T3 )
=> ( ( F4 @ T3 )
= ( partia1461432607296028304lambda @ B4 ) ) )
=> ( ! [T3: list_lambda_lambda,U4: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ A2 @ T3 @ U4 )
=> ( con_lambda_lambda @ B4 @ ( F4 @ T3 ) @ ( F4 @ U4 ) ) )
=> ( ! [T3: list_lambda_lambda,U4: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ A2 @ T3 @ U4 )
=> ( ( F4 @ ( A2 @ T3 @ U4 ) )
= ( B4 @ ( F4 @ T3 ) @ ( F4 @ U4 ) ) ) )
=> ( simula5056032608672110112lambda @ A2 @ B4 @ F4 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_1023_simulation__axioms_Ointro,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A2 @ T3 )
=> ( ( F4 @ T3 )
= ( partia6873046715171720854lambda @ B4 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ A2 @ T3 @ U4 )
=> ( con_li79537494059641180lambda @ B4 @ ( F4 @ T3 ) @ ( F4 @ U4 ) ) )
=> ( ! [T3: lambda_lambda,U4: lambda_lambda] :
( ( con_lambda_lambda @ A2 @ T3 @ U4 )
=> ( ( F4 @ ( A2 @ T3 @ U4 ) )
= ( B4 @ ( F4 @ T3 ) @ ( F4 @ U4 ) ) ) )
=> ( simula6289889787749269804lambda @ A2 @ B4 @ F4 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_1024_simulation__axioms_Ointro,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A2 @ T3 )
=> ( ( F4 @ T3 )
= ( partia6873046715171720854lambda @ B4 ) ) )
=> ( ! [T3: list_lambda_lambda,U4: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ A2 @ T3 @ U4 )
=> ( con_li79537494059641180lambda @ B4 @ ( F4 @ T3 ) @ ( F4 @ U4 ) ) )
=> ( ! [T3: list_lambda_lambda,U4: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ A2 @ T3 @ U4 )
=> ( ( F4 @ ( A2 @ T3 @ U4 ) )
= ( B4 @ ( F4 @ T3 ) @ ( F4 @ U4 ) ) ) )
=> ( simula6307967448222309926lambda @ A2 @ B4 @ F4 ) ) ) ) ).
% simulation_axioms.intro
thf(fact_1025_Resid__cons__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,N: nat] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T4 ) @ ( size_s4208335251162658524lambda @ U2 ) ) @ N )
=> ( ! [T2: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T2 @ T4 ) @ U2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T2 @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T2 @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) )
& ! [U10: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U10 @ U2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U10 @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U10 @ nil_lambda_lambda ) ) @ U2 )
!= nil_lambda_lambda ) ) )
& ! [T2: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T2 @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T2 @ T4 ) @ U2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T2 @ nil_lambda_lambda ) @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T2 @ nil_lambda_lambda ) ) ) ) ) )
& ! [U10: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U10 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U10 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U10 @ nil_lambda_lambda ) ) @ U2 ) ) ) ) ) ) ) ).
% Resid_cons_ind
thf(fact_1026_length__Resid__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T4 ) @ ( size_s4208335251162658524lambda @ U2 ) ) @ N )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( size_s4208335251162658524lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( size_s4208335251162658524lambda @ T4 ) ) ) ) ).
% length_Resid_ind
thf(fact_1027_Con__sym__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T4 ) @ ( size_s4208335251162658524lambda @ U2 ) ) @ N )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 )
!= nil_lambda_lambda ) ) ) ).
% Con_sym_ind
thf(fact_1028_Cube__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda,N: nat] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 )
!= nil_lambda_lambda )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( size_s4208335251162658524lambda @ T4 ) @ ( size_s4208335251162658524lambda @ U2 ) ) @ ( size_s4208335251162658524lambda @ V3 ) ) @ N )
=> ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
!= nil_lambda_lambda ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) ) ) ) ) ) ) ) ).
% Cube_ind
thf(fact_1029_length__append,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( size_s4208335251162658524lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s4208335251162658524lambda @ Xs ) @ ( size_s4208335251162658524lambda @ Ys ) ) ) ).
% length_append
thf(fact_1030_the__elem__set,axiom,
! [X2: lambda_lambda] :
( ( the_el5431214018478934239lambda @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) ) )
= X2 ) ).
% the_elem_set
thf(fact_1031_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_1032__092_060Lambda_062_Otargets__char,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ T ) @ B2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ ( lambda_resid @ T @ T ) ) ) ) ) ) ).
% \<Lambda>.targets_char
thf(fact_1033__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_1034__092_060Lambda_062_Otargets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ( trg_lambda_lambda @ lambda_resid @ T )
= B2 ) ) ) ) ).
% \<Lambda>.targets_char\<^sub>W\<^sub>E
thf(fact_1035__092_060Lambda_062_Osources__char,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [A5: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= A5 ) ) ) ) ).
% \<Lambda>.sources_char
thf(fact_1036_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_1037_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
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ B )
@ T4 ) ) ) ) ).
% Arr_map_App1
thf(fact_1038_targets__char,axiom,
! [T: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [B2: 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 ) @ B2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B2 @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ) ) ) ).
% targets_char
thf(fact_1039__092_060Lambda_062_Otargets__def,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B2 )
& ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B2 ) ) ) ) ).
% \<Lambda>.targets_def
thf(fact_1040_map__ident,axiom,
( ( map_la336867333220232516lambda
@ ^ [X3: lambda_lambda] : X3 )
= ( ^ [Xs5: list_lambda_lambda] : Xs5 ) ) ).
% map_ident
thf(fact_1041_cong__map__App2,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( ( 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 @ X2 )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X2 )
@ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X2 )
@ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X5: lambda_lambda] : ( lambda_App @ X5 @ X2 )
@ T4 ) ) ) ) ) ) ).
% cong_map_App2
thf(fact_1042_sources__char_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [A7: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A7 )
& ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ A7 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% sources_char\<^sub>P
thf(fact_1043_map__App2__Resid,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ X2 )
@ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ X2 )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [T6: lambda_lambda] : ( lambda_App @ T6 @ X2 )
@ U2 ) ) ) ) ) ).
% map_App2_Resid
thf(fact_1044_targets__def,axiom,
! [T: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [B2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B2 ) ) ) ) ).
% targets_def
thf(fact_1045_targets__char_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [B5: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ B5 )
& ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ B5 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% targets_char\<^sub>P
thf(fact_1046_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
@ ^ [A7: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A7 @ ( 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 ) @ A7 ) ) ) ) ) ).
% sources_char\<^sub>\<Lambda>\<^sub>P
thf(fact_1047_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
@ ^ [B5: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B5 @ ( 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 ) @ B5 ) ) ) ) ) ).
% targets_char\<^sub>\<Lambda>\<^sub>P
thf(fact_1048_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_1049_list_Omap__ident,axiom,
! [T: list_lambda_lambda] :
( ( map_la336867333220232516lambda
@ ^ [X3: lambda_lambda] : X3
@ T )
= T ) ).
% list.map_ident
thf(fact_1050__092_060Lambda_062_Osrc__def,axiom,
! [T: lambda_lambda] :
( ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( the_lambda_lambda
@ ^ [A5: lambda_lambda] : ( member_lambda_lambda @ A5 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) )
& ( ~ ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ).
% \<Lambda>.src_def
thf(fact_1051_img__Resid__Srcs,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ord_le3064231726521830256lambda
@ ( image_3692823078739859851lambda
@ ^ [A5: lambda_lambda] : ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A5 @ nil_lambda_lambda ) @ T4 )
@ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
@ ( image_3692823078739859851lambda
@ ^ [B2: lambda_lambda] : ( cons_lambda_lambda @ B2 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ).
% img_Resid_Srcs
thf(fact_1052_Resid__single__ide_I1_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 )
=> ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
@ ( image_3692823078739859851lambda
@ ^ [B2: lambda_lambda] : ( cons_lambda_lambda @ B2 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Resid_single_ide(1)
thf(fact_1053_list_Oset__map,axiom,
! [F: lambda_lambda > list_lambda_lambda,V: list_lambda_lambda] :
( ( set_li3846194476672328965lambda @ ( map_la6434941737452557130lambda @ F @ V ) )
= ( image_3692823078739859851lambda @ F @ ( set_lambda_lambda2 @ V ) ) ) ).
