TPTP Problem File: SLH0204^1.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_02201_085204__14393510_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1417 ( 522 unt; 133 typ; 0 def)
% Number of atoms : 3889 (1694 equ; 0 cnn)
% Maximal formula atoms : 38 ( 3 avg)
% Number of connectives : 14760 ( 577 ~; 28 |; 458 &;12056 @)
% ( 0 <=>;1641 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 962 ( 962 >; 0 *; 0 +; 0 <<)
% Number of symbols : 130 ( 127 usr; 10 con; 0-5 aty)
% Number of variables : 3464 ( 236 ^;3109 !; 119 ?;3464 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:44.629
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
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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_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_Set_Othe__elem_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
the_el4548681628688475493lambda: set_li7878742357367608272lambda > list_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_T,type,
t: list_lambda_lambda ).
thf(sy_v_b,type,
b: lambda_lambda ).
% Relevant facts (1278)
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_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_3_assms_I1_J,axiom,
lambda_Ide @ b ).
% assms(1)
thf(fact_4_paths__in__confluent__rts__axioms,axiom,
paths_9040249923598999359lambda @ lambda_resid ).
% paths_in_confluent_rts_axioms
thf(fact_5_lambda__calculus_OConD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T4: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T4 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ T4 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(1)
thf(fact_6_lambda__calculus_Oresid_Osimps_I31_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(31)
thf(fact_7_lambda__calculus_Oresid_Osimps_I26_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_resid @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(26)
thf(fact_8_lambda__calculus_Oresid_Osimps_I12_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(12)
thf(fact_9_lambda__calculus_Oresid_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(7)
thf(fact_10_lambda__calculus_Oresid_Osimps_I3_J,axiom,
! [T: lambda_lambda,T4: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T4 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T4 @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T4 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T4 @ U2 ) )
= ( lambda_App @ ( lambda_resid @ T @ T4 ) @ ( lambda_resid @ U @ U2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(3)
thf(fact_11_lambda__calculus_Olambda_Odistinct_I5_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( lambda_Nil
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(5)
thf(fact_12_lambda__calculus_OCube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ).
% lambda_calculus.Cube
thf(fact_13_lambda__calculus_OCon__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
= ( ( lambda_resid @ U @ T )
!= lambda_Nil ) ) ).
% lambda_calculus.Con_sym
thf(fact_14__092_060Lambda_062_Opartial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% \<Lambda>.partial_magma_axioms
thf(fact_15__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_16__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_17_simulation_Opreserves__reflects__arr,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ( arr_lambda_lambda @ B @ ( F @ T ) )
= ( arr_lambda_lambda @ A @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_18_simulation_Opreserves__reflects__arr,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ( arr_lambda_lambda @ B @ ( F @ T ) )
= ( arr_li3691443969012319837lambda @ A @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_19_simulation_Opreserves__reflects__arr,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ( arr_li3691443969012319837lambda @ B @ ( F @ T ) )
= ( arr_lambda_lambda @ A @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_20_simulation_Opreserves__reflects__arr,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ( arr_li3691443969012319837lambda @ B @ ( F @ T ) )
= ( arr_li3691443969012319837lambda @ A @ T ) ) ) ).
% simulation.preserves_reflects_arr
thf(fact_21_assms_I2_J,axiom,
paths_366302053746739058lambda @ lambda_resid @ t ).
% assms(2)
thf(fact_22_lambda__calculus_OApp__is__simulation1,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T5: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T5 ) @ ( lambda_App @ T5 @ A2 ) @ lambda_Nil ) ) ) ).
% lambda_calculus.App_is_simulation1
thf(fact_23_lambda__calculus_OApp__is__simulation2,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T5: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T5 ) @ ( lambda_App @ A2 @ T5 ) @ lambda_Nil ) ) ) ).
% lambda_calculus.App_is_simulation2
thf(fact_24__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_25_lambda__calculus_OLam__is__simulation,axiom,
( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T5: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T5 ) @ ( lambda_Lam @ T5 ) @ lambda_Nil ) ) ).
% lambda_calculus.Lam_is_simulation
thf(fact_26__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_27__092_060Lambda_062_Oide__implies__arr,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( arr_lambda_lambda @ lambda_resid @ A2 ) ) ).
% \<Lambda>.ide_implies_arr
thf(fact_28__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_29__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_30__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_31__092_060Lambda_062_Oide__backward__stable,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ A2 ) )
=> ( ide_lambda_lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.ide_backward_stable
thf(fact_32__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_33__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_34__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_35__092_060Lambda_062_Ojoin__of__un__upto__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V2: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ V2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V2 @ V ) ) ) ) ) ).
% \<Lambda>.join_of_un_upto_cong
thf(fact_36_lambda__calculus_Olambda_Oinject_I2_J,axiom,
! [X3: lambda_lambda,Y3: lambda_lambda] :
( ( ( lambda_Lam @ X3 )
= ( lambda_Lam @ Y3 ) )
= ( X3 = Y3 ) ) ).
% lambda_calculus.lambda.inject(2)
thf(fact_37_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_38_lambda__calculus_Oide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.ide_char
thf(fact_39_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_40_rts_Ojoin__of_Ocong,axiom,
join_o1386658102433174083lambda = join_o1386658102433174083lambda ).
% rts.join_of.cong
thf(fact_41_rts_Ojoin__of_Ocong,axiom,
join_o6956785509312998601lambda = join_o6956785509312998601lambda ).
% rts.join_of.cong
thf(fact_42_partial__magma_Ointro,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ? [X2: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP @ X2 @ T3 )
= X2 )
& ( ( OP @ T3 @ X2 )
= X2 ) )
& ! [Y2: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ Y2 @ T2 )
= Y2 )
& ( ( OP @ T2 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) )
=> ( partia3924075026459072775lambda @ OP ) ) ).
% partial_magma.intro
thf(fact_43_partial__magma_Ointro,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ? [X2: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP @ X2 @ T3 )
= X2 )
& ( ( OP @ T3 @ X2 )
= X2 ) )
& ! [Y2: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP @ Y2 @ T2 )
= Y2 )
& ( ( OP @ T2 @ Y2 )
= Y2 ) )
=> ( Y2 = X2 ) ) )
=> ( partia748546660539056525lambda @ OP ) ) ).
% partial_magma.intro
thf(fact_44_rts_Ocomposable_Ocong,axiom,
compos7127019122090555255lambda = compos7127019122090555255lambda ).
% rts.composable.cong
thf(fact_45_rts_Ocomposable_Ocong,axiom,
compos3359172601056258045lambda = compos3359172601056258045lambda ).
% rts.composable.cong
thf(fact_46_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_47_residuation_Oide_Ocong,axiom,
ide_li6773656263178250516lambda = ide_li6773656263178250516lambda ).
% residuation.ide.cong
thf(fact_48_paths__in__rts_OArr_Ocong,axiom,
paths_366302053746739058lambda = paths_366302053746739058lambda ).
% paths_in_rts.Arr.cong
thf(fact_49_partial__magma_Oex__un__null,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ X @ T2 )
= X )
& ( ( OP @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP @ Y @ T3 )
= Y )
& ( ( OP @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_50_partial__magma_Oex__un__null,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ? [X: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP @ X @ T2 )
= X )
& ( ( OP @ T2 @ X )
= X ) )
& ! [Y: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP @ Y @ T3 )
= Y )
& ( ( OP @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_51_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X4: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP2 @ X4 @ T5 )
= X4 )
& ( ( OP2 @ T5 @ X4 )
= X4 ) )
& ! [Y4: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP2 @ Y4 @ T5 )
= Y4 )
& ( ( OP2 @ T5 @ Y4 )
= Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% partial_magma_def
thf(fact_52_partial__magma__def,axiom,
( partia748546660539056525lambda
= ( ^ [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
? [X4: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( OP2 @ X4 @ T5 )
= X4 )
& ( ( OP2 @ T5 @ X4 )
= X4 ) )
& ! [Y4: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( OP2 @ Y4 @ T5 )
= Y4 )
& ( ( OP2 @ T5 @ Y4 )
= Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% partial_magma_def
thf(fact_53_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_54_mem__Collect__eq,axiom,
! [A2: lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ A2 @ ( collec621444948832755541lambda @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A2: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( member2667661554319814297lambda @ A2 @ ( collec8830145823237851355lambda @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A: set_lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A: set_li7878742357367608272lambda] :
( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_58_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_59_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_60_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_61_simulation_Opreserves__prfx,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ( ide_lambda_lambda @ A @ ( A @ T @ U ) )
=> ( ide_lambda_lambda @ B @ ( B @ ( F @ T ) @ ( F @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_62_simulation_Opreserves__prfx,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ( ide_lambda_lambda @ A @ ( A @ T @ U ) )
=> ( ide_li6773656263178250516lambda @ B @ ( B @ ( F @ T ) @ ( F @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_63_simulation_Opreserves__prfx,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ( ide_li6773656263178250516lambda @ A @ ( A @ T @ U ) )
=> ( ide_lambda_lambda @ B @ ( B @ ( F @ T ) @ ( F @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_64_simulation_Opreserves__prfx,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ( ide_li6773656263178250516lambda @ A @ ( A @ T @ U ) )
=> ( ide_li6773656263178250516lambda @ B @ ( B @ ( F @ T ) @ ( F @ U ) ) ) ) ) ).
% simulation.preserves_prfx
thf(fact_65_simulation_Opreserves__cong,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ( ( ide_lambda_lambda @ A @ ( A @ T @ U ) )
& ( ide_lambda_lambda @ A @ ( A @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B @ ( B @ ( F @ T ) @ ( F @ U ) ) )
& ( ide_lambda_lambda @ B @ ( B @ ( F @ U ) @ ( F @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_66_simulation_Opreserves__cong,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ( ( ide_lambda_lambda @ A @ ( A @ T @ U ) )
& ( ide_lambda_lambda @ A @ ( A @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ B @ ( B @ ( F @ T ) @ ( F @ U ) ) )
& ( ide_li6773656263178250516lambda @ B @ ( B @ ( F @ U ) @ ( F @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_67_simulation_Opreserves__cong,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ( ( ide_li6773656263178250516lambda @ A @ ( A @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A @ ( A @ U @ T ) ) )
=> ( ( ide_lambda_lambda @ B @ ( B @ ( F @ T ) @ ( F @ U ) ) )
& ( ide_lambda_lambda @ B @ ( B @ ( F @ U ) @ ( F @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_68_simulation_Opreserves__cong,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ( ( ide_li6773656263178250516lambda @ A @ ( A @ T @ U ) )
& ( ide_li6773656263178250516lambda @ A @ ( A @ U @ T ) ) )
=> ( ( ide_li6773656263178250516lambda @ B @ ( B @ ( F @ T ) @ ( F @ U ) ) )
& ( ide_li6773656263178250516lambda @ B @ ( B @ ( F @ U ) @ ( F @ T ) ) ) ) ) ) ).
% simulation.preserves_cong
thf(fact_69_simulation_Opreserves__ide,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,A2: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ( ide_lambda_lambda @ A @ A2 )
=> ( ide_lambda_lambda @ B @ ( F @ A2 ) ) ) ) ).
% simulation.preserves_ide
thf(fact_70_simulation_Opreserves__ide,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,A2: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ( ide_lambda_lambda @ A @ A2 )
=> ( ide_li6773656263178250516lambda @ B @ ( F @ A2 ) ) ) ) ).
% simulation.preserves_ide
thf(fact_71_simulation_Opreserves__ide,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,A2: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ( ide_li6773656263178250516lambda @ A @ A2 )
=> ( ide_lambda_lambda @ B @ ( F @ A2 ) ) ) ) ).
% simulation.preserves_ide
thf(fact_72_simulation_Opreserves__ide,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,A2: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ( ide_li6773656263178250516lambda @ A @ A2 )
=> ( ide_li6773656263178250516lambda @ B @ ( F @ A2 ) ) ) ) ).
% simulation.preserves_ide
thf(fact_73_simulation_Opreserves__joins,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ( join_o1386658102433174083lambda @ A @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B @ ( F @ T ) @ ( F @ U ) @ ( F @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_74_simulation_Opreserves__joins,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ( join_o1386658102433174083lambda @ A @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ B @ ( F @ T ) @ ( F @ U ) @ ( F @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_75_simulation_Opreserves__joins,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ( join_o6956785509312998601lambda @ A @ T @ U @ V )
=> ( join_o1386658102433174083lambda @ B @ ( F @ T ) @ ( F @ U ) @ ( F @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_76_simulation_Opreserves__joins,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ( join_o6956785509312998601lambda @ A @ T @ U @ V )
=> ( join_o6956785509312998601lambda @ B @ ( F @ T ) @ ( F @ U ) @ ( F @ V ) ) ) ) ).
% simulation.preserves_joins
thf(fact_77_lambda__calculus_Olambda_Odistinct_I3_J,axiom,
! [X3: lambda_lambda] :
( lambda_Nil
!= ( lambda_Lam @ X3 ) ) ).
% lambda_calculus.lambda.distinct(3)
thf(fact_78_lambda__calculus_Olambda_Odistinct_I15_J,axiom,
! [X3: lambda_lambda,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Lam @ X3 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(15)
thf(fact_79_lambda__calculus_OIde_Osimps_I1_J,axiom,
~ ( lambda_Ide @ lambda_Nil ) ).
% lambda_calculus.Ide.simps(1)
thf(fact_80_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_81_lambda__calculus_Oresid_Osimps_I2_J,axiom,
! [T: lambda_lambda,T4: lambda_lambda] :
( ( ( ( lambda_resid @ T @ T4 )
= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T4 ) )
= lambda_Nil ) )
& ( ( ( lambda_resid @ T @ T4 )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T4 ) )
= ( lambda_Lam @ ( lambda_resid @ T @ T4 ) ) ) ) ) ).
% lambda_calculus.resid.simps(2)
thf(fact_82_lambda__calculus_Oresid_Osimps_I8_J,axiom,
! [V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(8)
thf(fact_83_lambda__calculus_Oresid_Osimps_I27_J,axiom,
! [V: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(27)
thf(fact_84_lambda__calculus_Oresid_Osimps_I10_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_App @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(10)
thf(fact_85_lambda__calculus_Oresid_Osimps_I14_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(14)
thf(fact_86_lambda__calculus_Oresid_Osimps_I29_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(29)
thf(fact_87_lambda__calculus_Oresid_Osimps_I33_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(33)
thf(fact_88_residuation_Oarr_Ocong,axiom,
arr_lambda_lambda = arr_lambda_lambda ).
% residuation.arr.cong
thf(fact_89_residuation_Oarr_Ocong,axiom,
arr_li3691443969012319837lambda = arr_li3691443969012319837lambda ).
% residuation.arr.cong
thf(fact_90__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_91__092_060Lambda_062_Ocong__implies__coterminal,axiom,
! [U: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ U @ U2 ) ) ).
% \<Lambda>.cong_implies_coterminal
thf(fact_92__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_93_Arr__imp__arr__last,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( arr_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T6 ) ) ) ).
% Arr_imp_arr_last
thf(fact_94__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_95_Arr_Osimps_I1_J,axiom,
~ ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Arr.simps(1)
thf(fact_96__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_97_Arr__imp__arr__hd,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( arr_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) ) ) ).
% Arr_imp_arr_hd
thf(fact_98__092_060Lambda_062_Ojoin__of__arr__src_I1_J,axiom,
! [T: lambda_lambda,A2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ A2 @ T @ T ) ) ) ).
% \<Lambda>.join_of_arr_src(1)
thf(fact_99__092_060Lambda_062_Ojoin__of__arr__src_I2_J,axiom,
! [T: lambda_lambda,A2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ A2 @ T ) ) ) ).
% \<Lambda>.join_of_arr_src(2)
thf(fact_100__092_060Lambda_062_Ocong__implies__coinitial,axiom,
! [U: lambda_lambda,U2: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ U @ U2 ) ) ).
% \<Lambda>.cong_implies_coinitial
thf(fact_101__092_060Lambda_062_Otarget__is__ide,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ide_lambda_lambda @ lambda_resid @ A2 ) ) ).
% \<Lambda>.target_is_ide
thf(fact_102__092_060Lambda_062_Otargets__are__cong,axiom,
! [B3: lambda_lambda,T: lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B4 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ B4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B4 @ B3 ) ) ) ) ) ).
% \<Lambda>.targets_are_cong
thf(fact_103__092_060Lambda_062_Otargets__cong__closed,axiom,
! [B3: lambda_lambda,T: lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ B4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B4 @ B3 ) ) )
=> ( member_lambda_lambda @ B4 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.targets_cong_closed
thf(fact_104__092_060Lambda_062_Osource__is__ide,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ide_lambda_lambda @ lambda_resid @ A2 ) ) ).
% \<Lambda>.source_is_ide
thf(fact_105__092_060Lambda_062_Osources__are__cong,axiom,
! [A2: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A2 ) ) ) ) ) ).
% \<Lambda>.sources_are_cong
thf(fact_106__092_060Lambda_062_Osources__cong__closed,axiom,
! [A2: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A2 ) ) )
=> ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.sources_cong_closed
thf(fact_107__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_108__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_109__092_060Lambda_062_Oresid__source__in__targets,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( member_lambda_lambda @ ( lambda_resid @ A2 @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.resid_source_in_targets
thf(fact_110__092_060Lambda_062_Oresid__ide_I1_J,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ T )
=> ( ( lambda_resid @ T @ A2 )
= T ) ) ) ).
% \<Lambda>.resid_ide(1)
thf(fact_111__092_060Lambda_062_Ocoinitial__ide__are__cong,axiom,
! [A2: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ A3 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A3 @ A2 ) ) ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_cong
thf(fact_112__092_060Lambda_062_Ocoinitial__ide__are__eq,axiom,
! [A2: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ A3 )
=> ( A2 = A3 ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_eq
thf(fact_113__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_114__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_115__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_116__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_117__092_060Lambda_062_Ocong__respects__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda,T4: lambda_lambda,U2: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T4 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T4 @ U2 ) ) ) ) ).
% \<Lambda>.cong_respects_seq
thf(fact_118__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_119__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_120__092_060Lambda_062_Ocoinitial__iff,axiom,
! [T: lambda_lambda,T4: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T4 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T4 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T4 ) ) ) ) ).
% \<Lambda>.coinitial_iff
thf(fact_121__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_122__092_060Lambda_062_Ocoterminal__iff,axiom,
! [T: lambda_lambda,T4: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T4 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T4 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T4 ) ) ) ) ).
% \<Lambda>.coterminal_iff
thf(fact_123__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_124__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_125__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_126__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_127__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_128__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_129_rts_Oseq_Ocong,axiom,
seq_lambda_lambda = seq_lambda_lambda ).
% rts.seq.cong
thf(fact_130_rts_Oseq_Ocong,axiom,
seq_li8093550137657879445lambda = seq_li8093550137657879445lambda ).
% rts.seq.cong
thf(fact_131_rts_Osources_Ocong,axiom,
source5627170679743844886lambda = source5627170679743844886lambda ).
% rts.sources.cong
thf(fact_132_rts_Osources_Ocong,axiom,
source6119337877621565212lambda = source6119337877621565212lambda ).
% rts.sources.cong
thf(fact_133_rts_Otargets_Ocong,axiom,
target5490931411331770124lambda = target5490931411331770124lambda ).
% rts.targets.cong
thf(fact_134_rts_Otargets_Ocong,axiom,
target4026644918049360146lambda = target4026644918049360146lambda ).
% rts.targets.cong
thf(fact_135_rts_Ojoinable_Ocong,axiom,
joinab2519788767067900224lambda = joinab2519788767067900224lambda ).
% rts.joinable.cong
thf(fact_136_rts_Ojoinable_Ocong,axiom,
joinab677607494065406278lambda = joinab677607494065406278lambda ).
% rts.joinable.cong
thf(fact_137_rts_Ocoinitial_Ocong,axiom,
coinit1392077014771647542lambda = coinit1392077014771647542lambda ).
% rts.coinitial.cong
thf(fact_138_rts_Ocoinitial_Ocong,axiom,
coinit1063259437155584316lambda = coinit1063259437155584316lambda ).
% rts.coinitial.cong
thf(fact_139_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U2 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U )
= ( source5627170679743844886lambda @ Resid @ U2 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U )
= ( target5490931411331770124lambda @ Resid @ U2 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_140_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,U2: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( member2667661554319814297lambda @ U @ NN )
=> ( ( member2667661554319814297lambda @ U2 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ U )
= ( source6119337877621565212lambda @ Resid @ U2 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ U )
= ( target4026644918049360146lambda @ Resid @ U2 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( source6119337877621565212lambda @ Resid @ U ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U2 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U2 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_141_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,V: lambda_lambda,V2: lambda_lambda,W: lambda_lambda,W2: lambda_lambda,T: lambda_lambda,T4: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ V @ NN )
=> ( ( member_lambda_lambda @ V2 @ NN )
=> ( ( member_lambda_lambda @ W @ NN )
=> ( ( member_lambda_lambda @ W2 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ V )
= ( source5627170679743844886lambda @ Resid @ W ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ V2 )
= ( source5627170679743844886lambda @ Resid @ W2 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ W )
= ( target5490931411331770124lambda @ Resid @ W2 ) )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T4 @ V2 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T4 @ W2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T4 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_142_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,V: list_lambda_lambda,V2: list_lambda_lambda,W: list_lambda_lambda,W2: list_lambda_lambda,T: list_lambda_lambda,T4: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( member2667661554319814297lambda @ V @ NN )
=> ( ( member2667661554319814297lambda @ V2 @ NN )
=> ( ( member2667661554319814297lambda @ W @ NN )
=> ( ( member2667661554319814297lambda @ W2 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ V )
= ( source6119337877621565212lambda @ Resid @ W ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ V2 )
= ( source6119337877621565212lambda @ Resid @ W2 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ W )
= ( target4026644918049360146lambda @ Resid @ W2 ) )
=> ( ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T4 @ V2 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T4 @ V2 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T4 @ W2 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T4 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_143_rts_Ocoterminal_Ocong,axiom,
coterm4142657152830665724lambda = coterm4142657152830665724lambda ).
% rts.coterminal.cong
thf(fact_144_rts_Ocoterminal_Ocong,axiom,
coterm81126318419254786lambda = coterm81126318419254786lambda ).
% rts.coterminal.cong
thf(fact_145_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_146_hd__Nil__eq__last,axiom,
( ( hd_lambda_lambda @ nil_lambda_lambda )
= ( last_lambda_lambda @ nil_lambda_lambda ) ) ).
% hd_Nil_eq_last
thf(fact_147__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_148__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_149_Srcs__simp_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( source5627170679743844886lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) ) ) ) ).
% Srcs_simp\<^sub>P
thf(fact_150_Trgs__simp_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( target5490931411331770124lambda @ lambda_resid @ ( last_lambda_lambda @ T6 ) ) ) ) ).
% Trgs_simp\<^sub>P
thf(fact_151_Arr__imp__Arr__tl,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ( tl_lambda_lambda @ T6 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( tl_lambda_lambda @ T6 ) ) ) ) ).
% Arr_imp_Arr_tl
thf(fact_152__092_060Lambda_062_Osources__def,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A4 )
& ( con_lambda_lambda @ lambda_resid @ T @ A4 ) ) ) ) ).
% \<Lambda>.sources_def
thf(fact_153_Con__imp__Arr__Resid,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) ) ) ).
% Con_imp_Arr_Resid
thf(fact_154_Con__Arr__self,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T6 )
!= nil_lambda_lambda ) ) ).
% Con_Arr_self
thf(fact_155_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_156_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_157_coinitial__iff,axiom,
! [T: list_lambda_lambda,T4: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T4 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
& ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ) ).
% coinitial_iff
thf(fact_158_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_159_coterminal__iff,axiom,
! [T: list_lambda_lambda,T4: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T4 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ) ).
% coterminal_iff
thf(fact_160_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_161_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_162_resid__source__in__targets,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% resid_source_in_targets
thf(fact_163_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_164_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_165_cong__respects__seq_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,T7: list_lambda_lambda,U4: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T6 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U3 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 @ U4 ) ) ) ) ).
% cong_respects_seq\<^sub>P
thf(fact_166_cong__respects__seq,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% cong_respects_seq
thf(fact_167_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_168_composable__iff__seq,axiom,
! [G: list_lambda_lambda,F2: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F2 )
= ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F2 ) ) ).
% composable_iff_seq
thf(fact_169_targets__cong__closed,axiom,
! [B3: list_lambda_lambda,T: list_lambda_lambda,B4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ B4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B4 @ B3 ) ) )
=> ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% targets_cong_closed
thf(fact_170_targets__are__cong,axiom,
! [B3: list_lambda_lambda,T: list_lambda_lambda,B4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ B4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B4 @ B3 ) ) ) ) ) ).
% targets_are_cong
thf(fact_171_target__is__ide,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 ) ) ).
% target_is_ide
thf(fact_172_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_173_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_174_sources__cong__closed,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A2 ) ) )
=> ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% sources_cong_closed
thf(fact_175_sources__are__cong,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A2 ) ) ) ) ) ).
% sources_are_cong
thf(fact_176_source__is__ide,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 ) ) ).
% source_is_ide
thf(fact_177_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_178_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_179_join__of__arr__src_I1_J,axiom,
! [T: list_lambda_lambda,A2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ T @ T ) ) ) ).
% join_of_arr_src(1)
thf(fact_180_join__of__arr__src_I2_J,axiom,
! [T: list_lambda_lambda,A2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 @ T ) ) ) ).
% join_of_arr_src(2)
thf(fact_181_cong__implies__coinitial,axiom,
! [U: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U2 ) ) ).
% cong_implies_coinitial
thf(fact_182_coinitial__ide__are__cong,axiom,
! [A2: list_lambda_lambda,A3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A3 @ A2 ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_183_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_184_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_185_cong__implies__coterminal,axiom,
! [U: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
=> ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U2 ) ) ).
% cong_implies_coterminal
thf(fact_186_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_187_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_188_partial__magma__axioms,axiom,
partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% partial_magma_axioms
thf(fact_189_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_190_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_191_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_192_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_193_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_194_ide__implies__arr,axiom,
! [A2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 ) ) ).
% ide_implies_arr
thf(fact_195_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_196_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_197_ide__backward__stable,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ A2 ) )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% ide_backward_stable
thf(fact_198_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_199_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_200_join__of__un__upto__cong,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,V2: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ V2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V2 @ V ) ) ) ) ) ).
% join_of_un_upto_cong
thf(fact_201_Trgs_Ocases,axiom,
! [X6: list_lambda_lambda] :
( ( X6 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( X6
!= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( X6
!= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ).
% Trgs.cases
thf(fact_202__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_203__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_204_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_205_Con__sym,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T6 )
!= nil_lambda_lambda ) ) ).
% Con_sym
thf(fact_206_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_207_Cube_I1_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,V4: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ U3 ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V4 ) )
!= nil_lambda_lambda ) ) ).
% Cube(1)
thf(fact_208_Cube_I2_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,V4: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ U3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ U3 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V4 ) ) ) ) ).
% Cube(2)
thf(fact_209_arr__char,axiom,
! [T6: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( paths_366302053746739058lambda @ lambda_resid @ T6 ) ) ).
% arr_char
thf(fact_210__092_060Lambda_062_Oresid__ide__arr,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ T ) ) ) ) ).
% \<Lambda>.resid_ide_arr
thf(fact_211__092_060Lambda_062_Oresid__arr__ide,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ A2 )
=> ( ( lambda_resid @ T @ A2 )
= T ) ) ) ).
% \<Lambda>.resid_arr_ide
thf(fact_212__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_213__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_214__092_060Lambda_062_Oide__def,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
= ( ( con_lambda_lambda @ lambda_resid @ A2 @ A2 )
& ( ( lambda_resid @ A2 @ A2 )
= A2 ) ) ) ).
% \<Lambda>.ide_def
thf(fact_215__092_060Lambda_062_OideE,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ~ ( ( con_lambda_lambda @ lambda_resid @ A2 @ A2 )
=> ( ( lambda_resid @ A2 @ A2 )
!= A2 ) ) ) ).
% \<Lambda>.ideE
thf(fact_216__092_060Lambda_062_Ocon__transitive__on__ide,axiom,
! [A2: lambda_lambda,B3: lambda_lambda,C: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ B3 )
=> ( ( ide_lambda_lambda @ lambda_resid @ C )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ B3 )
=> ( ( con_lambda_lambda @ lambda_resid @ B3 @ C )
=> ( con_lambda_lambda @ lambda_resid @ A2 @ C ) ) ) ) ) ) ).
