TPTP Problem File: SLH0428^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : ResiduatedTransitionSystem/0001_LambdaCalculus/prob_07159_319223__14931188_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1407 ( 558 unt; 122 typ; 0 def)
% Number of atoms : 3991 (1932 equ; 0 cnn)
% Maximal formula atoms : 108 ( 3 avg)
% Number of connectives : 14566 ( 665 ~; 38 |; 558 &;11713 @)
% ( 0 <=>;1592 =>; 0 <=; 0 <~>)
% Maximal formula depth : 42 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 490 ( 490 >; 0 *; 0 +; 0 <<)
% Number of symbols : 118 ( 115 usr; 15 con; 0-4 aty)
% Number of variables : 3368 ( 74 ^;3048 !; 246 ?;3368 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:49:07.305
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
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% Explicit typings (115)
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source6119337877621565212lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
target5490931411331770124lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > set_lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
target4026644918049360146lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_wi3086218056445523938lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
rts_wi1553966559393445608lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_Set_OCollect_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
collec621444948832755541lambda: ( lambda_lambda > $o ) > set_lambda_lambda ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
collec8830145823237851355lambda: ( list_lambda_lambda > $o ) > set_li7878742357367608272lambda ).
thf(sy_c_Set_Oinsert_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
insert_lambda_lambda: lambda_lambda > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
insert8465602035467394112lambda: list_lambda_lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda ).
thf(sy_c_member_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
member_lambda_lambda: lambda_lambda > set_lambda_lambda > $o ).
thf(sy_c_member_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
member2667661554319814297lambda: list_lambda_lambda > set_li7878742357367608272lambda > $o ).
thf(sy_v_Ua,type,
ua: list_lambda_lambda ).
thf(sy_v_ta____,type,
ta: lambda_lambda ).
thf(sy_v_u____,type,
u: lambda_lambda ).
thf(sy_v_x41,type,
x41: lambda_lambda ).
thf(sy_v_x42,type,
x42: lambda_lambda ).
thf(sy_v_x51,type,
x51: lambda_lambda ).
thf(sy_v_x52,type,
x52: lambda_lambda ).
% Relevant facts (1277)
thf(fact_0_finite__developments,axiom,
! [T: lambda_lambda] : ( reduction_FD @ T ) ).
% finite_developments
thf(fact_1__092_060Lambda_062_Ocube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.cube
thf(fact_2__092_060Lambda_062_Oex__un__null,axiom,
? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( lambda_resid @ X @ T2 )
= X )
& ( ( lambda_resid @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( lambda_resid @ Y @ T3 )
= Y )
& ( ( lambda_resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% \<Lambda>.ex_un_null
thf(fact_3_Ide_Osimps_I1_J,axiom,
~ ( paths_7717490669901197737lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Ide.simps(1)
thf(fact_4_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_5_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_6__092_060open_062_092_060And_062t_O_AIde_A_091t_093_A_092_060Longrightarrow_062_Astdz__insert_At_A_091_093_A_061_A_091_093_092_060close_062,axiom,
! [T: lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
=> ( ( reduct5390047737487997688insert @ T @ nil_lambda_lambda )
= nil_lambda_lambda ) ) ).
% \<open>\<And>t. Ide [t] \<Longrightarrow> stdz_insert t [] = []\<close>
thf(fact_7_paths__in__confluent__rts__axioms,axiom,
paths_9040249923598999359lambda @ lambda_resid ).
% paths_in_confluent_rts_axioms
thf(fact_8_complete__development__cons,axiom,
! [T: lambda_lambda,U: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ ( cons_lambda_lambda @ U @ U2 ) )
=> ( reduct7282684938351565978opment @ ( lambda_resid @ T @ U ) @ U2 ) ) ).
% complete_development_cons
thf(fact_9_complete__development__Ide__iff,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ U2 )
=> ( ( lambda_Ide @ T )
= ( U2 = nil_lambda_lambda ) ) ) ).
% complete_development_Ide_iff
thf(fact_10_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_11_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_12_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_13_reduction__paths_OStd_Ocases,axiom,
! [X2: list_lambda_lambda] :
( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ).
% reduction_paths.Std.cases
thf(fact_14_reduction__paths_Ostandardize_Ocases,axiom,
! [X2: list_lambda_lambda] :
( ( X2 != nil_lambda_lambda )
=> ~ ! [V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ V2 @ Va ) ) ) ).
% reduction_paths.standardize.cases
thf(fact_15__092_060Lambda_062_Opartial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% \<Lambda>.partial_magma_axioms
thf(fact_16_Ide_Oelims_I3_J,axiom,
! [X2: list_lambda_lambda] :
( ~ ( paths_7717490669901197737lambda @ lambda_resid @ X2 )
=> ( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( ide_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ).
% Ide.elims(3)
thf(fact_17_Ide_Oelims_I2_J,axiom,
! [X2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ X2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Ide.elims(2)
thf(fact_18_Ide_Oelims_I1_J,axiom,
! [X2: list_lambda_lambda,Y2: $o] :
( ( ( paths_7717490669901197737lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> Y2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
= ( ~ ( ide_lambda_lambda @ lambda_resid @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( ide_lambda_lambda @ lambda_resid @ T3 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Ide.elims(1)
thf(fact_19_Ide_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( ( ide_lambda_lambda @ lambda_resid @ T )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ) ) ).
% Ide.simps(3)
thf(fact_20_Srcs__Resid__Arr__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( target5490931411331770124lambda @ lambda_resid @ U ) ) ) ).
% Srcs_Resid_Arr_single
thf(fact_21_development__Ide,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ T )
=> ( ( reduct9208763521642324371opment @ T @ U2 )
= ( U2 = nil_lambda_lambda ) ) ) ).
% development_Ide
thf(fact_22_Srcs_Osimps_I1_J,axiom,
( ( paths_2897053694880971696lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Srcs.simps(1)
thf(fact_23_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_24_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_25_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_26_subset__antisym,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_27_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_28_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_29_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_30_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_31__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_32__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_33__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_34__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_35__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_36_Con__sym,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 )
!= nil_lambda_lambda ) ) ).
% Con_sym
thf(fact_37_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_38_Cube_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
!= nil_lambda_lambda )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) ).
% Cube(1)
thf(fact_39_Cube_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) ) ) ) ).
% Cube(2)
thf(fact_40__092_060Lambda_062_Otargets__cong__closed,axiom,
! [B2: lambda_lambda,T: lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ B2 ) ) )
=> ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.targets_cong_closed
thf(fact_41__092_060Lambda_062_Otargets__are__cong,axiom,
! [B2: lambda_lambda,T: lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ B2 ) ) ) ) ) ).
% \<Lambda>.targets_are_cong
thf(fact_42__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_43_mem__Collect__eq,axiom,
! [A2: lambda_lambda,P: lambda_lambda > $o] :
( ( member_lambda_lambda @ A2 @ ( collec621444948832755541lambda @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A2: list_lambda_lambda,P: list_lambda_lambda > $o] :
( ( member2667661554319814297lambda @ A2 @ ( collec8830145823237851355lambda @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: set_lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: set_li7878742357367608272lambda] :
( ( collec8830145823237851355lambda
@ ^ [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_47_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_48_Collect__cong,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ! [X: list_lambda_lambda] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec8830145823237851355lambda @ P )
= ( collec8830145823237851355lambda @ Q ) ) ) ).
% Collect_cong
thf(fact_49__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_50__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_51__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_52__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_53_Con__initial__right,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ).
% Con_initial_right
thf(fact_54_Con__initial__left,axiom,
! [T: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ).
% Con_initial_left
thf(fact_55_Resid__rec_I2_J,axiom,
! [T4: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) ) ) ) ) ).
% Resid_rec(2)
thf(fact_56_Resid__rec_I3_J,axiom,
! [U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U2 ) ) ) ) ).
% Resid_rec(3)
thf(fact_57_Resid__cons_I2_J,axiom,
! [U2: list_lambda_lambda,T4: list_lambda_lambda,U: lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 ) ) ) ) ).
% Resid_cons(2)
thf(fact_58_Resid_Osimps_I2_J,axiom,
! [V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% Resid.simps(2)
thf(fact_59_Con__cons_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_cons(1)
thf(fact_60_Con__cons_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_cons(2)
thf(fact_61_Con__imp__eq__Srcs,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ).
% Con_imp_eq_Srcs
thf(fact_62_Resid__Ide__Arr__ind,axiom,
! [A: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T4 )
!= nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ T4 ) ) ) ) ).
% Resid_Ide_Arr_ind
thf(fact_63_Resid__Arr__Ide__ind,axiom,
! [A: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A )
= T4 ) ) ) ).
% Resid_Arr_Ide_ind
thf(fact_64_Resid__Ide_I1_J,axiom,
! [A: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A )
= T4 ) ) ) ).
% Resid_Ide(1)
thf(fact_65_Srcs_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( source5627170679743844886lambda @ lambda_resid @ T ) ) ).
% Srcs.simps(3)
thf(fact_66_Srcs__are__ide,axiom,
! [T4: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).
% Srcs_are_ide
thf(fact_67_empty__iff,axiom,
! [C: lambda_lambda] :
~ ( member_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ).
% empty_iff
thf(fact_68_empty__iff,axiom,
! [C: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ C @ bot_bo1397037886704484924lambda ) ).
% empty_iff
thf(fact_69_all__not__in__conv,axiom,
! [A: set_lambda_lambda] :
( ( ! [X3: lambda_lambda] :
~ ( member_lambda_lambda @ X3 @ A ) )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% all_not_in_conv
thf(fact_70_all__not__in__conv,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ! [X3: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ X3 @ A ) )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% all_not_in_conv
thf(fact_71_Collect__empty__eq,axiom,
! [P: lambda_lambda > $o] :
( ( ( collec621444948832755541lambda @ P )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_72_Collect__empty__eq,axiom,
! [P: list_lambda_lambda > $o] :
( ( ( collec8830145823237851355lambda @ P )
= bot_bo1397037886704484924lambda )
= ( ! [X3: list_lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_73_empty__Collect__eq,axiom,
! [P: lambda_lambda > $o] :
( ( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ P ) )
= ( ! [X3: lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_74_empty__Collect__eq,axiom,
! [P: list_lambda_lambda > $o] :
( ( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda @ P ) )
= ( ! [X3: list_lambda_lambda] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_75_Srcs_Oelims,axiom,
! [X2: list_lambda_lambda,Y2: set_lambda_lambda] :
( ( ( paths_2897053694880971696lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != bot_bo1271416176687173302lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
!= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda] :
( ? [V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( source5627170679743844886lambda @ lambda_resid @ T3 ) ) ) ) ) ) ).
% Srcs.elims
thf(fact_76_empty__subsetI,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A ) ).
% empty_subsetI
thf(fact_77_empty__subsetI,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A ) ).
% empty_subsetI
thf(fact_78_subset__empty,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ bot_bo1397037886704484924lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% subset_empty
thf(fact_79_subset__empty,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% subset_empty
thf(fact_80_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_81_lambda__calculus_Oide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.ide_char
thf(fact_82_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_83_Con__consI_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_consI(1)
thf(fact_84_Con__consI_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_consI(2)
thf(fact_85_Resid__single__ide_I2_J,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
= T4 ) ) ) ).
% Resid_single_ide(2)
thf(fact_86_is__confluent__rts,axiom,
conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% is_confluent_rts
thf(fact_87_emptyE,axiom,
! [A2: lambda_lambda] :
~ ( member_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ).
% emptyE
thf(fact_88_emptyE,axiom,
! [A2: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ A2 @ bot_bo1397037886704484924lambda ) ).
% emptyE
thf(fact_89_equals0D,axiom,
! [A: set_lambda_lambda,A2: lambda_lambda] :
( ( A = bot_bo1271416176687173302lambda )
=> ~ ( member_lambda_lambda @ A2 @ A ) ) ).
% equals0D
thf(fact_90_equals0D,axiom,
! [A: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
( ( A = bot_bo1397037886704484924lambda )
=> ~ ( member2667661554319814297lambda @ A2 @ A ) ) ).
% equals0D
thf(fact_91_equals0I,axiom,
! [A: set_lambda_lambda] :
( ! [Y3: lambda_lambda] :
~ ( member_lambda_lambda @ Y3 @ A )
=> ( A = bot_bo1271416176687173302lambda ) ) ).
% equals0I
thf(fact_92_equals0I,axiom,
! [A: set_li7878742357367608272lambda] :
( ! [Y3: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ Y3 @ A )
=> ( A = bot_bo1397037886704484924lambda ) ) ).
% equals0I
thf(fact_93_ex__in__conv,axiom,
! [A: set_lambda_lambda] :
( ( ? [X3: lambda_lambda] : ( member_lambda_lambda @ X3 @ A ) )
= ( A != bot_bo1271416176687173302lambda ) ) ).
% ex_in_conv
thf(fact_94_ex__in__conv,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ? [X3: list_lambda_lambda] : ( member2667661554319814297lambda @ X3 @ A ) )
= ( A != bot_bo1397037886704484924lambda ) ) ).
% ex_in_conv
thf(fact_95_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_96_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_97_Collect__mono__iff,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( ord_le3064231726521830256lambda @ ( collec8830145823237851355lambda @ P ) @ ( collec8830145823237851355lambda @ Q ) )
= ( ! [X3: list_lambda_lambda] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_98_Collect__mono__iff,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) )
= ( ! [X3: lambda_lambda] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_99_set__eq__subset,axiom,
( ( ^ [Y4: set_li7878742357367608272lambda,Z: set_li7878742357367608272lambda] : ( Y4 = Z ) )
= ( ^ [A4: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A4 @ B4 )
& ( ord_le3064231726521830256lambda @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_100_set__eq__subset,axiom,
( ( ^ [Y4: set_lambda_lambda,Z: set_lambda_lambda] : ( Y4 = Z ) )
= ( ^ [A4: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A4 @ B4 )
& ( ord_le1926335974945931754lambda @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_101_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_102_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_103_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_104_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_105_subset__refl,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ A @ A ) ).
% subset_refl
thf(fact_106_subset__refl,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A @ A ) ).
% subset_refl
thf(fact_107_subset__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A4: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
! [T5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ T5 @ A4 )
=> ( member2667661554319814297lambda @ T5 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_108_subset__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A4: set_lambda_lambda,B4: set_lambda_lambda] :
! [T5: lambda_lambda] :
( ( member_lambda_lambda @ T5 @ A4 )
=> ( member_lambda_lambda @ T5 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_109_equalityD2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A = B )
=> ( ord_le3064231726521830256lambda @ B @ A ) ) ).
% equalityD2
thf(fact_110_equalityD2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A = B )
=> ( ord_le1926335974945931754lambda @ B @ A ) ) ).
% equalityD2
thf(fact_111_equalityD1,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A = B )
=> ( ord_le3064231726521830256lambda @ A @ B ) ) ).
% equalityD1
thf(fact_112_equalityD1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A = B )
=> ( ord_le1926335974945931754lambda @ A @ B ) ) ).
% equalityD1
thf(fact_113_subset__eq,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A4: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ A4 )
=> ( member2667661554319814297lambda @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_114_subset__eq,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A4: set_lambda_lambda,B4: set_lambda_lambda] :
! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A4 )
=> ( member_lambda_lambda @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_115_equalityE,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A = B )
=> ~ ( ( ord_le3064231726521830256lambda @ A @ B )
=> ~ ( ord_le3064231726521830256lambda @ B @ A ) ) ) ).
% equalityE
thf(fact_116_equalityE,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( A = B )
=> ~ ( ( ord_le1926335974945931754lambda @ A @ B )
=> ~ ( ord_le1926335974945931754lambda @ B @ A ) ) ) ).
% equalityE
thf(fact_117_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_118_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_119_in__mono,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,X2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( member2667661554319814297lambda @ X2 @ A )
=> ( member2667661554319814297lambda @ X2 @ B ) ) ) ).
% in_mono
thf(fact_120_in__mono,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,X2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( member_lambda_lambda @ X2 @ A )
=> ( member_lambda_lambda @ X2 @ B ) ) ) ).
% in_mono
thf(fact_121_partial__magma__axioms,axiom,
partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% partial_magma_axioms
thf(fact_122_rts__with__composites__axioms,axiom,
rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% rts_with_composites_axioms
thf(fact_123_complete__development__def,axiom,
( reduct7282684938351565978opment
= ( ^ [T5: lambda_lambda,U3: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T5 @ U3 )
& ( ( lambda_Ide @ T5 )
| ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T5 @ nil_lambda_lambda ) @ U3 ) ) ) ) ) ) ).
% complete_development_def
thf(fact_124__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_125__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_126_complete__development__cong,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ U2 )
=> ( ~ ( lambda_Ide @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).
% complete_development_cong
thf(fact_127__092_060Lambda_062_Ocong__implies__coterminal,axiom,
! [U: lambda_lambda,U4: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ U @ U4 ) ) ).
% \<Lambda>.cong_implies_coterminal
thf(fact_128_Con__single__ide__ind,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Con_single_ide_ind
thf(fact_129_set__Ide__subset__ide,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ).
% set_Ide_subset_ide
thf(fact_130_Con__Ide__iff,axiom,
! [A: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T4 )
!= nil_lambda_lambda )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A ) ) ) ) ) ).
% Con_Ide_iff
thf(fact_131_Con__IdeI_I1_J,axiom,
! [A: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ A @ T4 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(1)
thf(fact_132_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_133_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_134_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_135_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_136_Arr_Osimps_I1_J,axiom,
~ ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Arr.simps(1)
thf(fact_137_ex__un__Src,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( Y = X ) ) ) ) ).
% ex_un_Src
thf(fact_138_Ide__implies__Arr,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% Ide_implies_Arr
thf(fact_139_Con__implies__Arr_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ U2 ) ) ).
% Con_implies_Arr(2)
thf(fact_140_Con__implies__Arr_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% Con_implies_Arr(1)
thf(fact_141_Arr__iff__Con__self,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T4 )
!= nil_lambda_lambda ) ) ).
% Arr_iff_Con_self
thf(fact_142_Con__Arr__self,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T4 )
!= nil_lambda_lambda ) ) ).
% Con_Arr_self
thf(fact_143_Con__imp__Arr__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) ) ) ).
% Con_imp_Arr_Resid
thf(fact_144_ide__char,axiom,
! [T4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ).
% ide_char
thf(fact_145_Resid__Arr__self,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T4 ) ) ) ).
% Resid_Arr_self
thf(fact_146_Arr__has__Src,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
!= bot_bo1271416176687173302lambda ) ) ).
% Arr_has_Src
thf(fact_147_Arr__Resid__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ).
% Arr_Resid_single
thf(fact_148_confluence__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda ) ) ) ) ).
% confluence_ind
thf(fact_149_complete__developments__cong,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ~ ( lambda_Ide @ T )
=> ( ( reduct7282684938351565978opment @ T @ U2 )
=> ( ( reduct7282684938351565978opment @ T @ V3 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) ) ) ) ) ) ).
% complete_developments_cong
thf(fact_150_Resid__Arr__Src,axiom,
! [T4: list_lambda_lambda,A2: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
= T4 ) ) ) ).