% list.set_map
thf(fact_1054_list_Oset__map,axiom,
! [F: lambda_lambda > lambda_lambda,V: list_lambda_lambda] :
( ( set_lambda_lambda2 @ ( map_la336867333220232516lambda @ F @ V ) )
= ( image_2356535832909760517lambda @ F @ ( set_lambda_lambda2 @ V ) ) ) ).
% list.set_map
thf(fact_1055_image__set,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda] :
( ( image_3692823078739859851lambda @ F @ ( set_lambda_lambda2 @ Xs ) )
= ( set_li3846194476672328965lambda @ ( map_la6434941737452557130lambda @ F @ Xs ) ) ) ).
% image_set
thf(fact_1056_image__set,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( image_2356535832909760517lambda @ F @ ( set_lambda_lambda2 @ Xs ) )
= ( set_lambda_lambda2 @ ( map_la336867333220232516lambda @ F @ Xs ) ) ) ).
% image_set
thf(fact_1057_set__Cons__sing__Nil,axiom,
! [A2: set_lambda_lambda] :
( ( set_Co4812318673604925728lambda @ A2 @ ( insert8465602035467394112lambda @ nil_lambda_lambda @ bot_bo1397037886704484924lambda ) )
= ( image_3692823078739859851lambda
@ ^ [X3: lambda_lambda] : ( cons_lambda_lambda @ X3 @ nil_lambda_lambda )
@ A2 ) ) ).
% set_Cons_sing_Nil
thf(fact_1058_incl__is__simulation,axiom,
( simula1175017789036123529lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ^ [T6: lambda_lambda] : ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T6 ) @ ( cons_lambda_lambda @ T6 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% incl_is_simulation
thf(fact_1059_simulation_Opreserves__composites,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ( compos7114946393191943382lambda @ A2 @ T @ U @ V )
=> ( compos7114946393191943382lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) @ ( F4 @ V ) ) ) ) ).
% simulation.preserves_composites
thf(fact_1060_simulation_Opreserves__composites,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ( compos7114946393191943382lambda @ A2 @ T @ U @ V )
=> ( compos9183582412961930448lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) @ ( F4 @ V ) ) ) ) ).
% simulation.preserves_composites
thf(fact_1061_simulation_Opreserves__composites,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( compos9183582412961930448lambda @ A2 @ T @ U @ V )
=> ( compos7114946393191943382lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) @ ( F4 @ V ) ) ) ) ).
% simulation.preserves_composites
thf(fact_1062_simulation_Opreserves__composites,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( compos9183582412961930448lambda @ A2 @ T @ U @ V )
=> ( compos9183582412961930448lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) @ ( F4 @ V ) ) ) ) ).
% simulation.preserves_composites
thf(fact_1063_simulation_Opreserves__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( paths_366302053746739058lambda @ A2 @ T4 )
=> ( paths_2015741907369688696lambda @ B4 @ ( map_la6434941737452557130lambda @ F4 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1064_simulation_Opreserves__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( paths_366302053746739058lambda @ A2 @ T4 )
=> ( paths_366302053746739058lambda @ B4 @ ( map_la336867333220232516lambda @ F4 @ T4 ) ) ) ) ).
% simulation.preserves_paths
thf(fact_1065_simulation_Oextensional,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ~ ( arr_lambda_lambda @ A2 @ T )
=> ( ( F4 @ T )
= ( partia1461432607296028304lambda @ B4 ) ) ) ) ).
% simulation.extensional
thf(fact_1066_simulation_Oextensional,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ~ ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F4 @ T )
= ( partia1461432607296028304lambda @ B4 ) ) ) ) ).
% simulation.extensional
thf(fact_1067_simulation_Oextensional,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ~ ( arr_lambda_lambda @ A2 @ T )
=> ( ( F4 @ T )
= ( partia6873046715171720854lambda @ B4 ) ) ) ) ).
% simulation.extensional
thf(fact_1068_simulation_Oextensional,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ~ ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F4 @ T )
= ( partia6873046715171720854lambda @ B4 ) ) ) ) ).
% simulation.extensional
thf(fact_1069_simulation_Opreserves__trg,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( arr_lambda_lambda @ A2 @ T )
=> ( ( F4 @ ( trg_lambda_lambda @ A2 @ T ) )
= ( trg_lambda_lambda @ B4 @ ( F4 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1070_simulation_Opreserves__trg,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( arr_lambda_lambda @ A2 @ T )
=> ( ( F4 @ ( trg_lambda_lambda @ A2 @ T ) )
= ( trg_li817795487149935061lambda @ B4 @ ( F4 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1071_simulation_Opreserves__trg,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F4 @ ( trg_li817795487149935061lambda @ A2 @ T ) )
= ( trg_lambda_lambda @ B4 @ ( F4 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1072_simulation_Opreserves__trg,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ( arr_li3691443969012319837lambda @ A2 @ T )
=> ( ( F4 @ ( trg_li817795487149935061lambda @ A2 @ T ) )
= ( trg_li817795487149935061lambda @ B4 @ ( F4 @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1073_simulation_Opreserves__joins,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ( join_o6956785509312998601lambda @ A2 @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) @ ( F4 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1074_simulation_Opreserves__joins,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ( join_o6956785509312998601lambda @ A2 @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) @ ( F4 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1075_simulation_Opreserves__joins,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( join_o1386658102433174083lambda @ A2 @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) @ ( F4 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1076_simulation_Opreserves__joins,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( join_o1386658102433174083lambda @ A2 @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) @ ( F4 @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_1077_simulation_Opreserves__reflects__arr,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( arr_lambda_lambda @ B4 @ ( F4 @ T ) )
= ( arr_lambda_lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1078_simulation_Opreserves__reflects__arr,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ( arr_lambda_lambda @ B4 @ ( F4 @ T ) )
= ( arr_li3691443969012319837lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1079_simulation_Opreserves__reflects__arr,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( arr_li3691443969012319837lambda @ B4 @ ( F4 @ T ) )
= ( arr_lambda_lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1080_simulation_Opreserves__reflects__arr,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ( arr_li3691443969012319837lambda @ B4 @ ( F4 @ T ) )
= ( arr_li3691443969012319837lambda @ A2 @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_1081_simulation_Opreserves__resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( ( F4 @ ( A2 @ T @ U ) )
= ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_1082_simulation_Opreserves__con,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( con_lambda_lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1083_simulation_Opreserves__con,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( con_lambda_lambda @ A2 @ T @ U )
=> ( con_li79537494059641180lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1084_simulation_Opreserves__con,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( con_lambda_lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1085_simulation_Opreserves__con,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ( con_li79537494059641180lambda @ A2 @ T @ U )
=> ( con_li79537494059641180lambda @ B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_1086_simulation_Opreserves__ide,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ A )
=> ( ide_li6773656263178250516lambda @ B4 @ ( F4 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1087_simulation_Opreserves__ide,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,A: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ A )
=> ( ide_lambda_lambda @ B4 @ ( F4 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1088_simulation_Opreserves__ide,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,A: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( ide_lambda_lambda @ A2 @ A )
=> ( ide_li6773656263178250516lambda @ B4 @ ( F4 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1089_simulation_Opreserves__ide,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,A: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( ide_lambda_lambda @ A2 @ A )
=> ( ide_lambda_lambda @ B4 @ ( F4 @ A ) ) ) ) ).
% simulation.preserves_ide
thf(fact_1090_simulation_Opreserves__cong,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) )
& ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F4 @ U ) @ ( F4 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1091_simulation_Opreserves__cong,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B4 @ ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) )
& ( ide_lambda_lambda @ B4 @ ( B4 @ ( F4 @ U ) @ ( F4 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1092_simulation_Opreserves__cong,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_lambda_lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) )
& ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F4 @ U ) @ ( F4 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1093_simulation_Opreserves__cong,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
& ( ide_lambda_lambda @ A2 @ ( A2 @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B4 @ ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) )
& ( ide_lambda_lambda @ B4 @ ( B4 @ ( F4 @ U ) @ ( F4 @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_1094_simulation_Opreserves__prfx,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1095_simulation_Opreserves__prfx,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ( ide_li6773656263178250516lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_lambda_lambda @ B4 @ ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1096_simulation_Opreserves__prfx,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_li6773656263178250516lambda @ B4 @ ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1097_simulation_Opreserves__prfx,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( ide_lambda_lambda @ A2 @ ( A2 @ T @ U ) )
=> ( ide_lambda_lambda @ B4 @ ( B4 @ ( F4 @ T ) @ ( F4 @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_1098_simulation_Oaxioms_I3_J,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( simula6289889787749269804lambda @ A2 @ B4 @ F4 ) ) ).