% \<Lambda>.con_transitive_on_ide
thf(fact_217__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_218__092_060Lambda_062_Ocon__imp__coinitial__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ? [A5: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A5 )
& ( con_lambda_lambda @ lambda_resid @ A5 @ T )
& ( con_lambda_lambda @ lambda_resid @ A5 @ U ) ) ) ).
% \<Lambda>.con_imp_coinitial_ax
thf(fact_219__092_060Lambda_062_Ocon__ide__are__eq,axiom,
! [A2: lambda_lambda,A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ A3 )
=> ( A2 = A3 ) ) ) ) ).
% \<Lambda>.con_ide_are_eq
thf(fact_220__092_060Lambda_062_Ocong__subst__right_I1_J,axiom,
! [U: lambda_lambda,U2: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U2 ) ) ) ).
% \<Lambda>.cong_subst_right(1)
thf(fact_221__092_060Lambda_062_Ocong__subst__right_I2_J,axiom,
! [U: lambda_lambda,U2: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U2 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U2 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_right(2)
thf(fact_222__092_060Lambda_062_Ocong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T4: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T4 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T4 @ U ) ) ) ).
% \<Lambda>.cong_subst_left(1)
thf(fact_223__092_060Lambda_062_Ocong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T4: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T4 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T4 @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T4 @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_left(2)
thf(fact_224_Trgs_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ).
% Trgs.simps(3)
thf(fact_225__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_226__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_227__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_228__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_229__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_230__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_231__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_232__092_060Lambda_062_Otargets__are__con,axiom,
! [B3: lambda_lambda,T: lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B4 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ B3 @ B4 ) ) ) ).
% \<Lambda>.targets_are_con
thf(fact_233_Srcs__are__con,axiom,
! [A2: lambda_lambda,T6: list_lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( con_lambda_lambda @ lambda_resid @ A2 @ A3 ) ) ) ).
% Srcs_are_con
thf(fact_234__092_060Lambda_062_Osources__are__con,axiom,
! [A2: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ A2 @ A3 ) ) ) ).
% \<Lambda>.sources_are_con
thf(fact_235_ex__un__Src,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( Y = X ) ) ) ) ).
% ex_un_Src
thf(fact_236_Trgs__are__con,axiom,
! [B3: lambda_lambda,T6: list_lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
=> ( ( member_lambda_lambda @ B4 @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
=> ( con_lambda_lambda @ lambda_resid @ B3 @ B4 ) ) ) ).
% Trgs_are_con
thf(fact_237_ex__un__Trg,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
=> ( Y = X ) ) ) ) ).
% ex_un_Trg
thf(fact_238__092_060Lambda_062_Ojoin__of__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ V @ W )
=> ( join_o1386658102433174083lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) @ ( lambda_resid @ W @ V ) ) ) ) ).
% \<Lambda>.join_of_resid
thf(fact_239__092_060Lambda_062_Ocon__with__join__of__iff_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W )
=> ( ( ( con_lambda_lambda @ lambda_resid @ U @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) )
=> ( con_lambda_lambda @ lambda_resid @ W @ V ) ) ) ).
% \<Lambda>.con_with_join_of_iff(1)
thf(fact_240__092_060Lambda_062_Ocon__with__join__of__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W )
=> ( ( con_lambda_lambda @ lambda_resid @ W @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) ) ) ) ) ).
% \<Lambda>.con_with_join_of_iff(2)
thf(fact_241__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_242__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_243__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_244__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_245_Con__cons_I2_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,U: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ U3 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U3 )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_cons(2)
thf(fact_246_Con__cons_I1_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U3 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_cons(1)
thf(fact_247_Resid_Osimps_I2_J,axiom,
! [V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% Resid.simps(2)
thf(fact_248_Resid__cons_I2_J,axiom,
! [U3: list_lambda_lambda,T6: list_lambda_lambda,U: lambda_lambda] :
( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ U3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ U3 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U3 ) ) ) ) ).
% Resid_cons(2)
thf(fact_249_Resid__rec_I3_J,axiom,
! [U3: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U3 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U3 ) ) ) ) ).
% Resid_rec(3)
thf(fact_250_Resid__rec_I2_J,axiom,
! [T6: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ).
% Resid_rec(2)
thf(fact_251_Con__initial__left,axiom,
! [T: lambda_lambda,T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
!= nil_lambda_lambda ) ) ).
% Con_initial_left
thf(fact_252_Con__initial__right,axiom,
! [T6: list_lambda_lambda,U: lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ U3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ).
% Con_initial_right
thf(fact_253_Con__implies__Arr_I2_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ U3 ) ) ).
% Con_implies_Arr(2)
thf(fact_254_Con__implies__Arr_I1_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ T6 ) ) ).
% Con_implies_Arr(1)
thf(fact_255_Arr__iff__Con__self,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T6 )
!= nil_lambda_lambda ) ) ).
% Arr_iff_Con_self
thf(fact_256_Con__imp__eq__Srcs,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ).
% Con_imp_eq_Srcs
thf(fact_257__092_060Lambda_062_Otargets__con__closed,axiom,
! [B3: lambda_lambda,T: lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B4 )
=> ( ( con_lambda_lambda @ lambda_resid @ B3 @ B4 )
=> ( member_lambda_lambda @ B4 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.targets_con_closed
thf(fact_258_Srcs_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( source5627170679743844886lambda @ lambda_resid @ T ) ) ).
% Srcs.simps(3)
thf(fact_259_Srcs__con__closed,axiom,
! [A2: lambda_lambda,T6: list_lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ A3 )
=> ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_260__092_060Lambda_062_Oin__sourcesE,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ~ ( con_lambda_lambda @ lambda_resid @ T @ A2 ) ) ) ).
% \<Lambda>.in_sourcesE
thf(fact_261__092_060Lambda_062_Osources__con__closed,axiom,
! [A2: lambda_lambda,T: lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ A3 )
=> ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.sources_con_closed
thf(fact_262_Srcs_Osimps_I1_J,axiom,
( ( paths_2897053694880971696lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Srcs.simps(1)
thf(fact_263_Trgs__con__closed,axiom,
! [B3: lambda_lambda,T6: list_lambda_lambda,B4: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B4 )
=> ( ( con_lambda_lambda @ lambda_resid @ B3 @ B4 )
=> ( member_lambda_lambda @ B4 @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% Trgs_con_closed
thf(fact_264_Trgs_Osimps_I1_J,axiom,
( ( paths_7594662086398683701lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Trgs.simps(1)
thf(fact_265_Arr__has__Src,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
!= bot_bo1271416176687173302lambda ) ) ).
% Arr_has_Src
thf(fact_266_Arr__has__Trg,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
!= bot_bo1271416176687173302lambda ) ) ).
% Arr_has_Trg
thf(fact_267_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_268_Con__rec_I4_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U @ U3 ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( 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 ) @ U3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_rec(4)
thf(fact_269_Con__rec_I3_J,axiom,
! [U3: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U3 ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U3 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_rec(3)
thf(fact_270_Con__rec_I2_J,axiom,
! [T6: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_rec(2)
thf(fact_271_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_272_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_273_sources__cons,axiom,
! [T: lambda_lambda,T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ T6 ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% sources_cons
thf(fact_274_Arr__Resid__single,axiom,
! [T6: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ).
% Arr_Resid_single
thf(fact_275_Trgs__Resid__sym__Arr__single,axiom,
! [T6: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T6 ) ) ) ) ).
% Trgs_Resid_sym_Arr_single
thf(fact_276_confluence__ind,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda ) ) ) ) ).
% confluence_ind
thf(fact_277_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_278_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_279_seq__char,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( paths_366302053746739058lambda @ lambda_resid @ U3 )
& ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ).
% seq_char
thf(fact_280_Resid__Arr__Src,axiom,
! [T6: list_lambda_lambda,A2: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
= T6 ) ) ) ).
% Resid_Arr_Src
thf(fact_281_Srcs__Resid__Arr__single,axiom,
! [T6: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ).
% Srcs_Resid_Arr_single
thf(fact_282_Srcs__Resid__single__Arr,axiom,
! [U: lambda_lambda,T6: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T6 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ).
% Srcs_Resid_single_Arr
thf(fact_283_Srcs_Oelims,axiom,
! [X6: list_lambda_lambda,Y5: set_lambda_lambda] :
( ( ( paths_2897053694880971696lambda @ lambda_resid @ X6 )
= Y5 )
=> ( ( ( X6 = nil_lambda_lambda )
=> ( Y5 != bot_bo1271416176687173302lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y5
!= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda] :
( ? [V3: lambda_lambda,Va2: list_lambda_lambda] :
( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y5
!= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) ) ) ) ) ).
% Srcs.elims
thf(fact_284_Trgs_Oelims,axiom,
! [X6: list_lambda_lambda,Y5: set_lambda_lambda] :
( ( ( paths_7594662086398683701lambda @ lambda_resid @ X6 )
= Y5 )
=> ( ( ( X6 = nil_lambda_lambda )
=> ( Y5 != bot_bo1271416176687173302lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y5
!= ( target5490931411331770124lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y5
!= ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Trgs.elims
thf(fact_285_Con__single__ide__ind,axiom,
! [A2: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% Con_single_ide_ind
thf(fact_286_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_287_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_288_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_289_confluence__single,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T3 @ U5 )
=> ( con_lambda_lambda @ lambda_resid @ T3 @ U5 ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
!= nil_lambda_lambda ) ) ) ) ) ).
% confluence_single
thf(fact_290_arrI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ).
% arrI\<^sub>P
thf(fact_291_lambda__calculus_Ocon__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ).
% lambda_calculus.con_char
thf(fact_292__092_060Lambda_062_OideI,axiom,
! [A2: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ A2 @ A2 )
=> ( ( ( lambda_resid @ A2 @ A2 )
= A2 )
=> ( ide_lambda_lambda @ lambda_resid @ A2 ) ) ) ).
% \<Lambda>.ideI
thf(fact_293__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_294_Con__consI_I2_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,U: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ U3 ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_consI(2)
thf(fact_295_Con__consI_I1_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U3 )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_consI(1)
thf(fact_296_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_297_Trgs__Resid__sym,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T6 ) ) ) ) ).
% Trgs_Resid_sym
thf(fact_298_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_299_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_300__092_060Lambda_062_Oin__sourcesI,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ A2 )
=> ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.in_sourcesI
thf(fact_301_lambda__calculus_Otargets__simps_I1_J,axiom,
( ( target5490931411331770124lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.targets_simps(1)
thf(fact_302_lambda__calculus_Osources__simps_I1_J,axiom,
( ( source5627170679743844886lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.sources_simps(1)
thf(fact_303__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_304_cong__cons,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda,T4: lambda_lambda,U4: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T4 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U3 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) @ ( cons_lambda_lambda @ T4 @ U4 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T4 @ U4 ) @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ) ) ).
% cong_cons
thf(fact_305_cong__cons__ideI_I1_J,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ) ).
% cong_cons_ideI(1)
thf(fact_306_cong__cons__ideI_I2_J,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ U3 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) @ U3 ) ) ) ) ) ).
% cong_cons_ideI(2)
thf(fact_307_Resid__single__ide_I2_J,axiom,
! [A2: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
= T6 ) ) ) ).
% Resid_single_ide(2)
thf(fact_308_Srcs__Resid,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ U3 ) ) ) ).
% Srcs_Resid
thf(fact_309_Con__single__ideI_I2_J,axiom,
! [A2: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(2)
thf(fact_310_Con__single__ideI_I1_J,axiom,
! [A2: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(1)
thf(fact_311_is__confluent__rts,axiom,
conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% is_confluent_rts
thf(fact_312_tl__Nil,axiom,
! [Xs: list_lambda_lambda] :
( ( ( tl_lambda_lambda @ Xs )
= nil_lambda_lambda )
= ( ( Xs = nil_lambda_lambda )
| ? [X4: lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ X4 @ nil_lambda_lambda ) ) ) ) ).
% tl_Nil
thf(fact_313_Nil__tl,axiom,
! [Xs: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( tl_lambda_lambda @ Xs ) )
= ( ( Xs = nil_lambda_lambda )
| ? [X4: lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ X4 @ nil_lambda_lambda ) ) ) ) ).
% Nil_tl
thf(fact_314_transpose_Ocases,axiom,
! [X6: list_l5168835686021156086lambda] :
( ( X6 != nil_li8379291187143545974lambda )
=> ( ! [Xss: list_l5168835686021156086lambda] :
( X6
!= ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ Xss ) )
=> ~ ! [X: lambda_lambda,Xs2: list_lambda_lambda,Xss: list_l5168835686021156086lambda] :
( X6
!= ( cons_l4558873032622161958lambda @ ( cons_lambda_lambda @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_315_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_316_not__Cons__self2,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( cons_lambda_lambda @ X6 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_317_paths__in__rts_OResid_Ocong,axiom,
paths_8607357831768207192lambda = paths_8607357831768207192lambda ).
% paths_in_rts.Resid.cong
thf(fact_318_paths__in__rts_OTrgs_Ocong,axiom,
paths_7594662086398683701lambda = paths_7594662086398683701lambda ).
% paths_in_rts.Trgs.cong
thf(fact_319_paths__in__rts_OSrcs_Ocong,axiom,
paths_2897053694880971696lambda = paths_2897053694880971696lambda ).
% paths_in_rts.Srcs.cong
thf(fact_320_residuation_Ocon_Ocong,axiom,
con_lambda_lambda = con_lambda_lambda ).
% residuation.con.cong
thf(fact_321_residuation_Ocon_Ocong,axiom,
con_li79537494059641180lambda = con_li79537494059641180lambda ).
% residuation.con.cong
thf(fact_322_list_Osel_I2_J,axiom,
( ( tl_lambda_lambda @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.sel(2)
thf(fact_323_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_324_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_325_list_Oexhaust,axiom,
! [Y5: list_lambda_lambda] :
( ( Y5 != nil_lambda_lambda )
=> ~ ! [X212: lambda_lambda,X222: list_lambda_lambda] :
( Y5
!= ( cons_lambda_lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_326_remdups__adj_Ocases,axiom,
! [X6: list_lambda_lambda] :
( ( X6 != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] :
( X6
!= ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ~ ! [X: lambda_lambda,Y2: lambda_lambda,Xs2: list_lambda_lambda] :
( X6
!= ( cons_lambda_lambda @ X @ ( cons_lambda_lambda @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_327_neq__Nil__conv,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
= ( ? [Y4: lambda_lambda,Ys: list_lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ Y4 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_328_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_lambda_lambda > $o,Xs: list_lambda_lambda,Ys2: 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 )
=> ( ! [Y2: lambda_lambda,Ys3: list_lambda_lambda] : ( P @ nil_lambda_lambda @ ( cons_lambda_lambda @ Y2 @ Ys3 ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y2: lambda_lambda,Ys3: list_lambda_lambda] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_329_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_330_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_331_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_332_simulation_Opreserves__resid,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ( con_lambda_lambda @ A @ T @ U )
=> ( ( F @ ( A @ T @ U ) )
= ( B @ ( F @ T ) @ ( F @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_333_simulation_Opreserves__resid,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ( con_lambda_lambda @ A @ T @ U )
=> ( ( F @ ( A @ T @ U ) )
= ( B @ ( F @ T ) @ ( F @ U ) ) ) ) ) ).
% simulation.preserves_resid
thf(fact_334_simulation_Opreserves__con,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ( con_lambda_lambda @ A @ T @ U )
=> ( con_lambda_lambda @ B @ ( F @ T ) @ ( F @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_335_simulation_Opreserves__con,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ( con_lambda_lambda @ A @ T @ U )
=> ( con_li79537494059641180lambda @ B @ ( F @ T ) @ ( F @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_336_simulation_Opreserves__con,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ( con_li79537494059641180lambda @ A @ T @ U )
=> ( con_lambda_lambda @ B @ ( F @ T ) @ ( F @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_337_simulation_Opreserves__con,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ( con_li79537494059641180lambda @ A @ T @ U )
=> ( con_li79537494059641180lambda @ B @ ( F @ T ) @ ( F @ U ) ) ) ) ).
% simulation.preserves_con
thf(fact_338_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_339_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U3: list_l5168835686021156086lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( ! [T3: list_lambda_lambda,U5: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ Resid @ T3 @ U5 )
=> ( con_li79537494059641180lambda @ Resid @ T3 @ U5 ) )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( paths_2015741907369688696lambda @ Resid @ U3 )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( paths_3172353640142224310lambda @ Resid @ U3 ) )
=> ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ U3 )
!= nil_li8379291187143545974lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_340_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U3: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid @ T3 @ U5 )
=> ( con_lambda_lambda @ Resid @ T3 @ U5 ) )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( paths_366302053746739058lambda @ Resid @ U3 )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( paths_2897053694880971696lambda @ Resid @ U3 ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_341_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_342_last_Osimps,axiom,
! [Xs: list_lambda_lambda,X6: lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X6 @ Xs ) )
= X6 ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X6 @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ) ).
% last.simps
thf(fact_343_last__ConsL,axiom,
! [Xs: list_lambda_lambda,X6: lambda_lambda] :
( ( Xs = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X6 @ Xs ) )
= X6 ) ) ).
% last_ConsL
thf(fact_344_last__ConsR,axiom,
! [Xs: list_lambda_lambda,X6: lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X6 @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_ConsR
thf(fact_345_paths__in__confluent__rts_Oconfluence__ind,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T6 )
=> ( ( paths_366302053746739058lambda @ Resid @ U3 )
=> ( ( ( paths_2897053694880971696lambda @ Resid @ T6 )
= ( paths_2897053694880971696lambda @ Resid @ U3 ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ T6 @ U3 )
!= nil_lambda_lambda ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_ind
thf(fact_346_Arr_Oelims_I3_J,axiom,
! [X6: list_lambda_lambda] :
( ~ ( paths_366302053746739058lambda @ lambda_resid @ X6 )
=> ( ( X6 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( arr_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).
% Arr.elims(3)
thf(fact_347_Arr_Oelims_I2_J,axiom,
! [X6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ X6 )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( arr_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Arr.elims(2)
thf(fact_348_Arr_Oelims_I1_J,axiom,
! [X6: list_lambda_lambda,Y5: $o] :
( ( ( paths_366302053746739058lambda @ lambda_resid @ X6 )
= Y5 )
=> ( ( ( X6 = nil_lambda_lambda )
=> Y5 )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y5
= ( ~ ( arr_lambda_lambda @ lambda_resid @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y5
= ( ~ ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% Arr.elims(1)
thf(fact_349_Arr_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ).
% Arr.simps(3)
thf(fact_350_rts__with__composites__axioms,axiom,
rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% rts_with_composites_axioms
thf(fact_351_seq__char_H,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( paths_366302053746739058lambda @ lambda_resid @ U3 )
& ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
!= bot_bo1271416176687173302lambda ) ) ) ).
% seq_char'
thf(fact_352_targets__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( collec8830145823237851355lambda
@ ^ [B5: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B5 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ B5 ) ) ) ) ) ).
% targets_char\<^sub>\<Lambda>\<^sub>P
thf(fact_353_Resid__single__ide_I1_J,axiom,
! [A2: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T6 )
!= nil_lambda_lambda )
=> ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T6 )
@ ( image_3692823078739859851lambda
@ ^ [B2: lambda_lambda] : ( cons_lambda_lambda @ B2 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% Resid_single_ide(1)
thf(fact_354_single__Trg__last__in__targets,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% single_Trg_last_in_targets
thf(fact_355_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_356_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_357_con__char,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda ) ) ).
% con_char
thf(fact_358_cong__subst__left_I2_J,axiom,
! [T: list_lambda_lambda,T4: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ 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 @ T4 @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_359_cong__subst__left_I1_J,axiom,
! [T: list_lambda_lambda,T4: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_360_cong__subst__right_I2_J,axiom,
! [U: list_lambda_lambda,U2: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ 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 @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_361_cong__subst__right_I1_J,axiom,
! [U: list_lambda_lambda,U2: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U2 ) ) ) ).
% cong_subst_right(1)
thf(fact_362_con__imp__coinitial__ax,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ? [A5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 @ T )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A5 @ U ) ) ) ).
% con_imp_coinitial_ax
thf(fact_363_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_364_con__transitive__on__ide,axiom,
! [A2: list_lambda_lambda,B3: list_lambda_lambda,C: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ C )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ B3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 @ C )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ C ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_365_ideE,axiom,
! [A2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ~ ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A2 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A2 )
!= A2 ) ) ) ).
% ideE
thf(fact_366_ide__def,axiom,
! [A2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
= ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A2 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A2 )
= A2 ) ) ) ).
% ide_def
thf(fact_367_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_368_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_369_resid__arr__ide,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T @ A2 )
= T ) ) ) ).
% resid_arr_ide
thf(fact_370_resid__ide__arr,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ T )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ T ) ) ) ) ).
% resid_ide_arr
thf(fact_371_sources__are__con,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A3 ) ) ) ).
% sources_are_con
thf(fact_372_targets__are__con,axiom,
! [B3: list_lambda_lambda,T: list_lambda_lambda,B4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 @ B4 ) ) ) ).
% targets_are_con
thf(fact_373_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_374_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_375_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_376_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_377_arrE,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).
% arrE
thf(fact_378_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_379_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_380_con__with__join__of__iff_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W @ 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_381_con__with__join__of__iff_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W )
=> ( ( ( 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 ) @ W @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_382_join__of__resid,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V @ W )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W @ V ) ) ) ) ).
% join_of_resid
thf(fact_383_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_384_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_385_confluence_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 ) ) ).
% confluence\<^sub>P
thf(fact_386_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_387_sources__con__closed,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda,A3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A3 )
=> ( member2667661554319814297lambda @ A3 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% sources_con_closed
thf(fact_388_in__sourcesE,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 ) ) ) ).
% in_sourcesE
thf(fact_389_targets__con__closed,axiom,
! [B3: list_lambda_lambda,T: list_lambda_lambda,B4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B4 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 @ B4 )
=> ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% targets_con_closed
thf(fact_390_Srcs__are__ide,axiom,
! [T6: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).
% Srcs_are_ide
thf(fact_391_Trgs__are__ide,axiom,
! [T6: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).
% Trgs_are_ide
thf(fact_392_sources__def,axiom,
! [T: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [A4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A4 ) ) ) ) ).
% sources_def
thf(fact_393__092_060Lambda_062_Otargets__eqI,axiom,
! [T: lambda_lambda,T4: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T4 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T4 ) ) ) ).
% \<Lambda>.targets_eqI
thf(fact_394_Srcs__eqI,axiom,
! [T6: list_lambda_lambda,T7: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ T7 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T7 ) ) ) ).
% Srcs_eqI
thf(fact_395__092_060Lambda_062_Osources__eqI,axiom,
! [T: lambda_lambda,T4: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T4 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T4 ) ) ) ).
% \<Lambda>.sources_eqI
thf(fact_396_Trgs__eqI,axiom,
! [T6: list_lambda_lambda,T7: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_7594662086398683701lambda @ lambda_resid @ T7 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T7 ) ) ) ).
% Trgs_eqI
thf(fact_397_Trg__last__eqI,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T6 ) ) )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ T6 ) )
= ( lambda_Trg @ ( last_lambda_lambda @ U3 ) ) ) ) ).
% Trg_last_eqI
thf(fact_398_seq__implies__Trgs__eq__Srcs,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ) ).
% seq_implies_Trgs_eq_Srcs
thf(fact_399__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_400__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_401__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_402_in__targets__iff,axiom,
! [T6: list_lambda_lambda,B: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ B ) ) ) ) ) ).
% in_targets_iff
thf(fact_403_conI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 ) ) ).
% conI\<^sub>P
thf(fact_404_ideI,axiom,
! [A2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A2 )
= A2 )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 ) ) ) ).
% ideI
thf(fact_405_arrI,axiom,
! [T: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% arrI
thf(fact_406_in__sourcesI,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 )
=> ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% in_sourcesI
thf(fact_407_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_408_Arr__consI_092_060_094sub_062P,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ).
% Arr_consI\<^sub>P
thf(fact_409_seqI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
!= bot_bo1271416176687173302lambda )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 ) ) ) ) ).
% seqI\<^sub>P
thf(fact_410_simulation_Opreserves__sources,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ ( source5627170679743844886lambda @ A @ T ) ) @ ( source5627170679743844886lambda @ B @ ( F @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_411_simulation_Opreserves__sources,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ ( source5627170679743844886lambda @ A @ T ) ) @ ( source6119337877621565212lambda @ B @ ( F @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_412_simulation_Opreserves__sources,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F @ ( source6119337877621565212lambda @ A @ T ) ) @ ( source5627170679743844886lambda @ B @ ( F @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_413_simulation_Opreserves__sources,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F @ ( source6119337877621565212lambda @ A @ T ) ) @ ( source6119337877621565212lambda @ B @ ( F @ T ) ) ) ) ).
% simulation.preserves_sources
thf(fact_414_simulation_Opreserves__targets,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ ( target5490931411331770124lambda @ A @ T ) ) @ ( target5490931411331770124lambda @ B @ ( F @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_415_simulation_Opreserves__targets,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ ( target5490931411331770124lambda @ A @ T ) ) @ ( target4026644918049360146lambda @ B @ ( F @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_416_simulation_Opreserves__targets,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F @ ( target4026644918049360146lambda @ A @ T ) ) @ ( target5490931411331770124lambda @ B @ ( F @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_417_simulation_Opreserves__targets,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F @ ( target4026644918049360146lambda @ A @ T ) ) @ ( target4026644918049360146lambda @ B @ ( F @ T ) ) ) ) ).
% simulation.preserves_targets
thf(fact_418_lambda__calculus_OTrg_Osimps_I5_J,axiom,
( ( lambda_Trg @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Trg.simps(5)
thf(fact_419_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_420_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_421_paths__in__confluent__rts_Oconfluence_092_060_094sub_062P,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T6 @ U3 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T6 @ U3 ) ) ) ).
% paths_in_confluent_rts.confluence\<^sub>P
thf(fact_422_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_423_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_424_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,G: lambda_lambda,F2: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( compos7127019122090555255lambda @ Resid @ G @ F2 )
= ( seq_lambda_lambda @ Resid @ G @ F2 ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_425_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda,F2: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( compos3359172601056258045lambda @ Resid @ G @ F2 )
= ( seq_li8093550137657879445lambda @ Resid @ G @ F2 ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_426_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_427_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_428_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_429_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_430_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_431_img__Resid__Srcs,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ord_le3064231726521830256lambda
@ ( image_3692823078739859851lambda
@ ^ [A4: lambda_lambda] : ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A4 @ nil_lambda_lambda ) @ T6 )
@ ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) )
@ ( image_3692823078739859851lambda
@ ^ [B2: lambda_lambda] : ( cons_lambda_lambda @ B2 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ) ).
% img_Resid_Srcs
thf(fact_432_sources__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( collec8830145823237851355lambda
@ ^ [A6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A6 @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ A6 ) ) ) ) ) ).
% sources_char\<^sub>\<Lambda>\<^sub>P
thf(fact_433_in__sources__iff,axiom,
! [T6: list_lambda_lambda,A: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ A ) ) ) ) ) ).
% in_sources_iff
thf(fact_434_Int__subset__iff,axiom,
! [C2: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ ( inf_in1766959559283430712lambda @ A @ B ) )
= ( ( ord_le1926335974945931754lambda @ C2 @ A )
& ( ord_le1926335974945931754lambda @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_435_Int__subset__iff,axiom,
! [C2: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C2 @ ( inf_in3354698378728726462lambda @ A @ B ) )
= ( ( ord_le3064231726521830256lambda @ C2 @ A )
& ( ord_le3064231726521830256lambda @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_436_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_437_boolean__algebra_Oconj__zero__right,axiom,
! [X6: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X6 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_438_boolean__algebra_Oconj__zero__right,axiom,
! [X6: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X6 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_439_sources__eqI,axiom,
! [T: list_lambda_lambda,T4: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% sources_eqI
thf(fact_440_targets__eqI,axiom,
! [T: list_lambda_lambda,T4: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% targets_eqI
thf(fact_441_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_442_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_443_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_444_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_445_image__eqI,axiom,
! [B3: lambda_lambda,F2: lambda_lambda > lambda_lambda,X6: lambda_lambda,A: set_lambda_lambda] :
( ( B3
= ( F2 @ X6 ) )
=> ( ( member_lambda_lambda @ X6 @ A )
=> ( member_lambda_lambda @ B3 @ ( image_2356535832909760517lambda @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_446_image__eqI,axiom,
! [B3: list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,X6: lambda_lambda,A: set_lambda_lambda] :
( ( B3
= ( F2 @ X6 ) )
=> ( ( member_lambda_lambda @ X6 @ A )
=> ( member2667661554319814297lambda @ B3 @ ( image_3692823078739859851lambda @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_447_image__eqI,axiom,
! [B3: lambda_lambda,F2: list_lambda_lambda > lambda_lambda,X6: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( B3
= ( F2 @ X6 ) )
=> ( ( member2667661554319814297lambda @ X6 @ A )
=> ( member_lambda_lambda @ B3 @ ( image_2458965899662700159lambda @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_448_image__eqI,axiom,
! [B3: list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,X6: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( B3
= ( F2 @ X6 ) )
=> ( ( member2667661554319814297lambda @ X6 @ A )
=> ( member2667661554319814297lambda @ B3 @ ( image_5886195720093285125lambda @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_449_empty__iff,axiom,
! [C: lambda_lambda] :
~ ( member_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ).