% Resid_Arr_Src
thf(fact_151_Con__IdeI_I2_J,axiom,
! [A: list_lambda_lambda,T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ A ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ A )
!= nil_lambda_lambda ) ) ) ) ).
% Con_IdeI(2)
thf(fact_152_Ide__char,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ) ) ).
% Ide_char
thf(fact_153_development__implies,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U2 )
=> ( ( reduct8314398545252454212n_path @ U2 )
& ( ( U2 != nil_lambda_lambda )
=> ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ).
% development_implies
thf(fact_154_IdeI,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ) ).
% IdeI
thf(fact_155_Con__single__ideI_I2_J,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(2)
thf(fact_156_Con__single__ideI_I1_J,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_single_ideI(1)
thf(fact_157_cong__implies__coterminal,axiom,
! [U: list_lambda_lambda,U4: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U4 ) ) ).
% cong_implies_coterminal
thf(fact_158_set__empty2,axiom,
! [Xs: list_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty2
thf(fact_159_set__empty2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ Xs ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty2
thf(fact_160_set__empty,axiom,
! [Xs: list_lambda_lambda] :
( ( ( set_lambda_lambda2 @ Xs )
= bot_bo1271416176687173302lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty
thf(fact_161_set__empty,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( ( set_li3846194476672328965lambda @ Xs )
= bot_bo1397037886704484924lambda )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty
thf(fact_162_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_163_paths__in__confluent__rts_Oconfluence__ind,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T4: list_l5168835686021156086lambda,U2: list_l5168835686021156086lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( ( paths_2015741907369688696lambda @ Resid @ T4 )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( ( paths_3172353640142224310lambda @ Resid @ T4 )
= ( paths_3172353640142224310lambda @ Resid @ U2 ) )
=> ( ( paths_3312059784463760222lambda @ Resid @ T4 @ U2 )
!= nil_li8379291187143545974lambda ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_ind
thf(fact_164_paths__in__confluent__rts_Oconfluence__ind,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ( paths_366302053746739058lambda @ Resid @ T4 )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( ( paths_2897053694880971696lambda @ Resid @ T4 )
= ( paths_2897053694880971696lambda @ Resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ T4 @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_ind
thf(fact_165_paths__in__confluent__rts_Ois__confluent__rts,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( conflu1940213901339057510lambda @ ( paths_3312059784463760222lambda @ Resid ) ) ) ).
% paths_in_confluent_rts.is_confluent_rts
thf(fact_166_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_167_Arr_Oelims_I3_J,axiom,
! [X2: list_lambda_lambda] :
( ~ ( paths_366302053746739058lambda @ lambda_resid @ X2 )
=> ( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( arr_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ).
% Arr.elims(3)
thf(fact_168_Arr_Oelims_I2_J,axiom,
! [X2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ X2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( arr_lambda_lambda @ lambda_resid @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Arr.elims(2)
thf(fact_169_Arr_Oelims_I1_J,axiom,
! [X2: list_lambda_lambda,Y2: $o] :
( ( ( paths_366302053746739058lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> Y2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
= ( ~ ( arr_lambda_lambda @ lambda_resid @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( arr_lambda_lambda @ lambda_resid @ T3 )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T3 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% Arr.elims(1)
thf(fact_170__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_171__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_172__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_173__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_174__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_175__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_176__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_177__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_178__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_179_list_Oinject,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda,Y21: list_lambda_lambda,Y22: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X21 @ X22 )
= ( cons_l4558873032622161958lambda @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_180_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_181__092_060Lambda_062_Ocoterminal__iff,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T6 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T6 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T6 ) ) ) ) ).
% \<Lambda>.coterminal_iff
thf(fact_182__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_183_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_184_set__Arr__subset__arr,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ ( arr_lambda_lambda @ lambda_resid ) ) ) ) ).
% set_Arr_subset_arr
thf(fact_185_Arr_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ) ) ).
% Arr.simps(3)
thf(fact_186__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_187_Arr__consI_092_060_094sub_062P,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ).
% Arr_consI\<^sub>P
thf(fact_188_residuation_Oarr_Ocong,axiom,
arr_lambda_lambda = arr_lambda_lambda ).
% residuation.arr.cong
thf(fact_189_residuation_Oarr_Ocong,axiom,
arr_li3691443969012319837lambda = arr_li3691443969012319837lambda ).
% residuation.arr.cong
thf(fact_190_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,U4: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U4 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U )
= ( source5627170679743844886lambda @ Resid @ U4 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U )
= ( target5490931411331770124lambda @ Resid @ U4 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U4 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_191_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,T: list_lambda_lambda,U: list_lambda_lambda,U4: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( member2667661554319814297lambda @ U @ NN )
=> ( ( member2667661554319814297lambda @ U4 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ U )
= ( source6119337877621565212lambda @ Resid @ U4 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ U )
= ( target4026644918049360146lambda @ Resid @ U4 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( source6119337877621565212lambda @ Resid @ U ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U4 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U4 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_192_not__Cons__self2,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( cons_l4558873032622161958lambda @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_193_not__Cons__self2,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( cons_lambda_lambda @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_194_paths__in__rts_OResid_Ocong,axiom,
paths_3312059784463760222lambda = paths_3312059784463760222lambda ).
% paths_in_rts.Resid.cong
thf(fact_195_paths__in__rts_OResid_Ocong,axiom,
paths_8607357831768207192lambda = paths_8607357831768207192lambda ).
% paths_in_rts.Resid.cong
thf(fact_196_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_197_residuation_Oide_Ocong,axiom,
ide_li6773656263178250516lambda = ide_li6773656263178250516lambda ).
% residuation.ide.cong
thf(fact_198_rts_Osources_Ocong,axiom,
source5627170679743844886lambda = source5627170679743844886lambda ).
% rts.sources.cong
thf(fact_199_rts_Osources_Ocong,axiom,
source6119337877621565212lambda = source6119337877621565212lambda ).
% rts.sources.cong
thf(fact_200_paths__in__rts_OArr_Ocong,axiom,
paths_2015741907369688696lambda = paths_2015741907369688696lambda ).
% paths_in_rts.Arr.cong
thf(fact_201_paths__in__rts_OArr_Ocong,axiom,
paths_366302053746739058lambda = paths_366302053746739058lambda ).
% paths_in_rts.Arr.cong
thf(fact_202_rts_Otargets_Ocong,axiom,
target5490931411331770124lambda = target5490931411331770124lambda ).
% rts.targets.cong
thf(fact_203_rts_Otargets_Ocong,axiom,
target4026644918049360146lambda = target4026644918049360146lambda ).
% rts.targets.cong
thf(fact_204_paths__in__rts_OSrcs_Ocong,axiom,
paths_3172353640142224310lambda = paths_3172353640142224310lambda ).
% paths_in_rts.Srcs.cong
thf(fact_205_paths__in__rts_OSrcs_Ocong,axiom,
paths_2897053694880971696lambda = paths_2897053694880971696lambda ).
% paths_in_rts.Srcs.cong
thf(fact_206_paths__in__rts_OIde_Ocong,axiom,
paths_7717490669901197737lambda = paths_7717490669901197737lambda ).
% paths_in_rts.Ide.cong
thf(fact_207_rts_Ocomposable_Ocong,axiom,
compos7127019122090555255lambda = compos7127019122090555255lambda ).
% rts.composable.cong
thf(fact_208_rts_Ocomposable_Ocong,axiom,
compos3359172601056258045lambda = compos3359172601056258045lambda ).
% rts.composable.cong
thf(fact_209_rts_Ocoterminal_Ocong,axiom,
coterm4142657152830665724lambda = coterm4142657152830665724lambda ).
% rts.coterminal.cong
thf(fact_210_rts_Ocoterminal_Ocong,axiom,
coterm81126318419254786lambda = coterm81126318419254786lambda ).
% rts.coterminal.cong
thf(fact_211_partial__magma_Ointro,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
( ? [X4: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP @ X4 @ T3 )
= X4 )
& ( ( OP @ T3 @ X4 )
= X4 ) )
& ! [Y3: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP @ Y3 @ T2 )
= Y3 )
& ( ( OP @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X4 ) ) )
=> ( partia3924075026459072775lambda @ OP ) ) ).
% partial_magma.intro
thf(fact_212_partial__magma_Ointro,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ? [X4: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP @ X4 @ T3 )
= X4 )
& ( ( OP @ T3 @ X4 )
= X4 ) )
& ! [Y3: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP @ Y3 @ T2 )
= Y3 )
& ( ( OP @ T2 @ Y3 )
= Y3 ) )
=> ( Y3 = X4 ) ) )
=> ( partia748546660539056525lambda @ OP ) ) ).
% partial_magma.intro
thf(fact_213_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_214_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_215_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X3: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP2 @ X3 @ T5 )
= X3 )
& ( ( OP2 @ T5 @ X3 )
= X3 ) )
& ! [Y5: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP2 @ Y5 @ T5 )
= Y5 )
& ( ( OP2 @ T5 @ Y5 )
= Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_216_partial__magma__def,axiom,
( partia748546660539056525lambda
= ( ^ [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
? [X3: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( OP2 @ X3 @ T5 )
= X3 )
& ( ( OP2 @ T5 @ X3 )
= X3 ) )
& ! [Y5: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( OP2 @ Y5 @ T5 )
= Y5 )
& ( ( OP2 @ T5 @ Y5 )
= Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% partial_magma_def
thf(fact_217_list__nonempty__induct,axiom,
! [Xs: list_l5168835686021156086lambda,P: list_l5168835686021156086lambda > $o] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( ( Xs2 != nil_li8379291187143545974lambda )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_218_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_219_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_l5168835686021156086lambda > $o,Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_lambda_lambda @ nil_li8379291187143545974lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ nil_li8379291187143545974lambda )
=> ( ! [Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_lambda_lambda @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_220_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_lambda_lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_lambda_lambda )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ nil_lambda_lambda )
=> ( ! [Y3: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_lambda_lambda @ Y3 @ Ys2 ) )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y3: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_221_list__induct2_H,axiom,
! [P: list_l5168835686021156086lambda > list_l5168835686021156086lambda > $o,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( P @ nil_li8379291187143545974lambda @ nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] : ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ nil_li8379291187143545974lambda )
=> ( ! [Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] : ( P @ nil_li8379291187143545974lambda @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) )
=> ( ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Y3: list_lambda_lambda,Ys2: list_l5168835686021156086lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ ( cons_l4558873032622161958lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_222_list__induct2_H,axiom,
! [P: list_lambda_lambda > list_lambda_lambda > $o,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( P @ nil_lambda_lambda @ nil_lambda_lambda )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda] : ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ nil_lambda_lambda )
=> ( ! [Y3: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_lambda_lambda @ ( cons_lambda_lambda @ Y3 @ Ys2 ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y3: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_223_neq__Nil__conv,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
= ( ? [Y5: list_lambda_lambda,Ys3: list_l5168835686021156086lambda] :
( Xs
= ( cons_l4558873032622161958lambda @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_224_neq__Nil__conv,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
= ( ? [Y5: lambda_lambda,Ys3: list_lambda_lambda] :
( Xs
= ( cons_lambda_lambda @ Y5 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_225_remdups__adj_Ocases,axiom,
! [X2: list_l5168835686021156086lambda] :
( ( X2 != nil_li8379291187143545974lambda )
=> ( ! [X: list_lambda_lambda] :
( X2
!= ( cons_l4558873032622161958lambda @ X @ nil_li8379291187143545974lambda ) )
=> ~ ! [X: list_lambda_lambda,Y3: list_lambda_lambda,Xs2: list_l5168835686021156086lambda] :
( X2
!= ( cons_l4558873032622161958lambda @ X @ ( cons_l4558873032622161958lambda @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_226_remdups__adj_Ocases,axiom,
! [X2: list_lambda_lambda] :
( ( X2 != nil_lambda_lambda )
=> ( ! [X: lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ X @ nil_lambda_lambda ) )
=> ~ ! [X: lambda_lambda,Y3: lambda_lambda,Xs2: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ X @ ( cons_lambda_lambda @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_227_transpose_Ocases,axiom,
! [X2: list_l4928420112573288188lambda] :
( ( X2 != nil_li6069905736445159420lambda )
=> ( ! [Xss: list_l4928420112573288188lambda] :
( X2
!= ( cons_l4910729503725484972lambda @ nil_li8379291187143545974lambda @ Xss ) )
=> ~ ! [X: list_lambda_lambda,Xs2: list_l5168835686021156086lambda,Xss: list_l4928420112573288188lambda] :
( X2
!= ( cons_l4910729503725484972lambda @ ( cons_l4558873032622161958lambda @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_228_transpose_Ocases,axiom,
! [X2: list_l5168835686021156086lambda] :
( ( X2 != nil_li8379291187143545974lambda )
=> ( ! [Xss: list_l5168835686021156086lambda] :
( X2
!= ( cons_l4558873032622161958lambda @ nil_lambda_lambda @ Xss ) )
=> ~ ! [X: lambda_lambda,Xs2: list_lambda_lambda,Xss: list_l5168835686021156086lambda] :
( X2
!= ( cons_l4558873032622161958lambda @ ( cons_lambda_lambda @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_229_list_Oexhaust,axiom,
! [Y2: list_l5168835686021156086lambda] :
( ( Y2 != nil_li8379291187143545974lambda )
=> ~ ! [X212: list_lambda_lambda,X222: list_l5168835686021156086lambda] :
( Y2
!= ( cons_l4558873032622161958lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_230_list_Oexhaust,axiom,
! [Y2: list_lambda_lambda] :
( ( Y2 != nil_lambda_lambda )
=> ~ ! [X212: lambda_lambda,X222: list_lambda_lambda] :
( Y2
!= ( cons_lambda_lambda @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_231_list_OdiscI,axiom,
! [List: list_l5168835686021156086lambda,X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( List
= ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
=> ( List != nil_li8379291187143545974lambda ) ) ).
% list.discI
thf(fact_232_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_233_list_Odistinct_I1_J,axiom,
! [X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( nil_li8379291187143545974lambda
!= ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_234_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_235_set__ConsD,axiom,
! [Y2: list_lambda_lambda,X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_236_set__ConsD,axiom,
! [Y2: lambda_lambda,X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_237_list_Oset__cases,axiom,
! [E: list_lambda_lambda,A2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ A2 ) )
=> ( ! [Z2: list_l5168835686021156086lambda] :
( A2
!= ( cons_l4558873032622161958lambda @ E @ Z2 ) )
=> ~ ! [Z1: list_lambda_lambda,Z2: list_l5168835686021156086lambda] :
( ( A2
= ( cons_l4558873032622161958lambda @ Z1 @ Z2 ) )
=> ~ ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_238_list_Oset__cases,axiom,
! [E: lambda_lambda,A2: list_lambda_lambda] :
( ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ A2 ) )
=> ( ! [Z2: list_lambda_lambda] :
( A2
!= ( cons_lambda_lambda @ E @ Z2 ) )
=> ~ ! [Z1: lambda_lambda,Z2: list_lambda_lambda] :
( ( A2
= ( cons_lambda_lambda @ Z1 @ Z2 ) )
=> ~ ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_239_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_240_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_241_list_Oset__intros_I2_J,axiom,
! [Y2: list_lambda_lambda,X22: list_l5168835686021156086lambda,X21: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ X22 ) )
=> ( member2667661554319814297lambda @ Y2 @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_242_list_Oset__intros_I2_J,axiom,
! [Y2: lambda_lambda,X22: list_lambda_lambda,X21: lambda_lambda] :
( ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ X22 ) )
=> ( member_lambda_lambda @ Y2 @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_243_subset__code_I1_J,axiom,
! [Xs: list_l5168835686021156086lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ B )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( member2667661554319814297lambda @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_244_subset__code_I1_J,axiom,
! [Xs: list_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ B )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
=> ( member_lambda_lambda @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_245_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_246_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_247_set__subset__Cons,axiom,
! [Xs: list_l5168835686021156086lambda,X2: list_lambda_lambda] : ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ ( set_li3846194476672328965lambda @ ( cons_l4558873032622161958lambda @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_248_set__subset__Cons,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] : ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_249_empty__set,axiom,
( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ nil_lambda_lambda ) ) ).
% empty_set
thf(fact_250_empty__set,axiom,
( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ nil_li8379291187143545974lambda ) ) ).
% empty_set
thf(fact_251_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,V: lambda_lambda,V4: lambda_lambda,W: lambda_lambda,W2: lambda_lambda,T: lambda_lambda,T6: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ V @ NN )
=> ( ( member_lambda_lambda @ V4 @ NN )
=> ( ( member_lambda_lambda @ W @ NN )
=> ( ( member_lambda_lambda @ W2 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ V )
= ( source5627170679743844886lambda @ Resid @ W ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ V4 )
= ( source5627170679743844886lambda @ Resid @ W2 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ W )
= ( target5490931411331770124lambda @ Resid @ W2 ) )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V4 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T6 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T6 @ W2 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T6 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_252_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,V: list_lambda_lambda,V4: list_lambda_lambda,W: list_lambda_lambda,W2: list_lambda_lambda,T: list_lambda_lambda,T6: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( member2667661554319814297lambda @ V @ NN )
=> ( ( member2667661554319814297lambda @ V4 @ NN )
=> ( ( member2667661554319814297lambda @ W @ NN )
=> ( ( member2667661554319814297lambda @ W2 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ V )
= ( source6119337877621565212lambda @ Resid @ W ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ V4 )
= ( source6119337877621565212lambda @ Resid @ W2 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ W )
= ( target4026644918049360146lambda @ Resid @ W2 ) )
=> ( ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T6 @ V4 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T6 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ W ) @ ( Resid @ T6 @ W2 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T6 @ W2 ) @ ( Resid @ T @ W ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_253_elementary__reduction__path__def,axiom,
( reduct8314398545252454212n_path
= ( ^ [T7: list_lambda_lambda] :
( ( T7 = nil_lambda_lambda )
| ( ( paths_366302053746739058lambda @ lambda_resid @ T7 )
& ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T7 ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ) ) ) ).
% elementary_reduction_path_def
thf(fact_254_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_255__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_256_Std__imp__Arr,axiom,
! [T4: list_lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( ( T4 != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ) ).
% Std_imp_Arr
thf(fact_257_cong__map__App1,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ T4 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ U2 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ T4 ) ) ) ) ) ) ).
% cong_map_App1
thf(fact_258_Srcs__simp_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( source5627170679743844886lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) ) ) ) ).
% Srcs_simp\<^sub>P
thf(fact_259_sources__cons,axiom,
! [T: lambda_lambda,T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ T4 ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% sources_cons
thf(fact_260_Resid__parallel,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T ) ) )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ U @ T6 ) ) ) ) ).
% Resid_parallel
thf(fact_261_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_262_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_263_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_264_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_265_targets__cong__closed,axiom,
! [B2: list_lambda_lambda,T: list_lambda_lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B2 @ B3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ B2 ) ) )
=> ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% targets_cong_closed
thf(fact_266_targets__are__cong,axiom,
! [B2: list_lambda_lambda,T: list_lambda_lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B2 @ B3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ B2 ) ) ) ) ) ).
% targets_are_cong
thf(fact_267_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_268_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_269_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_270_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_271_arr__char,axiom,
! [T4: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% arr_char
thf(fact_272_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_273_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_274_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_275_coterminal__iff,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T6 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ) ).