% simulation.axioms(3)
thf(fact_1099_simulation_Olifts__to__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( simula3219528783544830217lambda @ ( paths_8607357831768207192lambda @ A2 ) @ ( paths_3312059784463760222lambda @ B4 )
@ ^ [T7: list_lambda_lambda] : ( if_lis4115203008988617916lambda @ ( paths_366302053746739058lambda @ A2 @ T7 ) @ ( map_la6434941737452557130lambda @ F4 @ T7 ) @ nil_li8379291187143545974lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1100_simulation_Olifts__to__paths,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( simula3069223088535412739lambda @ ( paths_8607357831768207192lambda @ A2 ) @ ( paths_8607357831768207192lambda @ B4 )
@ ^ [T7: list_lambda_lambda] : ( if_lis7195930416592267574lambda @ ( paths_366302053746739058lambda @ A2 @ T7 ) @ ( map_la336867333220232516lambda @ F4 @ T7 ) @ nil_lambda_lambda ) ) ) ).
% simulation.lifts_to_paths
thf(fact_1101_simulation_Opreserves__sources,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F4 @ ( source6119337877621565212lambda @ A2 @ T ) ) @ ( source6119337877621565212lambda @ B4 @ ( F4 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1102_simulation_Opreserves__sources,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F4 @ ( source6119337877621565212lambda @ A2 @ T ) ) @ ( source5627170679743844886lambda @ B4 @ ( F4 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1103_simulation_Opreserves__sources,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F4 @ ( source5627170679743844886lambda @ A2 @ T ) ) @ ( source6119337877621565212lambda @ B4 @ ( F4 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1104_simulation_Opreserves__sources,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F4 @ ( source5627170679743844886lambda @ A2 @ T ) ) @ ( source5627170679743844886lambda @ B4 @ ( F4 @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_1105_simulation_Opreserves__targets,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A2 @ B4 @ F4 )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F4 @ ( target4026644918049360146lambda @ A2 @ T ) ) @ ( target4026644918049360146lambda @ B4 @ ( F4 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1106_simulation_Opreserves__targets,axiom,
! [A2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A2 @ B4 @ F4 )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F4 @ ( target4026644918049360146lambda @ A2 @ T ) ) @ ( target5490931411331770124lambda @ B4 @ ( F4 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1107_simulation_Opreserves__targets,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F4 @ ( target5490931411331770124lambda @ A2 @ T ) ) @ ( target4026644918049360146lambda @ B4 @ ( F4 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1108_simulation_Opreserves__targets,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F4 @ ( target5490931411331770124lambda @ A2 @ T ) ) @ ( target5490931411331770124lambda @ B4 @ ( F4 @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_1109_simulation_Omap__Resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_la6434941737452557130lambda @ F4 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ U2 ) )
= ( paths_3312059784463760222lambda @ B4 @ ( map_la6434941737452557130lambda @ F4 @ T4 ) @ ( map_la6434941737452557130lambda @ F4 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1110_simulation_Omap__Resid,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda @ F4 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ B4 @ ( map_la336867333220232516lambda @ F4 @ T4 ) @ ( map_la336867333220232516lambda @ F4 @ U2 ) ) ) ) ) ).
% simulation.map_Resid
thf(fact_1111_simulation_Omap__Resid__single,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A2 @ B4 @ F4 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
=> ( ( map_la6434941737452557130lambda @ F4 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_3312059784463760222lambda @ B4 @ ( map_la6434941737452557130lambda @ F4 @ T4 ) @ ( cons_l4558873032622161958lambda @ ( F4 @ U ) @ nil_li8379291187143545974lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1112_simulation_Omap__Resid__single,axiom,
! [A2: lambda_lambda > lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A2 @ B4 @ F4 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ A2 ) @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
=> ( ( map_la336867333220232516lambda @ F4 @ ( paths_8607357831768207192lambda @ A2 @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_8607357831768207192lambda @ B4 @ ( map_la336867333220232516lambda @ F4 @ T4 ) @ ( cons_lambda_lambda @ ( F4 @ U ) @ nil_lambda_lambda ) ) ) ) ) ).
% simulation.map_Resid_single
thf(fact_1113_incl__is__injective,axiom,
( inj_on3897920697455699703lambda
@ ^ [T6: lambda_lambda] : ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T6 ) @ ( cons_lambda_lambda @ T6 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
@ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ lambda_resid ) ) ) ).
% incl_is_injective
thf(fact_1114__092_060Lambda_062_Onull__def,axiom,
( ( partia1461432607296028304lambda @ lambda_resid )
= ( the_lambda_lambda
@ ^ [N2: lambda_lambda] :
! [T6: lambda_lambda] :
( ( ( lambda_resid @ N2 @ T6 )
= N2 )
& ( ( lambda_resid @ T6 @ N2 )
= N2 ) ) ) ) ).
% \<Lambda>.null_def
thf(fact_1115_partial__magma_Onull__def,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ( ( partia1461432607296028304lambda @ OP2 )
= ( the_lambda_lambda
@ ^ [N2: lambda_lambda] :
! [T6: lambda_lambda] :
( ( ( OP2 @ N2 @ T6 )
= N2 )
& ( ( OP2 @ T6 @ N2 )
= N2 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_1116_partial__magma_Onull__def,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ( ( partia6873046715171720854lambda @ OP2 )
= ( the_li7637651520389017975lambda
@ ^ [N2: list_lambda_lambda] :
! [T6: list_lambda_lambda] :
( ( ( OP2 @ N2 @ T6 )
= N2 )
& ( ( OP2 @ T6 @ N2 )
= N2 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_1117_listset_Osimps_I1_J,axiom,
( ( listse1500948262344051919lambda @ nil_se6914803783601386320lambda )
= ( insert8465602035467394112lambda @ nil_lambda_lambda @ bot_bo1397037886704484924lambda ) ) ).
% listset.simps(1)
thf(fact_1118_paths__in__rts_Oimg__Resid__Srcs,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ord_le3064231726521830256lambda
@ ( image_3692823078739859851lambda
@ ^ [A5: lambda_lambda] : ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ A5 @ nil_lambda_lambda ) @ T4 )
@ ( paths_2897053694880971696lambda @ Resid @ T4 ) )
@ ( image_3692823078739859851lambda
@ ^ [B2: lambda_lambda] : ( cons_lambda_lambda @ B2 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ Resid @ T4 ) ) ) ) ) ).
% paths_in_rts.img_Resid_Srcs
thf(fact_1119_paths__in__rts__axioms,axiom,
paths_6861597469809217359lambda @ lambda_resid ).
% paths_in_rts_axioms
thf(fact_1120_null__def,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= ( the_li7637651520389017975lambda
@ ^ [N2: list_lambda_lambda] :
! [T6: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ N2 @ T6 )
= N2 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ N2 )
= N2 ) ) ) ) ).
% null_def
thf(fact_1121_paths__in__rts_Ois__partial__magma,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ Resid ) ) ) ).
% paths_in_rts.is_partial_magma
thf(fact_1122_paths__in__rts_Ois__residuation,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( residu7698045915734027664lambda @ ( paths_8607357831768207192lambda @ Resid ) ) ) ).
% paths_in_rts.is_residuation
thf(fact_1123_paths__in__rts_OArr__imp__Arr__tl,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( ( tl_lambda_lambda @ T4 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ Resid @ ( tl_lambda_lambda @ T4 ) ) ) ) ) ).
% paths_in_rts.Arr_imp_Arr_tl
thf(fact_1124_paths__in__rts_OResid1x_Osimps_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_4460507232930246679lambda @ Resid @ T @ nil_li8379291187143545974lambda )
= ( partia6873046715171720854lambda @ Resid ) ) ) ).
% paths_in_rts.Resid1x.simps(1)
thf(fact_1125_paths__in__rts_OResid1x_Osimps_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_3319247187298047121lambda @ Resid @ T @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ Resid ) ) ) ).