% empty_iff
thf(fact_450_empty__iff,axiom,
! [C: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ C @ bot_bo1397037886704484924lambda ) ).
% empty_iff
thf(fact_451_all__not__in__conv,axiom,
! [A: set_lambda_lambda] :
( ( ! [X4: lambda_lambda] :
~ ( member_lambda_lambda @ X4 @ A ) )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% all_not_in_conv
thf(fact_452_all__not__in__conv,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ! [X4: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ X4 @ A ) )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% all_not_in_conv
thf(fact_453_Collect__empty__eq,axiom,
! [P: lambda_lambda > $o] :
( ( ( collec621444948832755541lambda @ P )
= bot_bo1271416176687173302lambda )
= ( ! [X4: lambda_lambda] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_454_Collect__empty__eq,axiom,
! [P: list_lambda_lambda > $o] :
( ( ( collec8830145823237851355lambda @ P )
= bot_bo1397037886704484924lambda )
= ( ! [X4: list_lambda_lambda] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_455_empty__Collect__eq,axiom,
! [P: lambda_lambda > $o] :
( ( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ P ) )
= ( ! [X4: lambda_lambda] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_456_empty__Collect__eq,axiom,
! [P: list_lambda_lambda > $o] :
( ( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda @ P ) )
= ( ! [X4: list_lambda_lambda] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_457_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_458_subsetI,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( member_lambda_lambda @ X @ B ) )
=> ( ord_le1926335974945931754lambda @ A @ B ) ) ).
% subsetI
thf(fact_459_subsetI,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( member2667661554319814297lambda @ X @ B ) )
=> ( ord_le3064231726521830256lambda @ A @ B ) ) ).
% subsetI
thf(fact_460_subset__antisym,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_461_subset__antisym,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_462_insertCI,axiom,
! [A2: lambda_lambda,B: set_lambda_lambda,B3: lambda_lambda] :
( ( ~ ( member_lambda_lambda @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B3 @ B ) ) ) ).
% insertCI
thf(fact_463_insertCI,axiom,
! [A2: list_lambda_lambda,B: set_li7878742357367608272lambda,B3: list_lambda_lambda] :
( ( ~ ( member2667661554319814297lambda @ A2 @ B )
=> ( A2 = B3 ) )
=> ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ B3 @ B ) ) ) ).
% insertCI
thf(fact_464_insert__iff,axiom,
! [A2: lambda_lambda,B3: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member_lambda_lambda @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_465_insert__iff,axiom,
! [A2: list_lambda_lambda,B3: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ B3 @ A ) )
= ( ( A2 = B3 )
| ( member2667661554319814297lambda @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_466_insert__absorb2,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda] :
( ( insert_lambda_lambda @ X6 @ ( insert_lambda_lambda @ X6 @ A ) )
= ( insert_lambda_lambda @ X6 @ A ) ) ).
% insert_absorb2
thf(fact_467_IntI,axiom,
! [C: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ A )
=> ( ( member_lambda_lambda @ C @ B )
=> ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% IntI
thf(fact_468_IntI,axiom,
! [C: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ A )
=> ( ( member2667661554319814297lambda @ C @ B )
=> ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% IntI
thf(fact_469_Int__iff,axiom,
! [C: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A @ B ) )
= ( ( member_lambda_lambda @ C @ A )
& ( member_lambda_lambda @ C @ B ) ) ) ).
% Int_iff
thf(fact_470_Int__iff,axiom,
! [C: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A @ B ) )
= ( ( member2667661554319814297lambda @ C @ A )
& ( member2667661554319814297lambda @ C @ B ) ) ) ).
% Int_iff
thf(fact_471_Src__hd__eqI,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T6 ) ) )
=> ( ( lambda_Src @ ( hd_lambda_lambda @ T6 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) ) ) ).
% Src_hd_eqI
thf(fact_472_Trg__last__Src__hd__eqI,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ T6 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) ) ) ).
% Trg_last_Src_hd_eqI
thf(fact_473_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_474_image__empty,axiom,
! [F2: lambda_lambda > lambda_lambda] :
( ( image_2356535832909760517lambda @ F2 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% image_empty
thf(fact_475_image__empty,axiom,
! [F2: lambda_lambda > list_lambda_lambda] :
( ( image_3692823078739859851lambda @ F2 @ bot_bo1271416176687173302lambda )
= bot_bo1397037886704484924lambda ) ).
% image_empty
thf(fact_476_image__empty,axiom,
! [F2: list_lambda_lambda > lambda_lambda] :
( ( image_2458965899662700159lambda @ F2 @ bot_bo1397037886704484924lambda )
= bot_bo1271416176687173302lambda ) ).
% image_empty
thf(fact_477_image__empty,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda] :
( ( image_5886195720093285125lambda @ F2 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% image_empty
thf(fact_478_empty__is__image,axiom,
! [F2: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( image_2356535832909760517lambda @ F2 @ A ) )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% empty_is_image
thf(fact_479_empty__is__image,axiom,
! [F2: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( bot_bo1271416176687173302lambda
= ( image_2458965899662700159lambda @ F2 @ A ) )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% empty_is_image
thf(fact_480_empty__is__image,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( bot_bo1397037886704484924lambda
= ( image_3692823078739859851lambda @ F2 @ A ) )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% empty_is_image
thf(fact_481_empty__is__image,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( image_5886195720093285125lambda @ F2 @ A ) )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% empty_is_image
thf(fact_482_image__is__empty,axiom,
! [F2: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( ( image_2356535832909760517lambda @ F2 @ A )
= bot_bo1271416176687173302lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% image_is_empty
thf(fact_483_image__is__empty,axiom,
! [F2: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( image_2458965899662700159lambda @ F2 @ A )
= bot_bo1271416176687173302lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% image_is_empty
thf(fact_484_image__is__empty,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( ( image_3692823078739859851lambda @ F2 @ A )
= bot_bo1397037886704484924lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% image_is_empty
thf(fact_485_image__is__empty,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( image_5886195720093285125lambda @ F2 @ A )
= bot_bo1397037886704484924lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% image_is_empty
thf(fact_486_subset__empty,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% subset_empty
thf(fact_487_subset__empty,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ bot_bo1397037886704484924lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% subset_empty
thf(fact_488_empty__subsetI,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A ) ).
% empty_subsetI
thf(fact_489_empty__subsetI,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A ) ).
% empty_subsetI
thf(fact_490_boolean__algebra_Oconj__zero__left,axiom,
! [X6: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X6 )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_491_boolean__algebra_Oconj__zero__left,axiom,
! [X6: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X6 )
= bot_bo1397037886704484924lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_492_insert__image,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,F2: lambda_lambda > list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ( ( insert8465602035467394112lambda @ ( F2 @ X6 ) @ ( image_3692823078739859851lambda @ F2 @ A ) )
= ( image_3692823078739859851lambda @ F2 @ A ) ) ) ).
% insert_image
thf(fact_493_insert__image,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,F2: lambda_lambda > lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ( ( insert_lambda_lambda @ ( F2 @ X6 ) @ ( image_2356535832909760517lambda @ F2 @ A ) )
= ( image_2356535832909760517lambda @ F2 @ A ) ) ) ).
% insert_image
thf(fact_494_insert__image,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,F2: list_lambda_lambda > lambda_lambda] :
( ( member2667661554319814297lambda @ X6 @ A )
=> ( ( insert_lambda_lambda @ ( F2 @ X6 ) @ ( image_2458965899662700159lambda @ F2 @ A ) )
= ( image_2458965899662700159lambda @ F2 @ A ) ) ) ).
% insert_image
thf(fact_495_image__insert,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A2: lambda_lambda,B: set_lambda_lambda] :
( ( image_3692823078739859851lambda @ F2 @ ( insert_lambda_lambda @ A2 @ B ) )
= ( insert8465602035467394112lambda @ ( F2 @ A2 ) @ ( image_3692823078739859851lambda @ F2 @ B ) ) ) ).
% image_insert
thf(fact_496_image__insert,axiom,
! [F2: lambda_lambda > lambda_lambda,A2: lambda_lambda,B: set_lambda_lambda] :
( ( image_2356535832909760517lambda @ F2 @ ( insert_lambda_lambda @ A2 @ B ) )
= ( insert_lambda_lambda @ ( F2 @ A2 ) @ ( image_2356535832909760517lambda @ F2 @ B ) ) ) ).
% image_insert
thf(fact_497_singletonI,axiom,
! [A2: lambda_lambda] : ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).
% singletonI
thf(fact_498_singletonI,axiom,
! [A2: list_lambda_lambda] : ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).
% singletonI
thf(fact_499_insert__subset,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ X6 @ A ) @ B )
= ( ( member_lambda_lambda @ X6 @ B )
& ( ord_le1926335974945931754lambda @ A @ B ) ) ) ).
% insert_subset
thf(fact_500_insert__subset,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( insert8465602035467394112lambda @ X6 @ A ) @ B )
= ( ( member2667661554319814297lambda @ X6 @ B )
& ( ord_le3064231726521830256lambda @ A @ B ) ) ) ).
% insert_subset
thf(fact_501_Int__insert__right__if1,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ ( insert_lambda_lambda @ A2 @ B ) )
= ( insert_lambda_lambda @ A2 @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_502_Int__insert__right__if1,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ ( insert8465602035467394112lambda @ A2 @ B ) )
= ( insert8465602035467394112lambda @ A2 @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_503_Int__insert__right__if0,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A2 @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ ( insert_lambda_lambda @ A2 @ B ) )
= ( inf_in1766959559283430712lambda @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_504_Int__insert__right__if0,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ A2 @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ ( insert8465602035467394112lambda @ A2 @ B ) )
= ( inf_in3354698378728726462lambda @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_505_insert__inter__insert,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ A ) @ ( insert_lambda_lambda @ A2 @ B ) )
= ( insert_lambda_lambda @ A2 @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_506_insert__inter__insert,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ A ) @ ( insert8465602035467394112lambda @ A2 @ B ) )
= ( insert8465602035467394112lambda @ A2 @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_507_Int__insert__left__if1,axiom,
! [A2: lambda_lambda,C2: set_lambda_lambda,B: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ B ) @ C2 )
= ( insert_lambda_lambda @ A2 @ ( inf_in1766959559283430712lambda @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_508_Int__insert__left__if1,axiom,
! [A2: list_lambda_lambda,C2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ C2 )
=> ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ B ) @ C2 )
= ( insert8465602035467394112lambda @ A2 @ ( inf_in3354698378728726462lambda @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_509_Int__insert__left__if0,axiom,
! [A2: lambda_lambda,C2: set_lambda_lambda,B: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A2 @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ B ) @ C2 )
= ( inf_in1766959559283430712lambda @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_510_Int__insert__left__if0,axiom,
! [A2: list_lambda_lambda,C2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ A2 @ C2 )
=> ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ B ) @ C2 )
= ( inf_in3354698378728726462lambda @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_511_single__Src__hd__in__sources,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T6 ) ) @ nil_lambda_lambda ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% single_Src_hd_in_sources
thf(fact_512_seq__imp__cong__Trg__last__Src__hd,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) @ ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) @ ( lambda_Trg @ ( last_lambda_lambda @ T6 ) ) ) ) ) ) ).
% seq_imp_cong_Trg_last_Src_hd
thf(fact_513_singleton__conv,axiom,
! [A2: lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] : ( X4 = A2 ) )
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).
% singleton_conv
thf(fact_514_singleton__conv,axiom,
! [A2: list_lambda_lambda] :
( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] : ( X4 = A2 ) )
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).
% singleton_conv
thf(fact_515_singleton__conv2,axiom,
! [A2: lambda_lambda] :
( ( collec621444948832755541lambda
@ ( ^ [Y6: lambda_lambda,Z: lambda_lambda] : ( Y6 = Z )
@ A2 ) )
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).
% singleton_conv2
thf(fact_516_singleton__conv2,axiom,
! [A2: list_lambda_lambda] :
( ( collec8830145823237851355lambda
@ ( ^ [Y6: list_lambda_lambda,Z: list_lambda_lambda] : ( Y6 = Z )
@ A2 ) )
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).
% singleton_conv2
thf(fact_517_singleton__insert__inj__eq_H,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B3: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ A )
= ( insert_lambda_lambda @ B3 @ bot_bo1271416176687173302lambda ) )
= ( ( A2 = B3 )
& ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B3 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_518_singleton__insert__inj__eq_H,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B3: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ A )
= ( insert8465602035467394112lambda @ B3 @ bot_bo1397037886704484924lambda ) )
= ( ( A2 = B3 )
& ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B3 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_519_singleton__insert__inj__eq,axiom,
! [B3: lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
( ( ( insert_lambda_lambda @ B3 @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B3 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_520_singleton__insert__inj__eq,axiom,
! [B3: list_lambda_lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( insert8465602035467394112lambda @ B3 @ bot_bo1397037886704484924lambda )
= ( insert8465602035467394112lambda @ A2 @ A ) )
= ( ( A2 = B3 )
& ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B3 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_521_insert__disjoint_I1_J,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ A ) @ B )
= bot_bo1271416176687173302lambda )
= ( ~ ( member_lambda_lambda @ A2 @ B )
& ( ( inf_in1766959559283430712lambda @ A @ B )
= bot_bo1271416176687173302lambda ) ) ) ).
% insert_disjoint(1)
thf(fact_522_insert__disjoint_I1_J,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ A ) @ B )
= bot_bo1397037886704484924lambda )
= ( ~ ( member2667661554319814297lambda @ A2 @ B )
& ( ( inf_in3354698378728726462lambda @ A @ B )
= bot_bo1397037886704484924lambda ) ) ) ).
% insert_disjoint(1)
thf(fact_523_insert__disjoint_I2_J,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ A ) @ B ) )
= ( ~ ( member_lambda_lambda @ A2 @ B )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_524_insert__disjoint_I2_J,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ A ) @ B ) )
= ( ~ ( member2667661554319814297lambda @ A2 @ B )
& ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_525_disjoint__insert_I1_J,axiom,
! [B: set_lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ B @ ( insert_lambda_lambda @ A2 @ A ) )
= bot_bo1271416176687173302lambda )
= ( ~ ( member_lambda_lambda @ A2 @ B )
& ( ( inf_in1766959559283430712lambda @ B @ A )
= bot_bo1271416176687173302lambda ) ) ) ).
% disjoint_insert(1)
thf(fact_526_disjoint__insert_I1_J,axiom,
! [B: set_li7878742357367608272lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ B @ ( insert8465602035467394112lambda @ A2 @ A ) )
= bot_bo1397037886704484924lambda )
= ( ~ ( member2667661554319814297lambda @ A2 @ B )
& ( ( inf_in3354698378728726462lambda @ B @ A )
= bot_bo1397037886704484924lambda ) ) ) ).
% disjoint_insert(1)
thf(fact_527_disjoint__insert_I2_J,axiom,
! [A: set_lambda_lambda,B3: lambda_lambda,B: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A @ ( insert_lambda_lambda @ B3 @ B ) ) )
= ( ~ ( member_lambda_lambda @ B3 @ A )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_528_disjoint__insert_I2_J,axiom,
! [A: set_li7878742357367608272lambda,B3: list_lambda_lambda,B: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A @ ( insert8465602035467394112lambda @ B3 @ B ) ) )
= ( ~ ( member2667661554319814297lambda @ B3 @ A )
& ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_529_conI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U3 )
=> ( ( ( lambda_Src @ ( hd_lambda_lambda @ T6 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 ) ) ) ) ).
% conI\<^sub>\<Lambda>\<^sub>P
thf(fact_530_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_531_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_532_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_533_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_534_seqI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ( lambda_Trg @ ( last_lambda_lambda @ T6 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U3 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 ) ) ) ) ).
% seqI\<^sub>\<Lambda>\<^sub>P
thf(fact_535_Int__insert__right,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ( member_lambda_lambda @ A2 @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ ( insert_lambda_lambda @ A2 @ B ) )
= ( insert_lambda_lambda @ A2 @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) )
& ( ~ ( member_lambda_lambda @ A2 @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ ( insert_lambda_lambda @ A2 @ B ) )
= ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_536_Int__insert__right,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ( member2667661554319814297lambda @ A2 @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ ( insert8465602035467394112lambda @ A2 @ B ) )
= ( insert8465602035467394112lambda @ A2 @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) )
& ( ~ ( member2667661554319814297lambda @ A2 @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ ( insert8465602035467394112lambda @ A2 @ B ) )
= ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_537_subset__insertI2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,B3: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_538_subset__insertI2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,B3: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B3 @ B ) ) ) ).
% subset_insertI2
thf(fact_539_Int__insert__left,axiom,
! [A2: lambda_lambda,C2: set_lambda_lambda,B: set_lambda_lambda] :
( ( ( member_lambda_lambda @ A2 @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ B ) @ C2 )
= ( insert_lambda_lambda @ A2 @ ( inf_in1766959559283430712lambda @ B @ C2 ) ) ) )
& ( ~ ( member_lambda_lambda @ A2 @ C2 )
=> ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ B ) @ C2 )
= ( inf_in1766959559283430712lambda @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_540_Int__insert__left,axiom,
! [A2: list_lambda_lambda,C2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ( member2667661554319814297lambda @ A2 @ C2 )
=> ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ B ) @ C2 )
= ( insert8465602035467394112lambda @ A2 @ ( inf_in3354698378728726462lambda @ B @ C2 ) ) ) )
& ( ~ ( member2667661554319814297lambda @ A2 @ C2 )
=> ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ B ) @ C2 )
= ( inf_in3354698378728726462lambda @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_541_subset__insertI,axiom,
! [B: set_lambda_lambda,A2: lambda_lambda] : ( ord_le1926335974945931754lambda @ B @ ( insert_lambda_lambda @ A2 @ B ) ) ).
% subset_insertI
thf(fact_542_subset__insertI,axiom,
! [B: set_li7878742357367608272lambda,A2: list_lambda_lambda] : ( ord_le3064231726521830256lambda @ B @ ( insert8465602035467394112lambda @ A2 @ B ) ) ).
% subset_insertI
thf(fact_543_subset__insert,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X6 @ A )
=> ( ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ X6 @ B ) )
= ( ord_le1926335974945931754lambda @ A @ B ) ) ) ).
% subset_insert
thf(fact_544_subset__insert,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ X6 @ A )
=> ( ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ X6 @ B ) )
= ( ord_le3064231726521830256lambda @ A @ B ) ) ) ).
% subset_insert
thf(fact_545_insert__mono,axiom,
! [C2: set_lambda_lambda,D: set_lambda_lambda,A2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ D )
=> ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ A2 @ C2 ) @ ( insert_lambda_lambda @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_546_insert__mono,axiom,
! [C2: set_li7878742357367608272lambda,D: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ C2 @ D )
=> ( ord_le3064231726521830256lambda @ ( insert8465602035467394112lambda @ A2 @ C2 ) @ ( insert8465602035467394112lambda @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_547_inf__set__def,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( collec621444948832755541lambda
@ ( inf_in2068736301489225125mbda_o
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ A6 )
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_548_inf__set__def,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( collec8830145823237851355lambda
@ ( inf_in6319970306715391135mbda_o
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ A6 )
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_549_Collect__conv__if2,axiom,
! [P: lambda_lambda > $o,A2: lambda_lambda] :
( ( ( P @ A2 )
=> ( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bo1271416176687173302lambda ) ) ) ).
% Collect_conv_if2
thf(fact_550_Collect__conv__if2,axiom,
! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
( ( ( P @ A2 )
=> ( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bo1397037886704484924lambda ) ) ) ).
% Collect_conv_if2
thf(fact_551_Collect__conv__if,axiom,
! [P: lambda_lambda > $o,A2: lambda_lambda] :
( ( ( P @ A2 )
=> ( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bo1271416176687173302lambda ) ) ) ).
% Collect_conv_if
thf(fact_552_Collect__conv__if,axiom,
! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
( ( ( P @ A2 )
=> ( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bo1397037886704484924lambda ) ) ) ).
% Collect_conv_if
thf(fact_553_insertE,axiom,
! [A2: lambda_lambda,B3: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member_lambda_lambda @ A2 @ A ) ) ) ).
% insertE
thf(fact_554_insertE,axiom,
! [A2: list_lambda_lambda,B3: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ B3 @ A ) )
=> ( ( A2 != B3 )
=> ( member2667661554319814297lambda @ A2 @ A ) ) ) ).
% insertE
thf(fact_555_insertI1,axiom,
! [A2: lambda_lambda,B: set_lambda_lambda] : ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ A2 @ B ) ) ).
% insertI1
thf(fact_556_insertI1,axiom,
! [A2: list_lambda_lambda,B: set_li7878742357367608272lambda] : ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ A2 @ B ) ) ).
% insertI1
thf(fact_557_insertI2,axiom,
! [A2: lambda_lambda,B: set_lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ B )
=> ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B3 @ B ) ) ) ).
% insertI2
thf(fact_558_insertI2,axiom,
! [A2: list_lambda_lambda,B: set_li7878742357367608272lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ B )
=> ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ B3 @ B ) ) ) ).
% insertI2
thf(fact_559_Set_Oset__insert,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ~ ! [B6: set_lambda_lambda] :
( ( A
= ( insert_lambda_lambda @ X6 @ B6 ) )
=> ( member_lambda_lambda @ X6 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_560_Set_Oset__insert,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ X6 @ A )
=> ~ ! [B6: set_li7878742357367608272lambda] :
( ( A
= ( insert8465602035467394112lambda @ X6 @ B6 ) )
=> ( member2667661554319814297lambda @ X6 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_561_insert__compr,axiom,
( insert_lambda_lambda
= ( ^ [A4: lambda_lambda,B5: set_lambda_lambda] :
( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( X4 = A4 )
| ( member_lambda_lambda @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_562_insert__compr,axiom,
( insert8465602035467394112lambda
= ( ^ [A4: list_lambda_lambda,B5: set_li7878742357367608272lambda] :
( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( X4 = A4 )
| ( member2667661554319814297lambda @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_563_insert__ident,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X6 @ A )
=> ( ~ ( member_lambda_lambda @ X6 @ B )
=> ( ( ( insert_lambda_lambda @ X6 @ A )
= ( insert_lambda_lambda @ X6 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_564_insert__ident,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ X6 @ A )
=> ( ~ ( member2667661554319814297lambda @ X6 @ B )
=> ( ( ( insert8465602035467394112lambda @ X6 @ A )
= ( insert8465602035467394112lambda @ X6 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_565_insert__absorb,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ A )
=> ( ( insert_lambda_lambda @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_566_insert__absorb,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ A )
=> ( ( insert8465602035467394112lambda @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_567_insert__eq__iff,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B3: lambda_lambda,B: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A2 @ A )
=> ( ~ ( member_lambda_lambda @ B3 @ B )
=> ( ( ( insert_lambda_lambda @ A2 @ A )
= ( insert_lambda_lambda @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_lambda_lambda] :
( ( A
= ( insert_lambda_lambda @ B3 @ C3 ) )
& ~ ( member_lambda_lambda @ B3 @ C3 )
& ( B
= ( insert_lambda_lambda @ A2 @ C3 ) )
& ~ ( member_lambda_lambda @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_568_insert__eq__iff,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B3: list_lambda_lambda,B: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ A2 @ A )
=> ( ~ ( member2667661554319814297lambda @ B3 @ B )
=> ( ( ( insert8465602035467394112lambda @ A2 @ A )
= ( insert8465602035467394112lambda @ B3 @ B ) )
= ( ( ( A2 = B3 )
=> ( A = B ) )
& ( ( A2 != B3 )
=> ? [C3: set_li7878742357367608272lambda] :
( ( A
= ( insert8465602035467394112lambda @ B3 @ C3 ) )
& ~ ( member2667661554319814297lambda @ B3 @ C3 )
& ( B
= ( insert8465602035467394112lambda @ A2 @ C3 ) )
& ~ ( member2667661554319814297lambda @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_569_insert__Collect,axiom,
! [A2: lambda_lambda,P: lambda_lambda > $o] :
( ( insert_lambda_lambda @ A2 @ ( collec621444948832755541lambda @ P ) )
= ( collec621444948832755541lambda
@ ^ [U6: lambda_lambda] :
( ( U6 != A2 )
=> ( P @ U6 ) ) ) ) ).
% insert_Collect
thf(fact_570_insert__Collect,axiom,
! [A2: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( insert8465602035467394112lambda @ A2 @ ( collec8830145823237851355lambda @ P ) )
= ( collec8830145823237851355lambda
@ ^ [U6: list_lambda_lambda] :
( ( U6 != A2 )
=> ( P @ U6 ) ) ) ) ).
% insert_Collect
thf(fact_571_insert__commute,axiom,
! [X6: lambda_lambda,Y5: lambda_lambda,A: set_lambda_lambda] :
( ( insert_lambda_lambda @ X6 @ ( insert_lambda_lambda @ Y5 @ A ) )
= ( insert_lambda_lambda @ Y5 @ ( insert_lambda_lambda @ X6 @ A ) ) ) ).
% insert_commute
thf(fact_572_mk__disjoint__insert,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ A )
=> ? [B6: set_lambda_lambda] :
( ( A
= ( insert_lambda_lambda @ A2 @ B6 ) )
& ~ ( member_lambda_lambda @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_573_mk__disjoint__insert,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ A )
=> ? [B6: set_li7878742357367608272lambda] :
( ( A
= ( insert8465602035467394112lambda @ A2 @ B6 ) )
& ~ ( member2667661554319814297lambda @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_574_singleton__inject,axiom,
! [A2: lambda_lambda,B3: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ B3 @ bot_bo1271416176687173302lambda ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_575_singleton__inject,axiom,
! [A2: list_lambda_lambda,B3: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda )
= ( insert8465602035467394112lambda @ B3 @ bot_bo1397037886704484924lambda ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_576_insert__not__empty,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda] :
( ( insert_lambda_lambda @ A2 @ A )
!= bot_bo1271416176687173302lambda ) ).
% insert_not_empty
thf(fact_577_insert__not__empty,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( insert8465602035467394112lambda @ A2 @ A )
!= bot_bo1397037886704484924lambda ) ).
% insert_not_empty
thf(fact_578_doubleton__eq__iff,axiom,
! [A2: lambda_lambda,B3: lambda_lambda,C: lambda_lambda,D2: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B3 @ bot_bo1271416176687173302lambda ) )
= ( insert_lambda_lambda @ C @ ( insert_lambda_lambda @ D2 @ bot_bo1271416176687173302lambda ) ) )
= ( ( ( A2 = C )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_579_doubleton__eq__iff,axiom,
! [A2: list_lambda_lambda,B3: list_lambda_lambda,C: list_lambda_lambda,D2: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ ( insert8465602035467394112lambda @ B3 @ bot_bo1397037886704484924lambda ) )
= ( insert8465602035467394112lambda @ C @ ( insert8465602035467394112lambda @ D2 @ bot_bo1397037886704484924lambda ) ) )
= ( ( ( A2 = C )
& ( B3 = D2 ) )
| ( ( A2 = D2 )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_580_singleton__iff,axiom,
! [B3: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_581_singleton__iff,axiom,
! [B3: list_lambda_lambda,A2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B3 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_582_bot__set__def,axiom,
( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ bot_bo3555140721347798439mbda_o ) ) ).
% bot_set_def
thf(fact_583_bot__set__def,axiom,
( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda @ bot_bo430095131446760353mbda_o ) ) ).