% coterminal_iff
thf(fact_276_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_277_Con__append_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(2)
thf(fact_278_Con__append_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% Con_append(1)
thf(fact_279_Resid__append__ind,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
!= nil_lambda_lambda )
= ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ U2 )
!= nil_lambda_lambda ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ V3 @ T4 ) @ U2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ V3 ) ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ V3 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ U2 ) ) ) ) ) ) ) ).
% Resid_append_ind
thf(fact_280_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_281_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_282_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_283_Arr__imp__arr__hd,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( arr_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) ) ) ).
% Arr_imp_arr_hd
thf(fact_284_Ide__imp__Ide__hd,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) ) ) ).
% Ide_imp_Ide_hd
thf(fact_285_Ide__imp__sources__eq__targets,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% Ide_imp_sources_eq_targets
thf(fact_286_cong__implies__coinitial,axiom,
! [U: list_lambda_lambda,U4: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U4 ) ) ).
% cong_implies_coinitial
thf(fact_287_coinitial__iff,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T6 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 )
& ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ) ).
% coinitial_iff
thf(fact_288_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_289__092_060Lambda_062_Otargets__eqI,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T6 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T6 ) ) ) ).
% \<Lambda>.targets_eqI
thf(fact_290_Srcs__eqI,axiom,
! [T4: list_lambda_lambda,T8: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ T8 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T8 ) ) ) ).
% Srcs_eqI
thf(fact_291__092_060Lambda_062_Osources__eqI,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T6 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T6 ) ) ) ).
% \<Lambda>.sources_eqI
thf(fact_292_Std__implies__set__subset__elementary__reduction,axiom,
! [U2: list_lambda_lambda] :
( ( reduction_Std @ U2 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ).
% Std_implies_set_subset_elementary_reduction
thf(fact_293_append_Oassoc,axiom,
! [A2: list_l5168835686021156086lambda,B2: list_l5168835686021156086lambda,C: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( append1017360404025248779lambda @ A2 @ B2 ) @ C )
= ( append1017360404025248779lambda @ A2 @ ( append1017360404025248779lambda @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_294_append_Oassoc,axiom,
! [A2: list_lambda_lambda,B2: list_lambda_lambda,C: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ A2 @ B2 ) @ C )
= ( append_lambda_lambda @ A2 @ ( append_lambda_lambda @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_295_append__assoc,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) @ Zs )
= ( append1017360404025248779lambda @ Xs @ ( append1017360404025248779lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_296_append__assoc,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) @ Zs )
= ( append_lambda_lambda @ Xs @ ( append_lambda_lambda @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_297_append__same__eq,axiom,
! [Ys: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Xs )
= ( append1017360404025248779lambda @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_298_append__same__eq,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Xs )
= ( append_lambda_lambda @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_299_same__append__eq,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_300_same__append__eq,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_301_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_302_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_303_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_304_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_305_Resid__cons_I1_J,axiom,
! [U2: list_lambda_lambda,T: lambda_lambda,T4: list_lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% Resid_cons(1)
thf(fact_306_Resid__rec_I4_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) ) ) ) ) ) ) ).
% Resid_rec(4)
thf(fact_307_Arr__map__App2,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda] :
( ( lambda_Ide @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ A2 ) @ T4 ) ) ) ) ).
% Arr_map_App2
thf(fact_308_map__App1__Resid,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ T4 ) @ ( map_la336867333220232516lambda @ ( lambda_App @ X2 ) @ U2 ) ) ) ) ) ).
% map_App1_Resid
thf(fact_309_list_Omap__disc__iff,axiom,
! [F: list_lambda_lambda > lambda_lambda,A2: list_l5168835686021156086lambda] :
( ( ( map_li5201084558375397438lambda @ F @ A2 )
= nil_lambda_lambda )
= ( A2 = nil_li8379291187143545974lambda ) ) ).
% list.map_disc_iff
thf(fact_310_list_Omap__disc__iff,axiom,
! [F: lambda_lambda > list_lambda_lambda,A2: list_lambda_lambda] :
( ( ( map_la6434941737452557130lambda @ F @ A2 )
= nil_li8379291187143545974lambda )
= ( A2 = nil_lambda_lambda ) ) ).
% list.map_disc_iff
thf(fact_311_list_Omap__disc__iff,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,A2: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ A2 )
= nil_li8379291187143545974lambda )
= ( A2 = nil_li8379291187143545974lambda ) ) ).
% list.map_disc_iff
thf(fact_312_list_Omap__disc__iff,axiom,
! [F: lambda_lambda > lambda_lambda,A2: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ A2 )
= nil_lambda_lambda )
= ( A2 = nil_lambda_lambda ) ) ).
% list.map_disc_iff
thf(fact_313_Nil__is__map__conv,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( nil_lambda_lambda
= ( map_li5201084558375397438lambda @ F @ Xs ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% Nil_is_map_conv
thf(fact_314_Nil__is__map__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda] :
( ( nil_li8379291187143545974lambda
= ( map_la6434941737452557130lambda @ F @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% Nil_is_map_conv
thf(fact_315_Nil__is__map__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( nil_li8379291187143545974lambda
= ( map_li385728805257398596lambda @ F @ Xs ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% Nil_is_map_conv
thf(fact_316_Nil__is__map__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( map_la336867333220232516lambda @ F @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% Nil_is_map_conv
thf(fact_317_map__is__Nil__conv,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( ( map_li5201084558375397438lambda @ F @ Xs )
= nil_lambda_lambda )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% map_is_Nil_conv
thf(fact_318_map__is__Nil__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda] :
( ( ( map_la6434941737452557130lambda @ F @ Xs )
= nil_li8379291187143545974lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% map_is_Nil_conv
thf(fact_319_map__is__Nil__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs )
= nil_li8379291187143545974lambda )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% map_is_Nil_conv
thf(fact_320_map__is__Nil__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= nil_lambda_lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% map_is_Nil_conv
thf(fact_321_append_Oright__neutral,axiom,
! [A2: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ A2 @ nil_li8379291187143545974lambda )
= A2 ) ).
% append.right_neutral
thf(fact_322_append_Oright__neutral,axiom,
! [A2: list_lambda_lambda] :
( ( append_lambda_lambda @ A2 @ nil_lambda_lambda )
= A2 ) ).
% append.right_neutral
thf(fact_323_append__Nil2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( append1017360404025248779lambda @ Xs @ nil_li8379291187143545974lambda )
= Xs ) ).
% append_Nil2
thf(fact_324_append__Nil2,axiom,
! [Xs: list_lambda_lambda] :
( ( append_lambda_lambda @ Xs @ nil_lambda_lambda )
= Xs ) ).
% append_Nil2
thf(fact_325_append__self__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= Xs )
= ( Ys = nil_li8379291187143545974lambda ) ) ).
% append_self_conv
thf(fact_326_append__self__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= Xs )
= ( Ys = nil_lambda_lambda ) ) ).
% append_self_conv
thf(fact_327_self__append__conv,axiom,
! [Y2: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Y2
= ( append1017360404025248779lambda @ Y2 @ Ys ) )
= ( Ys = nil_li8379291187143545974lambda ) ) ).
% self_append_conv
thf(fact_328_self__append__conv,axiom,
! [Y2: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Y2
= ( append_lambda_lambda @ Y2 @ Ys ) )
= ( Ys = nil_lambda_lambda ) ) ).
% self_append_conv
thf(fact_329_append__self__conv2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= Ys )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% append_self_conv2
thf(fact_330_append__self__conv2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= Ys )
= ( Xs = nil_lambda_lambda ) ) ).
% append_self_conv2
thf(fact_331_self__append__conv2,axiom,
! [Y2: list_l5168835686021156086lambda,Xs: list_l5168835686021156086lambda] :
( ( Y2
= ( append1017360404025248779lambda @ Xs @ Y2 ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% self_append_conv2
thf(fact_332_self__append__conv2,axiom,
! [Y2: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Y2
= ( append_lambda_lambda @ Xs @ Y2 ) )
= ( Xs = nil_lambda_lambda ) ) ).
% self_append_conv2
thf(fact_333_Nil__is__append__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( nil_li8379291187143545974lambda
= ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( ( Xs = nil_li8379291187143545974lambda )
& ( Ys = nil_li8379291187143545974lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_334_Nil__is__append__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( nil_lambda_lambda
= ( append_lambda_lambda @ Xs @ Ys ) )
= ( ( Xs = nil_lambda_lambda )
& ( Ys = nil_lambda_lambda ) ) ) ).
% Nil_is_append_conv
thf(fact_335_append__is__Nil__conv,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= nil_li8379291187143545974lambda )
= ( ( Xs = nil_li8379291187143545974lambda )
& ( Ys = nil_li8379291187143545974lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_336_append__is__Nil__conv,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= nil_lambda_lambda )
= ( ( Xs = nil_lambda_lambda )
& ( Ys = nil_lambda_lambda ) ) ) ).
% append_is_Nil_conv
thf(fact_337_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_338_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_339_map__eq__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs: list_l5168835686021156086lambda,G: list_lambda_lambda > list_lambda_lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs )
= ( map_li385728805257398596lambda @ G @ Xs ) )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_340_map__eq__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Xs ) )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_341_map__append,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( map_la6434941737452557130lambda @ F @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( append1017360404025248779lambda @ ( map_la6434941737452557130lambda @ F @ Xs ) @ ( map_la6434941737452557130lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_342_map__append,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( map_li5201084558375397438lambda @ F @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( append_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ Xs ) @ ( map_li5201084558375397438lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_343_map__append,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( map_li385728805257398596lambda @ F @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( append1017360404025248779lambda @ ( map_li385728805257398596lambda @ F @ Xs ) @ ( map_li385728805257398596lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_344_map__append,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( append_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) @ ( map_la336867333220232516lambda @ F @ Ys ) ) ) ).
% map_append
thf(fact_345_append1__eq__conv,axiom,
! [Xs: list_l5168835686021156086lambda,X2: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Y2: list_lambda_lambda] :
( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X2 @ nil_li8379291187143545974lambda ) )
= ( append1017360404025248779lambda @ Ys @ ( cons_l4558873032622161958lambda @ Y2 @ nil_li8379291187143545974lambda ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_346_append1__eq__conv,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda,Ys: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) )
= ( append_lambda_lambda @ Ys @ ( cons_lambda_lambda @ Y2 @ nil_lambda_lambda ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_347_hd__append2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Xs ) ) ) ).
% hd_append2
thf(fact_348_hd__append2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Xs ) ) ) ).
% hd_append2
thf(fact_349_arrI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ).
% arrI\<^sub>P
thf(fact_350_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_351_Con__appendI_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(2)
thf(fact_352_Con__appendI_I1_J,axiom,
! [T4: list_lambda_lambda,V3: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 )
!= nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda ) ) ) ).
% Con_appendI(1)
thf(fact_353_Resid__append_I2_J,axiom,
! [U2: list_lambda_lambda,V3: list_lambda_lambda,T4: list_lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ U2 @ V3 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) @ V3 ) ) ) ) ) ).
% Resid_append(2)
thf(fact_354_Resid__append_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ V3 )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) ) ) ) ) ) ).
% Resid_append(1)
thf(fact_355_Resid__append2,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda,W3: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( W3 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V3 @ W3 ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V3 @ W3 ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ W3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W3 @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) ) ) ) ) ) ) ) ) ) ).
% Resid_append2
thf(fact_356_Srcs__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ).
% Srcs_append
thf(fact_357_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_358_list_Omap__sel_I1_J,axiom,
! [A2: list_lambda_lambda,F: lambda_lambda > list_lambda_lambda] :
( ( A2 != nil_lambda_lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( map_la6434941737452557130lambda @ F @ A2 ) )
= ( F @ ( hd_lambda_lambda @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_359_list_Omap__sel_I1_J,axiom,
! [A2: list_l5168835686021156086lambda,F: list_lambda_lambda > lambda_lambda] :
( ( A2 != nil_li8379291187143545974lambda )
=> ( ( hd_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ A2 ) )
= ( F @ ( hd_lis2183642875561147995lambda @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_360_list_Omap__sel_I1_J,axiom,
! [A2: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( A2 != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( map_li385728805257398596lambda @ F @ A2 ) )
= ( F @ ( hd_lis2183642875561147995lambda @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_361_list_Omap__sel_I1_J,axiom,
! [A2: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( A2 != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ F @ A2 ) )
= ( F @ ( hd_lambda_lambda @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_362_rts_Ocoinitial_Ocong,axiom,
coinit1063259437155584316lambda = coinit1063259437155584316lambda ).
% rts.coinitial.cong
thf(fact_363_rts_Ocoinitial_Ocong,axiom,
coinit1392077014771647542lambda = coinit1392077014771647542lambda ).
% rts.coinitial.cong
thf(fact_364_hd__map,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( map_la6434941737452557130lambda @ F @ Xs ) )
= ( F @ ( hd_lambda_lambda @ Xs ) ) ) ) ).
% hd_map
thf(fact_365_hd__map,axiom,
! [Xs: list_l5168835686021156086lambda,F: list_lambda_lambda > lambda_lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ( hd_lambda_lambda @ ( map_li5201084558375397438lambda @ F @ Xs ) )
= ( F @ ( hd_lis2183642875561147995lambda @ Xs ) ) ) ) ).
% hd_map
thf(fact_366_hd__map,axiom,
! [Xs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( map_li385728805257398596lambda @ F @ Xs ) )
= ( F @ ( hd_lis2183642875561147995lambda @ Xs ) ) ) ) ).
% hd_map
thf(fact_367_hd__map,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) )
= ( F @ ( hd_lambda_lambda @ Xs ) ) ) ) ).
% hd_map
thf(fact_368_hd__append,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
( ( ( Xs = nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Ys ) ) )
& ( ( Xs != nil_li8379291187143545974lambda )
=> ( ( hd_lis2183642875561147995lambda @ ( append1017360404025248779lambda @ Xs @ Ys ) )
= ( hd_lis2183642875561147995lambda @ Xs ) ) ) ) ).
% hd_append
thf(fact_369_hd__append,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Ys ) ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( hd_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( hd_lambda_lambda @ Xs ) ) ) ) ).
% hd_append
thf(fact_370_append__eq__appendI,axiom,
! [Xs: list_l5168835686021156086lambda,Xs1: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Us: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append1017360404025248779lambda @ Xs1 @ Us ) )
=> ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_371_append__eq__appendI,axiom,
! [Xs: list_lambda_lambda,Xs1: list_lambda_lambda,Zs: list_lambda_lambda,Ys: list_lambda_lambda,Us: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_lambda_lambda @ Xs1 @ Us ) )
=> ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_372_append__eq__map__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,F: list_lambda_lambda > lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( map_li5201084558375397438lambda @ F @ Xs ) )
= ( ? [Us2: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_li5201084558375397438lambda @ F @ Us2 ) )
& ( Zs
= ( map_li5201084558375397438lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_373_append__eq__map__conv,axiom,
! [Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Zs )
= ( map_la6434941737452557130lambda @ F @ Xs ) )
= ( ? [Us2: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_la6434941737452557130lambda @ F @ Us2 ) )
& ( Zs
= ( map_la6434941737452557130lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_374_append__eq__map__conv,axiom,
! [Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Ys @ Zs )
= ( map_li385728805257398596lambda @ F @ Xs ) )
= ( ? [Us2: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_li385728805257398596lambda @ F @ Us2 ) )
& ( Zs
= ( map_li385728805257398596lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_375_append__eq__map__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( map_la336867333220232516lambda @ F @ Xs ) )
= ( ? [Us2: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us2 ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_376_map__eq__append__conv,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( map_li5201084558375397438lambda @ F @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ? [Us2: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_li5201084558375397438lambda @ F @ Us2 ) )
& ( Zs
= ( map_li5201084558375397438lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_377_map__eq__append__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( map_la6434941737452557130lambda @ F @ Xs )
= ( append1017360404025248779lambda @ Ys @ Zs ) )
= ( ? [Us2: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_la6434941737452557130lambda @ F @ Us2 ) )
& ( Zs
= ( map_la6434941737452557130lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_378_map__eq__append__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs )
= ( append1017360404025248779lambda @ Ys @ Zs ) )
= ( ? [Us2: list_l5168835686021156086lambda,Vs: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_li385728805257398596lambda @ F @ Us2 ) )
& ( Zs
= ( map_li385728805257398596lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_379_map__eq__append__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ? [Us2: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us2 ) )
& ( Zs
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_380_longest__common__prefix,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda] :
? [Ps: list_l5168835686021156086lambda,Xs3: list_l5168835686021156086lambda,Ys4: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ps @ Xs3 ) )
& ( Ys
= ( append1017360404025248779lambda @ Ps @ Ys4 ) )
& ( ( Xs3 = nil_li8379291187143545974lambda )
| ( Ys4 = nil_li8379291187143545974lambda )
| ( ( hd_lis2183642875561147995lambda @ Xs3 )
!= ( hd_lis2183642875561147995lambda @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_381_longest__common__prefix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ps: list_lambda_lambda,Xs3: list_lambda_lambda,Ys4: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ps @ Xs3 ) )
& ( Ys
= ( append_lambda_lambda @ Ps @ Ys4 ) )
& ( ( Xs3 = nil_lambda_lambda )
| ( Ys4 = nil_lambda_lambda )
| ( ( hd_lambda_lambda @ Xs3 )
!= ( hd_lambda_lambda @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_382_append__eq__append__conv2,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda,Ts: list_l5168835686021156086lambda] :
( ( ( append1017360404025248779lambda @ Xs @ Ys )
= ( append1017360404025248779lambda @ Zs @ Ts ) )
= ( ? [Us2: list_l5168835686021156086lambda] :
( ( ( Xs
= ( append1017360404025248779lambda @ Zs @ Us2 ) )
& ( ( append1017360404025248779lambda @ Us2 @ Ys )
= Ts ) )
| ( ( ( append1017360404025248779lambda @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append1017360404025248779lambda @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_383_append__eq__append__conv2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda,Ts: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs @ Ts ) )
= ( ? [Us2: list_lambda_lambda] :
( ( ( Xs
= ( append_lambda_lambda @ Zs @ Us2 ) )
& ( ( append_lambda_lambda @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_lambda_lambda @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_lambda_lambda @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_384_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_385_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_386_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_387_IntD1,axiom,
! [C: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A @ B ) )
=> ( member2667661554319814297lambda @ C @ A ) ) ).
% IntD1
thf(fact_388_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_389_IntD2,axiom,
! [C: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ C @ ( inf_in3354698378728726462lambda @ A @ B ) )
=> ( member2667661554319814297lambda @ C @ B ) ) ).
% IntD2
thf(fact_390_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_391_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_392_Int__absorb,axiom,
! [A: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ A )
= A ) ).
% Int_absorb
thf(fact_393_Int__absorb,axiom,
! [A: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ A )
= A ) ).
% Int_absorb
thf(fact_394_Int__commute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A4: set_lambda_lambda,B4: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_395_Int__commute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A4: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_396_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_397_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_398_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_399_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_400_reduction__paths_OStd_Osimps_I2_J,axiom,
! [T: lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
= ( lambda2670285437103676873uction @ T ) ) ).