% paths_in_rts.Resid1x.simps(1)
thf(fact_1126_paths__in__rts_Ocong__respects__seq_092_060_094sub_062P,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,T8: list_lambda_lambda,U5: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ T4 @ T8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ T8 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ U5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ U5 @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T8 @ U5 ) ) ) ) ) ).
% paths_in_rts.cong_respects_seq\<^sub>P
thf(fact_1127_paths__in__rts_OarrI_092_060_094sub_062P,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 ) ) ) ).
% paths_in_rts.arrI\<^sub>P
thf(fact_1128_paths__in__rts_Oarr__char,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 )
= ( paths_366302053746739058lambda @ Resid @ T4 ) ) ) ).
% paths_in_rts.arr_char
thf(fact_1129_paths__in__rts_OArr__imp__arr__hd,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( arr_li3691443969012319837lambda @ Resid @ ( hd_lis2183642875561147995lambda @ T4 ) ) ) ) ).
% paths_in_rts.Arr_imp_arr_hd
thf(fact_1130_paths__in__rts_OArr__imp__arr__hd,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( arr_lambda_lambda @ Resid @ ( hd_lambda_lambda @ T4 ) ) ) ) ).
% paths_in_rts.Arr_imp_arr_hd
thf(fact_1131_paths__in__rts_Onull__char,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ Resid ) )
= nil_lambda_lambda ) ) ).
% paths_in_rts.null_char
thf(fact_1132_paths__in__rts_OIde__imp__Ide__hd,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_5097954201535619375lambda @ Resid @ T4 )
=> ( ide_li6773656263178250516lambda @ Resid @ ( hd_lis2183642875561147995lambda @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_hd
thf(fact_1133_paths__in__rts_OIde__imp__Ide__hd,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ T4 )
=> ( ide_lambda_lambda @ Resid @ ( hd_lambda_lambda @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_hd
thf(fact_1134_paths__in__rts_OIde__imp__Ide__tl,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ T4 )
=> ( ( ( tl_lambda_lambda @ T4 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ Resid @ ( tl_lambda_lambda @ T4 ) ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_tl
thf(fact_1135_paths__in__rts_OResid1x__ide,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A )
=> ( ( ( paths_4460507232930246679lambda @ Resid @ A @ T4 )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( ide_li6773656263178250516lambda @ Resid @ ( paths_4460507232930246679lambda @ Resid @ A @ T4 ) ) ) ) ) ).
% paths_in_rts.Resid1x_ide
thf(fact_1136_paths__in__rts_OResid1x__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( ( paths_3319247187298047121lambda @ Resid @ A @ T4 )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ide_lambda_lambda @ Resid @ ( paths_3319247187298047121lambda @ Resid @ A @ T4 ) ) ) ) ) ).
% paths_in_rts.Resid1x_ide
thf(fact_1137_paths__in__rts_OArr__imp__arr__last,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( arr_li3691443969012319837lambda @ Resid @ ( last_l1874196399739661071lambda @ T4 ) ) ) ) ).
% paths_in_rts.Arr_imp_arr_last
thf(fact_1138_paths__in__rts_OArr__imp__arr__last,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( arr_lambda_lambda @ Resid @ ( last_lambda_lambda @ T4 ) ) ) ) ).
% paths_in_rts.Arr_imp_arr_last
thf(fact_1139_paths__in__rts_OIde__imp__Ide__last,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_5097954201535619375lambda @ Resid @ T4 )
=> ( ide_li6773656263178250516lambda @ Resid @ ( last_l1874196399739661071lambda @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_last
thf(fact_1140_paths__in__rts_OIde__imp__Ide__last,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ T4 )
=> ( ide_lambda_lambda @ Resid @ ( last_lambda_lambda @ T4 ) ) ) ) ).
% paths_in_rts.Ide_imp_Ide_last
thf(fact_1141_paths__in__rts_OCon__imp__eq__Srcs,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ Resid @ T4 )
= ( paths_2897053694880971696lambda @ Resid @ U2 ) ) ) ) ).
% paths_in_rts.Con_imp_eq_Srcs
thf(fact_1142_paths__in__rts_OResid__Ide_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ A2 @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ A2 )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Ide(1)
thf(fact_1143_paths__in__rts_OResid__Arr__Ide__ind,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ A2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ A2 )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_Ide_ind
thf(fact_1144_paths__in__rts_OResid__Ide__Arr__ind,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ A2 @ T4 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ A2 @ T4 ) ) ) ) ) ).
% paths_in_rts.Resid_Ide_Arr_ind
thf(fact_1145_paths__in__rts_OTrgs__Resid__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ U2 @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_Resid_sym
thf(fact_1146_paths__in__rts_OResid1x_Osimps_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_3319247187298047121lambda @ Resid @ T @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( Resid @ T @ U ) ) ) ).
% paths_in_rts.Resid1x.simps(2)
thf(fact_1147_paths__in__rts_OSrcs__append,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_2897053694880971696lambda @ Resid @ T4 ) ) ) ) ).
% paths_in_rts.Srcs_append
thf(fact_1148_paths__in__rts_Ocon__char,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
= ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda ) ) ) ).
% paths_in_rts.con_char
thf(fact_1149_paths__in__rts_OconI_092_060_094sub_062P,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 ) ) ) ).
% paths_in_rts.conI\<^sub>P
thf(fact_1150_paths__in__rts_OSrcs__are__ide,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ord_le3064231726521830256lambda @ ( paths_3172353640142224310lambda @ Resid @ T4 ) @ ( collec8830145823237851355lambda @ ( ide_li6773656263178250516lambda @ Resid ) ) ) ) ).
% paths_in_rts.Srcs_are_ide
thf(fact_1151_paths__in__rts_OSrcs__are__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ord_le1926335974945931754lambda @ ( paths_2897053694880971696lambda @ Resid @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ Resid ) ) ) ) ).
% paths_in_rts.Srcs_are_ide
thf(fact_1152_paths__in__rts_OSrcs_Osimps_I3_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda,Va2: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_3172353640142224310lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ ( cons_l4558873032622161958lambda @ V @ Va2 ) ) )
= ( source6119337877621565212lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Srcs.simps(3)
thf(fact_1153_paths__in__rts_OSrcs_Osimps_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_2897053694880971696lambda @ Resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( source5627170679743844886lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Srcs.simps(3)
thf(fact_1154_paths__in__rts_OTrgs__append,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ Resid @ U2 ) ) ) ) ).
% paths_in_rts.Trgs_append
thf(fact_1155_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda,T4: list_l5168835686021156086lambda,A3: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( member2667661554319814297lambda @ A @ ( paths_3172353640142224310lambda @ Resid @ T4 ) )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A3 )
=> ( ( con_li79537494059641180lambda @ Resid @ A @ A3 )
=> ( member2667661554319814297lambda @ A3 @ ( paths_3172353640142224310lambda @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_1156_paths__in__rts_OSrcs__con__closed,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T4: list_lambda_lambda,A3: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ Resid @ T4 ) )
=> ( ( ide_lambda_lambda @ Resid @ A3 )
=> ( ( con_lambda_lambda @ Resid @ A @ A3 )
=> ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Srcs_con_closed
thf(fact_1157_paths__in__rts_Olength__Residx1,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ord_less_eq_nat @ ( size_s4208335251162658524lambda @ ( paths_2112734093361750239lambda @ Resid @ T4 @ U ) ) @ ( size_s4208335251162658524lambda @ T4 ) ) ) ).
% paths_in_rts.length_Residx1
thf(fact_1158_paths__in__rts_OTrgs__are__ide,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ord_le3064231726521830256lambda @ ( paths_8725155050933150651lambda @ Resid @ T4 ) @ ( collec8830145823237851355lambda @ ( ide_li6773656263178250516lambda @ Resid ) ) ) ) ).
% paths_in_rts.Trgs_are_ide
thf(fact_1159_paths__in__rts_OTrgs__are__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ Resid @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ Resid ) ) ) ) ).
% paths_in_rts.Trgs_are_ide
thf(fact_1160_paths__in__rts_Olength__Resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( size_s4208335251162658524lambda @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) )
= ( size_s4208335251162658524lambda @ T4 ) ) ) ) ).
% paths_in_rts.length_Resid
thf(fact_1161_paths__in__rts_OCon__append_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda ) ) ) ) ) ) ) ).