% bot_set_def
thf(fact_584_singletonD,axiom,
! [B3: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_585_singletonD,axiom,
! [B3: list_lambda_lambda,A2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B3 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_586_lambda__calculus_OSrc_Osimps_I1_J,axiom,
( ( lambda_Src @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Src.simps(1)
thf(fact_587_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T4: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_Src @ ( lambda_App @ T4 @ U2 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T4 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U2 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_588_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_589_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_590_subset__singleton__iff,axiom,
! [X7: set_lambda_lambda,A2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X7 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
= ( ( X7 = bot_bo1271416176687173302lambda )
| ( X7
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_591_subset__singleton__iff,axiom,
! [X7: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ X7 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
= ( ( X7 = bot_bo1397037886704484924lambda )
| ( X7
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_592_subset__singletonD,axiom,
! [A: set_lambda_lambda,X6: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ X6 @ bot_bo1271416176687173302lambda ) )
=> ( ( A = bot_bo1271416176687173302lambda )
| ( A
= ( insert_lambda_lambda @ X6 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singletonD
thf(fact_593_subset__singletonD,axiom,
! [A: set_li7878742357367608272lambda,X6: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ X6 @ bot_bo1397037886704484924lambda ) )
=> ( ( A = bot_bo1397037886704484924lambda )
| ( A
= ( insert8465602035467394112lambda @ X6 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% subset_singletonD
thf(fact_594_image__constant__conv,axiom,
! [A: set_lambda_lambda,C: lambda_lambda] :
( ( ( A = bot_bo1271416176687173302lambda )
=> ( ( image_2356535832909760517lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= bot_bo1271416176687173302lambda ) )
& ( ( A != bot_bo1271416176687173302lambda )
=> ( ( image_2356535832909760517lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ) ).
% image_constant_conv
thf(fact_595_image__constant__conv,axiom,
! [A: set_lambda_lambda,C: list_lambda_lambda] :
( ( ( A = bot_bo1271416176687173302lambda )
=> ( ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= bot_bo1397037886704484924lambda ) )
& ( ( A != bot_bo1271416176687173302lambda )
=> ( ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ) ).
% image_constant_conv
thf(fact_596_image__constant__conv,axiom,
! [A: set_li7878742357367608272lambda,C: lambda_lambda] :
( ( ( A = bot_bo1397037886704484924lambda )
=> ( ( image_2458965899662700159lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= bot_bo1271416176687173302lambda ) )
& ( ( A != bot_bo1397037886704484924lambda )
=> ( ( image_2458965899662700159lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ) ).
% image_constant_conv
thf(fact_597_image__constant__conv,axiom,
! [A: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( ( A = bot_bo1397037886704484924lambda )
=> ( ( image_5886195720093285125lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= bot_bo1397037886704484924lambda ) )
& ( ( A != bot_bo1397037886704484924lambda )
=> ( ( image_5886195720093285125lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ) ).
% image_constant_conv
thf(fact_598_image__constant,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,C: lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ( ( image_2356535832909760517lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ).
% image_constant
thf(fact_599_image__constant,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,C: lambda_lambda] :
( ( member2667661554319814297lambda @ X6 @ A )
=> ( ( image_2458965899662700159lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ).
% image_constant
thf(fact_600_image__constant,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,C: list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ( ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ).
% image_constant
thf(fact_601_image__constant,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X6 @ A )
=> ( ( image_5886195720093285125lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ).
% image_constant
thf(fact_602_lambda__calculus_Ois__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% lambda_calculus.is_confluent_rts
thf(fact_603_lambda__calculus_OSrc__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_Src @ ( lambda_resid @ T @ U ) )
= ( lambda_Trg @ U ) ) ) ).
% lambda_calculus.Src_resid
thf(fact_604_imageI,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,F2: lambda_lambda > lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ( member_lambda_lambda @ ( F2 @ X6 ) @ ( image_2356535832909760517lambda @ F2 @ A ) ) ) ).
% imageI
thf(fact_605_imageI,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,F2: lambda_lambda > list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ( member2667661554319814297lambda @ ( F2 @ X6 ) @ ( image_3692823078739859851lambda @ F2 @ A ) ) ) ).
% imageI
thf(fact_606_imageI,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,F2: list_lambda_lambda > lambda_lambda] :
( ( member2667661554319814297lambda @ X6 @ A )
=> ( member_lambda_lambda @ ( F2 @ X6 ) @ ( image_2458965899662700159lambda @ F2 @ A ) ) ) ).
% imageI
thf(fact_607_imageI,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,F2: list_lambda_lambda > list_lambda_lambda] :
( ( member2667661554319814297lambda @ X6 @ A )
=> ( member2667661554319814297lambda @ ( F2 @ X6 ) @ ( image_5886195720093285125lambda @ F2 @ A ) ) ) ).
% imageI
thf(fact_608_image__iff,axiom,
! [Z2: list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( member2667661554319814297lambda @ Z2 @ ( image_3692823078739859851lambda @ F2 @ A ) )
= ( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
& ( Z2
= ( F2 @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_609_bex__imageD,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,P: list_lambda_lambda > $o] :
( ? [X2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ ( image_3692823078739859851lambda @ F2 @ A ) )
& ( P @ X2 ) )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
& ( P @ ( F2 @ X ) ) ) ) ).
% bex_imageD
thf(fact_610_image__cong,axiom,
! [M: set_lambda_lambda,N: set_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,G: lambda_lambda > list_lambda_lambda] :
( ( M = N )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ N )
=> ( ( F2 @ X )
= ( G @ X ) ) )
=> ( ( image_3692823078739859851lambda @ F2 @ M )
= ( image_3692823078739859851lambda @ G @ N ) ) ) ) ).
% image_cong
thf(fact_611_ball__imageD,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,P: list_lambda_lambda > $o] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ ( image_3692823078739859851lambda @ F2 @ A ) )
=> ( P @ X ) )
=> ! [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ A )
=> ( P @ ( F2 @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_612_rev__image__eqI,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,B3: lambda_lambda,F2: lambda_lambda > lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ( ( B3
= ( F2 @ X6 ) )
=> ( member_lambda_lambda @ B3 @ ( image_2356535832909760517lambda @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_613_rev__image__eqI,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,B3: list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ A )
=> ( ( B3
= ( F2 @ X6 ) )
=> ( member2667661554319814297lambda @ B3 @ ( image_3692823078739859851lambda @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_614_rev__image__eqI,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,B3: lambda_lambda,F2: list_lambda_lambda > lambda_lambda] :
( ( member2667661554319814297lambda @ X6 @ A )
=> ( ( B3
= ( F2 @ X6 ) )
=> ( member_lambda_lambda @ B3 @ ( image_2458965899662700159lambda @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_615_rev__image__eqI,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,B3: list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda] :
( ( member2667661554319814297lambda @ X6 @ A )
=> ( ( B3
= ( F2 @ X6 ) )
=> ( member2667661554319814297lambda @ B3 @ ( image_5886195720093285125lambda @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_616_emptyE,axiom,
! [A2: lambda_lambda] :
~ ( member_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ).
% emptyE
thf(fact_617_emptyE,axiom,
! [A2: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ A2 @ bot_bo1397037886704484924lambda ) ).
% emptyE
thf(fact_618_equals0D,axiom,
! [A: set_lambda_lambda,A2: lambda_lambda] :
( ( A = bot_bo1271416176687173302lambda )
=> ~ ( member_lambda_lambda @ A2 @ A ) ) ).
% equals0D
thf(fact_619_equals0D,axiom,
! [A: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
( ( A = bot_bo1397037886704484924lambda )
=> ~ ( member2667661554319814297lambda @ A2 @ A ) ) ).
% equals0D
thf(fact_620_equals0I,axiom,
! [A: set_lambda_lambda] :
( ! [Y2: lambda_lambda] :
~ ( member_lambda_lambda @ Y2 @ A )
=> ( A = bot_bo1271416176687173302lambda ) ) ).
% equals0I
thf(fact_621_equals0I,axiom,
! [A: set_li7878742357367608272lambda] :
( ! [Y2: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ Y2 @ A )
=> ( A = bot_bo1397037886704484924lambda ) ) ).
% equals0I
thf(fact_622_ex__in__conv,axiom,
! [A: set_lambda_lambda] :
( ( ? [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ A ) )
= ( A != bot_bo1271416176687173302lambda ) ) ).
% ex_in_conv
thf(fact_623_ex__in__conv,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ? [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ A ) )
= ( A != bot_bo1397037886704484924lambda ) ) ).
% ex_in_conv
thf(fact_624_in__mono,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,X6: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( member_lambda_lambda @ X6 @ A )
=> ( member_lambda_lambda @ X6 @ B ) ) ) ).
% in_mono
thf(fact_625_in__mono,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,X6: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( member2667661554319814297lambda @ X6 @ A )
=> ( member2667661554319814297lambda @ X6 @ B ) ) ) ).
% in_mono
thf(fact_626_subsetD,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( member_lambda_lambda @ C @ A )
=> ( member_lambda_lambda @ C @ B ) ) ) ).
% subsetD
thf(fact_627_subsetD,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( member2667661554319814297lambda @ C @ A )
=> ( member2667661554319814297lambda @ C @ B ) ) ) ).
% subsetD
thf(fact_628_equalityE,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A = B )
=> ~ ( ( ord_le1926335974945931754lambda @ A @ B )
=> ~ ( ord_le1926335974945931754lambda @ B @ A ) ) ) ).
% equalityE
thf(fact_629_equalityE,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A = B )
=> ~ ( ( ord_le3064231726521830256lambda @ A @ B )
=> ~ ( ord_le3064231726521830256lambda @ B @ A ) ) ) ).
% equalityE
thf(fact_630_subset__eq,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A6 )
=> ( member_lambda_lambda @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_631_subset__eq,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
! [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A6 )
=> ( member2667661554319814297lambda @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_632_equalityD1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A = B )
=> ( ord_le1926335974945931754lambda @ A @ B ) ) ).
% equalityD1
thf(fact_633_equalityD1,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A = B )
=> ( ord_le3064231726521830256lambda @ A @ B ) ) ).
% equalityD1
thf(fact_634_equalityD2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A = B )
=> ( ord_le1926335974945931754lambda @ B @ A ) ) ).
% equalityD2
thf(fact_635_equalityD2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A = B )
=> ( ord_le3064231726521830256lambda @ B @ A ) ) ).
% equalityD2
thf(fact_636_subset__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
! [T5: lambda_lambda] :
( ( member_lambda_lambda @ T5 @ A6 )
=> ( member_lambda_lambda @ T5 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_637_subset__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
! [T5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ T5 @ A6 )
=> ( member2667661554319814297lambda @ T5 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_638_subset__refl,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A @ A ) ).
% subset_refl
thf(fact_639_subset__refl,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ A @ A ) ).
% subset_refl
thf(fact_640_Collect__mono,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ! [X: lambda_lambda] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le1926335974945931754lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) ) ) ).
% Collect_mono
thf(fact_641_Collect__mono,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ! [X: list_lambda_lambda] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3064231726521830256lambda @ ( collec8830145823237851355lambda @ P ) @ ( collec8830145823237851355lambda @ Q ) ) ) ).
% Collect_mono
thf(fact_642_subset__trans,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( ord_le1926335974945931754lambda @ B @ C2 )
=> ( ord_le1926335974945931754lambda @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_643_subset__trans,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ C2 )
=> ( ord_le3064231726521830256lambda @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_644_set__eq__subset,axiom,
( ( ^ [Y6: set_lambda_lambda,Z: set_lambda_lambda] : ( Y6 = Z ) )
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A6 @ B5 )
& ( ord_le1926335974945931754lambda @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_645_set__eq__subset,axiom,
( ( ^ [Y6: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] : ( Y6 = Z ) )
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A6 @ B5 )
& ( ord_le3064231726521830256lambda @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_646_Collect__mono__iff,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) )
= ( ! [X4: lambda_lambda] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_647_Collect__mono__iff,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( ord_le3064231726521830256lambda @ ( collec8830145823237851355lambda @ P ) @ ( collec8830145823237851355lambda @ Q ) )
= ( ! [X4: list_lambda_lambda] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_648_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_lambda_lambda,K: set_lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( A
= ( inf_in1766959559283430712lambda @ K @ A2 ) )
=> ( ( inf_in1766959559283430712lambda @ A @ B3 )
= ( inf_in1766959559283430712lambda @ K @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_649_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_li7878742357367608272lambda,K: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( A
= ( inf_in3354698378728726462lambda @ K @ A2 ) )
=> ( ( inf_in3354698378728726462lambda @ A @ B3 )
= ( inf_in3354698378728726462lambda @ K @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_650_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_lambda_lambda,K: set_lambda_lambda,B3: set_lambda_lambda,A2: set_lambda_lambda] :
( ( B
= ( inf_in1766959559283430712lambda @ K @ B3 ) )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B )
= ( inf_in1766959559283430712lambda @ K @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_651_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_li7878742357367608272lambda,K: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( B
= ( inf_in3354698378728726462lambda @ K @ B3 ) )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B )
= ( inf_in3354698378728726462lambda @ K @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_652_IntE,axiom,
! [C: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A @ B ) )
=> ~ ( ( member_lambda_lambda @ C @ A )
=> ~ ( member_lambda_lambda @ C @ B ) ) ) ).
% IntE
thf(fact_653_IntE,axiom,
! [C: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A @ B ) )
=> ~ ( ( member2667661554319814297lambda @ C @ A )
=> ~ ( member2667661554319814297lambda @ C @ B ) ) ) ).
% IntE
thf(fact_654_IntD1,axiom,
! [C: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A @ B ) )
=> ( member_lambda_lambda @ C @ A ) ) ).
% IntD1
thf(fact_655_IntD1,axiom,
! [C: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A @ B ) )
=> ( member2667661554319814297lambda @ C @ A ) ) ).
% IntD1
thf(fact_656_IntD2,axiom,
! [C: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( member_lambda_lambda @ C @ ( inf_in1766959559283430712lambda @ A @ B ) )
=> ( member_lambda_lambda @ C @ B ) ) ).
% IntD2
thf(fact_657_IntD2,axiom,
! [C: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A @ B ) )
=> ( member2667661554319814297lambda @ C @ B ) ) ).
% IntD2
thf(fact_658_Int__assoc,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ C2 )
= ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_659_Int__assoc,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ C2 )
= ( inf_in3354698378728726462lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_660_Int__absorb,axiom,
! [A: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ A )
= A ) ).
% Int_absorb
thf(fact_661_Int__absorb,axiom,
! [A: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ A )
= A ) ).
% Int_absorb
thf(fact_662_Int__commute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ B5 @ A6 ) ) ) ).
% Int_commute
thf(fact_663_Int__commute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ B5 @ A6 ) ) ) ).
% Int_commute
thf(fact_664_Int__left__absorb,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ A @ B ) )
= ( inf_in1766959559283430712lambda @ A @ B ) ) ).
% Int_left_absorb
thf(fact_665_Int__left__absorb,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ ( inf_in3354698378728726462lambda @ A @ B ) )
= ( inf_in3354698378728726462lambda @ A @ B ) ) ).
% Int_left_absorb
thf(fact_666_Int__left__commute,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,C2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ ( inf_in1766959559283430712lambda @ B @ C2 ) )
= ( inf_in1766959559283430712lambda @ B @ ( inf_in1766959559283430712lambda @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_667_Int__left__commute,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ ( inf_in3354698378728726462lambda @ B @ C2 ) )
= ( inf_in3354698378728726462lambda @ B @ ( inf_in3354698378728726462lambda @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_668_imageE,axiom,
! [B3: lambda_lambda,F2: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( image_2356535832909760517lambda @ F2 @ A ) )
=> ~ ! [X: lambda_lambda] :
( ( B3
= ( F2 @ X ) )
=> ~ ( member_lambda_lambda @ X @ A ) ) ) ).
% imageE
thf(fact_669_imageE,axiom,
! [B3: lambda_lambda,F2: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member_lambda_lambda @ B3 @ ( image_2458965899662700159lambda @ F2 @ A ) )
=> ~ ! [X: list_lambda_lambda] :
( ( B3
= ( F2 @ X ) )
=> ~ ( member2667661554319814297lambda @ X @ A ) ) ) ).
% imageE
thf(fact_670_imageE,axiom,
! [B3: list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( member2667661554319814297lambda @ B3 @ ( image_3692823078739859851lambda @ F2 @ A ) )
=> ~ ! [X: lambda_lambda] :
( ( B3
= ( F2 @ X ) )
=> ~ ( member_lambda_lambda @ X @ A ) ) ) ).
% imageE
thf(fact_671_imageE,axiom,
! [B3: list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ B3 @ ( image_5886195720093285125lambda @ F2 @ A ) )
=> ~ ! [X: list_lambda_lambda] :
( ( B3
= ( F2 @ X ) )
=> ~ ( member2667661554319814297lambda @ X @ A ) ) ) ).
% imageE
thf(fact_672_image__image,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda,G: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( image_5886195720093285125lambda @ F2 @ ( image_3692823078739859851lambda @ G @ A ) )
= ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : ( F2 @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_673_image__image,axiom,
! [F2: lambda_lambda > list_lambda_lambda,G: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( image_3692823078739859851lambda @ F2 @ ( image_2356535832909760517lambda @ G @ A ) )
= ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : ( F2 @ ( G @ X4 ) )
@ A ) ) ).
% image_image
thf(fact_674_Compr__image__eq,axiom,
! [F2: lambda_lambda > lambda_lambda,A: set_lambda_lambda,P: lambda_lambda > $o] :
( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( image_2356535832909760517lambda @ F2 @ A ) )
& ( P @ X4 ) ) )
= ( image_2356535832909760517lambda @ F2
@ ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
& ( P @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_675_Compr__image__eq,axiom,
! [F2: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda,P: lambda_lambda > $o] :
( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( image_2458965899662700159lambda @ F2 @ A ) )
& ( P @ X4 ) ) )
= ( image_2458965899662700159lambda @ F2
@ ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A )
& ( P @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_676_Compr__image__eq,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,P: list_lambda_lambda > $o] :
( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ ( image_3692823078739859851lambda @ F2 @ A ) )
& ( P @ X4 ) ) )
= ( image_3692823078739859851lambda @ F2
@ ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
& ( P @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_677_Compr__image__eq,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ ( image_5886195720093285125lambda @ F2 @ A ) )
& ( P @ X4 ) ) )
= ( image_5886195720093285125lambda @ F2
@ ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A )
& ( P @ ( F2 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_678_empty__def,axiom,
( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] : $false ) ) ).
% empty_def
thf(fact_679_empty__def,axiom,
( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] : $false ) ) ).
% empty_def
thf(fact_680_Collect__subset,axiom,
! [A: set_lambda_lambda,P: lambda_lambda > $o] :
( ord_le1926335974945931754lambda
@ ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_681_Collect__subset,axiom,
! [A: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
( ord_le3064231726521830256lambda
@ ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_682_Int__def,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A6 )
& ( member_lambda_lambda @ X4 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_683_Int__def,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A6 )
& ( member2667661554319814297lambda @ X4 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_684_Int__Collect,axiom,
! [X6: lambda_lambda,A: set_lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ X6 @ ( inf_in1766959559283430712lambda @ A @ ( collec621444948832755541lambda @ P ) ) )
= ( ( member_lambda_lambda @ X6 @ A )
& ( P @ X6 ) ) ) ).
% Int_Collect
thf(fact_685_Int__Collect,axiom,
! [X6: list_lambda_lambda,A: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
( ( member2667661554319814297lambda @ X6 @ ( inf_in3354698378728726462lambda @ A @ ( collec8830145823237851355lambda @ P ) ) )
= ( ( member2667661554319814297lambda @ X6 @ A )
& ( P @ X6 ) ) ) ).
% Int_Collect
thf(fact_686_Collect__conj__eq,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_in1766959559283430712lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_687_Collect__conj__eq,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( P @ X4 )
& ( Q @ X4 ) ) )
= ( inf_in3354698378728726462lambda @ ( collec8830145823237851355lambda @ P ) @ ( collec8830145823237851355lambda @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_688_image__mono,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F2: lambda_lambda > lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F2 @ A ) @ ( image_2356535832909760517lambda @ F2 @ B ) ) ) ).
% image_mono
thf(fact_689_image__mono,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,F2: lambda_lambda > list_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F2 @ A ) @ ( image_3692823078739859851lambda @ F2 @ B ) ) ) ).
% image_mono
thf(fact_690_image__mono,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F2: list_lambda_lambda > lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F2 @ A ) @ ( image_2458965899662700159lambda @ F2 @ B ) ) ) ).
% image_mono
thf(fact_691_image__mono,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F2: list_lambda_lambda > list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F2 @ A ) @ ( image_5886195720093285125lambda @ F2 @ B ) ) ) ).
% image_mono
thf(fact_692_image__subsetI,axiom,
! [A: set_lambda_lambda,F2: lambda_lambda > lambda_lambda,B: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( member_lambda_lambda @ ( F2 @ X ) @ B ) )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_693_image__subsetI,axiom,
! [A: set_li7878742357367608272lambda,F2: list_lambda_lambda > lambda_lambda,B: set_lambda_lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( member_lambda_lambda @ ( F2 @ X ) @ B ) )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_694_image__subsetI,axiom,
! [A: set_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,B: set_li7878742357367608272lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( member2667661554319814297lambda @ ( F2 @ X ) @ B ) )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_695_image__subsetI,axiom,
! [A: set_li7878742357367608272lambda,F2: list_lambda_lambda > list_lambda_lambda,B: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( member2667661554319814297lambda @ ( F2 @ X ) @ B ) )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F2 @ A ) @ B ) ) ).
% image_subsetI
thf(fact_696_subset__imageE,axiom,
! [B: set_lambda_lambda,F2: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ ( image_2356535832909760517lambda @ F2 @ A ) )
=> ~ ! [C4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C4 @ A )
=> ( B
!= ( image_2356535832909760517lambda @ F2 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_697_subset__imageE,axiom,
! [B: set_lambda_lambda,F2: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ B @ ( image_2458965899662700159lambda @ F2 @ A ) )
=> ~ ! [C4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C4 @ A )
=> ( B
!= ( image_2458965899662700159lambda @ F2 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_698_subset__imageE,axiom,
! [B: set_li7878742357367608272lambda,F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ B @ ( image_3692823078739859851lambda @ F2 @ A ) )
=> ~ ! [C4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C4 @ A )
=> ( B
!= ( image_3692823078739859851lambda @ F2 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_699_subset__imageE,axiom,
! [B: set_li7878742357367608272lambda,F2: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ ( image_5886195720093285125lambda @ F2 @ A ) )
=> ~ ! [C4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C4 @ A )
=> ( B
!= ( image_5886195720093285125lambda @ F2 @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_700_image__subset__iff,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F2 @ A ) @ B )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
=> ( member2667661554319814297lambda @ ( F2 @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_701_subset__image__iff,axiom,
! [B: set_lambda_lambda,F2: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ ( image_2356535832909760517lambda @ F2 @ A ) )
= ( ? [AA: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ AA @ A )
& ( B
= ( image_2356535832909760517lambda @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_702_subset__image__iff,axiom,
! [B: set_lambda_lambda,F2: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ B @ ( image_2458965899662700159lambda @ F2 @ A ) )
= ( ? [AA: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ AA @ A )
& ( B
= ( image_2458965899662700159lambda @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_703_subset__image__iff,axiom,
! [B: set_li7878742357367608272lambda,F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ B @ ( image_3692823078739859851lambda @ F2 @ A ) )
= ( ? [AA: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ AA @ A )
& ( B
= ( image_3692823078739859851lambda @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_704_subset__image__iff,axiom,
! [B: set_li7878742357367608272lambda,F2: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ ( image_5886195720093285125lambda @ F2 @ A ) )
= ( ? [AA: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ AA @ A )
& ( B
= ( image_5886195720093285125lambda @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_705_disjoint__iff__not__equal,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A @ B )
= bot_bo1271416176687173302lambda )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
=> ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_706_disjoint__iff__not__equal,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ A @ B )
= bot_bo1397037886704484924lambda )
= ( ! [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A )
=> ! [Y4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y4 @ B )
=> ( X4 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_707_Int__empty__right,axiom,
! [A: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_right
thf(fact_708_Int__empty__right,axiom,
! [A: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% Int_empty_right
thf(fact_709_Int__empty__left,axiom,
! [B: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ B )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_left
thf(fact_710_Int__empty__left,axiom,
! [B: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ B )
= bot_bo1397037886704484924lambda ) ).
% Int_empty_left
thf(fact_711_disjoint__iff,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A @ B )
= bot_bo1271416176687173302lambda )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
=> ~ ( member_lambda_lambda @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_712_disjoint__iff,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ A @ B )
= bot_bo1397037886704484924lambda )
= ( ! [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A )
=> ~ ( member2667661554319814297lambda @ X4 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_713_Int__emptyI,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ~ ( member_lambda_lambda @ X @ B ) )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= bot_bo1271416176687173302lambda ) ) ).
% Int_emptyI
thf(fact_714_Int__emptyI,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ~ ( member2667661554319814297lambda @ X @ B ) )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= bot_bo1397037886704484924lambda ) ) ).
% Int_emptyI
thf(fact_715_Int__mono,axiom,
! [A: set_lambda_lambda,C2: set_lambda_lambda,B: set_lambda_lambda,D: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ C2 )
=> ( ( ord_le1926335974945931754lambda @ B @ D )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ ( inf_in1766959559283430712lambda @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_716_Int__mono,axiom,
! [A: set_li7878742357367608272lambda,C2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,D: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ C2 )
=> ( ( ord_le3064231726521830256lambda @ B @ D )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ ( inf_in3354698378728726462lambda @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_717_Int__lower1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_718_Int__lower1,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_719_Int__lower2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_720_Int__lower2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_721_Int__absorb1,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_722_Int__absorb1,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_723_Int__absorb2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_724_Int__absorb2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_725_Int__greatest,axiom,
! [C2: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C2 @ A )
=> ( ( ord_le1926335974945931754lambda @ C2 @ B )
=> ( ord_le1926335974945931754lambda @ C2 @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_726_Int__greatest,axiom,
! [C2: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C2 @ A )
=> ( ( ord_le3064231726521830256lambda @ C2 @ B )
=> ( ord_le3064231726521830256lambda @ C2 @ ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_727_Int__Collect__mono,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ ( collec621444948832755541lambda @ P ) ) @ ( inf_in1766959559283430712lambda @ B @ ( collec621444948832755541lambda @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_728_Int__Collect__mono,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ ( collec8830145823237851355lambda @ P ) ) @ ( inf_in3354698378728726462lambda @ B @ ( collec8830145823237851355lambda @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_729_image__Int__subset,axiom,
! [F2: lambda_lambda > lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F2 @ ( inf_in1766959559283430712lambda @ A @ B ) ) @ ( inf_in1766959559283430712lambda @ ( image_2356535832909760517lambda @ F2 @ A ) @ ( image_2356535832909760517lambda @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_730_image__Int__subset,axiom,
! [F2: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F2 @ ( inf_in3354698378728726462lambda @ A @ B ) ) @ ( inf_in1766959559283430712lambda @ ( image_2458965899662700159lambda @ F2 @ A ) @ ( image_2458965899662700159lambda @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_731_image__Int__subset,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F2 @ ( inf_in1766959559283430712lambda @ A @ B ) ) @ ( inf_in3354698378728726462lambda @ ( image_3692823078739859851lambda @ F2 @ A ) @ ( image_3692823078739859851lambda @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_732_image__Int__subset,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F2 @ ( inf_in3354698378728726462lambda @ A @ B ) ) @ ( inf_in3354698378728726462lambda @ ( image_5886195720093285125lambda @ F2 @ A ) @ ( image_5886195720093285125lambda @ F2 @ B ) ) ) ).
% image_Int_subset
thf(fact_733_Srcs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( insert_lambda_lambda @ ( paths_1651982994781619320lambda @ lambda_resid @ T6 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Srcs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_734_inf__bot__left,axiom,
! [X6: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X6 )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_left
thf(fact_735_inf__bot__left,axiom,
! [X6: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X6 )
= bot_bo1397037886704484924lambda ) ).