% reduction_paths.Std.simps(2)
thf(fact_401_map__eq__Cons__conv,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs: list_l5168835686021156086lambda,Y2: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_li5201084558375397438lambda @ F @ Xs )
= ( cons_lambda_lambda @ Y2 @ Ys ) )
= ( ? [Z3: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( cons_l4558873032622161958lambda @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_li5201084558375397438lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_402_map__eq__Cons__conv,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda,Y2: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( map_la6434941737452557130lambda @ F @ Xs )
= ( cons_l4558873032622161958lambda @ Y2 @ Ys ) )
= ( ? [Z3: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_la6434941737452557130lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_403_map__eq__Cons__conv,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs: list_l5168835686021156086lambda,Y2: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs )
= ( cons_l4558873032622161958lambda @ Y2 @ Ys ) )
= ( ? [Z3: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( cons_l4558873032622161958lambda @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_li385728805257398596lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_404_map__eq__Cons__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Y2: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( cons_lambda_lambda @ Y2 @ Ys ) )
= ( ? [Z3: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_405_Cons__eq__map__conv,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,F: list_lambda_lambda > lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( map_li5201084558375397438lambda @ F @ Ys ) )
= ( ? [Z3: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_li5201084558375397438lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_406_Cons__eq__map__conv,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,F: lambda_lambda > list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_l4558873032622161958lambda @ X2 @ Xs )
= ( map_la6434941737452557130lambda @ F @ Ys ) )
= ( ? [Z3: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_la6434941737452557130lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_407_Cons__eq__map__conv,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X2 @ Xs )
= ( map_li385728805257398596lambda @ F @ Ys ) )
= ( ? [Z3: list_lambda_lambda,Zs2: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_li385728805257398596lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_408_Cons__eq__map__conv,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( map_la336867333220232516lambda @ F @ Ys ) )
= ( ? [Z3: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_409_map__eq__Cons__D,axiom,
! [F: list_lambda_lambda > lambda_lambda,Xs: list_l5168835686021156086lambda,Y2: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_li5201084558375397438lambda @ F @ Xs )
= ( cons_lambda_lambda @ Y2 @ Ys ) )
=> ? [Z4: list_lambda_lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( cons_l4558873032622161958lambda @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_li5201084558375397438lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_410_map__eq__Cons__D,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda,Y2: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( map_la6434941737452557130lambda @ F @ Xs )
= ( cons_l4558873032622161958lambda @ Y2 @ Ys ) )
=> ? [Z4: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_la6434941737452557130lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_411_map__eq__Cons__D,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,Xs: list_l5168835686021156086lambda,Y2: list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( map_li385728805257398596lambda @ F @ Xs )
= ( cons_l4558873032622161958lambda @ Y2 @ Ys ) )
=> ? [Z4: list_lambda_lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( cons_l4558873032622161958lambda @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_li385728805257398596lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_412_map__eq__Cons__D,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Y2: lambda_lambda,Ys: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( cons_lambda_lambda @ Y2 @ Ys ) )
=> ? [Z4: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_413_Cons__eq__map__D,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,F: list_lambda_lambda > lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( map_li5201084558375397438lambda @ F @ Ys ) )
=> ? [Z4: list_lambda_lambda,Zs3: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_li5201084558375397438lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_414_Cons__eq__map__D,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,F: lambda_lambda > list_lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_l4558873032622161958lambda @ X2 @ Xs )
= ( map_la6434941737452557130lambda @ F @ Ys ) )
=> ? [Z4: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_la6434941737452557130lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_415_Cons__eq__map__D,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,Ys: list_l5168835686021156086lambda] :
( ( ( cons_l4558873032622161958lambda @ X2 @ Xs )
= ( map_li385728805257398596lambda @ F @ Ys ) )
=> ? [Z4: list_lambda_lambda,Zs3: list_l5168835686021156086lambda] :
( ( Ys
= ( cons_l4558873032622161958lambda @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_li385728805257398596lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_416_Cons__eq__map__D,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Ys: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( map_la336867333220232516lambda @ F @ Ys ) )
=> ? [Z4: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z4 @ Zs3 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_417_list_Osimps_I9_J,axiom,
! [F: lambda_lambda > list_lambda_lambda,X21: lambda_lambda,X22: list_lambda_lambda] :
( ( map_la6434941737452557130lambda @ F @ ( cons_lambda_lambda @ X21 @ X22 ) )
= ( cons_l4558873032622161958lambda @ ( F @ X21 ) @ ( map_la6434941737452557130lambda @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_418_list_Osimps_I9_J,axiom,
! [F: list_lambda_lambda > lambda_lambda,X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( map_li5201084558375397438lambda @ F @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
= ( cons_lambda_lambda @ ( F @ X21 ) @ ( map_li5201084558375397438lambda @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_419_list_Osimps_I9_J,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,X21: list_lambda_lambda,X22: list_l5168835686021156086lambda] :
( ( map_li385728805257398596lambda @ F @ ( cons_l4558873032622161958lambda @ X21 @ X22 ) )
= ( cons_l4558873032622161958lambda @ ( F @ X21 ) @ ( map_li385728805257398596lambda @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_420_list_Osimps_I9_J,axiom,
! [F: lambda_lambda > lambda_lambda,X21: lambda_lambda,X22: list_lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ ( cons_lambda_lambda @ X21 @ X22 ) )
= ( cons_lambda_lambda @ ( F @ X21 ) @ ( map_la336867333220232516lambda @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_421_list_Osimps_I8_J,axiom,
! [F: lambda_lambda > list_lambda_lambda] :
( ( map_la6434941737452557130lambda @ F @ nil_lambda_lambda )
= nil_li8379291187143545974lambda ) ).
% list.simps(8)
thf(fact_422_list_Osimps_I8_J,axiom,
! [F: list_lambda_lambda > lambda_lambda] :
( ( map_li5201084558375397438lambda @ F @ nil_li8379291187143545974lambda )
= nil_lambda_lambda ) ).
% list.simps(8)
thf(fact_423_list_Osimps_I8_J,axiom,
! [F: list_lambda_lambda > list_lambda_lambda] :
( ( map_li385728805257398596lambda @ F @ nil_li8379291187143545974lambda )
= nil_li8379291187143545974lambda ) ).
% list.simps(8)
thf(fact_424_list_Osimps_I8_J,axiom,
! [F: lambda_lambda > lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.simps(8)
thf(fact_425_list_Omap__cong,axiom,
! [X2: list_l5168835686021156086lambda,Ya: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda > list_lambda_lambda] :
( ( X2 = Ya )
=> ( ! [Z4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z4 @ ( set_li3846194476672328965lambda @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_li385728805257398596lambda @ F @ X2 )
= ( map_li385728805257398596lambda @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_426_list_Omap__cong,axiom,
! [X2: list_lambda_lambda,Ya: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( X2 = Ya )
=> ( ! [Z4: lambda_lambda] :
( ( member_lambda_lambda @ Z4 @ ( set_lambda_lambda2 @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_427_list_Omap__cong0,axiom,
! [X2: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda > list_lambda_lambda] :
( ! [Z4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z4 @ ( set_li3846194476672328965lambda @ X2 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_li385728805257398596lambda @ F @ X2 )
= ( map_li385728805257398596lambda @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_428_list_Omap__cong0,axiom,
! [X2: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [Z4: lambda_lambda] :
( ( member_lambda_lambda @ Z4 @ ( set_lambda_lambda2 @ X2 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_429_list_Oinj__map__strong,axiom,
! [X2: list_l5168835686021156086lambda,Xa: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,Fa: list_lambda_lambda > list_lambda_lambda] :
( ! [Z4: list_lambda_lambda,Za: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z4 @ ( set_li3846194476672328965lambda @ X2 ) )
=> ( ( member2667661554319814297lambda @ Za @ ( set_li3846194476672328965lambda @ Xa ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_li385728805257398596lambda @ F @ X2 )
= ( map_li385728805257398596lambda @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_430_list_Oinj__map__strong,axiom,
! [X2: list_lambda_lambda,Xa: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Fa: lambda_lambda > lambda_lambda] :
( ! [Z4: lambda_lambda,Za: lambda_lambda] :
( ( member_lambda_lambda @ Z4 @ ( set_lambda_lambda2 @ X2 ) )
=> ( ( member_lambda_lambda @ Za @ ( set_lambda_lambda2 @ Xa ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_431_list_Omap__ident__strong,axiom,
! [T: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [Z4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z4 @ ( set_li3846194476672328965lambda @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_li385728805257398596lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_432_list_Omap__ident__strong,axiom,
! [T: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [Z4: lambda_lambda] :
( ( member_lambda_lambda @ Z4 @ ( set_lambda_lambda2 @ T ) )
=> ( ( F @ Z4 )
= Z4 ) )
=> ( ( map_la336867333220232516lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_433_map__ext,axiom,
! [Xs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda > list_lambda_lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_li385728805257398596lambda @ F @ Xs )
= ( map_li385728805257398596lambda @ G @ Xs ) ) ) ).
% map_ext
thf(fact_434_map__ext,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Xs ) ) ) ).
% map_ext
thf(fact_435_map__idI,axiom,
! [Xs: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_li385728805257398596lambda @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_436_map__idI,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X )
= X ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_437_map__cong,axiom,
! [Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda > list_lambda_lambda] :
( ( Xs = Ys )
=> ( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ ( set_li3846194476672328965lambda @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_li385728805257398596lambda @ F @ Xs )
= ( map_li385728805257398596lambda @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_438_map__cong,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( Xs = Ys )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( map_la336867333220232516lambda @ F @ Xs )
= ( map_la336867333220232516lambda @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_439_ex__map__conv,axiom,
! [Ys: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( ? [Xs4: list_l5168835686021156086lambda] :
( Ys
= ( map_li385728805257398596lambda @ F @ Xs4 ) ) )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ ( set_li3846194476672328965lambda @ Ys ) )
=> ? [Y5: list_lambda_lambda] :
( X3
= ( F @ Y5 ) ) ) ) ) ).
% ex_map_conv
thf(fact_440_ex__map__conv,axiom,
! [Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( ? [Xs4: list_lambda_lambda] :
( Ys
= ( map_la336867333220232516lambda @ F @ Xs4 ) ) )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Ys ) )
=> ? [Y5: lambda_lambda] :
( X3
= ( F @ Y5 ) ) ) ) ) ).
% ex_map_conv
thf(fact_441_Cons__eq__appendI,axiom,
! [X2: lambda_lambda,Xs1: list_lambda_lambda,Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_lambda_lambda @ Xs1 @ Zs ) )
=> ( ( cons_lambda_lambda @ X2 @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_442_append__Cons,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) @ Ys )
= ( cons_lambda_lambda @ X2 @ ( append_lambda_lambda @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_443_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_444_eq__Nil__appendI,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
( ( Xs = Ys )
=> ( Xs
= ( append_lambda_lambda @ nil_lambda_lambda @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_445_append_Oleft__neutral,axiom,
! [A2: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_446_append__Nil,axiom,
! [Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_447_disjoint__iff__not__equal,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A @ B )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A )
=> ! [Y5: lambda_lambda] :
( ( member_lambda_lambda @ Y5 @ B )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_448_disjoint__iff__not__equal,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ A @ B )
= bot_bo1397037886704484924lambda )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ A )
=> ! [Y5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y5 @ B )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_449_Int__empty__right,axiom,
! [A: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_right
thf(fact_450_Int__empty__right,axiom,
! [A: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% Int_empty_right
thf(fact_451_Int__empty__left,axiom,
! [B: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ B )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_left
thf(fact_452_Int__empty__left,axiom,
! [B: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ B )
= bot_bo1397037886704484924lambda ) ).
% Int_empty_left
thf(fact_453_disjoint__iff,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A @ B )
= bot_bo1271416176687173302lambda )
= ( ! [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ A )
=> ~ ( member_lambda_lambda @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_454_disjoint__iff,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ A @ B )
= bot_bo1397037886704484924lambda )
= ( ! [X3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X3 @ A )
=> ~ ( member2667661554319814297lambda @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_455_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_456_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_457_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_458_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_459_Int__lower1,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_460_Int__lower1,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_461_Int__lower2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_462_Int__lower2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_463_Int__absorb1,axiom,
! [B: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_464_Int__absorb1,axiom,
! [B: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_465_Int__absorb2,axiom,
! [A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B )
=> ( ( inf_in3354698378728726462lambda @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_466_Int__absorb2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ( inf_in1766959559283430712lambda @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_467_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_468_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_469_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_470_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_471_reduction__paths_OStd_Osimps_I1_J,axiom,
reduction_Std @ nil_lambda_lambda ).
% reduction_paths.Std.simps(1)
thf(fact_472_lambda__calculus_Oelementary__reduction_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_App @ T @ U ) )
= ( ( ( lambda2670285437103676873uction @ T )
& ( lambda_Ide @ U ) )
| ( ( lambda_Ide @ T )
& ( lambda2670285437103676873uction @ U ) ) ) ) ).
% lambda_calculus.elementary_reduction.simps(4)
thf(fact_473_lambda__calculus_Oelementary__reduction_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Beta @ T @ U ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.elementary_reduction.simps(5)
thf(fact_474_lambda__calculus_Oelementary__reduction__not__ide,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ~ ( ide_lambda_lambda @ lambda_resid @ T ) ) ).
% lambda_calculus.elementary_reduction_not_ide
thf(fact_475_lambda__calculus_Ocong__elementary__reductions__are__equal,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ( ( lambda2670285437103676873uction @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) )
=> ( T = U ) ) ) ) ).
% lambda_calculus.cong_elementary_reductions_are_equal
thf(fact_476_lambda__calculus_Oelementary__reduction__is__arr,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ T )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% lambda_calculus.elementary_reduction_is_arr
thf(fact_477_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_478_rev__exhaust,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ~ ! [Ys2: list_lambda_lambda,Y3: lambda_lambda] :
( Xs
!= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ Y3 @ nil_lambda_lambda ) ) ) ) ).
% rev_exhaust
thf(fact_479_Cons__eq__append__conv,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( append_lambda_lambda @ Ys @ Zs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( ( cons_lambda_lambda @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_lambda_lambda @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_480_append__eq__Cons__conv,axiom,
! [Ys: list_lambda_lambda,Zs: list_lambda_lambda,X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs )
= ( cons_lambda_lambda @ X2 @ Xs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( Zs
= ( cons_lambda_lambda @ X2 @ Xs ) ) )
| ? [Ys5: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ X2 @ Ys5 ) )
& ( ( append_lambda_lambda @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_481_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_482_split__list__first__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_lambda_lambda,X3: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y5: lambda_lambda] :
( ( member_lambda_lambda @ Y5 @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_483_split__list__last__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X3: lambda_lambda] :
( ( member_lambda_lambda @ X3 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_lambda_lambda,X3: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y5: lambda_lambda] :
( ( member_lambda_lambda @ Y5 @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_484_in__set__conv__decomp__first,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_485_in__set__conv__decomp__first,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_486_in__set__conv__decomp__last,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_487_in__set__conv__decomp__last,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_488_split__list__first__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_489_split__list__last__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_490_split__list__first__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_491_split__list__last__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Xa2: lambda_lambda] :
( ( member_lambda_lambda @ Xa2 @ ( set_lambda_lambda2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_492_in__set__conv__decomp,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_493_in__set__conv__decomp,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_494_append__Cons__eq__iff,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs5: list_l5168835686021156086lambda,Ys6: list_l5168835686021156086lambda] :
( ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys ) )
=> ( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X2 @ Ys ) )
= ( append1017360404025248779lambda @ Xs5 @ ( cons_l4558873032622161958lambda @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_495_append__Cons__eq__iff,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Xs5: list_lambda_lambda,Ys6: list_lambda_lambda] :
( ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys ) )
=> ( ( ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X2 @ Ys ) )
= ( append_lambda_lambda @ Xs5 @ ( cons_lambda_lambda @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_496_split__list__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_497_split__list__first,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_498_split__list__first,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_499_split__list__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs3 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_500_split__list__last,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_501_split__list__last,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_502_split__list,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs3: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_503_split__list,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs3: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_504_lambda__calculus_Ois__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% lambda_calculus.is_confluent_rts
thf(fact_505_hd__in__set,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ Xs ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ).
% hd_in_set
thf(fact_506_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_507_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_508_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_509_const__ide__is__Ide,axiom,
! [T4: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T4 ) @ bot_bo1271416176687173302lambda ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ) ) ).
% const_ide_is_Ide
thf(fact_510_in__sources__iff,axiom,
! [T4: list_lambda_lambda,A: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member2667661554319814297lambda @ A @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ A ) ) ) ) ) ).
% in_sources_iff
thf(fact_511_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_512_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_513_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X2 )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_514_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X2 )
= bot_bo1397037886704484924lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_515_inf__bot__right,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_right
thf(fact_516_inf__bot__right,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% inf_bot_right
thf(fact_517_inf__bot__left,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X2 )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_left
thf(fact_518_inf__bot__left,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X2 )
= bot_bo1397037886704484924lambda ) ).
% inf_bot_left
thf(fact_519_le__inf__iff,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z5 ) )
= ( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
& ( ord_le3064231726521830256lambda @ X2 @ Z5 ) ) ) ).
% le_inf_iff
thf(fact_520_le__inf__iff,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z5 ) )
= ( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
& ( ord_le1926335974945931754lambda @ X2 @ Z5 ) ) ) ).
% le_inf_iff
thf(fact_521_inf_Obounded__iff,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B2 @ C ) )
= ( ( ord_le3064231726521830256lambda @ A2 @ B2 )
& ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_522_inf_Obounded__iff,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B2 @ C ) )
= ( ( ord_le1926335974945931754lambda @ A2 @ B2 )
& ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_523__092_060Lambda_062_Ocong__implies__coinitial,axiom,
! [U: lambda_lambda,U4: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ U @ U4 ) ) ).
% \<Lambda>.cong_implies_coinitial
thf(fact_524__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_525__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_526__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_527__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_528__092_060Lambda_062_Ocoinitial__iff,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T6 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T6 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T6 ) ) ) ) ).
% \<Lambda>.coinitial_iff
thf(fact_529_sources__eqI,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% sources_eqI
thf(fact_530_targets__eqI,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 ) ) ) ).
% targets_eqI
thf(fact_531_insert__absorb2,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda] :
( ( insert_lambda_lambda @ X2 @ ( insert_lambda_lambda @ X2 @ A ) )
= ( insert_lambda_lambda @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_532_insert__iff,axiom,
! [A2: lambda_lambda,B2: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_lambda_lambda @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_533_insert__iff,axiom,
! [A2: list_lambda_lambda,B2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member2667661554319814297lambda @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_534_insertCI,axiom,
! [A2: lambda_lambda,B: set_lambda_lambda,B2: lambda_lambda] :
( ( ~ ( member_lambda_lambda @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B2 @ B ) ) ) ).
% insertCI
thf(fact_535_insertCI,axiom,
! [A2: list_lambda_lambda,B: set_li7878742357367608272lambda,B2: list_lambda_lambda] :
( ( ~ ( member2667661554319814297lambda @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ B2 @ B ) ) ) ).