% paths_in_rts.Con_append(2)
thf(fact_1162_paths__in__rts_OCon__append_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ Resid @ U2 @ ( paths_8607357831768207192lambda @ Resid @ V3 @ T4 ) )
!= nil_lambda_lambda ) ) ) ) ) ) ) ).
% paths_in_rts.Con_append(1)
thf(fact_1163_paths__in__rts_OCon__appendI_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) ) ) ).
% paths_in_rts.Con_appendI(2)
thf(fact_1164_paths__in__rts_OCon__appendI_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,V3: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ U2 @ ( paths_8607357831768207192lambda @ Resid @ V3 @ T4 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda ) ) ) ) ).
% paths_in_rts.Con_appendI(1)
thf(fact_1165_paths__in__rts_OResid__append_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
= ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) @ V3 ) ) ) ) ) ) ).
% paths_in_rts.Resid_append(2)
thf(fact_1166_paths__in__rts_OResid__append_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( paths_8607357831768207192lambda @ Resid @ V3 @ T4 ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_append(1)
thf(fact_1167_paths__in__rts_OResid__append2,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda,W: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( W != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V3 @ W ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V3 @ W ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 ) @ W ) @ ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( paths_8607357831768207192lambda @ Resid @ V3 @ T4 ) ) @ ( paths_8607357831768207192lambda @ Resid @ W @ ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 ) ) ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_append2
thf(fact_1168_paths__in__rts_OResid__append__ind,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ Resid @ V3 @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( paths_8607357831768207192lambda @ Resid @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 ) @ U2 )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ Resid @ V3 @ U2 ) @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( paths_8607357831768207192lambda @ Resid @ U2 @ V3 ) ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
= ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 ) @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_append_ind
thf(fact_1169_paths__in__confluent__rts__def,axiom,
( paths_3043240146134862789lambda
= ( ^ [Resid2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid2 )
& ( conflu4036547884742428384lambda @ Resid2 ) ) ) ) ).
% paths_in_confluent_rts_def
thf(fact_1170_paths__in__confluent__rts__def,axiom,
( paths_9040249923598999359lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid2 )
& ( conflu4927472611300549594lambda @ Resid2 ) ) ) ) ).
% paths_in_confluent_rts_def
thf(fact_1171_paths__in__confluent__rts_Ointro,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( conflu4036547884742428384lambda @ Resid )
=> ( paths_3043240146134862789lambda @ Resid ) ) ) ).
% paths_in_confluent_rts.intro
thf(fact_1172_paths__in__confluent__rts_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( conflu4927472611300549594lambda @ Resid )
=> ( paths_9040249923598999359lambda @ Resid ) ) ) ).
% paths_in_confluent_rts.intro
thf(fact_1173_paths__in__rts_Ois__rts__with__composites,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ Resid ) ) ) ).
% paths_in_rts.is_rts_with_composites
thf(fact_1174_paths__in__rts_OTrgs_Ocases,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,X2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ).
% paths_in_rts.Trgs.cases
thf(fact_1175_paths__in__rts_OCube_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ Resid @ V3 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ Resid @ V3 @ U2 ) )
= ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ V3 ) ) ) ) ) ).
% paths_in_rts.Cube(2)
thf(fact_1176_paths__in__rts_OCube_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ Resid @ V3 @ U2 ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) ) ).
% paths_in_rts.Cube(1)
thf(fact_1177_paths__in__rts_OResid_Osimps_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,Uu: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_8607357831768207192lambda @ Resid @ nil_lambda_lambda @ Uu )
= nil_lambda_lambda ) ) ).
% paths_in_rts.Resid.simps(1)
thf(fact_1178_paths__in__rts_OCon__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ Resid @ U2 @ T4 )
!= nil_lambda_lambda ) ) ) ).
% paths_in_rts.Con_sym
thf(fact_1179_paths__in__rts_OArr_Osimps_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ~ ( paths_366302053746739058lambda @ Resid @ nil_lambda_lambda ) ) ).
% paths_in_rts.Arr.simps(1)
thf(fact_1180_paths__in__rts_OResidx1_Osimps_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_2112734093361750239lambda @ Resid @ nil_lambda_lambda @ U )
= nil_lambda_lambda ) ) ).
% paths_in_rts.Residx1.simps(1)
thf(fact_1181_paths__in__rts_OIde_Osimps_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ~ ( paths_7717490669901197737lambda @ Resid @ nil_lambda_lambda ) ) ).
% paths_in_rts.Ide.simps(1)
thf(fact_1182_paths__in__rts_OTrgs_Osimps_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7594662086398683701lambda @ Resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( paths_7594662086398683701lambda @ Resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ) ).
% paths_in_rts.Trgs.simps(3)
thf(fact_1183_paths__in__confluent__rts_Oaxioms_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( paths_6861597469809217359lambda @ Resid ) ) ).
% paths_in_confluent_rts.axioms(1)
thf(fact_1184_paths__in__rts_OResid1x__null,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_4460507232930246679lambda @ Resid @ ( partia6873046715171720854lambda @ Resid ) @ T4 )
= ( partia6873046715171720854lambda @ Resid ) ) ) ).
% paths_in_rts.Resid1x_null
thf(fact_1185_paths__in__rts_OResid1x__null,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_3319247187298047121lambda @ Resid @ ( partia1461432607296028304lambda @ Resid ) @ T4 )
= ( partia1461432607296028304lambda @ Resid ) ) ) ).
% paths_in_rts.Resid1x_null
thf(fact_1186_paths__in__rts_OIde__implies__Arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ T4 )
=> ( paths_366302053746739058lambda @ Resid @ T4 ) ) ) ).
% paths_in_rts.Ide_implies_Arr
thf(fact_1187_paths__in__rts_OTrgs__are__con,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda,T4: list_l5168835686021156086lambda,B3: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( member2667661554319814297lambda @ B @ ( paths_8725155050933150651lambda @ Resid @ T4 ) )
=> ( ( member2667661554319814297lambda @ B3 @ ( paths_8725155050933150651lambda @ Resid @ T4 ) )
=> ( con_li79537494059641180lambda @ Resid @ B @ B3 ) ) ) ) ).
% paths_in_rts.Trgs_are_con
thf(fact_1188_paths__in__rts_OTrgs__are__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda,T4: list_lambda_lambda,B3: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ Resid @ T4 ) )
=> ( ( member_lambda_lambda @ B3 @ ( paths_7594662086398683701lambda @ Resid @ T4 ) )
=> ( con_lambda_lambda @ Resid @ B @ B3 ) ) ) ) ).
% paths_in_rts.Trgs_are_con
thf(fact_1189_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda,T4: list_l5168835686021156086lambda,A3: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( member2667661554319814297lambda @ A @ ( paths_3172353640142224310lambda @ Resid @ T4 ) )
=> ( ( member2667661554319814297lambda @ A3 @ ( paths_3172353640142224310lambda @ Resid @ T4 ) )
=> ( con_li79537494059641180lambda @ Resid @ A @ A3 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_1190_paths__in__rts_OSrcs__are__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T4: list_lambda_lambda,A3: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( member_lambda_lambda @ A @ ( paths_2897053694880971696lambda @ Resid @ T4 ) )
=> ( ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ Resid @ T4 ) )
=> ( con_lambda_lambda @ Resid @ A @ A3 ) ) ) ) ).
% paths_in_rts.Srcs_are_con
thf(fact_1191_paths__in__rts_OResid1x_Osimps_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_3319247187298047121lambda @ Resid @ T @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( paths_3319247187298047121lambda @ Resid @ ( Resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ) ).
% paths_in_rts.Resid1x.simps(3)
thf(fact_1192_paths__in__rts_OCon__cons_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_rts.Con_cons(2)
thf(fact_1193_paths__in__rts_OCon__cons_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_rts.Con_cons(1)
thf(fact_1194_paths__in__rts_OCon__consI_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_rts.Con_consI(2)
thf(fact_1195_paths__in__rts_OCon__consI_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_rts.Con_consI(1)
thf(fact_1196_paths__in__rts_OResid__rec_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
= ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) @ U2 ) ) ) ) ) ).