% inf_bot_left
thf(fact_736_inf__bot__right,axiom,
! [X6: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X6 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_right
thf(fact_737_inf__bot__right,axiom,
! [X6: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X6 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% inf_bot_right
thf(fact_738_le__inf__iff,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X6 @ ( inf_in1766959559283430712lambda @ Y5 @ Z2 ) )
= ( ( ord_le1926335974945931754lambda @ X6 @ Y5 )
& ( ord_le1926335974945931754lambda @ X6 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_739_le__inf__iff,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X6 @ ( inf_in3354698378728726462lambda @ Y5 @ Z2 ) )
= ( ( ord_le3064231726521830256lambda @ X6 @ Y5 )
& ( ord_le3064231726521830256lambda @ X6 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_740_inf_Obounded__iff,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B3 @ C ) )
= ( ( ord_le1926335974945931754lambda @ A2 @ B3 )
& ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_741_inf_Obounded__iff,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B3 @ C ) )
= ( ( ord_le3064231726521830256lambda @ A2 @ B3 )
& ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_742_inf__right__idem,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) @ Y5 )
= ( inf_in1766959559283430712lambda @ X6 @ Y5 ) ) ).
% inf_right_idem
thf(fact_743_inf__right__idem,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) @ Y5 )
= ( inf_in3354698378728726462lambda @ X6 @ Y5 ) ) ).
% inf_right_idem
thf(fact_744_inf_Oright__idem,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ B3 )
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ).
% inf.right_idem
thf(fact_745_inf_Oright__idem,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ B3 )
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ).
% inf.right_idem
thf(fact_746_inf__left__idem,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X6 @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) )
= ( inf_in1766959559283430712lambda @ X6 @ Y5 ) ) ).
% inf_left_idem
thf(fact_747_inf__left__idem,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X6 @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) )
= ( inf_in3354698378728726462lambda @ X6 @ Y5 ) ) ).
% inf_left_idem
thf(fact_748_inf_Oleft__idem,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ).
% inf.left_idem
thf(fact_749_inf_Oleft__idem,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ).
% inf.left_idem
thf(fact_750_inf__idem,axiom,
! [X6: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X6 @ X6 )
= X6 ) ).
% inf_idem
thf(fact_751_inf__idem,axiom,
! [X6: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X6 @ X6 )
= X6 ) ).
% inf_idem
thf(fact_752_inf_Oidem,axiom,
! [A2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_753_inf_Oidem,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_754_paths__in__weakly__extensional__rts_OSrc_Ocong,axiom,
paths_1651982994781619320lambda = paths_1651982994781619320lambda ).
% paths_in_weakly_extensional_rts.Src.cong
thf(fact_755_less__eq__set__def,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A6: set_lambda_lambda,B5: set_lambda_lambda] :
( ord_le5488116190248881011mbda_o
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ A6 )
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_756_less__eq__set__def,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A6: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( ord_le4030711025213088621mbda_o
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ A6 )
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_757_inf__left__commute,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X6 @ ( inf_in1766959559283430712lambda @ Y5 @ Z2 ) )
= ( inf_in1766959559283430712lambda @ Y5 @ ( inf_in1766959559283430712lambda @ X6 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_758_inf__left__commute,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X6 @ ( inf_in3354698378728726462lambda @ Y5 @ Z2 ) )
= ( inf_in3354698378728726462lambda @ Y5 @ ( inf_in3354698378728726462lambda @ X6 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_759_inf_Oleft__commute,axiom,
! [B3: set_lambda_lambda,A2: set_lambda_lambda,C: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ B3 @ ( inf_in1766959559283430712lambda @ A2 @ C ) )
= ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_760_inf_Oleft__commute,axiom,
! [B3: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ B3 @ ( inf_in3354698378728726462lambda @ A2 @ C ) )
= ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ B3 @ C ) ) ) ).
% inf.left_commute
thf(fact_761_inf__commute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [X4: set_lambda_lambda,Y4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_762_inf__commute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [X4: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ Y4 @ X4 ) ) ) ).
% inf_commute
thf(fact_763_inf_Ocommute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A4: set_lambda_lambda,B2: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ B2 @ A4 ) ) ) ).
% inf.commute
thf(fact_764_inf_Ocommute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A4: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ B2 @ A4 ) ) ) ).
% inf.commute
thf(fact_765_inf__assoc,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) @ Z2 )
= ( inf_in1766959559283430712lambda @ X6 @ ( inf_in1766959559283430712lambda @ Y5 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_766_inf__assoc,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) @ Z2 )
= ( inf_in3354698378728726462lambda @ X6 @ ( inf_in3354698378728726462lambda @ Y5 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_767_inf_Oassoc,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ C )
= ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_768_inf_Oassoc,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ C )
= ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ B3 @ C ) ) ) ).
% inf.assoc
thf(fact_769_inf__sup__aci_I1_J,axiom,
( inf_in1766959559283430712lambda
= ( ^ [X4: set_lambda_lambda,Y4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_770_inf__sup__aci_I1_J,axiom,
( inf_in3354698378728726462lambda
= ( ^ [X4: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_771_inf__sup__aci_I2_J,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) @ Z2 )
= ( inf_in1766959559283430712lambda @ X6 @ ( inf_in1766959559283430712lambda @ Y5 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_772_inf__sup__aci_I2_J,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) @ Z2 )
= ( inf_in3354698378728726462lambda @ X6 @ ( inf_in3354698378728726462lambda @ Y5 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_773_inf__sup__aci_I3_J,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X6 @ ( inf_in1766959559283430712lambda @ Y5 @ Z2 ) )
= ( inf_in1766959559283430712lambda @ Y5 @ ( inf_in1766959559283430712lambda @ X6 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_774_inf__sup__aci_I3_J,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X6 @ ( inf_in3354698378728726462lambda @ Y5 @ Z2 ) )
= ( inf_in3354698378728726462lambda @ Y5 @ ( inf_in3354698378728726462lambda @ X6 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_775_inf__sup__aci_I4_J,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X6 @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) )
= ( inf_in1766959559283430712lambda @ X6 @ Y5 ) ) ).
% inf_sup_aci(4)
thf(fact_776_inf__sup__aci_I4_J,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X6 @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) )
= ( inf_in3354698378728726462lambda @ X6 @ Y5 ) ) ).
% inf_sup_aci(4)
thf(fact_777_inf_OcoboundedI2,axiom,
! [B3: set_lambda_lambda,C: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_778_inf_OcoboundedI2,axiom,
! [B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_779_inf_OcoboundedI1,axiom,
! [A2: set_lambda_lambda,C: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_780_inf_OcoboundedI1,axiom,
! [A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_781_inf_Oabsorb__iff2,axiom,
( ord_le1926335974945931754lambda
= ( ^ [B2: set_lambda_lambda,A4: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A4 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_782_inf_Oabsorb__iff2,axiom,
( ord_le3064231726521830256lambda
= ( ^ [B2: set_li7878742357367608272lambda,A4: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A4 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_783_inf_Oabsorb__iff1,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A4: set_lambda_lambda,B2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A4 @ B2 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_784_inf_Oabsorb__iff1,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A4: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A4 @ B2 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_785_inf_Ocobounded2,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_786_inf_Ocobounded2,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ B3 ) ).
% inf.cobounded2
thf(fact_787_inf_Ocobounded1,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_788_inf_Ocobounded1,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ A2 ) ).
% inf.cobounded1
thf(fact_789_inf_Oorder__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A4: set_lambda_lambda,B2: set_lambda_lambda] :
( A4
= ( inf_in1766959559283430712lambda @ A4 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_790_inf_Oorder__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A4: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( A4
= ( inf_in3354698378728726462lambda @ A4 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_791_inf__greatest,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X6 @ Y5 )
=> ( ( ord_le1926335974945931754lambda @ X6 @ Z2 )
=> ( ord_le1926335974945931754lambda @ X6 @ ( inf_in1766959559283430712lambda @ Y5 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_792_inf__greatest,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X6 @ Y5 )
=> ( ( ord_le3064231726521830256lambda @ X6 @ Z2 )
=> ( ord_le3064231726521830256lambda @ X6 @ ( inf_in3354698378728726462lambda @ Y5 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_793_inf_OboundedI,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_794_inf_OboundedI,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B3 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_795_inf_OboundedE,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B3 @ C ) )
=> ~ ( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ~ ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_796_inf_OboundedE,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B3 @ C ) )
=> ~ ( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ~ ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_797_inf__absorb2,axiom,
! [Y5: set_lambda_lambda,X6: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y5 @ X6 )
=> ( ( inf_in1766959559283430712lambda @ X6 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_798_inf__absorb2,axiom,
! [Y5: set_li7878742357367608272lambda,X6: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ Y5 @ X6 )
=> ( ( inf_in3354698378728726462lambda @ X6 @ Y5 )
= Y5 ) ) ).
% inf_absorb2
thf(fact_799_inf__absorb1,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X6 @ Y5 )
=> ( ( inf_in1766959559283430712lambda @ X6 @ Y5 )
= X6 ) ) ).
% inf_absorb1
thf(fact_800_inf__absorb1,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X6 @ Y5 )
=> ( ( inf_in3354698378728726462lambda @ X6 @ Y5 )
= X6 ) ) ).
% inf_absorb1
thf(fact_801_inf_Oabsorb2,axiom,
! [B3: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ A2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_802_inf_Oabsorb2,axiom,
! [B3: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ A2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B3 )
= B3 ) ) ).
% inf.absorb2
thf(fact_803_inf_Oabsorb1,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_804_inf_Oabsorb1,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B3 )
= A2 ) ) ).
% inf.absorb1
thf(fact_805_le__iff__inf,axiom,
( ord_le1926335974945931754lambda
= ( ^ [X4: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_806_le__iff__inf,axiom,
( ord_le3064231726521830256lambda
= ( ^ [X4: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X4 @ Y4 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_807_inf__unique,axiom,
! [F2: set_lambda_lambda > set_lambda_lambda > set_lambda_lambda,X6: set_lambda_lambda,Y5: set_lambda_lambda] :
( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F2 @ X @ Y2 ) @ X )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F2 @ X @ Y2 ) @ Y2 )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda,Z3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ X @ Z3 )
=> ( ord_le1926335974945931754lambda @ X @ ( F2 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_in1766959559283430712lambda @ X6 @ Y5 )
= ( F2 @ X6 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_808_inf__unique,axiom,
! [F2: set_li7878742357367608272lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda,X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F2 @ X @ Y2 ) @ X )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F2 @ X @ Y2 ) @ Y2 )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ X @ Z3 )
=> ( ord_le3064231726521830256lambda @ X @ ( F2 @ Y2 @ Z3 ) ) ) )
=> ( ( inf_in3354698378728726462lambda @ X6 @ Y5 )
= ( F2 @ X6 @ Y5 ) ) ) ) ) ).
% inf_unique
thf(fact_809_inf_OorderI,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( A2
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
=> ( ord_le1926335974945931754lambda @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_810_inf_OorderI,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( A2
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
=> ( ord_le3064231726521830256lambda @ A2 @ B3 ) ) ).
% inf.orderI
thf(fact_811_inf_OorderE,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( A2
= ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_812_inf_OorderE,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( A2
= ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ).
% inf.orderE
thf(fact_813_le__infI2,axiom,
! [B3: set_lambda_lambda,X6: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ X6 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ X6 ) ) ).
% le_infI2
thf(fact_814_le__infI2,axiom,
! [B3: set_li7878742357367608272lambda,X6: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ X6 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ X6 ) ) ).
% le_infI2
thf(fact_815_le__infI1,axiom,
! [A2: set_lambda_lambda,X6: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ X6 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ X6 ) ) ).
% le_infI1
thf(fact_816_le__infI1,axiom,
! [A2: set_li7878742357367608272lambda,X6: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ X6 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ X6 ) ) ).
% le_infI1
thf(fact_817_inf__mono,axiom,
! [A2: set_lambda_lambda,C: set_lambda_lambda,B3: set_lambda_lambda,D2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ( ord_le1926335974945931754lambda @ B3 @ D2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) @ ( inf_in1766959559283430712lambda @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_818_inf__mono,axiom,
! [A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,D2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ( ord_le3064231726521830256lambda @ B3 @ D2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) @ ( inf_in3354698378728726462lambda @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_819_le__infI,axiom,
! [X6: set_lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X6 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ X6 @ B3 )
=> ( ord_le1926335974945931754lambda @ X6 @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_820_le__infI,axiom,
! [X6: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X6 @ A2 )
=> ( ( ord_le3064231726521830256lambda @ X6 @ B3 )
=> ( ord_le3064231726521830256lambda @ X6 @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) ) ) ) ).
% le_infI
thf(fact_821_le__infE,axiom,
! [X6: set_lambda_lambda,A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X6 @ ( inf_in1766959559283430712lambda @ A2 @ B3 ) )
=> ~ ( ( ord_le1926335974945931754lambda @ X6 @ A2 )
=> ~ ( ord_le1926335974945931754lambda @ X6 @ B3 ) ) ) ).
% le_infE
thf(fact_822_le__infE,axiom,
! [X6: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X6 @ ( inf_in3354698378728726462lambda @ A2 @ B3 ) )
=> ~ ( ( ord_le3064231726521830256lambda @ X6 @ A2 )
=> ~ ( ord_le3064231726521830256lambda @ X6 @ B3 ) ) ) ).
% le_infE
thf(fact_823_inf__le2,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_824_inf__le2,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) @ Y5 ) ).
% inf_le2
thf(fact_825_inf__le1,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) @ X6 ) ).
% inf_le1
thf(fact_826_inf__le1,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) @ X6 ) ).
% inf_le1
thf(fact_827_inf__sup__ord_I1_J,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) @ X6 ) ).
% inf_sup_ord(1)
thf(fact_828_inf__sup__ord_I1_J,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) @ X6 ) ).
% inf_sup_ord(1)
thf(fact_829_inf__sup__ord_I2_J,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X6 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_830_inf__sup__ord_I2_J,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X6 @ Y5 ) @ Y5 ) ).
% inf_sup_ord(2)
thf(fact_831_set__Cons__sing__Nil,axiom,
! [A: set_lambda_lambda] :
( ( set_Co4812318673604925728lambda @ A @ ( insert8465602035467394112lambda @ nil_lambda_lambda @ bot_bo1397037886704484924lambda ) )
= ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : ( cons_lambda_lambda @ X4 @ nil_lambda_lambda )
@ A ) ) ).
% set_Cons_sing_Nil
thf(fact_832_Arr__append__iff_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( paths_366302053746739058lambda @ lambda_resid @ U3 )
& ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P
thf(fact_833_Ide_Oelims_I3_J,axiom,
! [X6: list_lambda_lambda] :
( ~ ( paths_7717490669901197737lambda @ lambda_resid @ X6 )
=> ( ( X6 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( ide_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ).
% Ide.elims(3)
thf(fact_834_Ide_Osimps_I1_J,axiom,
~ ( paths_7717490669901197737lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Ide.simps(1)
thf(fact_835_Ide__implies__Arr,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( paths_366302053746739058lambda @ lambda_resid @ T6 ) ) ).
% Ide_implies_Arr
thf(fact_836_Con__append_I2_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,V4: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( V4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ U3 @ V4 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) @ V4 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(2)
thf(fact_837_Con__append_I1_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,V4: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( V4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ V4 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ T6 ) )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(1)
thf(fact_838_Resid__append__ind,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,V4: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( V4 != nil_lambda_lambda )
=> ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V4 @ T6 ) @ U3 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ U3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V4 ) )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ V4 @ U3 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 ) @ U3 )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V4 @ T6 ) @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V4 @ T6 ) @ U3 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ V4 ) ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ V4 @ U3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ V4 @ U3 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 ) @ U3 ) ) ) ) ) ) ) ).
% Resid_append_ind
thf(fact_839_Resid__Ide_I1_J,axiom,
! [A: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T6 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ A )
= T6 ) ) ) ).
% Resid_Ide(1)
thf(fact_840_Resid__Arr__Ide__ind,axiom,
! [A: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ A )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ A )
= T6 ) ) ) ).
% Resid_Arr_Ide_ind
thf(fact_841_Resid__Ide__Arr__ind,axiom,
! [A: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T6 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ T6 ) ) ) ) ).
% Resid_Ide_Arr_ind
thf(fact_842_ide__char,axiom,
! [T6: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( paths_7717490669901197737lambda @ lambda_resid @ T6 ) ) ).
% ide_char
thf(fact_843_Resid__Arr__self,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T6 ) ) ) ).
% Resid_Arr_self
thf(fact_844_Ide__imp__Ide__hd,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) ) ) ).
% Ide_imp_Ide_hd
thf(fact_845_Ide__imp__Ide__last,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ide_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T6 ) ) ) ).
% Ide_imp_Ide_last
thf(fact_846_order__refl,axiom,
! [X6: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ X6 @ X6 ) ).
% order_refl
thf(fact_847_order__refl,axiom,
! [X6: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ X6 @ X6 ) ).
% order_refl
thf(fact_848_dual__order_Orefl,axiom,
! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_849_dual__order_Orefl,axiom,
! [A2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_850_same__append__eq,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys2 )
= ( append_lambda_lambda @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_851_append__same__eq,axiom,
! [Ys2: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys2 @ Xs )
= ( append_lambda_lambda @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_852_append__assoc,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) @ Zs )
= ( append_lambda_lambda @ Xs @ ( append_lambda_lambda @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_853_append_Oassoc,axiom,
! [A2: list_lambda_lambda,B3: list_lambda_lambda,C: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ A2 @ B3 ) @ C )
= ( append_lambda_lambda @ A2 @ ( append_lambda_lambda @ B3 @ C ) ) ) ).
% append.assoc
thf(fact_854_Ide__imp__Ide__tl,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( ( tl_lambda_lambda @ T6 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( tl_lambda_lambda @ T6 ) ) ) ) ).
% Ide_imp_Ide_tl
thf(fact_855_Resid__cons_I1_J,axiom,
! [U3: list_lambda_lambda,T: lambda_lambda,T6: list_lambda_lambda] :
( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U3 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% Resid_cons(1)
thf(fact_856_Resid__rec_I4_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U @ U3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ ( cons_lambda_lambda @ U @ U3 ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) ) ) ) ) ) ) ).
% Resid_rec(4)
thf(fact_857_Ide_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( ide_lambda_lambda @ lambda_resid @ T ) ) ).
% Ide.simps(2)
thf(fact_858_Ide__imp__sources__eq__targets,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% Ide_imp_sources_eq_targets
thf(fact_859_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_860_Con__IdeI_I2_J,axiom,
! [A: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ A )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(2)
thf(fact_861_Con__IdeI_I1_J,axiom,
! [A: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T6 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(1)
thf(fact_862_Con__Ide__iff,axiom,
! [A: list_lambda_lambda,T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T6 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ T6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A ) ) ) ) ) ).
% Con_Ide_iff
thf(fact_863_arr__append__imp__seq,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T6 @ U3 ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 ) ) ) ) ).
% arr_append_imp_seq
thf(fact_864_Arr__appendE_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
=> ( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
!= ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P
thf(fact_865_sources__char_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( collec8830145823237851355lambda
@ ^ [A6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A6 )
& ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ A6 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% sources_char\<^sub>P
thf(fact_866_Ide_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( ( ide_lambda_lambda @ lambda_resid @ T )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ) ) ).
% Ide.simps(3)
thf(fact_867_Ide__append__iff_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
& ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
& ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P
thf(fact_868_targets__char_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
= ( collec8830145823237851355lambda
@ ^ [B5: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ B5 )
& ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ B5 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) ) ) ) ) ).
% targets_char\<^sub>P
thf(fact_869_append__is__Nil__conv,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys2 )
= nil_lambda_lambda )
= ( ( Xs = nil_lambda_lambda )
& ( Ys2 = nil_lambda_lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_870_Nil__is__append__conv,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( ( Xs = nil_lambda_lambda )
& ( Ys2 = nil_lambda_lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_871_self__append__conv2,axiom,
! [Y5: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Y5
= ( append_lambda_lambda @ Xs @ Y5 ) )
= ( Xs = nil_lambda_lambda ) ) ).
% self_append_conv2
thf(fact_872_append__self__conv2,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_lambda_lambda ) ) ).
% append_self_conv2
thf(fact_873_self__append__conv,axiom,
! [Y5: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( Y5
= ( append_lambda_lambda @ Y5 @ Ys2 ) )
= ( Ys2 = nil_lambda_lambda ) ) ).
% self_append_conv
thf(fact_874_append__self__conv,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_lambda_lambda ) ) ).
% append_self_conv
thf(fact_875_append__Nil2,axiom,
! [Xs: list_lambda_lambda] :
( ( append_lambda_lambda @ Xs @ nil_lambda_lambda )
= Xs ) ).
% append_Nil2
thf(fact_876_append_Oright__neutral,axiom,
! [A2: list_lambda_lambda] :
( ( append_lambda_lambda @ A2 @ nil_lambda_lambda )
= A2 ) ).
% append.right_neutral
thf(fact_877_Ide_Oelims_I1_J,axiom,
! [X6: list_lambda_lambda,Y5: $o] :
( ( ( paths_7717490669901197737lambda @ lambda_resid @ X6 )
= Y5 )
=> ( ( ( X6 = nil_lambda_lambda )
=> Y5 )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y5
= ( ~ ( ide_lambda_lambda @ lambda_resid @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y5
= ( ~ ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% Ide.elims(1)
thf(fact_878_Ide_Oelims_I2_J,axiom,
! [X6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ X6 )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Ide.elims(2)
thf(fact_879_append1__eq__conv,axiom,
! [Xs: list_lambda_lambda,X6: lambda_lambda,Ys2: list_lambda_lambda,Y5: lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X6 @ nil_lambda_lambda ) )
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ Y5 @ nil_lambda_lambda ) ) )
= ( ( Xs = Ys2 )
& ( X6 = Y5 ) ) ) ).
% append1_eq_conv
thf(fact_880_hd__append2,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( hd_lambda_lambda @ Xs ) ) ) ).
% hd_append2
thf(fact_881_tl__append2,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( append_lambda_lambda @ ( tl_lambda_lambda @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_882_last__appendL,axiom,
! [Ys2: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Ys2 = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_appendL
thf(fact_883_last__appendR,axiom,
! [Ys2: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Ys2 != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( last_lambda_lambda @ Ys2 ) ) ) ).
% last_appendR
thf(fact_884_last__snoc,axiom,
! [Xs: list_lambda_lambda,X6: lambda_lambda] :
( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X6 @ nil_lambda_lambda ) ) )
= X6 ) ).
% last_snoc
thf(fact_885_Con__appendI_I2_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,V4: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) @ V4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ U3 @ V4 ) )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(2)
thf(fact_886_Con__appendI_I1_J,axiom,
! [T6: list_lambda_lambda,V4: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ T6 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ V4 )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(1)
thf(fact_887_Resid__append_I2_J,axiom,
! [U3: list_lambda_lambda,V4: list_lambda_lambda,T6: list_lambda_lambda] :
( ( U3 != nil_lambda_lambda )
=> ( ( V4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ U3 @ V4 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ U3 @ V4 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) @ V4 ) ) ) ) ) ).
% Resid_append(2)
thf(fact_888_Resid__append_I1_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,V4: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ V4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ V4 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ T6 ) ) ) ) ) ) ).
% Resid_append(1)
thf(fact_889_Resid__append2,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,V4: list_lambda_lambda,W3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( V4 != nil_lambda_lambda )
=> ( ( W3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ ( append_lambda_lambda @ V4 @ W3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ ( append_lambda_lambda @ V4 @ W3 ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 ) @ W3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ T6 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W3 @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ V4 ) ) ) ) ) ) ) ) ) ) ).
% Resid_append2
thf(fact_890_Srcs__append,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( paths_2897053694880971696lambda @ lambda_resid @ T6 ) ) ) ).
% Srcs_append
thf(fact_891_Trgs__append,axiom,
! [U3: list_lambda_lambda,T6: list_lambda_lambda] :
( ( U3 != nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ U3 ) ) ) ).
% Trgs_append
thf(fact_892_path__from__NF__is__Ide,axiom,
! [A2: lambda_lambda,U3: list_lambda_lambda] :
( ( lambda_NF @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ( paths_1651982994781619320lambda @ lambda_resid @ U3 )
= A2 )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U3 ) ) ) ) ).
% path_from_NF_is_Ide
thf(fact_893_cong__append__ideI_I4_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ T6 @ U3 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ T6 ) ) ) ) ) ).
% cong_append_ideI(4)
thf(fact_894_cong__append__ideI_I3_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( append_lambda_lambda @ T6 @ U3 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ U3 ) ) ) ) ) ).
% cong_append_ideI(3)
thf(fact_895_cong__append__ideI_I2_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( append_lambda_lambda @ T6 @ U3 ) ) ) ) ) ) ).
% cong_append_ideI(2)
thf(fact_896_cong__append__ideI_I1_J,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( append_lambda_lambda @ T6 @ U3 ) ) ) ) ) ) ).
% cong_append_ideI(1)
thf(fact_897_cong__append,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda,T7: list_lambda_lambda,U4: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T6 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U3 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) @ ( append_lambda_lambda @ T7 @ U4 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T7 @ U4 ) @ ( append_lambda_lambda @ T6 @ U3 ) ) ) ) ) ) ) ).
% cong_append
thf(fact_898_sources__append,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% sources_append
thf(fact_899_targets__append,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U3 ) ) ) ).
% targets_append
thf(fact_900_arr__append,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T6 @ U3 ) ) ) ).
% arr_append
thf(fact_901_Src__append,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( paths_1651982994781619320lambda @ lambda_resid @ T6 ) ) ) ).
% Src_append
thf(fact_902_Arr__appendI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) ) ) ) ) ).
% Arr_appendI\<^sub>P
thf(fact_903_Ide__appendI_092_060_094sub_062P,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T6 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U3 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) ) ) ) ) ).
% Ide_appendI\<^sub>P
thf(fact_904_paths__in__rts_OIde_Ocong,axiom,
paths_7717490669901197737lambda = paths_7717490669901197737lambda ).
% paths_in_rts.Ide.cong
thf(fact_905_append__eq__append__conv2,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda,Zs: list_lambda_lambda,Ts: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys2 )
= ( append_lambda_lambda @ Zs @ Ts ) )
= ( ? [Us: list_lambda_lambda] :
( ( ( Xs
= ( append_lambda_lambda @ Zs @ Us ) )
& ( ( append_lambda_lambda @ Us @ Ys2 )
= Ts ) )
| ( ( ( append_lambda_lambda @ Xs @ Us )
= Zs )
& ( Ys2
= ( append_lambda_lambda @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_906_append__eq__appendI,axiom,
! [Xs: list_lambda_lambda,Xs1: list_lambda_lambda,Zs: list_lambda_lambda,Ys2: list_lambda_lambda,Us2: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_lambda_lambda @ Xs1 @ Us2 ) )
=> ( ( append_lambda_lambda @ Xs @ Ys2 )
= ( append_lambda_lambda @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_907_eq__Nil__appendI,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_lambda_lambda @ nil_lambda_lambda @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_908_append_Oleft__neutral,axiom,
! [A2: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_909_append__Nil,axiom,
! [Ys2: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_910_Cons__eq__appendI,axiom,
! [X6: lambda_lambda,Xs1: list_lambda_lambda,Ys2: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X6 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_lambda_lambda @ Xs1 @ Zs ) )
=> ( ( cons_lambda_lambda @ X6 @ Xs )
= ( append_lambda_lambda @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_911_append__Cons,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( append_lambda_lambda @ ( cons_lambda_lambda @ X6 @ Xs ) @ Ys2 )
= ( cons_lambda_lambda @ X6 @ ( append_lambda_lambda @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_912_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_913_append__eq__Cons__conv,axiom,
! [Ys2: list_lambda_lambda,Zs: list_lambda_lambda,X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys2 @ Zs )
= ( cons_lambda_lambda @ X6 @ Xs ) )
= ( ( ( Ys2 = nil_lambda_lambda )
& ( Zs
= ( cons_lambda_lambda @ X6 @ Xs ) ) )
| ? [Ys4: list_lambda_lambda] :
( ( Ys2
= ( cons_lambda_lambda @ X6 @ Ys4 ) )
& ( ( append_lambda_lambda @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_914_Cons__eq__append__conv,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda,Ys2: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X6 @ Xs )
= ( append_lambda_lambda @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_lambda_lambda )
& ( ( cons_lambda_lambda @ X6 @ Xs )
= Zs ) )
| ? [Ys4: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X6 @ Ys4 )
= Ys2 )
& ( Xs
= ( append_lambda_lambda @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_915_rev__exhaust,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ~ ! [Ys3: list_lambda_lambda,Y2: lambda_lambda] :
( Xs
!= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ Y2 @ nil_lambda_lambda ) ) ) ) ).
% rev_exhaust
thf(fact_916_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_917_hd__append,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( hd_lambda_lambda @ Ys2 ) ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( hd_lambda_lambda @ Xs ) ) ) ) ).