% insertCI
thf(fact_536_inf_Oidem,axiom,
! [A2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_537_inf_Oidem,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_538_inf__idem,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_539_inf__idem,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_540_inf_Oleft__idem,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) )
= ( inf_in1766959559283430712lambda @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_541_inf_Oleft__idem,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) )
= ( inf_in3354698378728726462lambda @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_542_inf__left__idem,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) )
= ( inf_in1766959559283430712lambda @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_543_inf__left__idem,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) )
= ( inf_in3354698378728726462lambda @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_544_inf_Oright__idem,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ B2 )
= ( inf_in1766959559283430712lambda @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_545_inf_Oright__idem,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ B2 )
= ( inf_in3354698378728726462lambda @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_546_inf__right__idem,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Y2 )
= ( inf_in1766959559283430712lambda @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_547_inf__right__idem,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Y2 )
= ( inf_in3354698378728726462lambda @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_548__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_549_Src__hd__eqI,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( lambda_Src @ ( hd_lambda_lambda @ T4 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) ) ).
% Src_hd_eqI
thf(fact_550_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_551_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_552_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_553_set__Ide__subset__single__hd,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( insert_lambda_lambda @ ( hd_lambda_lambda @ T4 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% set_Ide_subset_single_hd
thf(fact_554_single__Src__hd__in__sources,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% single_Src_hd_in_sources
thf(fact_555_singletonI,axiom,
! [A2: lambda_lambda] : ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).
% singletonI
thf(fact_556_singletonI,axiom,
! [A2: list_lambda_lambda] : ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).
% singletonI
thf(fact_557_insert__subset,axiom,
! [X2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( insert8465602035467394112lambda @ X2 @ A ) @ B )
= ( ( member2667661554319814297lambda @ X2 @ B )
& ( ord_le3064231726521830256lambda @ A @ B ) ) ) ).
% insert_subset
thf(fact_558_insert__subset,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ X2 @ A ) @ B )
= ( ( member_lambda_lambda @ X2 @ B )
& ( ord_le1926335974945931754lambda @ A @ B ) ) ) ).
% insert_subset
thf(fact_559_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_560_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_561_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_562_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_563_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_564_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_565_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_566_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_567_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_568_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_569_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_570_singleton__insert__inj__eq,axiom,
! [B2: list_lambda_lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( insert8465602035467394112lambda @ B2 @ bot_bo1397037886704484924lambda )
= ( insert8465602035467394112lambda @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B2 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_571_singleton__insert__inj__eq,axiom,
! [B2: lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
( ( ( insert_lambda_lambda @ B2 @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B2 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_572_singleton__insert__inj__eq_H,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B2: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ A )
= ( insert8465602035467394112lambda @ B2 @ bot_bo1397037886704484924lambda ) )
= ( ( A2 = B2 )
& ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B2 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_573_singleton__insert__inj__eq_H,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B2: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ A )
= ( insert_lambda_lambda @ B2 @ bot_bo1271416176687173302lambda ) )
= ( ( A2 = B2 )
& ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B2 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_574_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_575_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_576_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_577_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_578_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_579_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_580_disjoint__insert_I2_J,axiom,
! [A: set_lambda_lambda,B2: lambda_lambda,B: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A @ ( insert_lambda_lambda @ B2 @ B ) ) )
= ( ~ ( member_lambda_lambda @ B2 @ A )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_581_disjoint__insert_I2_J,axiom,
! [A: set_li7878742357367608272lambda,B2: list_lambda_lambda,B: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A @ ( insert8465602035467394112lambda @ B2 @ B ) ) )
= ( ~ ( member2667661554319814297lambda @ B2 @ A )
& ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_582__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_583_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_584_mk__disjoint__insert,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ A )
=> ? [B5: set_lambda_lambda] :
( ( A
= ( insert_lambda_lambda @ A2 @ B5 ) )
& ~ ( member_lambda_lambda @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_585_mk__disjoint__insert,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ A )
=> ? [B5: set_li7878742357367608272lambda] :
( ( A
= ( insert8465602035467394112lambda @ A2 @ B5 ) )
& ~ ( member2667661554319814297lambda @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_586_insert__commute,axiom,
! [X2: lambda_lambda,Y2: lambda_lambda,A: set_lambda_lambda] :
( ( insert_lambda_lambda @ X2 @ ( insert_lambda_lambda @ Y2 @ A ) )
= ( insert_lambda_lambda @ Y2 @ ( insert_lambda_lambda @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_587_insert__eq__iff,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B2: lambda_lambda,B: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ A2 @ A )
=> ( ~ ( member_lambda_lambda @ B2 @ B )
=> ( ( ( insert_lambda_lambda @ A2 @ A )
= ( insert_lambda_lambda @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_lambda_lambda] :
( ( A
= ( insert_lambda_lambda @ B2 @ C3 ) )
& ~ ( member_lambda_lambda @ B2 @ C3 )
& ( B
= ( insert_lambda_lambda @ A2 @ C3 ) )
& ~ ( member_lambda_lambda @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_588_insert__eq__iff,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B2: list_lambda_lambda,B: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ A2 @ A )
=> ( ~ ( member2667661554319814297lambda @ B2 @ B )
=> ( ( ( insert8465602035467394112lambda @ A2 @ A )
= ( insert8465602035467394112lambda @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_li7878742357367608272lambda] :
( ( A
= ( insert8465602035467394112lambda @ B2 @ C3 ) )
& ~ ( member2667661554319814297lambda @ B2 @ C3 )
& ( B
= ( insert8465602035467394112lambda @ A2 @ C3 ) )
& ~ ( member2667661554319814297lambda @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_589_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_590_insert__absorb,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ A )
=> ( ( insert8465602035467394112lambda @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_591_insert__ident,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X2 @ A )
=> ( ~ ( member_lambda_lambda @ X2 @ B )
=> ( ( ( insert_lambda_lambda @ X2 @ A )
= ( insert_lambda_lambda @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_592_insert__ident,axiom,
! [X2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ X2 @ A )
=> ( ~ ( member2667661554319814297lambda @ X2 @ B )
=> ( ( ( insert8465602035467394112lambda @ X2 @ A )
= ( insert8465602035467394112lambda @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_593_Set_Oset__insert,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ A )
=> ~ ! [B5: set_lambda_lambda] :
( ( A
= ( insert_lambda_lambda @ X2 @ B5 ) )
=> ( member_lambda_lambda @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_594_Set_Oset__insert,axiom,
! [X2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ X2 @ A )
=> ~ ! [B5: set_li7878742357367608272lambda] :
( ( A
= ( insert8465602035467394112lambda @ X2 @ B5 ) )
=> ( member2667661554319814297lambda @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_595_insertI2,axiom,
! [A2: lambda_lambda,B: set_lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ B )
=> ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B2 @ B ) ) ) ).
% insertI2
thf(fact_596_insertI2,axiom,
! [A2: list_lambda_lambda,B: set_li7878742357367608272lambda,B2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ B )
=> ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ B2 @ B ) ) ) ).
% insertI2
thf(fact_597_insertI1,axiom,
! [A2: lambda_lambda,B: set_lambda_lambda] : ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ A2 @ B ) ) ).
% insertI1
thf(fact_598_insertI1,axiom,
! [A2: list_lambda_lambda,B: set_li7878742357367608272lambda] : ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ A2 @ B ) ) ).
% insertI1
thf(fact_599_insertE,axiom,
! [A2: lambda_lambda,B2: lambda_lambda,A: set_lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_lambda_lambda @ A2 @ A ) ) ) ).
% insertE
thf(fact_600_insertE,axiom,
! [A2: list_lambda_lambda,B2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member2667661554319814297lambda @ A2 @ A ) ) ) ).
% insertE
thf(fact_601_bot__set__def,axiom,
( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ bot_bo3555140721347798439mbda_o ) ) ).
% bot_set_def
thf(fact_602_bot__set__def,axiom,
( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda @ bot_bo430095131446760353mbda_o ) ) ).
% bot_set_def
thf(fact_603_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_Src @ ( lambda_App @ T6 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T6 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_604_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_605_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T6 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T6 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(3)
thf(fact_606_singletonD,axiom,
! [B2: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_607_singletonD,axiom,
! [B2: list_lambda_lambda,A2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B2 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_608_singleton__iff,axiom,
! [B2: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_609_singleton__iff,axiom,
! [B2: list_lambda_lambda,A2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B2 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_610_doubleton__eq__iff,axiom,
! [A2: lambda_lambda,B2: lambda_lambda,C: lambda_lambda,D2: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B2 @ bot_bo1271416176687173302lambda ) )
= ( insert_lambda_lambda @ C @ ( insert_lambda_lambda @ D2 @ bot_bo1271416176687173302lambda ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_611_doubleton__eq__iff,axiom,
! [A2: list_lambda_lambda,B2: list_lambda_lambda,C: list_lambda_lambda,D2: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ ( insert8465602035467394112lambda @ B2 @ bot_bo1397037886704484924lambda ) )
= ( insert8465602035467394112lambda @ C @ ( insert8465602035467394112lambda @ D2 @ bot_bo1397037886704484924lambda ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_612_insert__not__empty,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda] :
( ( insert_lambda_lambda @ A2 @ A )
!= bot_bo1271416176687173302lambda ) ).
% insert_not_empty
thf(fact_613_insert__not__empty,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( insert8465602035467394112lambda @ A2 @ A )
!= bot_bo1397037886704484924lambda ) ).
% insert_not_empty
thf(fact_614_singleton__inject,axiom,
! [A2: lambda_lambda,B2: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ B2 @ bot_bo1271416176687173302lambda ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_615_singleton__inject,axiom,
! [A2: list_lambda_lambda,B2: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda )
= ( insert8465602035467394112lambda @ B2 @ bot_bo1397037886704484924lambda ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_616_subset__insertI2,axiom,
! [A: set_lambda_lambda,B: set_lambda_lambda,B2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B )
=> ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_617_subset__insertI,axiom,
! [B: set_lambda_lambda,A2: lambda_lambda] : ( ord_le1926335974945931754lambda @ B @ ( insert_lambda_lambda @ A2 @ B ) ) ).
% subset_insertI
thf(fact_618_subset__insert,axiom,
! [X2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ X2 @ A )
=> ( ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ X2 @ B ) )
= ( ord_le3064231726521830256lambda @ A @ B ) ) ) ).
% subset_insert
thf(fact_619_subset__insert,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda,B: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X2 @ A )
=> ( ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ X2 @ B ) )
= ( ord_le1926335974945931754lambda @ A @ B ) ) ) ).
% subset_insert
thf(fact_620_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_621_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_622_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_623_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_624_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_625_subset__singletonD,axiom,
! [A: set_li7878742357367608272lambda,X2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ X2 @ bot_bo1397037886704484924lambda ) )
=> ( ( A = bot_bo1397037886704484924lambda )
| ( A
= ( insert8465602035467394112lambda @ X2 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% subset_singletonD
thf(fact_626_subset__singletonD,axiom,
! [A: set_lambda_lambda,X2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ X2 @ bot_bo1271416176687173302lambda ) )
=> ( ( A = bot_bo1271416176687173302lambda )
| ( A
= ( insert_lambda_lambda @ X2 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singletonD
thf(fact_627_subset__singleton__iff,axiom,
! [X5: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ X5 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
= ( ( X5 = bot_bo1397037886704484924lambda )
| ( X5
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_628_subset__singleton__iff,axiom,
! [X5: set_lambda_lambda,A2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X5 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
= ( ( X5 = bot_bo1271416176687173302lambda )
| ( X5
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_629_inf__sup__aci_I4_J,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) )
= ( inf_in1766959559283430712lambda @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_630_inf__sup__aci_I4_J,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) )
= ( inf_in3354698378728726462lambda @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_631_inf__sup__aci_I3_J,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z5 ) )
= ( inf_in1766959559283430712lambda @ Y2 @ ( inf_in1766959559283430712lambda @ X2 @ Z5 ) ) ) ).
% inf_sup_aci(3)
thf(fact_632_inf__sup__aci_I3_J,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z5 ) )
= ( inf_in3354698378728726462lambda @ Y2 @ ( inf_in3354698378728726462lambda @ X2 @ Z5 ) ) ) ).
% inf_sup_aci(3)
thf(fact_633_inf__sup__aci_I2_J,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Z5 )
= ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z5 ) ) ) ).
% inf_sup_aci(2)
thf(fact_634_inf__sup__aci_I2_J,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Z5 )
= ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z5 ) ) ) ).
% inf_sup_aci(2)
thf(fact_635_inf__sup__aci_I1_J,axiom,
( inf_in1766959559283430712lambda
= ( ^ [X3: set_lambda_lambda,Y5: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_636_inf__sup__aci_I1_J,axiom,
( inf_in3354698378728726462lambda
= ( ^ [X3: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_637_inf_Oassoc,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ C )
= ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_638_inf_Oassoc,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ C )
= ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_639_inf__assoc,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Z5 )
= ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z5 ) ) ) ).
% inf_assoc
thf(fact_640_inf__assoc,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Z5 )
= ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z5 ) ) ) ).
% inf_assoc
thf(fact_641_inf_Ocommute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [A5: set_lambda_lambda,B6: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ B6 @ A5 ) ) ) ).
% inf.commute
thf(fact_642_inf_Ocommute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [A5: set_li7878742357367608272lambda,B6: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ B6 @ A5 ) ) ) ).
% inf.commute
thf(fact_643_inf__commute,axiom,
( inf_in1766959559283430712lambda
= ( ^ [X3: set_lambda_lambda,Y5: set_lambda_lambda] : ( inf_in1766959559283430712lambda @ Y5 @ X3 ) ) ) ).
% inf_commute
thf(fact_644_inf__commute,axiom,
( inf_in3354698378728726462lambda
= ( ^ [X3: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] : ( inf_in3354698378728726462lambda @ Y5 @ X3 ) ) ) ).
% inf_commute
thf(fact_645_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_lambda_lambda,K: set_lambda_lambda,A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( A
= ( inf_in1766959559283430712lambda @ K @ A2 ) )
=> ( ( inf_in1766959559283430712lambda @ A @ B2 )
= ( inf_in1766959559283430712lambda @ K @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_646_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_li7878742357367608272lambda,K: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( A
= ( inf_in3354698378728726462lambda @ K @ A2 ) )
=> ( ( inf_in3354698378728726462lambda @ A @ B2 )
= ( inf_in3354698378728726462lambda @ K @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_647_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_lambda_lambda,K: set_lambda_lambda,B2: set_lambda_lambda,A2: set_lambda_lambda] :
( ( B
= ( inf_in1766959559283430712lambda @ K @ B2 ) )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B )
= ( inf_in1766959559283430712lambda @ K @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_648_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_li7878742357367608272lambda,K: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( B
= ( inf_in3354698378728726462lambda @ K @ B2 ) )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B )
= ( inf_in3354698378728726462lambda @ K @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_649_inf_Oleft__commute,axiom,
! [B2: set_lambda_lambda,A2: set_lambda_lambda,C: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ B2 @ ( inf_in1766959559283430712lambda @ A2 @ C ) )
= ( inf_in1766959559283430712lambda @ A2 @ ( inf_in1766959559283430712lambda @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_650_inf_Oleft__commute,axiom,
! [B2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ B2 @ ( inf_in3354698378728726462lambda @ A2 @ C ) )
= ( inf_in3354698378728726462lambda @ A2 @ ( inf_in3354698378728726462lambda @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_651_inf__left__commute,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z5 ) )
= ( inf_in1766959559283430712lambda @ Y2 @ ( inf_in1766959559283430712lambda @ X2 @ Z5 ) ) ) ).
% inf_left_commute
thf(fact_652_inf__left__commute,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z5 ) )
= ( inf_in3354698378728726462lambda @ Y2 @ ( inf_in3354698378728726462lambda @ X2 @ Z5 ) ) ) ).
% inf_left_commute
thf(fact_653_inf__sup__ord_I2_J,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_654_inf__sup__ord_I2_J,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_655_inf__sup__ord_I1_J,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_656_inf__sup__ord_I1_J,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_657_inf__le1,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_658_inf__le1,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_659_inf__le2,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_660_inf__le2,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_661_le__infE,axiom,
! [X2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) )
=> ~ ( ( ord_le3064231726521830256lambda @ X2 @ A2 )
=> ~ ( ord_le3064231726521830256lambda @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_662_le__infE,axiom,
! [X2: set_lambda_lambda,A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) )
=> ~ ( ( ord_le1926335974945931754lambda @ X2 @ A2 )
=> ~ ( ord_le1926335974945931754lambda @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_663_le__infI,axiom,
! [X2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ A2 )
=> ( ( ord_le3064231726521830256lambda @ X2 @ B2 )
=> ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_664_le__infI,axiom,
! [X2: set_lambda_lambda,A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ X2 @ B2 )
=> ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_665_inf__mono,axiom,
! [A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda,D2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ( ord_le3064231726521830256lambda @ B2 @ D2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ ( inf_in3354698378728726462lambda @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_666_inf__mono,axiom,
! [A2: set_lambda_lambda,C: set_lambda_lambda,B2: set_lambda_lambda,D2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ( ord_le1926335974945931754lambda @ B2 @ D2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ ( inf_in1766959559283430712lambda @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_667_le__infI1,axiom,
! [A2: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ X2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_668_le__infI1,axiom,
! [A2: set_lambda_lambda,X2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ X2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_669_le__infI2,axiom,
! [B2: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B2 @ X2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_670_le__infI2,axiom,
! [B2: set_lambda_lambda,X2: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B2 @ X2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_671_inf_OorderE,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B2 )
=> ( A2
= ( inf_in3354698378728726462lambda @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_672_inf_OorderE,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( A2
= ( inf_in1766959559283430712lambda @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_673_inf_OorderI,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( A2
= ( inf_in3354698378728726462lambda @ A2 @ B2 ) )
=> ( ord_le3064231726521830256lambda @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_674_inf_OorderI,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( A2
= ( inf_in1766959559283430712lambda @ A2 @ B2 ) )
=> ( ord_le1926335974945931754lambda @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_675_inf__unique,axiom,
! [F: set_li7878742357367608272lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_li7878742357367608272lambda,Y3: set_li7878742357367608272lambda,Z4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y3 )
=> ( ( ord_le3064231726521830256lambda @ X @ Z4 )
=> ( ord_le3064231726521830256lambda @ X @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_676_inf__unique,axiom,
! [F: set_lambda_lambda > set_lambda_lambda > set_lambda_lambda,X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F @ X @ Y3 ) @ X )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F @ X @ Y3 ) @ Y3 )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda,Z4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ( ord_le1926335974945931754lambda @ X @ Z4 )
=> ( ord_le1926335974945931754lambda @ X @ ( F @ Y3 @ Z4 ) ) ) )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_677_le__iff__inf,axiom,
( ord_le3064231726521830256lambda
= ( ^ [X3: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_678_le__iff__inf,axiom,
( ord_le1926335974945931754lambda
= ( ^ [X3: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_679_inf_Oabsorb1,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_680_inf_Oabsorb1,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_681_inf_Oabsorb2,axiom,
! [B2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B2 @ A2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_682_inf_Oabsorb2,axiom,
! [B2: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B2 @ A2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_683_inf__absorb1,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_684_inf__absorb1,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_685_inf__absorb2,axiom,
! [Y2: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ Y2 @ X2 )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_686_inf__absorb2,axiom,
! [Y2: set_lambda_lambda,X2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y2 @ X2 )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_687_inf_OboundedE,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B2 @ C ) )
=> ~ ( ( ord_le3064231726521830256lambda @ A2 @ B2 )
=> ~ ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_688_inf_OboundedE,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B2 @ C ) )
=> ~ ( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ~ ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_689_inf_OboundedI,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B2 )
=> ( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_690_inf_OboundedI,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_691_inf__greatest,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ X2 @ Z5 )
=> ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z5 ) ) ) ) ).