% paths_in_rts.Resid_rec(3)
thf(fact_1197_paths__in__rts_OResid__rec_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( Resid @ T @ U ) @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ ( Resid @ U @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_rec(2)
thf(fact_1198_paths__in__rts_OResid__rec_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) ) ) ) ).
% paths_in_rts.Resid_rec(1)
thf(fact_1199_paths__in__rts_OResid__cons_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U2: list_lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
= ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 ) ) ) ) ) ).
% paths_in_rts.Resid_cons(2)
thf(fact_1200_paths__in__rts_OResid_Osimps_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ V @ Va2 ) @ nil_lambda_lambda )
= nil_lambda_lambda ) ) ).
% paths_in_rts.Resid.simps(2)
thf(fact_1201_paths__in__rts_OCon__initial__left,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ).
% paths_in_rts.Con_initial_left
thf(fact_1202_paths__in__rts_OCon__initial__right,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ).
% paths_in_rts.Con_initial_right
thf(fact_1203_paths__in__rts_OCon__implies__Arr_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ Resid @ U2 ) ) ) ).
% paths_in_rts.Con_implies_Arr(2)
thf(fact_1204_paths__in__rts_OCon__implies__Arr_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ Resid @ T4 ) ) ) ).
% paths_in_rts.Con_implies_Arr(1)
thf(fact_1205_paths__in__rts_OCon__Arr__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ T4 )
!= nil_lambda_lambda ) ) ) ).
% paths_in_rts.Con_Arr_self
thf(fact_1206_paths__in__rts_OArr__iff__Con__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
= ( ( paths_8607357831768207192lambda @ Resid @ T4 @ T4 )
!= nil_lambda_lambda ) ) ) ).
% paths_in_rts.Arr_iff_Con_self
thf(fact_1207_paths__in__rts_OCon__imp__Arr__Resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) ) ) ) ).
% paths_in_rts.Con_imp_Arr_Resid
thf(fact_1208_paths__in__rts_OArr__has__Trg,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_8725155050933150651lambda @ Resid @ T4 )
!= bot_bo1397037886704484924lambda ) ) ) ).
% paths_in_rts.Arr_has_Trg
thf(fact_1209_paths__in__rts_OArr__has__Trg,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_7594662086398683701lambda @ Resid @ T4 )
!= bot_bo1271416176687173302lambda ) ) ) ).
% paths_in_rts.Arr_has_Trg
thf(fact_1210_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda,T4: list_l5168835686021156086lambda,B3: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( member2667661554319814297lambda @ B @ ( paths_8725155050933150651lambda @ Resid @ T4 ) )
=> ( ( ide_li6773656263178250516lambda @ Resid @ B3 )
=> ( ( con_li79537494059641180lambda @ Resid @ B @ B3 )
=> ( member2667661554319814297lambda @ B3 @ ( paths_8725155050933150651lambda @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_1211_paths__in__rts_OTrgs__con__closed,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda,T4: list_lambda_lambda,B3: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ Resid @ T4 ) )
=> ( ( ide_lambda_lambda @ Resid @ B3 )
=> ( ( con_lambda_lambda @ Resid @ B @ B3 )
=> ( member_lambda_lambda @ B3 @ ( paths_7594662086398683701lambda @ Resid @ T4 ) ) ) ) ) ) ).
% paths_in_rts.Trgs_con_closed
thf(fact_1212_paths__in__rts_Oide__char,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 )
= ( paths_7717490669901197737lambda @ Resid @ T4 ) ) ) ).
% paths_in_rts.ide_char
thf(fact_1213_paths__in__rts_OResid__Arr__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( paths_7717490669901197737lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ T4 ) ) ) ) ).
% paths_in_rts.Resid_Arr_self
thf(fact_1214_paths__in__rts_OArr__has__Src,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_3172353640142224310lambda @ Resid @ T4 )
!= bot_bo1397037886704484924lambda ) ) ) ).
% paths_in_rts.Arr_has_Src
thf(fact_1215_paths__in__rts_OArr__has__Src,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ Resid @ T4 )
!= bot_bo1271416176687173302lambda ) ) ) ).
% paths_in_rts.Arr_has_Src
thf(fact_1216_paths__in__rts_OTrgs__eqI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,T8: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( inf_in3354698378728726462lambda @ ( paths_8725155050933150651lambda @ Resid @ T4 ) @ ( paths_8725155050933150651lambda @ Resid @ T8 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( paths_8725155050933150651lambda @ Resid @ T4 )
= ( paths_8725155050933150651lambda @ Resid @ T8 ) ) ) ) ).
% paths_in_rts.Trgs_eqI
thf(fact_1217_paths__in__rts_OTrgs__eqI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,T8: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ Resid @ T4 ) @ ( paths_7594662086398683701lambda @ Resid @ T8 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_7594662086398683701lambda @ Resid @ T4 )
= ( paths_7594662086398683701lambda @ Resid @ T8 ) ) ) ) ).
% paths_in_rts.Trgs_eqI
thf(fact_1218_paths__in__rts_OSrcs__eqI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,T8: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( inf_in3354698378728726462lambda @ ( paths_3172353640142224310lambda @ Resid @ T4 ) @ ( paths_3172353640142224310lambda @ Resid @ T8 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( paths_3172353640142224310lambda @ Resid @ T4 )
= ( paths_3172353640142224310lambda @ Resid @ T8 ) ) ) ) ).
% paths_in_rts.Srcs_eqI
thf(fact_1219_paths__in__rts_OSrcs__eqI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,T8: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( inf_in1766959559283430712lambda @ ( paths_2897053694880971696lambda @ Resid @ T4 ) @ ( paths_2897053694880971696lambda @ Resid @ T8 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_2897053694880971696lambda @ Resid @ T4 )
= ( paths_2897053694880971696lambda @ Resid @ T8 ) ) ) ) ).
% paths_in_rts.Srcs_eqI
thf(fact_1220_paths__in__rts_OTrgs_Osimps_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_8725155050933150651lambda @ Resid @ nil_li8379291187143545974lambda )
= bot_bo1397037886704484924lambda ) ) ).
% paths_in_rts.Trgs.simps(1)
thf(fact_1221_paths__in__rts_OTrgs_Osimps_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7594662086398683701lambda @ Resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ) ).
% paths_in_rts.Trgs.simps(1)
thf(fact_1222_paths__in__rts_OSrcs_Osimps_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_3172353640142224310lambda @ Resid @ nil_li8379291187143545974lambda )
= bot_bo1397037886704484924lambda ) ) ).
% paths_in_rts.Srcs.simps(1)
thf(fact_1223_paths__in__rts_OSrcs_Osimps_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_2897053694880971696lambda @ Resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ) ).
% paths_in_rts.Srcs.simps(1)
thf(fact_1224_paths__in__rts_OResid__single__ide_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A: list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A )
=> ( ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ A @ nil_li8379291187143545974lambda ) @ T4 )
!= nil_li8379291187143545974lambda )
=> ( ( paths_3312059784463760222lambda @ Resid @ T4 @ ( cons_l4558873032622161958lambda @ A @ nil_li8379291187143545974lambda ) )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_single_ide(2)
thf(fact_1225_paths__in__rts_OResid__single__ide_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A: lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ A @ nil_lambda_lambda ) )
= T4 ) ) ) ) ).
% paths_in_rts.Resid_single_ide(2)
thf(fact_1226_paths__in__rts_OResid__cons_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 ) @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons(1)
thf(fact_1227_paths__in__rts_OResid__rec_I4_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) @ U2 ) @ ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ ( Resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_rec(4)
thf(fact_1228_paths__in__rts_OResid_Osimps_I3_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) )
= ( cons_l4558873032622161958lambda @ ( Resid @ T @ U ) @ nil_li8379291187143545974lambda ) ) )
& ( ~ ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) )
= nil_li8379291187143545974lambda ) ) ) ) ).
% paths_in_rts.Resid.simps(3)
thf(fact_1229_paths__in__rts_OResid_Osimps_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= nil_lambda_lambda ) ) ) ) ).
% paths_in_rts.Resid.simps(3)
thf(fact_1230_paths__in__rts_OCon__rec_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) )
!= nil_li8379291187143545974lambda )
= ( con_li79537494059641180lambda @ Resid @ T @ U ) ) ) ).