% hd_append
thf(fact_918_longest__common__prefix,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
? [Ps: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ps @ Xs3 ) )
& ( Ys2
= ( 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_919_tl__append__if,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( tl_lambda_lambda @ Ys2 ) ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( tl_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( append_lambda_lambda @ ( tl_lambda_lambda @ Xs ) @ Ys2 ) ) ) ) ).
% tl_append_if
thf(fact_920_last__append,axiom,
! [Ys2: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( ( Ys2 = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( last_lambda_lambda @ Xs ) ) )
& ( ( Ys2 != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys2 ) )
= ( last_lambda_lambda @ Ys2 ) ) ) ) ).
% last_append
thf(fact_921_longest__common__suffix,axiom,
! [Xs: list_lambda_lambda,Ys2: list_lambda_lambda] :
? [Ss: list_lambda_lambda,Xs3: list_lambda_lambda,Ys5: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Xs3 @ Ss ) )
& ( Ys2
= ( 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_922_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_lambda_lambda,Z: set_lambda_lambda] : ( Y6 = Z ) )
= ( ^ [X4: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X4 @ Y4 )
& ( ord_le1926335974945931754lambda @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_923_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] : ( Y6 = Z ) )
= ( ^ [X4: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X4 @ Y4 )
& ( ord_le3064231726521830256lambda @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_924_ord__eq__le__trans,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( A2 = B3 )
=> ( ( ord_le1926335974945931754lambda @ B3 @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_925_ord__eq__le__trans,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A2 = B3 )
=> ( ( ord_le3064231726521830256lambda @ B3 @ C )
=> ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_926_ord__le__eq__trans,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_927_ord__le__eq__trans,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_928_order__antisym,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X6 @ Y5 )
=> ( ( ord_le1926335974945931754lambda @ Y5 @ X6 )
=> ( X6 = Y5 ) ) ) ).
% order_antisym
thf(fact_929_order__antisym,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X6 @ Y5 )
=> ( ( ord_le3064231726521830256lambda @ Y5 @ X6 )
=> ( X6 = Y5 ) ) ) ).
% order_antisym
thf(fact_930_order_Otrans,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ord_le1926335974945931754lambda @ B3 @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% order.trans
thf(fact_931_order_Otrans,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ord_le3064231726521830256lambda @ B3 @ C )
=> ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% order.trans
thf(fact_932_order__trans,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X6 @ Y5 )
=> ( ( ord_le1926335974945931754lambda @ Y5 @ Z2 )
=> ( ord_le1926335974945931754lambda @ X6 @ Z2 ) ) ) ).
% order_trans
thf(fact_933_order__trans,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X6 @ Y5 )
=> ( ( ord_le3064231726521830256lambda @ Y5 @ Z2 )
=> ( ord_le3064231726521830256lambda @ X6 @ Z2 ) ) ) ).
% order_trans
thf(fact_934_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_lambda_lambda,Z: set_lambda_lambda] : ( Y6 = Z ) )
= ( ^ [A4: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B2 @ A4 )
& ( ord_le1926335974945931754lambda @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_935_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] : ( Y6 = Z ) )
= ( ^ [A4: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B2 @ A4 )
& ( ord_le3064231726521830256lambda @ A4 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_936_dual__order_Oantisym,axiom,
! [B3: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_937_dual__order_Oantisym,axiom,
! [B3: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ A2 )
=> ( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_938_dual__order_Otrans,axiom,
! [B3: set_lambda_lambda,A2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ C @ B3 )
=> ( ord_le1926335974945931754lambda @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_939_dual__order_Otrans,axiom,
! [B3: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ A2 )
=> ( ( ord_le3064231726521830256lambda @ C @ B3 )
=> ( ord_le3064231726521830256lambda @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_940_antisym,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ord_le1926335974945931754lambda @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_941_antisym,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ord_le3064231726521830256lambda @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_942_order__eq__iff,axiom,
( ( ^ [Y6: set_lambda_lambda,Z: set_lambda_lambda] : ( Y6 = Z ) )
= ( ^ [A4: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A4 @ B2 )
& ( ord_le1926335974945931754lambda @ B2 @ A4 ) ) ) ) ).
% order_eq_iff
thf(fact_943_order__eq__iff,axiom,
( ( ^ [Y6: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] : ( Y6 = Z ) )
= ( ^ [A4: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A4 @ B2 )
& ( ord_le3064231726521830256lambda @ B2 @ A4 ) ) ) ) ).
% order_eq_iff
thf(fact_944_order__subst1,axiom,
! [A2: set_lambda_lambda,F2: set_lambda_lambda > set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( F2 @ B3 ) )
=> ( ( ord_le1926335974945931754lambda @ B3 @ C )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ord_le1926335974945931754lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_945_order__subst1,axiom,
! [A2: set_lambda_lambda,F2: set_li7878742357367608272lambda > set_lambda_lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( F2 @ B3 ) )
=> ( ( ord_le3064231726521830256lambda @ B3 @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ord_le1926335974945931754lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_946_order__subst1,axiom,
! [A2: set_li7878742357367608272lambda,F2: set_lambda_lambda > set_li7878742357367608272lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( F2 @ B3 ) )
=> ( ( ord_le1926335974945931754lambda @ B3 @ C )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ord_le3064231726521830256lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le3064231726521830256lambda @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_947_order__subst1,axiom,
! [A2: set_li7878742357367608272lambda,F2: set_li7878742357367608272lambda > set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( F2 @ B3 ) )
=> ( ( ord_le3064231726521830256lambda @ B3 @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ord_le3064231726521830256lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le3064231726521830256lambda @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_948_order__subst2,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,F2: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ord_le1926335974945931754lambda @ ( F2 @ B3 ) @ C )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ord_le1926335974945931754lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_949_order__subst2,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,F2: set_lambda_lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ord_le3064231726521830256lambda @ ( F2 @ B3 ) @ C )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ord_le3064231726521830256lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_950_order__subst2,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,F2: set_li7878742357367608272lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ord_le1926335974945931754lambda @ ( F2 @ B3 ) @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ord_le1926335974945931754lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_951_order__subst2,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,F2: set_li7878742357367608272lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ord_le3064231726521830256lambda @ ( F2 @ B3 ) @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ord_le3064231726521830256lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_952_order__eq__refl,axiom,
! [X6: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( X6 = Y5 )
=> ( ord_le1926335974945931754lambda @ X6 @ Y5 ) ) ).
% order_eq_refl
thf(fact_953_order__eq__refl,axiom,
! [X6: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( X6 = Y5 )
=> ( ord_le3064231726521830256lambda @ X6 @ Y5 ) ) ).
% order_eq_refl
thf(fact_954_ord__eq__le__subst,axiom,
! [A2: set_lambda_lambda,F2: set_lambda_lambda > set_lambda_lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( A2
= ( F2 @ B3 ) )
=> ( ( ord_le1926335974945931754lambda @ B3 @ C )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ord_le1926335974945931754lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_955_ord__eq__le__subst,axiom,
! [A2: set_li7878742357367608272lambda,F2: set_lambda_lambda > set_li7878742357367608272lambda,B3: set_lambda_lambda,C: set_lambda_lambda] :
( ( A2
= ( F2 @ B3 ) )
=> ( ( ord_le1926335974945931754lambda @ B3 @ C )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ord_le3064231726521830256lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le3064231726521830256lambda @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_956_ord__eq__le__subst,axiom,
! [A2: set_lambda_lambda,F2: set_li7878742357367608272lambda > set_lambda_lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A2
= ( F2 @ B3 ) )
=> ( ( ord_le3064231726521830256lambda @ B3 @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ord_le1926335974945931754lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_957_ord__eq__le__subst,axiom,
! [A2: set_li7878742357367608272lambda,F2: set_li7878742357367608272lambda > set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A2
= ( F2 @ B3 ) )
=> ( ( ord_le3064231726521830256lambda @ B3 @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ord_le3064231726521830256lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le3064231726521830256lambda @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_958_ord__le__eq__subst,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,F2: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ord_le1926335974945931754lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_959_ord__le__eq__subst,axiom,
! [A2: set_lambda_lambda,B3: set_lambda_lambda,F2: set_lambda_lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y2 )
=> ( ord_le3064231726521830256lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_960_ord__le__eq__subst,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,F2: set_li7878742357367608272lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ord_le1926335974945931754lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_961_ord__le__eq__subst,axiom,
! [A2: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda,F2: set_li7878742357367608272lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B3 )
=> ( ( ( F2 @ B3 )
= C )
=> ( ! [X: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y2 )
=> ( ord_le3064231726521830256lambda @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_962_order__antisym__conv,axiom,
! [Y5: set_lambda_lambda,X6: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y5 @ X6 )
=> ( ( ord_le1926335974945931754lambda @ X6 @ Y5 )
= ( X6 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_963_order__antisym__conv,axiom,
! [Y5: set_li7878742357367608272lambda,X6: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ Y5 @ X6 )
=> ( ( ord_le3064231726521830256lambda @ X6 @ Y5 )
= ( X6 = Y5 ) ) ) ).
% order_antisym_conv
thf(fact_964_bot_Oextremum,axiom,
! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A2 ) ).
% bot.extremum
thf(fact_965_bot_Oextremum,axiom,
! [A2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A2 ) ).
% bot.extremum
thf(fact_966_bot_Oextremum__unique,axiom,
! [A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_unique
thf(fact_967_bot_Oextremum__unique,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_unique
thf(fact_968_bot_Oextremum__uniqueI,axiom,
! [A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
=> ( A2 = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_969_bot_Oextremum__uniqueI,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
=> ( A2 = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_970_const__ide__is__Ide,axiom,
! [T6: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T6 ) @ bot_bo1271416176687173302lambda ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ T6 ) ) ) ) ).
% const_ide_is_Ide
thf(fact_971_Ide__appendE,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
=> ( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ~ ( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
!= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ) ).
% Ide_appendE
thf(fact_972_Ide__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
& ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_973_Trg_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) ) ) ).
% Trg.simps(3)
thf(fact_974_Trg__resid__sym,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ T6 ) ) ) ) ).
% Trg_resid_sym
thf(fact_975_Ide__imp__Src__eq__Trg,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ T6 )
= ( paths_2695323870737867955lambda @ lambda_resid @ T6 ) ) ) ).
% Ide_imp_Src_eq_Trg
thf(fact_976_set__Arr__subset__arr,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ lambda_resid ) ) ) ) ).
% set_Arr_subset_arr
thf(fact_977_set__Ide__subset__ide,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ).
% set_Ide_subset_ide
thf(fact_978_Arr__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( paths_366302053746739058lambda @ lambda_resid @ U3 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_979_Arr__appendE_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
=> ( ( T6 != nil_lambda_lambda )
=> ( ( U3 != nil_lambda_lambda )
=> ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
!= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P\<^sub>W\<^sub>E
thf(fact_980_Ide__char,axiom,
! [T6: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ) ).
% Ide_char
thf(fact_981_set__empty,axiom,
! [Xs: list_lambda_lambda] :
( ( ( set_lambda_lambda2 @ Xs )
= bot_bo1271416176687173302lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty
thf(fact_982_set__empty,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( ( set_li3846194476672328965lambda @ Xs )
= bot_bo1397037886704484924lambda )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty
thf(fact_983_set__empty2,axiom,
! [Xs: list_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty2
thf(fact_984_set__empty2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ Xs ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty2
thf(fact_985_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_986_Src__resid,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U3 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ U3 ) ) ) ).
% Src_resid
thf(fact_987_Trg__append,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U3 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ U3 ) ) ) ).
% Trg_append
thf(fact_988_Arr__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) ) ) ) ) ).
% Arr_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_989_Ide__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda,U3: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T6 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
=> ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T6 @ U3 ) ) ) ) ) ).
% Ide_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_990_Trgs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T6 )
= ( insert_lambda_lambda @ ( paths_2695323870737867955lambda @ lambda_resid @ T6 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Trgs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_991_IdeI,axiom,
! [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T6 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ T6 ) ) ) ).
% IdeI
thf(fact_992_subset__code_I1_J,axiom,
! [Xs: list_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ B )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
=> ( member_lambda_lambda @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_993_subset__code_I1_J,axiom,
! [Xs: list_l5168835686021156086lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ B )
= ( ! [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( member2667661554319814297lambda @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_994_list_Oset__intros_I2_J,axiom,
! [Y5: list_lambda_lambda,X22: list_l5168835686021156086lambda,X21: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y5 @ ( set_li3846194476672328965lambda @ X22 ) )
=> ( member2667661554319814297lambda @ Y5 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_995_list_Oset__intros_I2_J,axiom,
! [Y5: lambda_lambda,X22: list_lambda_lambda,X21: lambda_lambda] :
( ( member_lambda_lambda @ Y5 @ ( set_lambda_lambda2 @ X22 ) )
=> ( member_lambda_lambda @ Y5 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_996_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_997_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_998_list_Oset__cases,axiom,
! [E: list_lambda_lambda,A2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ A2 ) )
=> ( ! [Z22: list_l5168835686021156086lambda] :
( A2
!= ( cons_l4558873032622161958lambda @ E @ Z22 ) )
=> ~ ! [Z1: list_lambda_lambda,Z22: list_l5168835686021156086lambda] :
( ( A2
= ( cons_l4558873032622161958lambda @ Z1 @ Z22 ) )
=> ~ ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_999_list_Oset__cases,axiom,
! [E: lambda_lambda,A2: list_lambda_lambda] :
( ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ A2 ) )
=> ( ! [Z22: list_lambda_lambda] :
( A2
!= ( cons_lambda_lambda @ E @ Z22 ) )
=> ~ ! [Z1: lambda_lambda,Z22: list_lambda_lambda] :
( ( A2
= ( cons_lambda_lambda @ Z1 @ Z22 ) )
=> ~ ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1000_set__ConsD,axiom,
! [Y5: list_lambda_lambda,X6: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ Y5 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X6 @ Xs ) ) )
=> ( ( Y5 = X6 )
| ( member2667661554319814297lambda @ Y5 @ ( set_li3846194476672328965lambda @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1001_set__ConsD,axiom,
! [Y5: lambda_lambda,X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ Y5 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X6 @ Xs ) ) )
=> ( ( Y5 = X6 )
| ( member_lambda_lambda @ Y5 @ ( set_lambda_lambda2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1002_paths__in__weakly__extensional__rts_OTrg_Ocong,axiom,
paths_2695323870737867955lambda = paths_2695323870737867955lambda ).
% paths_in_weakly_extensional_rts.Trg.cong
thf(fact_1003_empty__set,axiom,
( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ nil_lambda_lambda ) ) ).
% empty_set
thf(fact_1004_empty__set,axiom,
( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ nil_li8379291187143545974lambda ) ) ).
% empty_set
thf(fact_1005_set__subset__Cons,axiom,
! [Xs: list_lambda_lambda,X6: lambda_lambda] : ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X6 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1006_set__subset__Cons,axiom,
! [Xs: list_l5168835686021156086lambda,X6: list_lambda_lambda] : ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X6 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1007_split__list__first__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys: list_lambda_lambda,X4: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Ys ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1008_split__list__last__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys: list_lambda_lambda,X4: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1009_in__set__conv__decomp__first,axiom,
! [X6: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys @ ( cons_l4558873032622161958lambda @ X6 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Ys ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1010_in__set__conv__decomp__first,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ X6 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Ys ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1011_in__set__conv__decomp__last,axiom,
! [X6: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys @ ( cons_l4558873032622161958lambda @ X6 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1012_in__set__conv__decomp__last,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ X6 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1013_split__list__first__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: lambda_lambda] :
( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1014_split__list__last__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list_lambda_lambda,X: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: lambda_lambda] :
( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1015_split__list__first__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: lambda_lambda] :
( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_1016_split__list__last__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list_lambda_lambda,X: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa: lambda_lambda] :
( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_1017_in__set__conv__decomp,axiom,
! [X6: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys @ ( cons_l4558873032622161958lambda @ X6 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1018_in__set__conv__decomp,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys: list_lambda_lambda,Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ X6 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1019_append__Cons__eq__iff,axiom,
! [X6: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys2: list_l5168835686021156086lambda,Xs4: list_l5168835686021156086lambda,Ys6: list_l5168835686021156086lambda] :
( ~ ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ~ ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Ys2 ) )
=> ( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X6 @ Ys2 ) )
= ( append1017360404025248779lambda @ Xs4 @ ( cons_l4558873032622161958lambda @ X6 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys2 = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1020_append__Cons__eq__iff,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda,Ys2: list_lambda_lambda,Xs4: list_lambda_lambda,Ys6: list_lambda_lambda] :
( ~ ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ~ ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X6 @ Ys2 ) )
= ( append_lambda_lambda @ Xs4 @ ( cons_lambda_lambda @ X6 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys2 = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1021_split__list__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X2 ) )
=> ~ ! [Ys3: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_1022_split__list__first,axiom,
! [X6: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X6 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1023_split__list__first,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X6 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1024_split__list__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X2: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X2 ) )
=> ? [Ys3: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_1025_split__list__last,axiom,
! [X6: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X6 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_1026_split__list__last,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X6 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_1027_split__list,axiom,
! [X6: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys3: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X6 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_1028_split__list,axiom,
! [X6: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys3: list_lambda_lambda,Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X6 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_1029_hd__in__set,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ Xs ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ).
% hd_in_set
thf(fact_1030_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_1031_list_Oset__sel_I1_J,axiom,
! [A2: list_l5168835686021156086lambda] :
( ( A2 != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ A2 ) @ ( set_li3846194476672328965lambda @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1032_list_Oset__sel_I1_J,axiom,
! [A2: list_lambda_lambda] :
( ( A2 != nil_lambda_lambda )
=> ( member_lambda_lambda @ ( hd_lambda_lambda @ A2 ) @ ( set_lambda_lambda2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1033_list_Oset__sel_I2_J,axiom,
! [A2: list_l5168835686021156086lambda,X6: list_lambda_lambda] :
( ( A2 != nil_li8379291187143545974lambda )
=> ( ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ ( tl_lis5905685823335986783lambda @ A2 ) ) )
=> ( member2667661554319814297lambda @ X6 @ ( set_li3846194476672328965lambda @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_1034_list_Oset__sel_I2_J,axiom,
! [A2: list_lambda_lambda,X6: lambda_lambda] :
( ( A2 != nil_lambda_lambda )
=> ( ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ ( tl_lambda_lambda @ A2 ) ) )
=> ( member_lambda_lambda @ X6 @ ( set_lambda_lambda2 @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_1035_last__in__set,axiom,
! [As: list_l5168835686021156086lambda] :
( ( As != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( last_l1874196399739661071lambda @ As ) @ ( set_li3846194476672328965lambda @ As ) ) ) ).
% last_in_set
thf(fact_1036_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_1037_red__iff,axiom,
( lambda_red
= ( ^ [A4: lambda_lambda,B2: lambda_lambda] :
? [T8: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T8 )
& ( ( paths_1651982994781619320lambda @ lambda_resid @ T8 )
= A4 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T8 )
= B2 ) ) ) ) ).
% red_iff
thf(fact_1038_Ide__consE,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ( U3 != nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U3 ) )
=> ~ ( ( U3 != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).
% Ide_consE
thf(fact_1039_Arr__consE,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( U3 != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ U3 ) )
=> ~ ( ( U3 != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) ) ) ) ) ) ).
% Arr_consE
thf(fact_1040__092_060Lambda_062_Otrg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_resid @ T @ T ) ) ).
% \<Lambda>.trg_def
thf(fact_1041__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_1042__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_1043__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_1044__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_1045__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_1046__092_060Lambda_062_Oide__iff__trg__self,axiom,
! [A2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A2 )
= ( ( trg_lambda_lambda @ lambda_resid @ A2 )
= A2 ) ) ) ).
% \<Lambda>.ide_iff_trg_self
thf(fact_1047__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_1048__092_060Lambda_062_Oresid__ide_I2_J,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ T )
=> ( ( lambda_resid @ A2 @ T )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.resid_ide(2)
thf(fact_1049__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_1050__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_1051__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_1052__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_1053__092_060Lambda_062_Oin__targetsE,axiom,
! [B3: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ B3 )
=> ~ ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B3 ) ) ) ).
% \<Lambda>.in_targetsE
thf(fact_1054_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_1055__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_1056__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_1057__092_060Lambda_062_Otrg__ide,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( trg_lambda_lambda @ lambda_resid @ A2 )
= A2 ) ) ).
% \<Lambda>.trg_ide
thf(fact_1058__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_1059__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_1060__092_060Lambda_062_Oin__targetsI,axiom,
! [B3: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B3 )
=> ( ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B3 )
=> ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.in_targetsI
thf(fact_1061_Trg__simp,axiom,
! [T6: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= ( trg_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T6 ) ) ) ) ).
% Trg_simp
thf(fact_1062_Arr__consI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U3 )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ).
% Arr_consI\<^sub>P\<^sub>W\<^sub>E
thf(fact_1063_Ide__consI,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U3 )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U3 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U3 ) ) ) ) ) ).
% Ide_consI
thf(fact_1064_simulation_Opreserves__trg,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ( arr_lambda_lambda @ A @ T )
=> ( ( F @ ( trg_lambda_lambda @ A @ T ) )
= ( trg_lambda_lambda @ B @ ( F @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1065_simulation_Opreserves__trg,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ( arr_lambda_lambda @ A @ T )
=> ( ( F @ ( trg_lambda_lambda @ A @ T ) )
= ( trg_li817795487149935061lambda @ B @ ( F @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1066_simulation_Opreserves__trg,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ( arr_li3691443969012319837lambda @ A @ T )
=> ( ( F @ ( trg_li817795487149935061lambda @ A @ T ) )
= ( trg_lambda_lambda @ B @ ( F @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1067_simulation_Opreserves__trg,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ( arr_li3691443969012319837lambda @ A @ T )
=> ( ( F @ ( trg_li817795487149935061lambda @ A @ T ) )
= ( trg_li817795487149935061lambda @ B @ ( F @ T ) ) ) ) ) ).
% simulation.preserves_trg
thf(fact_1068_lambda__calculus_Ored_Ointros_I2_J,axiom,
! [A2: lambda_lambda,B3: lambda_lambda,C: lambda_lambda] :
( ( lambda_red @ A2 @ B3 )
=> ( ( lambda_red @ B3 @ C )
=> ( lambda_red @ A2 @ C ) ) ) ).
% lambda_calculus.red.intros(2)
thf(fact_1069_lambda__calculus_Oweak__diamond,axiom,
! [A2: lambda_lambda,B3: lambda_lambda,B4: lambda_lambda] :
( ( lambda_red @ A2 @ B3 )
=> ( ( lambda_red @ A2 @ B4 )
=> ~ ! [C5: lambda_lambda] :
( ( lambda_red @ B3 @ C5 )
=> ~ ( lambda_red @ B4 @ C5 ) ) ) ) ).
% lambda_calculus.weak_diamond
thf(fact_1070_residuation_Otrg_Ocong,axiom,
trg_lambda_lambda = trg_lambda_lambda ).
% residuation.trg.cong
thf(fact_1071_residuation_Otrg_Ocong,axiom,
trg_li817795487149935061lambda = trg_li817795487149935061lambda ).
% residuation.trg.cong
thf(fact_1072_lambda__calculus_ONF__unique,axiom,
! [T: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( lambda_red @ T @ U )
=> ( ( lambda_red @ T @ U2 )
=> ( ( lambda_NF @ U )
=> ( ( lambda_NF @ U2 )
=> ( U = U2 ) ) ) ) ) ).
% lambda_calculus.NF_unique
thf(fact_1073_lambda__calculus_ONF__reduct__is__trivial,axiom,
! [A2: lambda_lambda,B3: lambda_lambda] :
( ( lambda_NF @ A2 )
=> ( ( lambda_red @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% lambda_calculus.NF_reduct_is_trivial
thf(fact_1074_lambda__calculus_Onormalizable__def,axiom,
( lambda_normalizable
= ( ^ [A4: lambda_lambda] :
( ( lambda_Ide @ A4 )
& ? [B2: lambda_lambda] :
( ( lambda_red @ A4 @ B2 )
& ( lambda_NF @ B2 ) ) ) ) ) ).
% lambda_calculus.normalizable_def
thf(fact_1075_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_1076_trg__def,axiom,
! [T: list_lambda_lambda] :
( ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ).
% trg_def
thf(fact_1077_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_1078_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_1079_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_1080_null__eqI,axiom,
! [N2: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ N2 @ T3 )
= N2 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ N2 )
= N2 ) )
=> ( N2
= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% null_eqI
thf(fact_1081_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_1082_null__char,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= nil_lambda_lambda ) ).
% null_char
thf(fact_1083_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_1084_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_1085_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_1086_not__arr__null,axiom,
~ ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% not_arr_null
thf(fact_1087_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_1088_in__targetsE,axiom,
! [B3: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 )
=> ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B3 ) ) ) ).
% in_targetsE
thf(fact_1089_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_1090_incl__is__simulation,axiom,
( simula1175017789036123529lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ^ [T5: lambda_lambda] : ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T5 ) @ ( cons_lambda_lambda @ T5 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% incl_is_simulation
thf(fact_1091_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_1092_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_1093_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_1094_in__targetsI,axiom,
! [B3: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B3 )
=> ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% in_targetsI
thf(fact_1095_partial__magma_Onull_Ocong,axiom,
partia6873046715171720854lambda = partia6873046715171720854lambda ).
% partial_magma.null.cong
thf(fact_1096_partial__magma_Onull_Ocong,axiom,
partia1461432607296028304lambda = partia1461432607296028304lambda ).
% partial_magma.null.cong
thf(fact_1097_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ( ( OP @ T @ ( partia6873046715171720854lambda @ OP ) )
= ( partia6873046715171720854lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_1098_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ( OP @ T @ ( partia1461432607296028304lambda @ OP ) )
= ( partia1461432607296028304lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_1099_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ( ( OP @ ( partia6873046715171720854lambda @ OP ) @ T )
= ( partia6873046715171720854lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_1100_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ( OP @ ( partia1461432607296028304lambda @ OP ) @ T )
= ( partia1461432607296028304lambda @ OP ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_1101_partial__magma_Onull__eqI,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,N2: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ( ! [T3: list_lambda_lambda] :
( ( ( OP @ N2 @ T3 )
= N2 )
& ( ( OP @ T3 @ N2 )
= N2 ) )
=> ( N2
= ( partia6873046715171720854lambda @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_1102_partial__magma_Onull__eqI,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,N2: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ! [T3: lambda_lambda] :
( ( ( OP @ N2 @ T3 )
= N2 )
& ( ( OP @ T3 @ N2 )
= N2 ) )
=> ( N2
= ( partia1461432607296028304lambda @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_1103_simulation_Oextensional,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,T: lambda_lambda] :
( ( simula1175017789036123529lambda @ A @ B @ F )
=> ( ~ ( arr_lambda_lambda @ A @ T )
=> ( ( F @ T )
= ( partia6873046715171720854lambda @ B ) ) ) ) ).
% simulation.extensional
thf(fact_1104_simulation_Oextensional,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( simula3069223088535412739lambda @ A @ B @ F )
=> ( ~ ( arr_li3691443969012319837lambda @ A @ T )
=> ( ( F @ T )
= ( partia6873046715171720854lambda @ B ) ) ) ) ).
% simulation.extensional
thf(fact_1105_simulation_Oextensional,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( simula5235733974575307523lambda @ A @ B @ F )
=> ( ~ ( arr_lambda_lambda @ A @ T )
=> ( ( F @ T )
= ( partia1461432607296028304lambda @ B ) ) ) ) ).
% simulation.extensional
thf(fact_1106_simulation_Oextensional,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,T: list_lambda_lambda] :
( ( simula9164532646813739645lambda @ A @ B @ F )
=> ( ~ ( arr_li3691443969012319837lambda @ A @ T )
=> ( ( F @ T )
= ( partia1461432607296028304lambda @ B ) ) ) ) ).
% simulation.extensional
thf(fact_1107_incl__is__injective,axiom,
( inj_on3897920697455699703lambda
@ ^ [T5: lambda_lambda] : ( if_lis7195930416592267574lambda @ ( arr_lambda_lambda @ lambda_resid @ T5 ) @ ( cons_lambda_lambda @ T5 @ nil_lambda_lambda ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) )
@ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ lambda_resid ) ) ) ).