% inf_greatest
thf(fact_692_inf__greatest,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ X2 @ Z5 )
=> ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z5 ) ) ) ) ).
% inf_greatest
thf(fact_693_inf_Oorder__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A5: set_li7878742357367608272lambda,B6: set_li7878742357367608272lambda] :
( A5
= ( inf_in3354698378728726462lambda @ A5 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_694_inf_Oorder__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A5: set_lambda_lambda,B6: set_lambda_lambda] :
( A5
= ( inf_in1766959559283430712lambda @ A5 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_695_inf_Ocobounded1,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_696_inf_Ocobounded1,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_697_inf_Ocobounded2,axiom,
! [A2: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_698_inf_Ocobounded2,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_699_inf_Oabsorb__iff1,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A5: set_li7878742357367608272lambda,B6: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A5 @ B6 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_700_inf_Oabsorb__iff1,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A5: set_lambda_lambda,B6: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A5 @ B6 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_701_inf_Oabsorb__iff2,axiom,
( ord_le3064231726521830256lambda
= ( ^ [B6: set_li7878742357367608272lambda,A5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A5 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_702_inf_Oabsorb__iff2,axiom,
( ord_le1926335974945931754lambda
= ( ^ [B6: set_lambda_lambda,A5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A5 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_703_inf_OcoboundedI1,axiom,
! [A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_704_inf_OcoboundedI1,axiom,
! [A2: set_lambda_lambda,C: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_705_inf_OcoboundedI2,axiom,
! [B2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B2 @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_706_inf_OcoboundedI2,axiom,
! [B2: set_lambda_lambda,C: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B2 @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_707_confluence__single,axiom,
! [T: lambda_lambda,U2: 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 @ U2 )
=> ( ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ).
% confluence_single
thf(fact_708_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_709_Srcs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ T4 )
= ( insert_lambda_lambda @ ( paths_1651982994781619320lambda @ lambda_resid @ T4 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Srcs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_710_Ide__append__iff_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
& ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
& ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P
thf(fact_711_Arr__append__iff_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ U2 )
& ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P
thf(fact_712__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_713__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_714_Trgs_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ).
% Trgs.simps(3)
thf(fact_715__092_060Lambda_062_Ocong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T6 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T6 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T6 @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T6 @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_left(2)
thf(fact_716__092_060Lambda_062_Ocong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T6 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T6 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T6 @ U ) ) ) ).
% \<Lambda>.cong_subst_left(1)
thf(fact_717__092_060Lambda_062_Ocong__subst__right_I2_J,axiom,
! [U: lambda_lambda,U4: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U4 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U4 ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_right(2)
thf(fact_718__092_060Lambda_062_Ocong__subst__right_I1_J,axiom,
! [U: lambda_lambda,U4: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U4 ) ) ) ).
% \<Lambda>.cong_subst_right(1)
thf(fact_719__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_720__092_060Lambda_062_Ocon__imp__coinitial__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ? [A6: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A6 )
& ( con_lambda_lambda @ lambda_resid @ A6 @ T )
& ( con_lambda_lambda @ lambda_resid @ A6 @ U ) ) ) ).
% \<Lambda>.con_imp_coinitial_ax
thf(fact_721__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_722__092_060Lambda_062_Ocon__transitive__on__ide,axiom,
! [A2: lambda_lambda,B2: lambda_lambda,C: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ B2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ C )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ B2 )
=> ( ( con_lambda_lambda @ lambda_resid @ B2 @ C )
=> ( con_lambda_lambda @ lambda_resid @ A2 @ C ) ) ) ) ) ) ).
% \<Lambda>.con_transitive_on_ide
thf(fact_723__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_724__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_725__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_726__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_727__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_728__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_729__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_730__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_731__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_732__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_733__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_734__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_735__092_060Lambda_062_Otargets__are__con,axiom,
! [B2: lambda_lambda,T: lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ B2 @ B3 ) ) ) ).
% \<Lambda>.targets_are_con
thf(fact_736__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_737_Srcs__are__con,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( con_lambda_lambda @ lambda_resid @ A2 @ A3 ) ) ) ).
% Srcs_are_con
thf(fact_738__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_739_ex__un__Trg,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ? [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
& ! [Y: lambda_lambda] :
( ( member_lambda_lambda @ Y @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( Y = X ) ) ) ) ).
% ex_un_Trg
thf(fact_740_Trgs__are__con,axiom,
! [B2: lambda_lambda,T4: list_lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( ( member_lambda_lambda @ B3 @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( con_lambda_lambda @ lambda_resid @ B2 @ B3 ) ) ) ).
% Trgs_are_con
thf(fact_741__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_742__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_743__092_060Lambda_062_Otargets__con__closed,axiom,
! [B2: lambda_lambda,T: lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B3 )
=> ( ( con_lambda_lambda @ lambda_resid @ B2 @ B3 )
=> ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.targets_con_closed
thf(fact_744_Srcs__con__closed,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda,A3: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A3 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ A3 )
=> ( member_lambda_lambda @ A3 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_745__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_746__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_747_Trgs__con__closed,axiom,
! [B2: lambda_lambda,T4: list_lambda_lambda,B3: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B3 )
=> ( ( con_lambda_lambda @ lambda_resid @ B2 @ B3 )
=> ( member_lambda_lambda @ B3 @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Trgs_con_closed
thf(fact_748_Trgs_Osimps_I1_J,axiom,
( ( paths_7594662086398683701lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Trgs.simps(1)
thf(fact_749_Trgs__are__ide,axiom,
! [T4: list_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( collec621444948832755541lambda @ ( ide_lambda_lambda @ lambda_resid ) ) ) ).
% Trgs_are_ide
thf(fact_750_Arr__has__Trg,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
!= bot_bo1271416176687173302lambda ) ) ).
% Arr_has_Trg
thf(fact_751_Trgs__eqI,axiom,
! [T4: list_lambda_lambda,T8: list_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_7594662086398683701lambda @ lambda_resid @ T8 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T8 ) ) ) ).
% Trgs_eqI
thf(fact_752_Con__rec_I4_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) ) )
!= nil_lambda_lambda ) ) ) ) ) ).
% Con_rec(4)
thf(fact_753_Con__rec_I3_J,axiom,
! [U2: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ U2 ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ).
% Con_rec(3)
thf(fact_754_Con__rec_I2_J,axiom,
! [T4: list_lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
= ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ ( lambda_resid @ U @ T ) @ nil_lambda_lambda ) )
!= nil_lambda_lambda ) ) ) ) ).
% Con_rec(2)
thf(fact_755_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_756_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_757_Trgs__Resid__sym__Arr__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T4 ) ) ) ) ).
% Trgs_Resid_sym_Arr_single
thf(fact_758_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_759_seq__implies__Trgs__eq__Srcs,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ) ).
% seq_implies_Trgs_eq_Srcs
thf(fact_760__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_761_Srcs__Resid__single__Arr,axiom,
! [U: lambda_lambda,T4: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T4 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ T4 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ).
% Srcs_Resid_single_Arr
thf(fact_762_Trgs_Oelims,axiom,
! [X2: list_lambda_lambda,Y2: set_lambda_lambda] :
( ( ( paths_7594662086398683701lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != bot_bo1271416176687173302lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
!= ( target5490931411331770124lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_7594662086398683701lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Trgs.elims
thf(fact_763_Arr__appendE_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
!= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P
thf(fact_764__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_765__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_766_Trgs__Resid__sym,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) ) ) ).
% Trgs_Resid_sym
thf(fact_767_Trgs__append,axiom,
! [U2: list_lambda_lambda,T4: list_lambda_lambda] :
( ( U2 != nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ U2 ) ) ) ).
% Trgs_append
thf(fact_768__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_769__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_770_Srcs__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_7594662086398683701lambda @ lambda_resid @ U2 ) ) ) ).
% Srcs_Resid
thf(fact_771_Arr__appendI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Arr_appendI\<^sub>P
thf(fact_772_Ide__appendI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Ide_appendI\<^sub>P
thf(fact_773_rts_Ojoinable_Ocong,axiom,
joinab677607494065406278lambda = joinab677607494065406278lambda ).
% rts.joinable.cong
thf(fact_774_rts_Ojoinable_Ocong,axiom,
joinab2519788767067900224lambda = joinab2519788767067900224lambda ).
% rts.joinable.cong
thf(fact_775_residuation_Ocon_Ocong,axiom,
con_lambda_lambda = con_lambda_lambda ).
% residuation.con.cong
thf(fact_776_residuation_Ocon_Ocong,axiom,
con_li79537494059641180lambda = con_li79537494059641180lambda ).
% residuation.con.cong
thf(fact_777_paths__in__rts_OTrgs_Ocong,axiom,
paths_7594662086398683701lambda = paths_7594662086398683701lambda ).
% paths_in_rts.Trgs.cong
thf(fact_778_paths__in__weakly__extensional__rts_OSrc_Ocong,axiom,
paths_1651982994781619320lambda = paths_1651982994781619320lambda ).
% paths_in_weakly_extensional_rts.Src.cong
thf(fact_779_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_780_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_781_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U2: list_l5168835686021156086lambda] :
( ( paths_3043240146134862789lambda @ Resid )
=> ( ! [T3: list_lambda_lambda,U5: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ Resid @ T3 @ U5 )
=> ( con_li79537494059641180lambda @ Resid @ T3 @ U5 ) )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( paths_2015741907369688696lambda @ Resid @ U2 )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( paths_3172353640142224310lambda @ Resid @ U2 ) )
=> ( ( paths_3312059784463760222lambda @ Resid @ ( cons_l4558873032622161958lambda @ T @ nil_li8379291187143545974lambda ) @ U2 )
!= nil_li8379291187143545974lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_782_paths__in__confluent__rts_Oconfluence__single,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( coinit1392077014771647542lambda @ Resid @ T3 @ U5 )
=> ( con_lambda_lambda @ Resid @ T3 @ U5 ) )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( paths_366302053746739058lambda @ Resid @ U2 )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( paths_2897053694880971696lambda @ Resid @ U2 ) )
=> ( ( paths_8607357831768207192lambda @ Resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda ) ) ) ) ) ) ).
% paths_in_confluent_rts.confluence_single
thf(fact_783_path__from__NF__is__Ide,axiom,
! [A2: lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_NF @ A2 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_1651982994781619320lambda @ lambda_resid @ U2 )
= A2 )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U2 ) ) ) ) ).
% path_from_NF_is_Ide
thf(fact_784_Trgs__complete__development,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct7282684938351565978opment @ T @ U2 )
=> ( ~ ( lambda_Ide @ T )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ U2 )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% Trgs_complete_development
thf(fact_785_seq__char_H,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ U2 )
& ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
!= bot_bo1271416176687173302lambda ) ) ) ).
% seq_char'
thf(fact_786_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_787_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_788__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_789__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_790__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_791_con__char,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
= ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda ) ) ).
% con_char
thf(fact_792_cong__subst__left_I2_J,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_793_cong__subst__left_I1_J,axiom,
! [T: list_lambda_lambda,T6: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_794_cong__subst__right_I2_J,axiom,
! [U: list_lambda_lambda,U4: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U4 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U4 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_795_cong__subst__right_I1_J,axiom,
! [U: list_lambda_lambda,U4: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U4 ) ) ) ).
% cong_subst_right(1)
thf(fact_796_con__imp__coinitial__ax,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ? [A6: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A6 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A6 @ T )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A6 @ U ) ) ) ).
% con_imp_coinitial_ax
thf(fact_797_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_798_con__transitive__on__ide,axiom,
! [A2: list_lambda_lambda,B2: list_lambda_lambda,C: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ C )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ B2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 @ C )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ C ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_targets__are__con,axiom,
! [B2: list_lambda_lambda,T: list_lambda_lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 @ B3 ) ) ) ).
% targets_are_con
thf(fact_807_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_808_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_809_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_810_arrE,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).
% arrE
thf(fact_811_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_812_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_813_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_814_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_815_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_816_confluence_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ).
% confluence\<^sub>P
thf(fact_817_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_818_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_819_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_820_targets__con__closed,axiom,
! [B2: list_lambda_lambda,T: list_lambda_lambda,B3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 @ B3 )
=> ( member2667661554319814297lambda @ B3 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% targets_con_closed
thf(fact_821_cong__respects__seq,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,T6: list_lambda_lambda,U4: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T6 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T6 @ U4 ) ) ) ) ).
% cong_respects_seq
thf(fact_822_cong__respects__seq_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T8: list_lambda_lambda,U6: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T8 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U6 @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T8 @ U6 ) ) ) ) ).
% cong_respects_seq\<^sub>P
thf(fact_823_dual__order_Orefl,axiom,
! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_824_order__refl,axiom,
! [X2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ X2 @ X2 ) ).
% order_refl
thf(fact_825_composable__iff__seq,axiom,
! [G: list_lambda_lambda,F: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F )
= ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F ) ) ).
% composable_iff_seq
thf(fact_826_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_827_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_828_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_829_arr__append__imp__seq,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% arr_append_imp_seq
thf(fact_830_seq__char,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ U2 )
& ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) ) ) ) ).
% seq_char
thf(fact_831_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_832_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_833_conI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ).
% conI\<^sub>P
thf(fact_834_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_835_arrI,axiom,
! [T: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% arrI
thf(fact_836_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_837_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_838_conI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
=> ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( ( lambda_Src @ ( hd_lambda_lambda @ T4 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% conI\<^sub>\<Lambda>\<^sub>P
thf(fact_839_cong__append__ideI_I4_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ T4 @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ T4 ) ) ) ) ) ).
% cong_append_ideI(4)
thf(fact_840_cong__append__ideI_I3_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( append_lambda_lambda @ T4 @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ U2 ) ) ) ) ) ).
% cong_append_ideI(3)
thf(fact_841_cong__append__ideI_I2_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ T4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ).
% cong_append_ideI(2)
thf(fact_842_cong__append__ideI_I1_J,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ).
% cong_append_ideI(1)
thf(fact_843_cong__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T8: list_lambda_lambda,U6: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ T8 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T8 @ T4 ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U6 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ T8 @ U6 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T8 @ U6 ) @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ) ).
% cong_append
thf(fact_844_sources__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% sources_append
thf(fact_845_targets__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U2 ) ) ) ).
% targets_append
thf(fact_846_arr__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ).
% arr_append
thf(fact_847_Src__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_1651982994781619320lambda @ lambda_resid @ T4 ) ) ) ).
% Src_append
thf(fact_848_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_849_cong__cons,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,T6: lambda_lambda,U6: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T6 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T6 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U6 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U6 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ ( cons_lambda_lambda @ T6 @ U6 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T6 @ U6 ) @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ) ).
% cong_cons
thf(fact_850_cong__cons__ideI_I1_J,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ).
% cong_cons_ideI(1)
thf(fact_851_cong__cons__ideI_I2_J,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ U2 ) ) ) ) ) ).
% cong_cons_ideI(2)
thf(fact_852_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_853_seqI_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( inf_in1766959559283430712lambda @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) @ ( paths_2897053694880971696lambda @ lambda_resid @ U2 ) )
!= bot_bo1271416176687173302lambda )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% seqI\<^sub>P
thf(fact_854_rts_Oseq_Ocong,axiom,
seq_li8093550137657879445lambda = seq_li8093550137657879445lambda ).
% rts.seq.cong
thf(fact_855_rts_Oseq_Ocong,axiom,
seq_lambda_lambda = seq_lambda_lambda ).
% rts.seq.cong
thf(fact_856_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_857_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_858_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_859_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,G: lambda_lambda,F: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( compos7127019122090555255lambda @ Resid @ G @ F )
= ( seq_lambda_lambda @ Resid @ G @ F ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_860_rts__with__composites_Ocomposable__iff__seq,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda,F: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( compos3359172601056258045lambda @ Resid @ G @ F )
= ( seq_li8093550137657879445lambda @ Resid @ G @ F ) ) ) ).
% rts_with_composites.composable_iff_seq
thf(fact_861_order__antisym__conv,axiom,
! [Y2: set_lambda_lambda,X2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y2 @ X2 )
=> ( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_862_ord__le__eq__subst,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_863_ord__eq__le__subst,axiom,
! [A2: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le1926335974945931754lambda @ B2 @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_864_order__eq__refl,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( X2 = Y2 )
=> ( ord_le1926335974945931754lambda @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_865_order__subst2,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( ( ord_le1926335974945931754lambda @ ( F @ B2 ) @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_866_order__subst1,axiom,
! [A2: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( F @ B2 ) )
=> ( ( ord_le1926335974945931754lambda @ B2 @ C )
=> ( ! [X: set_lambda_lambda,Y3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y3 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_867_order__eq__iff,axiom,
( ( ^ [Y4: set_lambda_lambda,Z: set_lambda_lambda] : ( Y4 = Z ) )
= ( ^ [A5: set_lambda_lambda,B6: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A5 @ B6 )
& ( ord_le1926335974945931754lambda @ B6 @ A5 ) ) ) ) ).
% order_eq_iff
thf(fact_868_antisym,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( ( ord_le1926335974945931754lambda @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_869_dual__order_Otrans,axiom,
! [B2: set_lambda_lambda,A2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B2 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ C @ B2 )
=> ( ord_le1926335974945931754lambda @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_870_dual__order_Oantisym,axiom,
! [B2: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B2 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_871_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_lambda_lambda,Z: set_lambda_lambda] : ( Y4 = Z ) )
= ( ^ [A5: set_lambda_lambda,B6: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B6 @ A5 )
& ( ord_le1926335974945931754lambda @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_872_order__trans,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ Y2 @ Z5 )
=> ( ord_le1926335974945931754lambda @ X2 @ Z5 ) ) ) ).
% order_trans
thf(fact_873_order_Otrans,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( ( ord_le1926335974945931754lambda @ B2 @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% order.trans
thf(fact_874_order__antisym,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_875_ord__le__eq__trans,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_876_ord__eq__le__trans,axiom,
! [A2: set_lambda_lambda,B2: set_lambda_lambda,C: set_lambda_lambda] :
( ( A2 = B2 )
=> ( ( ord_le1926335974945931754lambda @ B2 @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_877_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_lambda_lambda,Z: set_lambda_lambda] : ( Y4 = Z ) )
= ( ^ [X3: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X3 @ Y5 )
& ( ord_le1926335974945931754lambda @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_878_paths__in__confluent__rts_Oconfluence_092_060_094sub_062P,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_9040249923598999359lambda @ Resid )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ Resid ) @ T4 @ U2 ) ) ) ).