% paths_in_rts.Con_rec(1)
thf(fact_1231_paths__in__rts_OCon__rec_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
= ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% paths_in_rts.Con_rec(1)
thf(fact_1232_paths__in__rts_OCon__rec_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( T4 != nil_li8379291187143545974lambda )
=> ( ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ T4 ) @ ( cons_l4558873032622161958lambda @ U @ nil_li8379291187143545974lambda ) )
!= nil_li8379291187143545974lambda )
= ( ( con_li79537494059641180lambda @ Resid @ T @ U )
& ( ( paths_3312059784463760222lambda @ Resid @ T4 @ ( cons_l4558873032622161958lambda @ ( Resid @ U @ T ) @ nil_li8379291187143545974lambda ) )
!= nil_li8379291187143545974lambda ) ) ) ) ) ).
% paths_in_rts.Con_rec(2)
thf(fact_1233_paths__in__rts_OCon__rec_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ Resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ ( Resid @ U @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% paths_in_rts.Con_rec(2)
thf(fact_1234_paths__in__rts_OCon__rec_I3_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,U2: list_l5168835686021156086lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( U2 != nil_li8379291187143545974lambda )
=> ( ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ ( cons_l4558873032622161958lambda @ U @ U2 ) )
!= nil_li8379291187143545974lambda )
= ( ( con_li79537494059641180lambda @ Resid @ T @ U )
& ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ ( Resid @ T @ U ) @ nil_li8379291187143545974lambda ) @ U2 )
!= nil_li8379291187143545974lambda ) ) ) ) ) ).
% paths_in_rts.Con_rec(3)
thf(fact_1235_paths__in__rts_OCon__rec_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ Resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% paths_in_rts.Con_rec(3)
thf(fact_1236_paths__in__rts_OCon__rec_I4_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( T4 != nil_li8379291187143545974lambda )
=> ( ( U2 != nil_li8379291187143545974lambda )
=> ( ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ T4 ) @ ( cons_l4558873032622161958lambda @ U @ U2 ) )
!= nil_li8379291187143545974lambda )
= ( ( con_li79537494059641180lambda @ Resid @ T @ U )
& ( ( paths_3312059784463760222lambda @ Resid @ T4 @ ( cons_l4558873032622161958lambda @ ( Resid @ U @ T ) @ nil_li8379291187143545974lambda ) )
!= nil_li8379291187143545974lambda )
& ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ ( Resid @ T @ U ) @ nil_li8379291187143545974lambda ) @ U2 )
!= nil_li8379291187143545974lambda )
& ( ( paths_3312059784463760222lambda @ Resid @ ( paths_3312059784463760222lambda @ Resid @ T4 @ ( cons_l4558873032622161958lambda @ ( Resid @ U @ T ) @ nil_li8379291187143545974lambda ) ) @ ( paths_3312059784463760222lambda @ Resid @ U2 @ ( cons_l4558873032622161958lambda @ ( Resid @ T @ U ) @ nil_li8379291187143545974lambda ) ) )
!= nil_li8379291187143545974lambda ) ) ) ) ) ) ).
% paths_in_rts.Con_rec(4)
thf(fact_1237_paths__in__rts_OCon__rec_I4_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ Resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ ( Resid @ U @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ ( Resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_rts.Con_rec(4)
thf(fact_1238_paths__in__rts_OArr__Resid__single,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ) ).
% paths_in_rts.Arr_Resid_single
thf(fact_1239_paths__in__rts_OResidx1__as__Resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_2112734093361750239lambda @ Resid @ T4 @ U )
= ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ).
% paths_in_rts.Residx1_as_Resid
thf(fact_1240_paths__in__rts_OTrgs__Resid__sym__Arr__single,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_7594662086398683701lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T4 ) ) ) ) ) ).
% paths_in_rts.Trgs_Resid_sym_Arr_single
thf(fact_1241_paths__in__rts_OArr_Osimps_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) )
= ( arr_li3691443969012319837lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Arr.simps(2)
thf(fact_1242_paths__in__rts_OArr_Osimps_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Arr.simps(2)
thf(fact_1243_paths__in__rts_OIde_Osimps_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_5097954201535619375lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) )
= ( ide_li6773656263178250516lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Ide.simps(2)
thf(fact_1244_paths__in__rts_OIde_Osimps_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( ide_lambda_lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Ide.simps(2)
thf(fact_1245_paths__in__rts_OResid1x__as__Resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ Resid @ T @ U2 ) @ nil_lambda_lambda ) ) ) ) ).
% paths_in_rts.Resid1x_as_Resid
thf(fact_1246_paths__in__rts_OResid__cons_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ Resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid_cons'
thf(fact_1247_paths__in__rts_OResidx1_Oelims,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,X2: list_l5168835686021156086lambda,Xa: list_lambda_lambda,Y2: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( paths_7088367479088905573lambda @ Resid @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_li8379291187143545974lambda )
=> ( Y2 != nil_li8379291187143545974lambda ) )
=> ( ! [T3: list_lambda_lambda] :
( ( X2
= ( cons_l4558873032622161958lambda @ T3 @ nil_li8379291187143545974lambda ) )
=> ~ ( ( ( con_li79537494059641180lambda @ Resid @ T3 @ Xa )
=> ( Y2
= ( cons_l4558873032622161958lambda @ ( Resid @ T3 @ Xa ) @ nil_li8379291187143545974lambda ) ) )
& ( ~ ( con_li79537494059641180lambda @ Resid @ T3 @ Xa )
=> ( Y2 = nil_li8379291187143545974lambda ) ) ) )
=> ~ ! [T3: list_lambda_lambda,V2: list_lambda_lambda,Va: list_l5168835686021156086lambda] :
( ( X2
= ( cons_l4558873032622161958lambda @ T3 @ ( cons_l4558873032622161958lambda @ V2 @ Va ) ) )
=> ~ ( ( ( ( con_li79537494059641180lambda @ Resid @ T3 @ Xa )
& ( ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ V2 @ Va ) @ ( Resid @ Xa @ T3 ) )
!= nil_li8379291187143545974lambda ) )
=> ( Y2
= ( cons_l4558873032622161958lambda @ ( Resid @ T3 @ Xa ) @ ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ V2 @ Va ) @ ( Resid @ Xa @ T3 ) ) ) ) )
& ( ~ ( ( con_li79537494059641180lambda @ Resid @ T3 @ Xa )
& ( ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ V2 @ Va ) @ ( Resid @ Xa @ T3 ) )
!= nil_li8379291187143545974lambda ) )
=> ( Y2 = nil_li8379291187143545974lambda ) ) ) ) ) ) ) ) ).
% paths_in_rts.Residx1.elims
thf(fact_1248_paths__in__rts_OResidx1_Oelims,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,X2: list_lambda_lambda,Xa: lambda_lambda,Y2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_2112734093361750239lambda @ Resid @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ Resid @ T3 @ Xa )
=> ( Y2
= ( cons_lambda_lambda @ ( Resid @ T3 @ Xa ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ Resid @ T3 @ Xa )
=> ( Y2 = nil_lambda_lambda ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ Resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( Resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( Resid @ T3 @ Xa ) @ ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( Resid @ Xa @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ Resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( Resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ) ).
% paths_in_rts.Residx1.elims
thf(fact_1249_paths__in__rts_OResidx1_Osimps_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ U )
= ( cons_l4558873032622161958lambda @ ( Resid @ T @ U ) @ nil_li8379291187143545974lambda ) ) )
& ( ~ ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ U )
= nil_li8379291187143545974lambda ) ) ) ) ).
% paths_in_rts.Residx1.simps(2)
thf(fact_1250_paths__in__rts_OResidx1_Osimps_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
= ( cons_lambda_lambda @ ( Resid @ T @ U ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U )
= nil_lambda_lambda ) ) ) ) ).
% paths_in_rts.Residx1.simps(2)
thf(fact_1251_paths__in__rts_OResidx1_Osimps_I3_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,Va2: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( ( con_li79537494059641180lambda @ Resid @ T @ U )
& ( ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_li8379291187143545974lambda ) )
=> ( ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ ( cons_l4558873032622161958lambda @ V @ Va2 ) ) @ U )
= ( cons_l4558873032622161958lambda @ ( Resid @ T @ U ) @ ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ V @ Va2 ) @ ( Resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_li79537494059641180lambda @ Resid @ T @ U )
& ( ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_li8379291187143545974lambda ) )
=> ( ( paths_7088367479088905573lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ ( cons_l4558873032622161958lambda @ V @ Va2 ) ) @ U )
= nil_li8379291187143545974lambda ) ) ) ) ).