% incl_is_injective
thf(fact_1108_Trg_Oelims,axiom,
! [X6: list_lambda_lambda,Y5: lambda_lambda] :
( ( ( paths_2695323870737867955lambda @ lambda_resid @ X6 )
= Y5 )
=> ( ( ( X6 = nil_lambda_lambda )
=> ( Y5
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y5
!= ( trg_lambda_lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y5
!= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Trg.elims
thf(fact_1109__092_060Lambda_062_Onull__eqI,axiom,
! [N2: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( lambda_resid @ N2 @ T3 )
= N2 )
& ( ( lambda_resid @ T3 @ N2 )
= N2 ) )
=> ( N2
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.null_eqI
thf(fact_1110__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_1111__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_1112__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_1113__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_1114__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_1115__092_060Lambda_062_Onot__arr__null,axiom,
~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.not_arr_null
thf(fact_1116_Trg_Osimps_I1_J,axiom,
( ( paths_2695323870737867955lambda @ lambda_resid @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Trg.simps(1)
thf(fact_1117__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_1118__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_1119_lambda__calculus_Onull__char,axiom,
( ( partia1461432607296028304lambda @ lambda_resid )
= lambda_Nil ) ).
% lambda_calculus.null_char
thf(fact_1120__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_1121_inj__on__empty,axiom,
! [F2: lambda_lambda > list_lambda_lambda] : ( inj_on3897920697455699703lambda @ F2 @ bot_bo1271416176687173302lambda ) ).
% inj_on_empty
thf(fact_1122_inj__on__image__Int,axiom,
! [F2: lambda_lambda > lambda_lambda,C2: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inj_on287184477777394545lambda @ F2 @ C2 )
=> ( ( ord_le1926335974945931754lambda @ A @ C2 )
=> ( ( ord_le1926335974945931754lambda @ B @ C2 )
=> ( ( image_2356535832909760517lambda @ F2 @ ( inf_in1766959559283430712lambda @ A @ B ) )
= ( inf_in1766959559283430712lambda @ ( image_2356535832909760517lambda @ F2 @ A ) @ ( image_2356535832909760517lambda @ F2 @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1123_inj__on__image__Int,axiom,
! [F2: lambda_lambda > list_lambda_lambda,C2: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inj_on3897920697455699703lambda @ F2 @ C2 )
=> ( ( ord_le1926335974945931754lambda @ A @ C2 )
=> ( ( ord_le1926335974945931754lambda @ B @ C2 )
=> ( ( image_3692823078739859851lambda @ F2 @ ( inf_in1766959559283430712lambda @ A @ B ) )
= ( inf_in3354698378728726462lambda @ ( image_3692823078739859851lambda @ F2 @ A ) @ ( image_3692823078739859851lambda @ F2 @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1124_inj__on__image__Int,axiom,
! [F2: list_lambda_lambda > lambda_lambda,C2: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( inj_on2664063518378540011lambda @ F2 @ C2 )
=> ( ( ord_le3064231726521830256lambda @ A @ C2 )
=> ( ( ord_le3064231726521830256lambda @ B @ C2 )
=> ( ( image_2458965899662700159lambda @ F2 @ ( inf_in3354698378728726462lambda @ A @ B ) )
= ( inf_in1766959559283430712lambda @ ( image_2458965899662700159lambda @ F2 @ A ) @ ( image_2458965899662700159lambda @ F2 @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1125_inj__on__image__Int,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda,C2: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( inj_on8159307203718685297lambda @ F2 @ C2 )
=> ( ( ord_le3064231726521830256lambda @ A @ C2 )
=> ( ( ord_le3064231726521830256lambda @ B @ C2 )
=> ( ( image_5886195720093285125lambda @ F2 @ ( inf_in3354698378728726462lambda @ A @ B ) )
= ( inf_in3354698378728726462lambda @ ( image_5886195720093285125lambda @ F2 @ A ) @ ( image_5886195720093285125lambda @ F2 @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1126_inj__on__Cons1,axiom,
! [X6: lambda_lambda,A: set_li7878742357367608272lambda] : ( inj_on8159307203718685297lambda @ ( cons_lambda_lambda @ X6 ) @ A ) ).
% inj_on_Cons1
thf(fact_1127_inj__singleton,axiom,
! [A: set_lambda_lambda] :
( inj_on1689892504205702737lambda
@ ^ [X4: lambda_lambda] : ( insert_lambda_lambda @ X4 @ bot_bo1271416176687173302lambda )
@ A ) ).
% inj_singleton
thf(fact_1128_inj__singleton,axiom,
! [A: set_li7878742357367608272lambda] :
( inj_on4344131335932687313lambda
@ ^ [X4: list_lambda_lambda] : ( insert8465602035467394112lambda @ X4 @ bot_bo1397037886704484924lambda )
@ A ) ).
% inj_singleton
thf(fact_1129_inj__on__image__iff,axiom,
! [A: set_lambda_lambda,G: lambda_lambda > list_lambda_lambda,F2: lambda_lambda > lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ A )
=> ( ( ( G @ ( F2 @ X ) )
= ( G @ ( F2 @ Xa2 ) ) )
= ( ( G @ X )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on287184477777394545lambda @ F2 @ A )
=> ( ( inj_on3897920697455699703lambda @ G @ ( image_2356535832909760517lambda @ F2 @ A ) )
= ( inj_on3897920697455699703lambda @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_1130_inj__on__subset,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inj_on3897920697455699703lambda @ F2 @ A )
=> ( ( ord_le1926335974945931754lambda @ B @ A )
=> ( inj_on3897920697455699703lambda @ F2 @ B ) ) ) ).
% inj_on_subset
thf(fact_1131_subset__inj__on,axiom,
! [F2: lambda_lambda > list_lambda_lambda,B: set_lambda_lambda,A: set_lambda_lambda] :
( ( inj_on3897920697455699703lambda @ F2 @ B )
=> ( ( ord_le1926335974945931754lambda @ A @ B )
=> ( inj_on3897920697455699703lambda @ F2 @ A ) ) ) ).
% subset_inj_on
thf(fact_1132_inj__on__Int,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ( inj_on3897920697455699703lambda @ F2 @ A )
| ( inj_on3897920697455699703lambda @ F2 @ B ) )
=> ( inj_on3897920697455699703lambda @ F2 @ ( inf_in1766959559283430712lambda @ A @ B ) ) ) ).
% inj_on_Int
thf(fact_1133_inj__on__image__eq__iff,axiom,
! [F2: lambda_lambda > list_lambda_lambda,C2: set_lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( inj_on3897920697455699703lambda @ F2 @ C2 )
=> ( ( ord_le1926335974945931754lambda @ A @ C2 )
=> ( ( ord_le1926335974945931754lambda @ B @ C2 )
=> ( ( ( image_3692823078739859851lambda @ F2 @ A )
= ( image_3692823078739859851lambda @ F2 @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1134_inj__on__image__mem__iff,axiom,
! [F2: lambda_lambda > lambda_lambda,B: set_lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
( ( inj_on287184477777394545lambda @ F2 @ B )
=> ( ( member_lambda_lambda @ A2 @ B )
=> ( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( member_lambda_lambda @ ( F2 @ A2 ) @ ( image_2356535832909760517lambda @ F2 @ A ) )
= ( member_lambda_lambda @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1135_inj__on__image__mem__iff,axiom,
! [F2: lambda_lambda > list_lambda_lambda,B: set_lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
( ( inj_on3897920697455699703lambda @ F2 @ B )
=> ( ( member_lambda_lambda @ A2 @ B )
=> ( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( member2667661554319814297lambda @ ( F2 @ A2 ) @ ( image_3692823078739859851lambda @ F2 @ A ) )
= ( member_lambda_lambda @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1136_inj__on__image__mem__iff,axiom,
! [F2: list_lambda_lambda > lambda_lambda,B: set_li7878742357367608272lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( inj_on2664063518378540011lambda @ F2 @ B )
=> ( ( member2667661554319814297lambda @ A2 @ B )
=> ( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( member_lambda_lambda @ ( F2 @ A2 ) @ ( image_2458965899662700159lambda @ F2 @ A ) )
= ( member2667661554319814297lambda @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1137_inj__on__image__mem__iff,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda,B: set_li7878742357367608272lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( inj_on8159307203718685297lambda @ F2 @ B )
=> ( ( member2667661554319814297lambda @ A2 @ B )
=> ( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( member2667661554319814297lambda @ ( F2 @ A2 ) @ ( image_5886195720093285125lambda @ F2 @ A ) )
= ( member2667661554319814297lambda @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1138_inj__img__insertE,axiom,
! [F2: lambda_lambda > lambda_lambda,A: set_lambda_lambda,X6: lambda_lambda,B: set_lambda_lambda] :
( ( inj_on287184477777394545lambda @ F2 @ A )
=> ( ~ ( member_lambda_lambda @ X6 @ B )
=> ( ( ( insert_lambda_lambda @ X6 @ B )
= ( image_2356535832909760517lambda @ F2 @ A ) )
=> ~ ! [X8: lambda_lambda,A7: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X8 @ A7 )
=> ( ( A
= ( insert_lambda_lambda @ X8 @ A7 ) )
=> ( ( X6
= ( F2 @ X8 ) )
=> ( B
!= ( image_2356535832909760517lambda @ F2 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1139_inj__img__insertE,axiom,
! [F2: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda,X6: lambda_lambda,B: set_lambda_lambda] :
( ( inj_on2664063518378540011lambda @ F2 @ A )
=> ( ~ ( member_lambda_lambda @ X6 @ B )
=> ( ( ( insert_lambda_lambda @ X6 @ B )
= ( image_2458965899662700159lambda @ F2 @ A ) )
=> ~ ! [X8: list_lambda_lambda,A7: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ X8 @ A7 )
=> ( ( A
= ( insert8465602035467394112lambda @ X8 @ A7 ) )
=> ( ( X6
= ( F2 @ X8 ) )
=> ( B
!= ( image_2458965899662700159lambda @ F2 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1140_inj__img__insertE,axiom,
! [F2: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,X6: list_lambda_lambda,B: set_li7878742357367608272lambda] :
( ( inj_on3897920697455699703lambda @ F2 @ A )
=> ( ~ ( member2667661554319814297lambda @ X6 @ B )
=> ( ( ( insert8465602035467394112lambda @ X6 @ B )
= ( image_3692823078739859851lambda @ F2 @ A ) )
=> ~ ! [X8: lambda_lambda,A7: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X8 @ A7 )
=> ( ( A
= ( insert_lambda_lambda @ X8 @ A7 ) )
=> ( ( X6
= ( F2 @ X8 ) )
=> ( B
!= ( image_3692823078739859851lambda @ F2 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1141_inj__img__insertE,axiom,
! [F2: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda,X6: list_lambda_lambda,B: set_li7878742357367608272lambda] :
( ( inj_on8159307203718685297lambda @ F2 @ A )
=> ( ~ ( member2667661554319814297lambda @ X6 @ B )
=> ( ( ( insert8465602035467394112lambda @ X6 @ B )
= ( image_5886195720093285125lambda @ F2 @ A ) )
=> ~ ! [X8: list_lambda_lambda,A7: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ X8 @ A7 )
=> ( ( A
= ( insert8465602035467394112lambda @ X8 @ A7 ) )
=> ( ( X6
= ( F2 @ X8 ) )
=> ( B
!= ( image_5886195720093285125lambda @ F2 @ A7 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1142_inj__on__iff__surj,axiom,
! [A: set_lambda_lambda,A8: set_lambda_lambda] :
( ( A != bot_bo1271416176687173302lambda )
=> ( ( ? [F3: lambda_lambda > lambda_lambda] :
( ( inj_on287184477777394545lambda @ F3 @ A )
& ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F3 @ A ) @ A8 ) ) )
= ( ? [G2: lambda_lambda > lambda_lambda] :
( ( image_2356535832909760517lambda @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1143_inj__on__iff__surj,axiom,
! [A: set_li7878742357367608272lambda,A8: set_lambda_lambda] :
( ( A != bot_bo1397037886704484924lambda )
=> ( ( ? [F3: list_lambda_lambda > lambda_lambda] :
( ( inj_on2664063518378540011lambda @ F3 @ A )
& ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F3 @ A ) @ A8 ) ) )
= ( ? [G2: lambda_lambda > list_lambda_lambda] :
( ( image_3692823078739859851lambda @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1144_inj__on__iff__surj,axiom,
! [A: set_lambda_lambda,A8: set_li7878742357367608272lambda] :
( ( A != bot_bo1271416176687173302lambda )
=> ( ( ? [F3: lambda_lambda > list_lambda_lambda] :
( ( inj_on3897920697455699703lambda @ F3 @ A )
& ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F3 @ A ) @ A8 ) ) )
= ( ? [G2: list_lambda_lambda > lambda_lambda] :
( ( image_2458965899662700159lambda @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1145_inj__on__iff__surj,axiom,
! [A: set_li7878742357367608272lambda,A8: set_li7878742357367608272lambda] :
( ( A != bot_bo1397037886704484924lambda )
=> ( ( ? [F3: list_lambda_lambda > list_lambda_lambda] :
( ( inj_on8159307203718685297lambda @ F3 @ A )
& ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F3 @ A ) @ A8 ) ) )
= ( ? [G2: list_lambda_lambda > list_lambda_lambda] :
( ( image_5886195720093285125lambda @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1146_Resid1x__as__Resid_H,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 )
= ( hd_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
= nil_lambda_lambda )
=> ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ).
% Resid1x_as_Resid'
thf(fact_1147_Resid1x_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) ) ).
% Resid1x.simps(3)
thf(fact_1148_Resid1x__null,axiom,
! [T6: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T6 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x_null
thf(fact_1149_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_1150_Resid1x_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x.simps(1)
thf(fact_1151_Resid1x__ide,axiom,
! [A2: lambda_lambda,T6: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ( paths_3319247187298047121lambda @ lambda_resid @ A2 @ T6 )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( paths_3319247187298047121lambda @ lambda_resid @ A2 @ T6 ) ) ) ) ).
% Resid1x_ide
thf(fact_1152_Resid1x__as__Resid,axiom,
! [T: lambda_lambda,U3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U3 )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 ) @ nil_lambda_lambda ) ) ) ).
% Resid1x_as_Resid
thf(fact_1153_Resid__cons_H,axiom,
! [T6: list_lambda_lambda,T: lambda_lambda,U3: list_lambda_lambda] :
( ( T6 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T6 ) @ U3 )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% Resid_cons'
thf(fact_1154_Resid1x_Oelims,axiom,
! [X6: lambda_lambda,Xa3: list_lambda_lambda,Y5: lambda_lambda] :
( ( ( paths_3319247187298047121lambda @ lambda_resid @ X6 @ Xa3 )
= Y5 )
=> ( ( ( Xa3 = nil_lambda_lambda )
=> ( Y5
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [U5: lambda_lambda] :
( ( Xa3
= ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) )
=> ( Y5
!= ( lambda_resid @ X6 @ U5 ) ) )
=> ~ ! [U5: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( Xa3
= ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ( Y5
!= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ X6 @ U5 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) ) ) ) ) ) ) ).
% Resid1x.elims
thf(fact_1155_Resid_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(4)
thf(fact_1156_paths__in__rts_OResid1x_Ocong,axiom,
paths_3319247187298047121lambda = paths_3319247187298047121lambda ).
% paths_in_rts.Resid1x.cong
thf(fact_1157_subset__image__inj,axiom,
! [S: set_lambda_lambda,F2: lambda_lambda > lambda_lambda,T6: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ S @ ( image_2356535832909760517lambda @ F2 @ T6 ) )
= ( ? [U7: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ U7 @ T6 )
& ( inj_on287184477777394545lambda @ F2 @ U7 )
& ( S
= ( image_2356535832909760517lambda @ F2 @ U7 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1158_subset__image__inj,axiom,
! [S: set_lambda_lambda,F2: list_lambda_lambda > lambda_lambda,T6: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ S @ ( image_2458965899662700159lambda @ F2 @ T6 ) )
= ( ? [U7: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ U7 @ T6 )
& ( inj_on2664063518378540011lambda @ F2 @ U7 )
& ( S
= ( image_2458965899662700159lambda @ F2 @ U7 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1159_subset__image__inj,axiom,
! [S: set_li7878742357367608272lambda,F2: lambda_lambda > list_lambda_lambda,T6: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ S @ ( image_3692823078739859851lambda @ F2 @ T6 ) )
= ( ? [U7: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ U7 @ T6 )
& ( inj_on3897920697455699703lambda @ F2 @ U7 )
& ( S
= ( image_3692823078739859851lambda @ F2 @ U7 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1160_subset__image__inj,axiom,
! [S: set_li7878742357367608272lambda,F2: list_lambda_lambda > list_lambda_lambda,T6: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ S @ ( image_5886195720093285125lambda @ F2 @ T6 ) )
= ( ? [U7: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ U7 @ T6 )
& ( inj_on8159307203718685297lambda @ F2 @ U7 )
& ( S
= ( image_5886195720093285125lambda @ F2 @ U7 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1161_Resid_Oelims,axiom,
! [X6: list_lambda_lambda,Xa3: list_lambda_lambda,Y5: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X6 @ Xa3 )
= Y5 )
=> ( ( ( X6 = nil_lambda_lambda )
=> ( Y5 != nil_lambda_lambda ) )
=> ( ( ? [V3: lambda_lambda,Va2: list_lambda_lambda] :
( X6
= ( cons_lambda_lambda @ V3 @ Va2 ) )
=> ( ( Xa3 = nil_lambda_lambda )
=> ( Y5 != nil_lambda_lambda ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U5: lambda_lambda] :
( ( Xa3
= ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
=> ( Y5
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U5 ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
=> ( Y5 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U5: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( Xa3
= ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y5
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ V3 @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y5 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ! [U5: lambda_lambda] :
( ( Xa3
= ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U5 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y5
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U5 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U5 @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U5 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y5 = nil_lambda_lambda ) ) ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ! [U5: lambda_lambda,Vb2: lambda_lambda,Vc: list_lambda_lambda] :
( ( Xa3
= ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ Vb2 @ Vc ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb2 @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U5 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc ) @ ( lambda_resid @ T3 @ U5 ) ) )
!= nil_lambda_lambda ) )
=> ( Y5
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb2 @ Vc ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U5 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc ) @ ( lambda_resid @ T3 @ U5 ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb2 @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ U5 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc ) @ ( lambda_resid @ T3 @ U5 ) ) )
!= nil_lambda_lambda ) )
=> ( Y5 = nil_lambda_lambda ) ) ) ) ) ) ) ) ) ) ) ).
% Resid.elims
thf(fact_1162_Residx1_Osimps_I1_J,axiom,
! [U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ nil_lambda_lambda @ U )
= nil_lambda_lambda ) ).
% Residx1.simps(1)
thf(fact_1163_Residx1__as__Resid,axiom,
! [T6: list_lambda_lambda,U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ T6 @ U )
= ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ).
% Residx1_as_Resid
thf(fact_1164_Residx1_Oelims,axiom,
! [X6: list_lambda_lambda,Xa3: lambda_lambda,Y5: list_lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ X6 @ Xa3 )
= Y5 )
=> ( ( ( X6 = nil_lambda_lambda )
=> ( Y5 != nil_lambda_lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa3 )
=> ( Y5
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa3 ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ Xa3 )
=> ( Y5 = nil_lambda_lambda ) ) ) )
=> ~ ! [T3: lambda_lambda,V3: lambda_lambda,Va2: list_lambda_lambda] :
( ( X6
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V3 @ Va2 ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa3 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa3 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y5
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa3 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa3 @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa3 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V3 @ Va2 ) @ ( lambda_resid @ Xa3 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y5 = nil_lambda_lambda ) ) ) ) ) ) ) ).
% Residx1.elims
thf(fact_1165_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_1166_Residx1_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ U )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ U )
= nil_lambda_lambda ) ) ) ).
% Residx1.simps(3)
thf(fact_1167_Con__sym1,axiom,
! [T6: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ T6 @ U )
!= nil_lambda_lambda )
= ( ( paths_3319247187298047121lambda @ lambda_resid @ U @ T6 )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% Con_sym1
thf(fact_1168_Resid_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(5)
thf(fact_1169_Resid_Osimps_I7_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda,Vb: 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 @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(7)
thf(fact_1170_Resid_Osimps_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb: lambda_lambda,Vc2: list_lambda_lambda,V: lambda_lambda,Va: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(6)
thf(fact_1171_paths__in__rts_OResidx1_Ocong,axiom,
paths_2112734093361750239lambda = paths_2112734093361750239lambda ).
% paths_in_rts.Residx1.cong
thf(fact_1172_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_1173_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 ) ) )
=> ( ! [A5: lambda_lambda,T3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A5 )
=> ( ( con_lambda_lambda @ Resid @ T3 @ A5 )
=> ( ( Resid @ T3 @ A5 )
= T3 ) ) )
=> ( ! [A5: lambda_lambda,T3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A5 )
=> ( ( con_lambda_lambda @ Resid @ A5 @ T3 )
=> ( ide_lambda_lambda @ Resid @ ( Resid @ A5 @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( con_lambda_lambda @ Resid @ T3 @ U5 )
=> ? [A9: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A9 )
& ( con_lambda_lambda @ Resid @ A9 @ T3 )
& ( con_lambda_lambda @ Resid @ A9 @ U5 ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ ( Resid @ T3 @ U5 ) )
=> ( ( con_lambda_lambda @ Resid @ U5 @ V3 )
=> ( con_lambda_lambda @ Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ V3 @ U5 ) ) ) )
=> ( rts_ax7298520571313827089lambda @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_1174_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 ) ) )
=> ( ! [A5: list_lambda_lambda,T3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A5 )
=> ( ( con_li79537494059641180lambda @ Resid @ T3 @ A5 )
=> ( ( Resid @ T3 @ A5 )
= T3 ) ) )
=> ( ! [A5: list_lambda_lambda,T3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A5 )
=> ( ( con_li79537494059641180lambda @ Resid @ A5 @ T3 )
=> ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ A5 @ T3 ) ) ) )
=> ( ! [T3: list_lambda_lambda,U5: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ Resid @ T3 @ U5 )
=> ? [A9: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A9 )
& ( con_li79537494059641180lambda @ Resid @ A9 @ T3 )
& ( con_li79537494059641180lambda @ Resid @ A9 @ U5 ) ) )
=> ( ! [T3: list_lambda_lambda,U5: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ T3 @ U5 ) )
=> ( ( con_li79537494059641180lambda @ Resid @ U5 @ V3 )
=> ( con_li79537494059641180lambda @ Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ V3 @ U5 ) ) ) )
=> ( rts_ax2283037410143324311lambda @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_1175_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_1176_lambda__calculus_Oresid_Osimps_I38_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(38)
thf(fact_1177_lambda__calculus_Oresid_Osimps_I19_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(19)
thf(fact_1178_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T4: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T4 @ U2 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T4 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U2 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(3)
thf(fact_1179_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_1180_lambda__calculus_Olambda_Odistinct_I17_J,axiom,
! [X3: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Lam @ X3 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(17)
thf(fact_1181_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_1182_lambda__calculus_Olambda_Odistinct_I7_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( lambda_Nil
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(7)
thf(fact_1183_lambda__calculus_Oresid_Osimps_I15_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,Vc2: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Va ) @ ( lambda_Beta @ Vb @ Vc2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(15)
thf(fact_1184_lambda__calculus_Oresid_Osimps_I17_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc2: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(17)
thf(fact_1185_lambda__calculus_Oresid_Osimps_I18_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc2: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(18)
thf(fact_1186_lambda__calculus_Oresid_Osimps_I22_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vc2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ lambda_Nil @ Vc2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(22)
thf(fact_1187_lambda__calculus_Oresid_Osimps_I24_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(24)
thf(fact_1188_lambda__calculus_Oresid_Osimps_I25_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(25)
thf(fact_1189_lambda__calculus_Oresid_Osimps_I34_J,axiom,
! [Vb: lambda_lambda,Vc2: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc2 ) @ ( lambda_App @ lambda_Nil @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(34)
thf(fact_1190_lambda__calculus_Oresid_Osimps_I36_J,axiom,
! [Vb: lambda_lambda,Vc2: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc2 ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(36)
thf(fact_1191_lambda__calculus_Oresid_Osimps_I37_J,axiom,
! [Vb: lambda_lambda,Vc2: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc2 ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(37)
thf(fact_1192_lambda__calculus_Oresid_Osimps_I41_J,axiom,
! [Vc2: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Vc2 ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(41)
thf(fact_1193_lambda__calculus_Oresid_Osimps_I43_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc2: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc2 ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(43)
thf(fact_1194_lambda__calculus_Oresid_Osimps_I44_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc2: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc2 ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(44)
thf(fact_1195_lambda__calculus_Oresid_Osimps_I11_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Beta @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(11)
thf(fact_1196_lambda__calculus_Oresid_Osimps_I21_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(21)
thf(fact_1197_lambda__calculus_Oresid_Osimps_I30_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(30)
thf(fact_1198_lambda__calculus_Oresid_Osimps_I40_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(40)
thf(fact_1199_lambda__calculus_OConD_I2_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,V2: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_Beta @ V2 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Lam @ V2 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(2)
thf(fact_1200_lambda__calculus_OSrc__eq__iff_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T4: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T4 @ U2 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T4 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U2 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(4)
thf(fact_1201_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_1202_lambda__calculus_Oresid_Osimps_I6_J,axiom,
! [T: lambda_lambda,T4: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T4 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T4 ) @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T4 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T4 ) @ U2 ) )
= ( lambda_Beta @ ( lambda_resid @ T @ T4 ) @ ( lambda_resid @ U @ U2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(6)
thf(fact_1203_lambda__calculus_OConD_I3_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,T4: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_App @ T4 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ T4 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(3)
thf(fact_1204_lambda__calculus_OConD_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V2: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_Beta @ V2 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ ( lambda_Lam @ V2 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(4)
thf(fact_1205_lambda__calculus_OIde_Oelims_I3_J,axiom,
! [X6: lambda_lambda] :
( ~ ( lambda_Ide @ X6 )
=> ( ( X6 != lambda_Nil )
=> ( ! [T3: lambda_lambda] :
( ( X6
= ( lambda_Lam @ T3 ) )
=> ( lambda_Ide @ T3 ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X6
= ( lambda_App @ T3 @ U5 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 ) ) )
=> ~ ! [T3: lambda_lambda,U5: lambda_lambda] :
( X6
!= ( lambda_Beta @ T3 @ U5 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(3)
thf(fact_1206_rts__axioms__def,axiom,
( rts_ax7298520571313827089lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( arr_lambda_lambda @ Resid2 @ T5 )
=> ( ide_lambda_lambda @ Resid2 @ ( trg_lambda_lambda @ Resid2 @ T5 ) ) )
& ! [A4: lambda_lambda,T5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A4 )
=> ( ( con_lambda_lambda @ Resid2 @ T5 @ A4 )
=> ( ( Resid2 @ T5 @ A4 )
= T5 ) ) )
& ! [A4: lambda_lambda,T5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A4 )
=> ( ( con_lambda_lambda @ Resid2 @ A4 @ T5 )
=> ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ A4 @ T5 ) ) ) )
& ! [T5: lambda_lambda,U6: lambda_lambda] :
( ( con_lambda_lambda @ Resid2 @ T5 @ U6 )
=> ? [A4: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A4 )
& ( con_lambda_lambda @ Resid2 @ A4 @ T5 )
& ( con_lambda_lambda @ Resid2 @ A4 @ U6 ) ) )
& ! [T5: lambda_lambda,U6: lambda_lambda,V5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ T5 @ U6 ) )
=> ( ( con_lambda_lambda @ Resid2 @ U6 @ V5 )
=> ( con_lambda_lambda @ Resid2 @ ( Resid2 @ T5 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_1207_rts__axioms__def,axiom,
( rts_ax2283037410143324311lambda
= ( ^ [Resid2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ Resid2 @ T5 )
=> ( ide_li6773656263178250516lambda @ Resid2 @ ( trg_li817795487149935061lambda @ Resid2 @ T5 ) ) )
& ! [A4: list_lambda_lambda,T5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A4 )
=> ( ( con_li79537494059641180lambda @ Resid2 @ T5 @ A4 )
=> ( ( Resid2 @ T5 @ A4 )
= T5 ) ) )
& ! [A4: list_lambda_lambda,T5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A4 )
=> ( ( con_li79537494059641180lambda @ Resid2 @ A4 @ T5 )
=> ( ide_li6773656263178250516lambda @ Resid2 @ ( Resid2 @ A4 @ T5 ) ) ) )
& ! [T5: list_lambda_lambda,U6: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ Resid2 @ T5 @ U6 )
=> ? [A4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A4 )
& ( con_li79537494059641180lambda @ Resid2 @ A4 @ T5 )
& ( con_li79537494059641180lambda @ Resid2 @ A4 @ U6 ) ) )
& ! [T5: list_lambda_lambda,U6: list_lambda_lambda,V5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ ( Resid2 @ T5 @ U6 ) )
=> ( ( con_li79537494059641180lambda @ Resid2 @ U6 @ V5 )
=> ( con_li79537494059641180lambda @ Resid2 @ ( Resid2 @ T5 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_1208_sub__rts__axioms__def,axiom,
( sub_rt3575465509250962105lambda
= ( ^ [R: lambda_lambda > lambda_lambda > lambda_lambda,Arr: lambda_lambda > $o] :
( ! [T5: lambda_lambda] :
( ( Arr @ T5 )
=> ( arr_lambda_lambda @ R @ T5 ) )
& ! [T5: lambda_lambda] :
( ( Arr @ T5 )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R @ T5 ) @ ( collec621444948832755541lambda @ Arr ) ) )
& ! [T5: lambda_lambda,U6: lambda_lambda] :
( ( Arr @ T5 )
=> ( ( Arr @ U6 )
=> ( ( con_lambda_lambda @ R @ T5 @ U6 )
=> ( Arr @ ( R @ T5 @ U6 ) ) ) ) ) ) ) ) ).