% paths_in_confluent_rts.confluence\<^sub>P
thf(fact_879_lambda__calculus_Onot__NF__elim,axiom,
! [T: lambda_lambda] :
( ~ ( lambda_NF @ T )
=> ( ( lambda_Ide @ T )
=> ~ ! [U5: lambda_lambda] :
~ ( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U5 )
& ~ ( lambda_Ide @ U5 ) ) ) ) ).
% lambda_calculus.not_NF_elim
thf(fact_880_bot_Oextremum__uniqueI,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
=> ( A2 = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_881_bot_Oextremum__uniqueI,axiom,
! [A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
=> ( A2 = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_882_bot_Oextremum__unique,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_unique
thf(fact_883_bot_Oextremum__unique,axiom,
! [A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_unique
thf(fact_884_bot_Oextremum,axiom,
! [A2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A2 ) ).
% bot.extremum
thf(fact_885_bot_Oextremum,axiom,
! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A2 ) ).
% bot.extremum
thf(fact_886_seq__Lam__Arr__implies,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
=> ( ( lambda_is_Lam @ T )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) ) ) ) ).
% seq_Lam_Arr_implies
thf(fact_887_orthogonal__App__single__Arr,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ T ) ) @ U2 ) @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) @ nil_lambda_lambda ) )
= ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ T ) ) @ U2 ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) @ nil_lambda_lambda ) @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ T ) ) @ U2 ) )
= ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) ) @ nil_lambda_lambda ) ) ) ) ) ).
% orthogonal_App_single_Arr
thf(fact_888_in__targets__iff,axiom,
! [T4: list_lambda_lambda,B: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ B ) ) ) ) ) ).
% in_targets_iff
thf(fact_889_seq__imp__cong__Trg__last__Src__hd,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) ) ) ) ).
% seq_imp_cong_Trg_last_Src_hd
thf(fact_890__092_060Lambda_062_Ocong__respects__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T6 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T6 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U4 @ U ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T6 @ U4 ) ) ) ) ).
% \<Lambda>.cong_respects_seq
thf(fact_891__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_892_Arr__imp__arr__last,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( arr_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T4 ) ) ) ).
% Arr_imp_arr_last
thf(fact_893_Ide__imp__Ide__last,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ide_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T4 ) ) ) ).
% Ide_imp_Ide_last
thf(fact_894__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_895__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_896_Trg__last__eqI,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ T4 ) )
= ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) ) ) ).
% Trg_last_eqI
thf(fact_897_Trgs__simp_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( target5490931411331770124lambda @ lambda_resid @ ( last_lambda_lambda @ T4 ) ) ) ) ).
% Trgs_simp\<^sub>P
thf(fact_898_Trg__last__Src__hd__eqI,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ T4 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) ) ) ).
% Trg_last_Src_hd_eqI
thf(fact_899_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_900_single__Trg__last__in__targets,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( member2667661554319814297lambda @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) ) ) ).
% single_Trg_last_in_targets
thf(fact_901_last__appendL,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Ys = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_appendL
thf(fact_902_last__appendR,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( Ys != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Ys ) ) ) ).
% last_appendR
thf(fact_903_last__snoc,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] :
( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) ) )
= X2 ) ).
% last_snoc
thf(fact_904__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_905_seqI_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( lambda_Trg @ ( last_lambda_lambda @ T4 ) )
= ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 ) ) ) ) ).
% seqI\<^sub>\<Lambda>\<^sub>P
thf(fact_906_lambda__calculus_Olambda_Odisc_I10_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
~ ( lambda_is_Lam @ ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.disc(10)
thf(fact_907_lambda__calculus_Olambda_Odisc_I9_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
~ ( lambda_is_Lam @ ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.disc(9)
thf(fact_908_last_Osimps,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] :
( ( ( Xs = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ) ).
% last.simps
thf(fact_909_last__ConsL,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] :
( ( Xs = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_910_last__ConsR,axiom,
! [Xs: list_lambda_lambda,X2: lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( cons_lambda_lambda @ X2 @ Xs ) )
= ( last_lambda_lambda @ Xs ) ) ) ).
% last_ConsR
thf(fact_911_last__in__set,axiom,
! [As: list_l5168835686021156086lambda] :
( ( As != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( last_l1874196399739661071lambda @ As ) @ ( set_li3846194476672328965lambda @ As ) ) ) ).
% last_in_set
thf(fact_912_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_913_last__map,axiom,
! [Xs: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( map_la336867333220232516lambda @ F @ Xs ) )
= ( F @ ( last_lambda_lambda @ Xs ) ) ) ) ).
% last_map
thf(fact_914_last__append,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda] :
( ( ( Ys = nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Xs ) ) )
& ( ( Ys != nil_lambda_lambda )
=> ( ( last_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) )
= ( last_lambda_lambda @ Ys ) ) ) ) ).
% last_append
thf(fact_915_longest__common__suffix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ss: list_lambda_lambda,Xs3: list_lambda_lambda,Ys4: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Xs3 @ Ss ) )
& ( Ys
= ( append_lambda_lambda @ Ys4 @ Ss ) )
& ( ( Xs3 = nil_lambda_lambda )
| ( Ys4 = nil_lambda_lambda )
| ( ( last_lambda_lambda @ Xs3 )
!= ( last_lambda_lambda @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_916_hd__Nil__eq__last,axiom,
( ( hd_lambda_lambda @ nil_lambda_lambda )
= ( last_lambda_lambda @ nil_lambda_lambda ) ) ).
% hd_Nil_eq_last
thf(fact_917_lambda__calculus_ONF__App__iff,axiom,
! [T1: lambda_lambda,T22: lambda_lambda] :
( ( lambda_NF @ ( lambda_App @ T1 @ T22 ) )
= ( ~ ( lambda_is_Lam @ T1 )
& ( lambda_NF @ T1 )
& ( lambda_NF @ T22 ) ) ) ).
% lambda_calculus.NF_App_iff
thf(fact_918_Arr__map__un__Lam,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_Lam @ T4 ) ) ) ) ).
% Arr_map_un_Lam
thf(fact_919_Std__map__un__Lam,axiom,
! [T4: list_lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ lambda_is_Lam ) )
=> ( reduction_Std @ ( map_la336867333220232516lambda @ lambda_un_Lam @ T4 ) ) ) ) ).
% Std_map_un_Lam
thf(fact_920_Ide__appendE,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ~ ( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
!= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ) ).
% Ide_appendE
thf(fact_921_Trg_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ).
% Trg.simps(3)
thf(fact_922_Trg__resid__sym,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) ) ) ).
% Trg_resid_sym
thf(fact_923_Ide__imp__Src__eq__Trg,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ T4 )
= ( paths_2695323870737867955lambda @ lambda_resid @ T4 ) ) ) ).
% Ide_imp_Src_eq_Trg
thf(fact_924_Arr__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( paths_366302053746739058lambda @ lambda_resid @ U2 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Arr_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_925_Arr__appendE_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ~ ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
!= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ) ).
% Arr_appendE\<^sub>P\<^sub>W\<^sub>E
thf(fact_926_Ide__append__iff_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
& ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Ide_append_iff\<^sub>P\<^sub>W\<^sub>E
thf(fact_927_Src__resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_1651982994781619320lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ U2 ) ) ) ).
% Src_resid
thf(fact_928_Trg__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) )
= ( paths_2695323870737867955lambda @ lambda_resid @ U2 ) ) ) ).
% Trg_append
thf(fact_929_Arr__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Arr_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_930_Ide__appendI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Ide_appendI\<^sub>P\<^sub>W\<^sub>E
thf(fact_931_Trgs__simp_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ T4 )
= ( insert_lambda_lambda @ ( paths_2695323870737867955lambda @ lambda_resid @ T4 ) @ bot_bo1271416176687173302lambda ) ) ) ).
% Trgs_simp\<^sub>P\<^sub>W\<^sub>E
thf(fact_932_paths__in__weakly__extensional__rts_OTrg_Ocong,axiom,
paths_2695323870737867955lambda = paths_2695323870737867955lambda ).
% paths_in_weakly_extensional_rts.Trg.cong
thf(fact_933_lambda__calculus_Oelementary__reduction__Lam__iff,axiom,
! [T: lambda_lambda] :
( ( lambda_is_Lam @ T )
=> ( ( lambda2670285437103676873uction @ T )
= ( lambda2670285437103676873uction @ ( lambda_un_Lam @ T ) ) ) ) ).
% lambda_calculus.elementary_reduction_Lam_iff
thf(fact_934_red__iff,axiom,
( lambda_red
= ( ^ [A5: lambda_lambda,B6: lambda_lambda] :
? [T7: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T7 )
& ( ( paths_1651982994781619320lambda @ lambda_resid @ T7 )
= A5 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T7 )
= B6 ) ) ) ) ).
% red_iff
thf(fact_935_seq__map__un__Lam,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Lam @ T ) @ nil_lambda_lambda ) @ U2 )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( map_la336867333220232516lambda @ lambda_un_Lam @ U2 ) ) ) ).
% seq_map_un_Lam
thf(fact_936_Std__imp__sseq__last__hd,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( reduction_Std @ ( append_lambda_lambda @ T4 @ U2 ) )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( lambda_sseq @ ( last_lambda_lambda @ T4 ) @ ( hd_lambda_lambda @ U2 ) ) ) ) ) ).
% Std_imp_sseq_last_hd
thf(fact_937_Std__map__Lam,axiom,
! [T4: list_lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( reduction_Std @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) ) ) ).
% Std_map_Lam
thf(fact_938_development__map__Lam,axiom,
! [T: lambda_lambda,T4: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ T4 )
=> ( reduct9208763521642324371opment @ ( lambda_Lam @ T ) @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) ) ) ).
% development_map_Lam
thf(fact_939_Arr__map__Lam,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) ) ) ).
% Arr_map_Lam
thf(fact_940_lambda__calculus_Olambda_Oinject_I2_J,axiom,
! [X32: lambda_lambda,Y32: lambda_lambda] :
( ( ( lambda_Lam @ X32 )
= ( lambda_Lam @ Y32 ) )
= ( X32 = Y32 ) ) ).
% lambda_calculus.lambda.inject(2)
thf(fact_941_cong__map__Lam,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) @ ( map_la336867333220232516lambda @ lambda_Lam @ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ U2 ) @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) ) ) ) ) ).
% cong_map_Lam
thf(fact_942_map__Lam__Resid,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda @ lambda_Lam @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_Lam @ T4 ) @ ( map_la336867333220232516lambda @ lambda_Lam @ U2 ) ) ) ) ).
% map_Lam_Resid
thf(fact_943_Std__append__single,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( ( T4 != nil_lambda_lambda )
=> ( ( lambda_sseq @ ( last_lambda_lambda @ T4 ) @ U )
=> ( reduction_Std @ ( append_lambda_lambda @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ) ) ).
% Std_append_single
thf(fact_944_Std__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( reduction_Std @ T4 )
=> ( ( reduction_Std @ U2 )
=> ( ( ( T4 = nil_lambda_lambda )
| ( U2 = nil_lambda_lambda )
| ( lambda_sseq @ ( last_lambda_lambda @ T4 ) @ ( hd_lambda_lambda @ U2 ) ) )
=> ( reduction_Std @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ).
% Std_append
thf(fact_945_lambda__calculus_Olambda_Ocollapse_I2_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ( ( lambda_Lam @ ( lambda_un_Lam @ Lambda ) )
= Lambda ) ) ).
% lambda_calculus.lambda.collapse(2)
thf(fact_946_lambda__calculus_Osseq__BetaI,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( ( lambda_Ide @ U )
=> ( ( lambda2670285437103676873uction @ V )
=> ( ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ V )
=> ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ V ) ) ) ) ) ).
% lambda_calculus.sseq_BetaI
thf(fact_947_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_948_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_949_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_950_lambda__calculus_Olambda_Osel_I2_J,axiom,
! [X32: lambda_lambda] :
( ( lambda_un_Lam @ ( lambda_Lam @ X32 ) )
= X32 ) ).
% lambda_calculus.lambda.sel(2)
thf(fact_951_lambda__calculus_ONF__unique,axiom,
! [T: lambda_lambda,U: lambda_lambda,U4: lambda_lambda] :
( ( lambda_red @ T @ U )
=> ( ( lambda_red @ T @ U4 )
=> ( ( lambda_NF @ U )
=> ( ( lambda_NF @ U4 )
=> ( U = U4 ) ) ) ) ) ).
% lambda_calculus.NF_unique
thf(fact_952_lambda__calculus_ONF__reduct__is__trivial,axiom,
! [A2: lambda_lambda,B2: lambda_lambda] :
( ( lambda_NF @ A2 )
=> ( ( lambda_red @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% lambda_calculus.NF_reduct_is_trivial
thf(fact_953_lambda__calculus_Osseq_Osimps_I33_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va2 ) ) ).
% lambda_calculus.sseq.simps(33)
thf(fact_954_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_955_lambda__calculus_ONF__Lam__iff,axiom,
! [T: lambda_lambda] :
( ( lambda_NF @ ( lambda_Lam @ T ) )
= ( lambda_NF @ T ) ) ).
% lambda_calculus.NF_Lam_iff
thf(fact_956_lambda__calculus_Osseq_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( lambda_Beta @ T6 @ U4 ) ) ).
% lambda_calculus.sseq.simps(5)
thf(fact_957_lambda__calculus_Oweak__diamond,axiom,
! [A2: lambda_lambda,B2: lambda_lambda,B3: lambda_lambda] :
( ( lambda_red @ A2 @ B2 )
=> ( ( lambda_red @ A2 @ B3 )
=> ~ ! [C4: lambda_lambda] :
( ( lambda_red @ B2 @ C4 )
=> ~ ( lambda_red @ B3 @ C4 ) ) ) ) ).
% lambda_calculus.weak_diamond
thf(fact_958_lambda__calculus_Osseq_Osimps_I3_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T6 ) )
= ( lambda_sseq @ T @ T6 ) ) ).
% lambda_calculus.sseq.simps(3)
thf(fact_959_lambda__calculus_Ored_Ointros_I2_J,axiom,
! [A2: lambda_lambda,B2: lambda_lambda,C: lambda_lambda] :
( ( lambda_red @ A2 @ B2 )
=> ( ( lambda_red @ B2 @ C )
=> ( lambda_red @ A2 @ C ) ) ) ).
% lambda_calculus.red.intros(2)
thf(fact_960_lambda__calculus_Osseq_Osimps_I30_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va2 ) ) ).
% lambda_calculus.sseq.simps(30)
thf(fact_961_lambda__calculus_Osseq_Osimps_I23_J,axiom,
! [Vc: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Lam @ Vc ) @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(23)
thf(fact_962_lambda__calculus_Osseq_Osimps_I24_J,axiom,
! [Vc: lambda_lambda,Vd: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vc @ Vd ) @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(24)
thf(fact_963_lambda__calculus_Osseq_Osimps_I27_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va2 ) ) ).
% lambda_calculus.sseq.simps(27)
thf(fact_964_lambda__calculus_Oelementary__reduction_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda2670285437103676873uction @ ( lambda_Lam @ T ) )
= ( lambda2670285437103676873uction @ T ) ) ).
% lambda_calculus.elementary_reduction.simps(3)
thf(fact_965_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_966_lambda__calculus_Osseq__imp__elementary__reduction1,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_sseq @ T @ U )
=> ( lambda2670285437103676873uction @ T ) ) ).
% lambda_calculus.sseq_imp_elementary_reduction1
thf(fact_967_lambda__calculus_Osseq__imp__elementary__reduction2,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_sseq @ T @ U )
=> ( lambda2670285437103676873uction @ U ) ) ).
% lambda_calculus.sseq_imp_elementary_reduction2
thf(fact_968_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_969_lambda__calculus_Olambda_Odistinct_I17_J,axiom,
! [X32: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Lam @ X32 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(17)
thf(fact_970_lambda__calculus_Olambda_Odistinct_I15_J,axiom,
! [X32: lambda_lambda,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Lam @ X32 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(15)
thf(fact_971_lambda__calculus_Olambda_Odisc_I8_J,axiom,
! [X32: lambda_lambda] : ( lambda_is_Lam @ ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.disc(8)
thf(fact_972_lambda__calculus_Olambda_OdiscI_I2_J,axiom,
! [Lambda: lambda_lambda,X32: lambda_lambda] :
( ( Lambda
= ( lambda_Lam @ X32 ) )
=> ( lambda_is_Lam @ Lambda ) ) ).
% lambda_calculus.lambda.discI(2)
thf(fact_973_lambda__calculus_Ois__Lam__def,axiom,
( lambda_is_Lam
= ( ^ [Lambda2: lambda_lambda] :
? [X33: lambda_lambda] :
( Lambda2
= ( lambda_Lam @ X33 ) ) ) ) ).
% lambda_calculus.is_Lam_def
thf(fact_974_lambda__calculus_Osseq__imp__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_sseq @ T @ U )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ).
% lambda_calculus.sseq_imp_seq
thf(fact_975_lambda__calculus_Osseq_Osimps_I10_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ ( lambda_Lam @ Va2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Lam @ Va2 ) ) ) ) ).
% lambda_calculus.sseq.simps(10)
thf(fact_976_lambda__calculus_Osseq_Osimps_I7_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_sseq @ ( lambda_App @ ( lambda_Beta @ T @ U ) @ V ) @ ( lambda_Lam @ Va2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U )
& ( lambda_Ide @ V )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T @ U ) @ V ) @ ( lambda_Lam @ Va2 ) ) ) ) ).
% lambda_calculus.sseq.simps(7)
thf(fact_977_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_978_lambda__calculus_OSrc__eq__iff_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T6 @ U4 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T6 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U4 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(4)
thf(fact_979_reduction__paths_OStd_Osimps_I3_J,axiom,
! [T: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( ( lambda_sseq @ T @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V @ Va2 ) ) )
& ( reduction_Std @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ) ).
% reduction_paths.Std.simps(3)
thf(fact_980_reduction__paths_Ostdz__insert_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Uv: lambda_lambda,Uw: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct5390047737487997688insert @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( cons_lambda_lambda @ ( lambda_Beta @ Uv @ Uw ) @ U2 ) )
= ( reduct5390047737487997688insert @ ( lambda_Beta @ T @ U ) @ U2 ) ) ).
% reduction_paths.stdz_insert.simps(4)
thf(fact_981_lambda__calculus_Oelementary__reduction_Oelims_I2_J,axiom,
! [X2: lambda_lambda] :
( ( lambda2670285437103676873uction @ X2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda2670285437103676873uction @ T3 ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_App @ T3 @ U5 ) )
=> ~ ( ( ( lambda2670285437103676873uction @ T3 )
& ( lambda_Ide @ U5 ) )
| ( ( lambda_Ide @ T3 )
& ( lambda2670285437103676873uction @ U5 ) ) ) )
=> ~ ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 ) ) ) ) ) ) ).
% lambda_calculus.elementary_reduction.elims(2)
thf(fact_982_lambda__calculus_Oprfx__Lam__iff,axiom,
! [U: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Lam @ T ) ) )
= ( ( lambda_is_Lam @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_Lam @ U ) @ T ) ) ) ) ).
% lambda_calculus.prfx_Lam_iff
thf(fact_983_lambda__calculus_Osseq_Osimps_I12_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ Va2 @ Vb ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va2 @ Vb ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ Va2 @ Vb ) ) ) ) ).