% paths_in_rts.Residx1.simps(3)
thf(fact_1252_paths__in__rts_OResidx1_Osimps_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( ( con_lambda_lambda @ Resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ U )
= ( cons_lambda_lambda @ ( Resid @ T @ U ) @ ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( Resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ Resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( Resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ Resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ U )
= nil_lambda_lambda ) ) ) ) ).
% paths_in_rts.Residx1.simps(3)
thf(fact_1253_paths__in__rts_OSrcs_Osimps_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_3172353640142224310lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) )
= ( source6119337877621565212lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Srcs.simps(2)
thf(fact_1254_paths__in__rts_OSrcs_Osimps_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_2897053694880971696lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( source5627170679743844886lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Srcs.simps(2)
thf(fact_1255_paths__in__rts_Oset__Arr__subset__arr,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ T4 ) @ ( collec8830145823237851355lambda @ ( arr_li3691443969012319837lambda @ Resid ) ) ) ) ) ).
% paths_in_rts.set_Arr_subset_arr
thf(fact_1256_paths__in__rts_Oset__Arr__subset__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ Resid ) ) ) ) ) ).
% paths_in_rts.set_Arr_subset_arr
thf(fact_1257_paths__in__rts_Oset__Ide__subset__ide,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_5097954201535619375lambda @ Resid @ T4 )
=> ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ T4 ) @ ( collec8830145823237851355lambda @ ( ide_li6773656263178250516lambda @ Resid ) ) ) ) ) ).
% paths_in_rts.set_Ide_subset_ide
thf(fact_1258_paths__in__rts_Oset__Ide__subset__ide,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7717490669901197737lambda @ Resid @ T4 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ Resid ) ) ) ) ) ).
% paths_in_rts.set_Ide_subset_ide
thf(fact_1259_paths__in__rts_OTrgs_Osimps_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_8725155050933150651lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) )
= ( target4026644918049360146lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Trgs.simps(2)
thf(fact_1260_paths__in__rts_OTrgs_Osimps_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_7594662086398683701lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( target5490931411331770124lambda @ Resid @ T ) ) ) ).
% paths_in_rts.Trgs.simps(2)
thf(fact_1261_paths__in__rts_OSrcs__Resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ Resid @ ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ Resid @ U2 ) ) ) ) ).
% paths_in_rts.Srcs_Resid
thf(fact_1262_paths__in__rts_OResid1x_Oelims,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,X2: list_lambda_lambda,Xa: list_l5168835686021156086lambda,Y2: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( paths_4460507232930246679lambda @ Resid @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_li8379291187143545974lambda )
=> ( Y2
!= ( partia6873046715171720854lambda @ Resid ) ) )
=> ( ! [U4: list_lambda_lambda] :
( ( Xa
= ( cons_l4558873032622161958lambda @ U4 @ nil_li8379291187143545974lambda ) )
=> ( Y2
!= ( Resid @ X2 @ U4 ) ) )
=> ~ ! [U4: list_lambda_lambda,V2: list_lambda_lambda,Va: list_l5168835686021156086lambda] :
( ( Xa
= ( cons_l4558873032622161958lambda @ U4 @ ( cons_l4558873032622161958lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_4460507232930246679lambda @ Resid @ ( Resid @ X2 @ U4 ) @ ( cons_l4558873032622161958lambda @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid1x.elims
thf(fact_1263_paths__in__rts_OResid1x_Oelims,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,X2: lambda_lambda,Xa: list_lambda_lambda,Y2: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_3319247187298047121lambda @ Resid @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_lambda_lambda )
=> ( Y2
!= ( partia1461432607296028304lambda @ Resid ) ) )
=> ( ! [U4: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ nil_lambda_lambda ) )
=> ( Y2
!= ( Resid @ X2 @ U4 ) ) )
=> ~ ! [U4: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U4 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_3319247187298047121lambda @ Resid @ ( Resid @ X2 @ U4 ) @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ).
% paths_in_rts.Resid1x.elims
thf(fact_1264_paths__in__rts_Ocong__append__ideI_I4_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ T4 @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ T4 ) ) ) ) ) ) ).
% paths_in_rts.cong_append_ideI(4)
thf(fact_1265_paths__in__rts_Ocong__append__ideI_I3_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( append_lambda_lambda @ T4 @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ U2 ) ) ) ) ) ) ).
% paths_in_rts.cong_append_ideI(3)
thf(fact_1266_paths__in__rts_Ocong__append__ideI_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ T4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ T4 @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ) ).
% paths_in_rts.cong_append_ideI(2)
thf(fact_1267_paths__in__rts_Ocong__append__ideI_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ) ).
% paths_in_rts.cong_append_ideI(1)
thf(fact_1268_paths__in__rts_Ocong__append,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda,T8: list_lambda_lambda,U5: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ T4 @ T8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ T8 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ U2 @ U5 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ U5 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ T8 @ U5 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ Resid ) @ ( paths_8607357831768207192lambda @ Resid @ ( append_lambda_lambda @ T8 @ U5 ) @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ) ) ).
% paths_in_rts.cong_append
thf(fact_1269_paths__in__rts_Oseq__implies__Trgs__eq__Srcs,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( ord_le3064231726521830256lambda @ ( paths_8725155050933150651lambda @ Resid @ T4 ) @ ( paths_3172353640142224310lambda @ Resid @ U2 ) )
=> ( ( paths_8725155050933150651lambda @ Resid @ T4 )
= ( paths_3172353640142224310lambda @ Resid @ U2 ) ) ) ) ) ) ).
% paths_in_rts.seq_implies_Trgs_eq_Srcs
thf(fact_1270_paths__in__rts_Oseq__implies__Trgs__eq__Srcs,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ Resid @ T4 ) @ ( paths_2897053694880971696lambda @ Resid @ U2 ) )
=> ( ( paths_7594662086398683701lambda @ Resid @ T4 )
= ( paths_2897053694880971696lambda @ Resid @ U2 ) ) ) ) ) ) ).
% paths_in_rts.seq_implies_Trgs_eq_Srcs
thf(fact_1271_paths__in__rts_OCon__sym1,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U: list_lambda_lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( ( paths_7088367479088905573lambda @ Resid @ T4 @ U )
!= nil_li8379291187143545974lambda )
= ( ( paths_4460507232930246679lambda @ Resid @ U @ T4 )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% paths_in_rts.Con_sym1
thf(fact_1272_paths__in__rts_OCon__sym1,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( ( paths_2112734093361750239lambda @ Resid @ T4 @ U )
!= nil_lambda_lambda )
= ( ( paths_3319247187298047121lambda @ Resid @ U @ T4 )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% paths_in_rts.Con_sym1
thf(fact_1273_paths__in__rts_OSrcs__simp_092_060_094sub_062P,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda] :
( ( paths_5254936386813293525lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_3172353640142224310lambda @ Resid @ T4 )
= ( source6119337877621565212lambda @ Resid @ ( hd_lis2183642875561147995lambda @ T4 ) ) ) ) ) ).
% paths_in_rts.Srcs_simp\<^sub>P
thf(fact_1274_paths__in__rts_OSrcs__simp_092_060_094sub_062P,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda] :
( ( paths_6861597469809217359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ Resid @ T4 )
= ( source5627170679743844886lambda @ Resid @ ( hd_lambda_lambda @ T4 ) ) ) ) ) ).
% paths_in_rts.Srcs_simp\<^sub>P
% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X2: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X2: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_T,axiom,
! [X2: list_l5168835686021156086lambda,Y2: list_l5168835686021156086lambda] :
( ( if_lis4115203008988617916lambda @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_J_T,axiom,
! [X2: list_l5168835686021156086lambda,Y2: list_l5168835686021156086lambda] :
( ( if_lis4115203008988617916lambda @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ u @ ua ) @ ( cons_lambda_lambda @ ta @ nil_lambda_lambda ) ) )
= ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_resid @ u @ ta ) @ nil_lambda_lambda ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ua @ ( cons_lambda_lambda @ ( lambda_resid @ ta @ u ) @ nil_lambda_lambda ) ) ) ) ) ).
%------------------------------------------------------------------------------