% sub_rts_axioms_def
thf(fact_1209_sub__rts__axioms__def,axiom,
( sub_rt3099298114314975807lambda
= ( ^ [R: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Arr: list_lambda_lambda > $o] :
( ! [T5: list_lambda_lambda] :
( ( Arr @ T5 )
=> ( arr_li3691443969012319837lambda @ R @ T5 ) )
& ! [T5: list_lambda_lambda] :
( ( Arr @ T5 )
=> ( ord_le3064231726521830256lambda @ ( source6119337877621565212lambda @ R @ T5 ) @ ( collec8830145823237851355lambda @ Arr ) ) )
& ! [T5: list_lambda_lambda,U6: list_lambda_lambda] :
( ( Arr @ T5 )
=> ( ( Arr @ U6 )
=> ( ( con_li79537494059641180lambda @ R @ T5 @ U6 )
=> ( Arr @ ( R @ T5 @ U6 ) ) ) ) ) ) ) ) ).
% sub_rts_axioms_def
thf(fact_1210_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,U5: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ( Arr2 @ U5 )
=> ( ( con_lambda_lambda @ R2 @ T3 @ U5 )
=> ( Arr2 @ ( R2 @ T3 @ U5 ) ) ) ) )
=> ( sub_rt3575465509250962105lambda @ R2 @ Arr2 ) ) ) ) ).
% sub_rts_axioms.intro
thf(fact_1211_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,U5: list_lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ( Arr2 @ U5 )
=> ( ( con_li79537494059641180lambda @ R2 @ T3 @ U5 )
=> ( Arr2 @ ( R2 @ T3 @ U5 ) ) ) ) )
=> ( sub_rt3099298114314975807lambda @ R2 @ Arr2 ) ) ) ) ).
% sub_rts_axioms.intro
thf(fact_1212_inf__Int__eq,axiom,
! [R2: set_lambda_lambda,S: set_lambda_lambda] :
( ( inf_in2068736301489225125mbda_o
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ R2 )
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ S ) )
= ( ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ ( inf_in1766959559283430712lambda @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1213_inf__Int__eq,axiom,
! [R2: set_li7878742357367608272lambda,S: set_li7878742357367608272lambda] :
( ( inf_in6319970306715391135mbda_o
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ R2 )
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ S ) )
= ( ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ ( inf_in3354698378728726462lambda @ R2 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_1214_pred__subset__eq,axiom,
! [R2: set_lambda_lambda,S: set_lambda_lambda] :
( ( ord_le5488116190248881011mbda_o
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ R2 )
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ S ) )
= ( ord_le1926335974945931754lambda @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1215_pred__subset__eq,axiom,
! [R2: set_li7878742357367608272lambda,S: set_li7878742357367608272lambda] :
( ( ord_le4030711025213088621mbda_o
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ R2 )
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ S ) )
= ( ord_le3064231726521830256lambda @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1216_bot__empty__eq,axiom,
( bot_bo3555140721347798439mbda_o
= ( ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ bot_bo1271416176687173302lambda ) ) ) ).
% bot_empty_eq
thf(fact_1217_bot__empty__eq,axiom,
( bot_bo430095131446760353mbda_o
= ( ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ bot_bo1397037886704484924lambda ) ) ) ).
% bot_empty_eq
thf(fact_1218_listset_Osimps_I1_J,axiom,
( ( listse1500948262344051919lambda @ nil_se6914803783601386320lambda )
= ( insert8465602035467394112lambda @ nil_lambda_lambda @ bot_bo1397037886704484924lambda ) ) ).
% listset.simps(1)
thf(fact_1219__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_1220__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_1221__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_1222__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_1223__092_060Lambda_062_Osrc__eqI,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= A2 ) ) ) ).
% \<Lambda>.src_eqI
thf(fact_1224__092_060Lambda_062_Oide__iff__src__self,axiom,
! [A2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A2 )
= ( ( weakly4554751870655413501lambda @ lambda_resid @ A2 )
= A2 ) ) ) ).
% \<Lambda>.ide_iff_src_self
thf(fact_1225__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_1226__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_1227__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_1228__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_1229_Src_Osimps,axiom,
! [T6: list_lambda_lambda] :
( ( paths_1651982994781619320lambda @ lambda_resid @ T6 )
= ( weakly4554751870655413501lambda @ lambda_resid @ ( hd_lambda_lambda @ T6 ) ) ) ).
% Src.simps
thf(fact_1230_Src_Oelims,axiom,
! [X6: list_lambda_lambda,Y5: lambda_lambda] :
( ( ( paths_1651982994781619320lambda @ lambda_resid @ X6 )
= Y5 )
=> ( Y5
= ( weakly4554751870655413501lambda @ lambda_resid @ ( hd_lambda_lambda @ X6 ) ) ) ) ).
% Src.elims
thf(fact_1231__092_060Lambda_062_Osources__char,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [A4: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= A4 ) ) ) ) ).
% \<Lambda>.sources_char
thf(fact_1232__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_1233__092_060Lambda_062_Osrc__ide,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ A2 )
= A2 ) ) ).
% \<Lambda>.src_ide
thf(fact_1234__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_1235__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_1236__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_1237__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_1238__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_1239__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_1240_weakly__extensional__rts_Osrc_Ocong,axiom,
weakly4554751870655413501lambda = weakly4554751870655413501lambda ).
% weakly_extensional_rts.src.cong
thf(fact_1241__092_060Lambda_062_Osrc__def,axiom,
! [T: lambda_lambda] :
( ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( the_lambda_lambda
@ ^ [A4: lambda_lambda] : ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) )
& ( ~ ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ).
% \<Lambda>.src_def
thf(fact_1242_transformation__axioms__def,axiom,
( transf8838427370478101288lambda
= ( ^ [A6: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,G3: list_lambda_lambda > list_lambda_lambda,Tau: list_lambda_lambda > list_lambda_lambda] :
( ! [F3: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A6 @ F3 )
=> ( ( Tau @ F3 )
= ( partia6873046715171720854lambda @ B5 ) ) )
& ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A6 @ F3 )
=> ( ( weakly2957279434024022659lambda @ B5 @ ( Tau @ F3 ) )
= ( F4 @ ( weakly2957279434024022659lambda @ A6 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A6 @ F3 )
=> ( ( trg_li817795487149935061lambda @ B5 @ ( Tau @ F3 ) )
= ( G3 @ ( trg_li817795487149935061lambda @ A6 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A6 @ F3 )
=> ( ( B5 @ ( Tau @ ( weakly2957279434024022659lambda @ A6 @ F3 ) ) @ ( F4 @ F3 ) )
= ( Tau @ ( trg_li817795487149935061lambda @ A6 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A6 @ F3 )
=> ( ( B5 @ ( F4 @ F3 ) @ ( Tau @ ( weakly2957279434024022659lambda @ A6 @ F3 ) ) )
= ( G3 @ F3 ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A6 @ F3 )
=> ( join_o6956785509312998601lambda @ B5 @ ( Tau @ ( weakly2957279434024022659lambda @ A6 @ F3 ) ) @ ( F4 @ F3 ) @ ( Tau @ F3 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_1243_transformation__axioms__def,axiom,
( transf4653517080217399086lambda
= ( ^ [A6: lambda_lambda > lambda_lambda > lambda_lambda,B5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,G3: lambda_lambda > list_lambda_lambda,Tau: lambda_lambda > list_lambda_lambda] :
( ! [F3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A6 @ F3 )
=> ( ( Tau @ F3 )
= ( partia6873046715171720854lambda @ B5 ) ) )
& ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A6 @ F3 )
=> ( ( weakly2957279434024022659lambda @ B5 @ ( Tau @ F3 ) )
= ( F4 @ ( weakly4554751870655413501lambda @ A6 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A6 @ F3 )
=> ( ( trg_li817795487149935061lambda @ B5 @ ( Tau @ F3 ) )
= ( G3 @ ( trg_lambda_lambda @ A6 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A6 @ F3 )
=> ( ( B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A6 @ F3 ) ) @ ( F4 @ F3 ) )
= ( Tau @ ( trg_lambda_lambda @ A6 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A6 @ F3 )
=> ( ( B5 @ ( F4 @ F3 ) @ ( Tau @ ( weakly4554751870655413501lambda @ A6 @ F3 ) ) )
= ( G3 @ F3 ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A6 @ F3 )
=> ( join_o6956785509312998601lambda @ B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A6 @ F3 ) ) @ ( F4 @ F3 ) @ ( Tau @ F3 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_1244_transformation__axioms__def,axiom,
( transf3419659901140239394lambda
= ( ^ [A6: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,G3: list_lambda_lambda > lambda_lambda,Tau: list_lambda_lambda > lambda_lambda] :
( ! [F3: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A6 @ F3 )
=> ( ( Tau @ F3 )
= ( partia1461432607296028304lambda @ B5 ) ) )
& ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A6 @ F3 )
=> ( ( weakly4554751870655413501lambda @ B5 @ ( Tau @ F3 ) )
= ( F4 @ ( weakly2957279434024022659lambda @ A6 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A6 @ F3 )
=> ( ( trg_lambda_lambda @ B5 @ ( Tau @ F3 ) )
= ( G3 @ ( trg_li817795487149935061lambda @ A6 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A6 @ F3 )
=> ( ( B5 @ ( Tau @ ( weakly2957279434024022659lambda @ A6 @ F3 ) ) @ ( F4 @ F3 ) )
= ( Tau @ ( trg_li817795487149935061lambda @ A6 @ F3 ) ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A6 @ F3 )
=> ( ( B5 @ ( F4 @ F3 ) @ ( Tau @ ( weakly2957279434024022659lambda @ A6 @ F3 ) ) )
= ( G3 @ F3 ) ) )
& ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A6 @ F3 )
=> ( join_o1386658102433174083lambda @ B5 @ ( Tau @ ( weakly2957279434024022659lambda @ A6 @ F3 ) ) @ ( F4 @ F3 ) @ ( Tau @ F3 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_1245_transformation__axioms__def,axiom,
( transf356316610583137064lambda
= ( ^ [A6: lambda_lambda > lambda_lambda > lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,G3: lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda] :
( ! [F3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A6 @ F3 )
=> ( ( Tau @ F3 )
= ( partia1461432607296028304lambda @ B5 ) ) )
& ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A6 @ F3 )
=> ( ( weakly4554751870655413501lambda @ B5 @ ( Tau @ F3 ) )
= ( F4 @ ( weakly4554751870655413501lambda @ A6 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A6 @ F3 )
=> ( ( trg_lambda_lambda @ B5 @ ( Tau @ F3 ) )
= ( G3 @ ( trg_lambda_lambda @ A6 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A6 @ F3 )
=> ( ( B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A6 @ F3 ) ) @ ( F4 @ F3 ) )
= ( Tau @ ( trg_lambda_lambda @ A6 @ F3 ) ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A6 @ F3 )
=> ( ( B5 @ ( F4 @ F3 ) @ ( Tau @ ( weakly4554751870655413501lambda @ A6 @ F3 ) ) )
= ( G3 @ F3 ) ) )
& ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A6 @ F3 )
=> ( join_o1386658102433174083lambda @ B5 @ ( Tau @ ( weakly4554751870655413501lambda @ A6 @ F3 ) ) @ ( F4 @ F3 ) @ ( Tau @ F3 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_1246_transformation__axioms_Ointro,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Tau2: list_lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: list_lambda_lambda > list_lambda_lambda,G4: list_lambda_lambda > list_lambda_lambda] :
( ! [F5: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia6873046715171720854lambda @ B ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A @ F5 )
=> ( ( weakly2957279434024022659lambda @ B @ ( Tau2 @ F5 ) )
= ( F @ ( weakly2957279434024022659lambda @ A @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A @ F5 )
=> ( ( trg_li817795487149935061lambda @ B @ ( Tau2 @ F5 ) )
= ( G4 @ ( trg_li817795487149935061lambda @ A @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F5 )
=> ( ( B @ ( Tau2 @ ( weakly2957279434024022659lambda @ A @ F5 ) ) @ ( F @ F5 ) )
= ( Tau2 @ ( trg_li817795487149935061lambda @ A @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F5 )
=> ( ( B @ ( F @ F5 ) @ ( Tau2 @ ( weakly2957279434024022659lambda @ A @ F5 ) ) )
= ( G4 @ F5 ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F5 )
=> ( join_o6956785509312998601lambda @ B @ ( Tau2 @ ( weakly2957279434024022659lambda @ A @ F5 ) ) @ ( F @ F5 ) @ ( Tau2 @ F5 ) ) )
=> ( transf8838427370478101288lambda @ A @ B @ F @ G4 @ Tau2 ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_1247_transformation__axioms_Ointro,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,Tau2: lambda_lambda > list_lambda_lambda,B: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F: lambda_lambda > list_lambda_lambda,G4: lambda_lambda > list_lambda_lambda] :
( ! [F5: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia6873046715171720854lambda @ B ) ) )
=> ( ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A @ F5 )
=> ( ( weakly2957279434024022659lambda @ B @ ( Tau2 @ F5 ) )
= ( F @ ( weakly4554751870655413501lambda @ A @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A @ F5 )
=> ( ( trg_li817795487149935061lambda @ B @ ( Tau2 @ F5 ) )
= ( G4 @ ( trg_lambda_lambda @ A @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F5 )
=> ( ( B @ ( Tau2 @ ( weakly4554751870655413501lambda @ A @ F5 ) ) @ ( F @ F5 ) )
= ( Tau2 @ ( trg_lambda_lambda @ A @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F5 )
=> ( ( B @ ( F @ F5 ) @ ( Tau2 @ ( weakly4554751870655413501lambda @ A @ F5 ) ) )
= ( G4 @ F5 ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F5 )
=> ( join_o6956785509312998601lambda @ B @ ( Tau2 @ ( weakly4554751870655413501lambda @ A @ F5 ) ) @ ( F @ F5 ) @ ( Tau2 @ F5 ) ) )
=> ( transf4653517080217399086lambda @ A @ B @ F @ G4 @ Tau2 ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_1248_transformation__axioms_Ointro,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Tau2: list_lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: list_lambda_lambda > lambda_lambda,G4: list_lambda_lambda > lambda_lambda] :
( ! [F5: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia1461432607296028304lambda @ B ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A @ F5 )
=> ( ( weakly4554751870655413501lambda @ B @ ( Tau2 @ F5 ) )
= ( F @ ( weakly2957279434024022659lambda @ A @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A @ F5 )
=> ( ( trg_lambda_lambda @ B @ ( Tau2 @ F5 ) )
= ( G4 @ ( trg_li817795487149935061lambda @ A @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F5 )
=> ( ( B @ ( Tau2 @ ( weakly2957279434024022659lambda @ A @ F5 ) ) @ ( F @ F5 ) )
= ( Tau2 @ ( trg_li817795487149935061lambda @ A @ F5 ) ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F5 )
=> ( ( B @ ( F @ F5 ) @ ( Tau2 @ ( weakly2957279434024022659lambda @ A @ F5 ) ) )
= ( G4 @ F5 ) ) )
=> ( ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F5 )
=> ( join_o1386658102433174083lambda @ B @ ( Tau2 @ ( weakly2957279434024022659lambda @ A @ F5 ) ) @ ( F @ F5 ) @ ( Tau2 @ F5 ) ) )
=> ( transf3419659901140239394lambda @ A @ B @ F @ G4 @ Tau2 ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_1249_transformation__axioms_Ointro,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,Tau2: lambda_lambda > lambda_lambda,B: lambda_lambda > lambda_lambda > lambda_lambda,F: lambda_lambda > lambda_lambda,G4: lambda_lambda > lambda_lambda] :
( ! [F5: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia1461432607296028304lambda @ B ) ) )
=> ( ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A @ F5 )
=> ( ( weakly4554751870655413501lambda @ B @ ( Tau2 @ F5 ) )
= ( F @ ( weakly4554751870655413501lambda @ A @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A @ F5 )
=> ( ( trg_lambda_lambda @ B @ ( Tau2 @ F5 ) )
= ( G4 @ ( trg_lambda_lambda @ A @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F5 )
=> ( ( B @ ( Tau2 @ ( weakly4554751870655413501lambda @ A @ F5 ) ) @ ( F @ F5 ) )
= ( Tau2 @ ( trg_lambda_lambda @ A @ F5 ) ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F5 )
=> ( ( B @ ( F @ F5 ) @ ( Tau2 @ ( weakly4554751870655413501lambda @ A @ F5 ) ) )
= ( G4 @ F5 ) ) )
=> ( ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F5 )
=> ( join_o1386658102433174083lambda @ B @ ( Tau2 @ ( weakly4554751870655413501lambda @ A @ F5 ) ) @ ( F @ F5 ) @ ( Tau2 @ F5 ) ) )
=> ( transf356316610583137064lambda @ A @ B @ F @ G4 @ Tau2 ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_1250_the__equality,axiom,
! [P: lambda_lambda > $o,A2: lambda_lambda] :
( ( P @ A2 )
=> ( ! [X: lambda_lambda] :
( ( P @ X )
=> ( X = A2 ) )
=> ( ( the_lambda_lambda @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_1251_the__equality,axiom,
! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
( ( P @ A2 )
=> ( ! [X: list_lambda_lambda] :
( ( P @ X )
=> ( X = A2 ) )
=> ( ( the_li7637651520389017975lambda @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_1252_the__eq__trivial,axiom,
! [A2: lambda_lambda] :
( ( the_lambda_lambda
@ ^ [X4: lambda_lambda] : ( X4 = A2 ) )
= A2 ) ).
% the_eq_trivial
thf(fact_1253_the__eq__trivial,axiom,
! [A2: list_lambda_lambda] :
( ( the_li7637651520389017975lambda
@ ^ [X4: list_lambda_lambda] : ( X4 = A2 ) )
= A2 ) ).
% the_eq_trivial
thf(fact_1254_the__sym__eq__trivial,axiom,
! [X6: lambda_lambda] :
( ( the_lambda_lambda
@ ( ^ [Y6: lambda_lambda,Z: lambda_lambda] : ( Y6 = Z )
@ X6 ) )
= X6 ) ).
% the_sym_eq_trivial
thf(fact_1255_the__sym__eq__trivial,axiom,
! [X6: list_lambda_lambda] :
( ( the_li7637651520389017975lambda
@ ( ^ [Y6: list_lambda_lambda,Z: list_lambda_lambda] : ( Y6 = Z )
@ X6 ) )
= X6 ) ).
% the_sym_eq_trivial
thf(fact_1256_the1__equality,axiom,
! [P: lambda_lambda > $o,A2: lambda_lambda] :
( ? [X2: lambda_lambda] :
( ( P @ X2 )
& ! [Y2: lambda_lambda] :
( ( P @ Y2 )
=> ( Y2 = X2 ) ) )
=> ( ( P @ A2 )
=> ( ( the_lambda_lambda @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_1257_the1__equality,axiom,
! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
( ? [X2: list_lambda_lambda] :
( ( P @ X2 )
& ! [Y2: list_lambda_lambda] :
( ( P @ Y2 )
=> ( Y2 = X2 ) ) )
=> ( ( P @ A2 )
=> ( ( the_li7637651520389017975lambda @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_1258_the1I2,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ? [X2: lambda_lambda] :
( ( P @ X2 )
& ! [Y2: lambda_lambda] :
( ( P @ Y2 )
=> ( Y2 = X2 ) ) )
=> ( ! [X: lambda_lambda] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_lambda_lambda @ P ) ) ) ) ).
% the1I2
thf(fact_1259_the1I2,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ? [X2: list_lambda_lambda] :
( ( P @ X2 )
& ! [Y2: list_lambda_lambda] :
( ( P @ Y2 )
=> ( Y2 = X2 ) ) )
=> ( ! [X: list_lambda_lambda] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_li7637651520389017975lambda @ P ) ) ) ) ).
% the1I2
thf(fact_1260_If__def,axiom,
( if_lambda_lambda
= ( ^ [P2: $o,X4: lambda_lambda,Y4: lambda_lambda] :
( the_lambda_lambda
@ ^ [Z4: lambda_lambda] :
( ( P2
=> ( Z4 = X4 ) )
& ( ~ P2
=> ( Z4 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_1261_If__def,axiom,
( if_lis7195930416592267574lambda
= ( ^ [P2: $o,X4: list_lambda_lambda,Y4: list_lambda_lambda] :
( the_li7637651520389017975lambda
@ ^ [Z4: list_lambda_lambda] :
( ( P2
=> ( Z4 = X4 ) )
& ( ~ P2
=> ( Z4 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_1262_theI2,axiom,
! [P: lambda_lambda > $o,A2: lambda_lambda,Q: lambda_lambda > $o] :
( ( P @ A2 )
=> ( ! [X: lambda_lambda] :
( ( P @ X )
=> ( X = A2 ) )
=> ( ! [X: lambda_lambda] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_lambda_lambda @ P ) ) ) ) ) ).
% theI2
thf(fact_1263_theI2,axiom,
! [P: list_lambda_lambda > $o,A2: list_lambda_lambda,Q: list_lambda_lambda > $o] :
( ( P @ A2 )
=> ( ! [X: list_lambda_lambda] :
( ( P @ X )
=> ( X = A2 ) )
=> ( ! [X: list_lambda_lambda] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the_li7637651520389017975lambda @ P ) ) ) ) ) ).
% theI2
thf(fact_1264_theI_H,axiom,
! [P: lambda_lambda > $o] :
( ? [X2: lambda_lambda] :
( ( P @ X2 )
& ! [Y2: lambda_lambda] :
( ( P @ Y2 )
=> ( Y2 = X2 ) ) )
=> ( P @ ( the_lambda_lambda @ P ) ) ) ).
% theI'
thf(fact_1265_theI_H,axiom,
! [P: list_lambda_lambda > $o] :
( ? [X2: list_lambda_lambda] :
( ( P @ X2 )
& ! [Y2: list_lambda_lambda] :
( ( P @ Y2 )
=> ( Y2 = X2 ) ) )
=> ( P @ ( the_li7637651520389017975lambda @ P ) ) ) ).
% theI'
thf(fact_1266_theI,axiom,
! [P: lambda_lambda > $o,A2: lambda_lambda] :
( ( P @ A2 )
=> ( ! [X: lambda_lambda] :
( ( P @ X )
=> ( X = A2 ) )
=> ( P @ ( the_lambda_lambda @ P ) ) ) ) ).
% theI
thf(fact_1267_theI,axiom,
! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
( ( P @ A2 )
=> ( ! [X: list_lambda_lambda] :
( ( P @ X )
=> ( X = A2 ) )
=> ( P @ ( the_li7637651520389017975lambda @ P ) ) ) ) ).
% theI
thf(fact_1268__092_060Lambda_062_Onull__def,axiom,
( ( partia1461432607296028304lambda @ lambda_resid )
= ( the_lambda_lambda
@ ^ [N3: lambda_lambda] :
! [T5: lambda_lambda] :
( ( ( lambda_resid @ N3 @ T5 )
= N3 )
& ( ( lambda_resid @ T5 @ N3 )
= N3 ) ) ) ) ).
% \<Lambda>.null_def
thf(fact_1269_the__elem__def,axiom,
( the_el5431214018478934239lambda
= ( ^ [X5: set_lambda_lambda] :
( the_lambda_lambda
@ ^ [X4: lambda_lambda] :
( X5
= ( insert_lambda_lambda @ X4 @ bot_bo1271416176687173302lambda ) ) ) ) ) ).
% the_elem_def
thf(fact_1270_the__elem__def,axiom,
( the_el4548681628688475493lambda
= ( ^ [X5: set_li7878742357367608272lambda] :
( the_li7637651520389017975lambda
@ ^ [X4: list_lambda_lambda] :
( X5
= ( insert8465602035467394112lambda @ X4 @ bot_bo1397037886704484924lambda ) ) ) ) ) ).
% the_elem_def
thf(fact_1271_the__elem__eq,axiom,
! [X6: lambda_lambda] :
( ( the_el5431214018478934239lambda @ ( insert_lambda_lambda @ X6 @ bot_bo1271416176687173302lambda ) )
= X6 ) ).
% the_elem_eq
thf(fact_1272_the__elem__eq,axiom,
! [X6: list_lambda_lambda] :
( ( the_el4548681628688475493lambda @ ( insert8465602035467394112lambda @ X6 @ bot_bo1397037886704484924lambda ) )
= X6 ) ).
% the_elem_eq
thf(fact_1273_partial__magma_Onull__def,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ( ( partia6873046715171720854lambda @ OP )
= ( the_li7637651520389017975lambda
@ ^ [N3: list_lambda_lambda] :
! [T5: list_lambda_lambda] :
( ( ( OP @ N3 @ T5 )
= N3 )
& ( ( OP @ T5 @ N3 )
= N3 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_1274_partial__magma_Onull__def,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ( partia1461432607296028304lambda @ OP )
= ( the_lambda_lambda
@ ^ [N3: lambda_lambda] :
! [T5: lambda_lambda] :
( ( ( OP @ N3 @ T5 )
= N3 )
& ( ( OP @ T5 @ N3 )
= N3 ) ) ) ) ) ).
% partial_magma.null_def
thf(fact_1275_the__elem__image__unique,axiom,
! [A: set_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,X6: lambda_lambda] :
( ( A != bot_bo1271416176687173302lambda )
=> ( ! [Y2: lambda_lambda] :
( ( member_lambda_lambda @ Y2 @ A )
=> ( ( F2 @ Y2 )
= ( F2 @ X6 ) ) )
=> ( ( the_el4548681628688475493lambda @ ( image_3692823078739859851lambda @ F2 @ A ) )
= ( F2 @ X6 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1276_paths__in__rts__axioms,axiom,
paths_6861597469809217359lambda @ lambda_resid ).
% paths_in_rts_axioms
thf(fact_1277_null__def,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= ( the_li7637651520389017975lambda
@ ^ [N3: list_lambda_lambda] :
! [T5: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ N3 @ T5 )
= N3 )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T5 @ N3 )
= N3 ) ) ) ) ).
% null_def
% Helper facts (5)
thf(help_If_2_1_If_001t__LambdaCalculus__Olambda____calculus__Olambda_T,axiom,
! [X6: lambda_lambda,Y5: lambda_lambda] :
( ( if_lambda_lambda @ $false @ X6 @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__LambdaCalculus__Olambda____calculus__Olambda_T,axiom,
! [X6: lambda_lambda,Y5: lambda_lambda] :
( ( if_lambda_lambda @ $true @ X6 @ Y5 )
= X6 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X6: list_lambda_lambda,Y5: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $false @ X6 @ Y5 )
= Y5 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X6: list_lambda_lambda,Y5: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $true @ X6 @ Y5 )
= X6 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( simula5235733974575307523lambda @ lambda_resid @ lambda_resid
@ ^ [T5: lambda_lambda] : ( if_lambda_lambda @ ( arr_lambda_lambda @ lambda_resid @ T5 ) @ ( lambda_App @ T5 @ b ) @ lambda_Nil ) ) ).
%------------------------------------------------------------------------------