% lambda_calculus.sseq.simps(12)
thf(fact_984_lambda__calculus_Osseq__Beta,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ V )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U )
& ( lambda2670285437103676873uction @ V )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ V ) ) ) ).
% lambda_calculus.sseq_Beta
thf(fact_985_lambda__calculus_Osseq_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( lambda_sseq @ ( lambda_App @ T @ U ) @ ( lambda_App @ T6 @ U4 ) )
= ( ( ( lambda_sseq @ T @ T6 )
& ( lambda_Ide @ U )
& ( U = U4 ) )
| ( ( lambda_Ide @ T )
& ( T = T6 )
& ( lambda_sseq @ U @ U4 ) )
| ( ( lambda2670285437103676873uction @ T )
& ( ( lambda_Trg @ T )
= T6 )
& ( U
= ( lambda_Src @ U4 ) )
& ( lambda2670285437103676873uction @ U4 ) ) ) ) ).
% lambda_calculus.sseq.simps(4)
thf(fact_986_reduction__paths_OStd_Oelims_I3_J,axiom,
! [X2: list_lambda_lambda] :
( ~ ( reduction_Std @ X2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( lambda2670285437103676873uction @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( ( lambda_sseq @ T3 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V2 @ Va ) ) )
& ( reduction_Std @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ).
% reduction_paths.Std.elims(3)
thf(fact_987_reduction__paths_OStd_Oelims_I2_J,axiom,
! [X2: list_lambda_lambda] :
( ( reduction_Std @ X2 )
=> ( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( lambda2670285437103676873uction @ T3 ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( lambda_sseq @ T3 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V2 @ Va ) ) )
& ( reduction_Std @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% reduction_paths.Std.elims(2)
thf(fact_988_reduction__paths_OStd_Oelims_I1_J,axiom,
! [X2: list_lambda_lambda,Y2: $o] :
( ( ( reduction_Std @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ~ Y2 )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
= ( ~ ( lambda2670285437103676873uction @ T3 ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
= ( ~ ( ( lambda_sseq @ T3 @ ( hd_lambda_lambda @ ( cons_lambda_lambda @ V2 @ Va ) ) )
& ( reduction_Std @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ) ) ).
% reduction_paths.Std.elims(1)
thf(fact_989_lambda__calculus_Osseq_Osimps_I11_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ Va2 @ Vb ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U )
& ( lambda2670285437103676873uction @ ( lambda_App @ Va2 @ Vb ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ Va2 @ Vb ) ) ) ) ).
% lambda_calculus.sseq.simps(11)
thf(fact_990_lambda__calculus_Osseq_Osimps_I8_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_sseq @ ( lambda_App @ ( lambda_Beta @ T @ U ) @ V ) @ ( lambda_Beta @ Va2 @ Vb ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U )
& ( lambda_Ide @ V )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va2 @ Vb ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T @ U ) @ V ) @ ( lambda_Beta @ Va2 @ Vb ) ) ) ) ).
% lambda_calculus.sseq.simps(8)
thf(fact_991_lambda__calculus_Onormalizable__def,axiom,
( lambda_normalizable
= ( ^ [A5: lambda_lambda] :
( ( lambda_Ide @ A5 )
& ? [B6: lambda_lambda] :
( ( lambda_red @ A5 @ B6 )
& ( lambda_NF @ B6 ) ) ) ) ) ).
% lambda_calculus.normalizable_def
thf(fact_992_Std__consE,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduction_Std @ ( cons_lambda_lambda @ T @ U2 ) )
=> ~ ( ( lambda_Arr @ T )
=> ( ( ( U2 != nil_lambda_lambda )
=> ( lambda_sseq @ T @ ( hd_lambda_lambda @ U2 ) ) )
=> ~ ( reduction_Std @ U2 ) ) ) ) ).
% Std_consE
thf(fact_993_Ide__consE,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( ( U2 != nil_lambda_lambda )
=> ( paths_7717490669901197737lambda @ lambda_resid @ U2 ) )
=> ~ ( ( U2 != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Ide_consE
thf(fact_994__092_060Lambda_062_Otrg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_resid @ T @ T ) ) ).
% \<Lambda>.trg_def
thf(fact_995__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_996_development__imp__Arr,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U2 )
=> ( lambda_Arr @ T ) ) ).
% development_imp_Arr
thf(fact_997__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_998__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_999__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_1000__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_1001__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_1002__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_1003__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_1004__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_1005_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_1006__092_060Lambda_062_Oin__targetsE,axiom,
! [B2: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ B2 )
=> ~ ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B2 ) ) ) ).
% \<Lambda>.in_targetsE
thf(fact_1007_Arr__consE,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( U2 != nil_lambda_lambda )
=> ( paths_366302053746739058lambda @ lambda_resid @ U2 ) )
=> ~ ( ( U2 != nil_lambda_lambda )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) ) ) ) ) ) ).
% Arr_consE
thf(fact_1008_lambda__calculus_OSrc__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_Src @ T ) )
= ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Src_Src
thf(fact_1009_lambda__calculus_OTrg__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Trg @ ( lambda_Trg @ T ) )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Trg_Trg
thf(fact_1010__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_1011_lambda__calculus_Oresid__Arr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ ( lambda_Src @ T ) )
= T ) ) ).
% lambda_calculus.resid_Arr_Src
thf(fact_1012_lambda__calculus_Oarr__char,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.arr_char
thf(fact_1013_lambda__calculus_OTrg__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Trg @ ( lambda_Src @ T ) )
= ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Trg_Src
thf(fact_1014_lambda__calculus_OSrc__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_Trg @ T ) )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Src_Trg
thf(fact_1015__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_1016__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_1017_lambda__calculus_Oresid__Src__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ ( lambda_Src @ T ) @ T )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.resid_Src_Arr
thf(fact_1018_lambda__calculus_Ocoinitial__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Src @ T )
= ( lambda_Src @ U ) ) ) ) ).
% lambda_calculus.coinitial_char
thf(fact_1019__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_1020_lambda__calculus_OseqI_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Trg @ T )
= ( lambda_Src @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% lambda_calculus.seqI\<^sub>\<Lambda>
thf(fact_1021__092_060Lambda_062_Oin__targetsI,axiom,
! [B2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B2 )
=> ( ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B2 )
=> ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.in_targetsI
thf(fact_1022_Trg__simp,axiom,
! [T4: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( paths_2695323870737867955lambda @ lambda_resid @ T4 )
= ( trg_lambda_lambda @ lambda_resid @ ( last_lambda_lambda @ T4 ) ) ) ) ).
% Trg_simp
thf(fact_1023_Arr__consI_092_060_094sub_062P_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ).
% Arr_consI\<^sub>P\<^sub>W\<^sub>E
thf(fact_1024_Ide__consI,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
=> ( ( paths_7717490669901197737lambda @ lambda_resid @ U2 )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( paths_1651982994781619320lambda @ lambda_resid @ U2 ) )
=> ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ).
% Ide_consI
thf(fact_1025_lambda__calculus_Osources__simp,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.sources_simp
thf(fact_1026_lambda__calculus_Otargets__simp,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.targets_simp
thf(fact_1027_sources__single__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Src @ T ) @ nil_lambda_lambda ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% sources_single_Src
thf(fact_1028_targets__single__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Trg @ T ) @ nil_lambda_lambda ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% targets_single_Trg
thf(fact_1029_targets__single__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Src @ T ) @ nil_lambda_lambda ) )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% targets_single_Src
thf(fact_1030_sources__single__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Trg @ T ) @ nil_lambda_lambda ) )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ).
% sources_single_Trg
thf(fact_1031_complete__development__bottom__up__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( reduct7282684938351565978opment @ T @ ( reduct6613843825294780173opment @ T ) ) ) ).
% complete_development_bottom_up_development
thf(fact_1032_lambda__calculus_OArr_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Lam @ T ) )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.Arr.simps(3)
thf(fact_1033_lambda__calculus_OArr_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_App @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(4)
thf(fact_1034_lambda__calculus_OArr_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Beta @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(5)
thf(fact_1035_lambda__calculus_OIde__implies__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Ide_implies_Arr
thf(fact_1036_lambda__calculus_OArr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Arr_Src
thf(fact_1037_residuation_Otrg_Ocong,axiom,
trg_lambda_lambda = trg_lambda_lambda ).
% residuation.trg.cong
thf(fact_1038_residuation_Otrg_Ocong,axiom,
trg_li817795487149935061lambda = trg_li817795487149935061lambda ).
% residuation.trg.cong
thf(fact_1039_lambda__calculus_OArr__Trg,axiom,
! [X5: lambda_lambda] :
( ( lambda_Arr @ X5 )
=> ( lambda_Arr @ ( lambda_Trg @ X5 ) ) ) ).
% lambda_calculus.Arr_Trg
thf(fact_1040_lambda__calculus_Oresid__Arr__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_resid @ T @ T )
= ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.resid_Arr_self
thf(fact_1041_lambda__calculus_OIde__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Ide_Src
thf(fact_1042_lambda__calculus_OIde__iff__Src__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( lambda_Src @ T )
= T ) ) ) ).
% lambda_calculus.Ide_iff_Src_self
thf(fact_1043_lambda__calculus_OIde__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Ide_Trg
thf(fact_1044_lambda__calculus_OIde__iff__Trg__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( lambda_Trg @ T )
= T ) ) ) ).
% lambda_calculus.Ide_iff_Trg_self
thf(fact_1045_reduction__paths_Odevelopment_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ nil_lambda_lambda )
= ( lambda_Arr @ T ) ) ).
% reduction_paths.development.simps(1)
thf(fact_1046_lambda__calculus_Oresid__Ide__Arr,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( lambda_Ide @ A2 )
=> ( ( ( lambda_Arr @ A2 )
& ( lambda_Arr @ T )
& ( ( lambda_Src @ A2 )
= ( lambda_Src @ T ) ) )
=> ( lambda_Ide @ ( lambda_resid @ A2 @ T ) ) ) ) ).
% lambda_calculus.resid_Ide_Arr
thf(fact_1047_lambda__calculus_Oresid__Arr__Ide,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( lambda_Ide @ A2 )
=> ( ( ( lambda_Arr @ T )
& ( lambda_Arr @ A2 )
& ( ( lambda_Src @ T )
= ( lambda_Src @ A2 ) ) )
=> ( ( lambda_resid @ T @ A2 )
= T ) ) ) ).
% lambda_calculus.resid_Arr_Ide
thf(fact_1048_lambda__calculus_Ored_Ointros_I1_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_red @ ( lambda_Src @ T ) @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.red.intros(1)
thf(fact_1049_lambda__calculus_Ored_Ocases,axiom,
! [A1: lambda_lambda,A22: lambda_lambda] :
( ( lambda_red @ A1 @ A22 )
=> ( ! [T3: lambda_lambda] :
( ( A1
= ( lambda_Src @ T3 ) )
=> ( ( A22
= ( lambda_Trg @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) ) )
=> ~ ! [B7: lambda_lambda] :
( ( lambda_red @ A1 @ B7 )
=> ~ ( lambda_red @ B7 @ A22 ) ) ) ) ).
% lambda_calculus.red.cases
thf(fact_1050_lambda__calculus_Ored_Osimps,axiom,
( lambda_red
= ( ^ [A12: lambda_lambda,A23: lambda_lambda] :
( ? [T5: lambda_lambda] :
( ( A12
= ( lambda_Src @ T5 ) )
& ( A23
= ( lambda_Trg @ T5 ) )
& ( lambda_Arr @ T5 ) )
| ? [A5: lambda_lambda,B6: lambda_lambda,C5: lambda_lambda] :
( ( A12 = A5 )
& ( A23 = C5 )
& ( lambda_red @ A5 @ B6 )
& ( lambda_red @ B6 @ C5 ) ) ) ) ) ).
% lambda_calculus.red.simps
thf(fact_1051_lambda__calculus_ONF__def,axiom,
( lambda_NF
= ( ^ [A5: lambda_lambda] :
( ( lambda_Ide @ A5 )
& ! [T5: lambda_lambda] :
( ( ( lambda_Arr @ T5 )
& ( ( lambda_Src @ T5 )
= A5 ) )
=> ( lambda_Ide @ T5 ) ) ) ) ) ).
% lambda_calculus.NF_def
thf(fact_1052_lambda__calculus_Oseq__char,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U )
& ( ( lambda_Trg @ T )
= ( lambda_Src @ U ) ) ) ) ).
% lambda_calculus.seq_char
thf(fact_1053_lambda__calculus_OseqE_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ~ ( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( lambda_Trg @ T )
!= ( lambda_Src @ U ) ) ) ) ) ).
% lambda_calculus.seqE\<^sub>\<Lambda>
thf(fact_1054_lambda__calculus_Osources__char_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( lambda_Arr @ T )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.sources_char\<^sub>\<Lambda>
thf(fact_1055_lambda__calculus_Otargets__char_092_060_094sub_062_092_060Lambda_062,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( lambda_Arr @ T )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= bot_bo1271416176687173302lambda ) ) ) ).
% lambda_calculus.targets_char\<^sub>\<Lambda>
thf(fact_1056_reduction__paths_Oorthogonal__App__single__single_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ U ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ U ) ) @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Trg @ T ) @ U ) @ nil_lambda_lambda ) ) ) ) ).
% reduction_paths.orthogonal_App_single_single(1)
thf(fact_1057_reduction__paths_Oorthogonal__App__single__single_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Src @ U ) ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Src @ T ) @ U ) @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_App @ T @ ( lambda_Trg @ U ) ) @ nil_lambda_lambda ) ) ) ) ).
% reduction_paths.orthogonal_App_single_single(2)
thf(fact_1058_reduction__paths_OBeta__decomp_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ T @ U ) @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_Beta @ ( lambda_Trg @ T ) @ ( lambda_Trg @ U ) ) @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% reduction_paths.Beta_decomp(2)
thf(fact_1059_Trgs__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( ( reduct1603535164923377302opment @ T )
!= nil_lambda_lambda )
=> ( ( paths_7594662086398683701lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) )
= ( insert_lambda_lambda @ ( lambda_Trg @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% Trgs_standard_development
thf(fact_1060_Srcs__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( ( reduct1603535164923377302opment @ T )
!= nil_lambda_lambda )
=> ( ( paths_2897053694880971696lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) )
= ( insert_lambda_lambda @ ( lambda_Src @ T ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% Srcs_standard_development
thf(fact_1061_trg__def,axiom,
! [T: list_lambda_lambda] :
( ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ).
% trg_def
thf(fact_1062_Std__standard__development,axiom,
! [T: lambda_lambda] : ( reduction_Std @ ( reduct1603535164923377302opment @ T ) ) ).
% Std_standard_development
thf(fact_1063_development__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( reduct9208763521642324371opment @ T @ ( reduct1603535164923377302opment @ T ) ) ) ).
% development_standard_development
thf(fact_1064_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_1065_Ide__iff__standard__development__empty,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( reduct1603535164923377302opment @ T )
= nil_lambda_lambda ) ) ) ).
% Ide_iff_standard_development_empty
thf(fact_1066_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_1067_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_1068_Src__hd__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ~ ( lambda_Ide @ T )
=> ( ( lambda_Src @ ( hd_lambda_lambda @ ( reduct1603535164923377302opment @ T ) ) )
= ( lambda_Src @ T ) ) ) ) ).
% Src_hd_standard_development
thf(fact_1069_in__targetsE,axiom,
! [B2: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 )
=> ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B2 ) ) ) ).
% in_targetsE
thf(fact_1070_Trg__last__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ~ ( lambda_Ide @ T )
=> ( ( lambda_Trg @ ( last_lambda_lambda @ ( reduct1603535164923377302opment @ T ) ) )
= ( lambda_Trg @ T ) ) ) ) ).
% Trg_last_standard_development
thf(fact_1071_set__standard__development,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ ( reduct1603535164923377302opment @ T ) ) @ ( collec621444948832755541lambda @ lambda2670285437103676873uction ) ) ) ).
% set_standard_development
thf(fact_1072_cong__standard__development,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ~ ( lambda_Ide @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( reduct1603535164923377302opment @ T ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( reduct1603535164923377302opment @ T ) ) ) ) ) ).
% cong_standard_development
thf(fact_1073_in__targetsI,axiom,
! [B2: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B2 )
=> ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% in_targetsI
thf(fact_1074_reduction__paths_Ostandard__development_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( reduct1603535164923377302opment @ ( lambda_Lam @ T ) )
= ( map_la336867333220232516lambda @ lambda_Lam @ ( reduct1603535164923377302opment @ T ) ) ) ).
% reduction_paths.standard_development.simps(3)
thf(fact_1075_reduction__paths_Ostdz__insert_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( reduct5390047737487997688insert @ T @ nil_lambda_lambda )
= ( reduct1603535164923377302opment @ T ) ) ).
% reduction_paths.stdz_insert.simps(1)
thf(fact_1076_head__redex__decomp,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_head_redex @ T ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ T @ ( lambda_head_redex @ T ) ) @ nil_lambda_lambda ) ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ ( lambda_head_redex @ T ) @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ ( lambda_resid @ T @ ( lambda_head_redex @ T ) ) @ nil_lambda_lambda ) ) ) ) ) ) ).
% head_redex_decomp
thf(fact_1077_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_1078_null__eqI,axiom,
! [N: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ N @ T3 )
= N )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ N )
= N ) )
=> ( N
= ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ) ).
% null_eqI
thf(fact_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_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_1081_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_1082_null__char,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= nil_lambda_lambda ) ).
% null_char
thf(fact_1083_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_1084_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_1085_not__arr__null,axiom,
~ ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% not_arr_null
thf(fact_1086_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_1087_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_1088_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_1089_lambda__calculus_Ohead__redex_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_head_redex @ T ) ) ) ).
% lambda_calculus.head_redex.simps(3)
thf(fact_1090_lambda__calculus_Ohead__redex_Osimps_I9_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_Beta @ T @ U ) )
= ( lambda_Beta @ ( lambda_Src @ T ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(9)
thf(fact_1091_partial__magma_Onull__eqI,axiom,
! [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,N: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP )
=> ( ! [T3: list_lambda_lambda] :
( ( ( OP @ N @ T3 )
= N )
& ( ( OP @ T3 @ N )
= N ) )
=> ( N
= ( partia6873046715171720854lambda @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_1092_partial__magma_Onull__eqI,axiom,
! [OP: lambda_lambda > lambda_lambda > lambda_lambda,N: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP )
=> ( ! [T3: lambda_lambda] :
( ( ( OP @ N @ T3 )
= N )
& ( ( OP @ T3 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ OP ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_1093_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_1094_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_1095_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_1096_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_1097_partial__magma_Onull_Ocong,axiom,
partia6873046715171720854lambda = partia6873046715171720854lambda ).
% partial_magma.null.cong
thf(fact_1098_partial__magma_Onull_Ocong,axiom,
partia1461432607296028304lambda = partia1461432607296028304lambda ).
% partial_magma.null.cong
thf(fact_1099_lambda__calculus_Ohead__redex_Osimps_I7_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_App @ V @ Va2 ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_App @ V @ Va2 ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(7)
thf(fact_1100_lambda__calculus_OSrc__head__redex,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Src @ ( lambda_head_redex @ T ) )
= ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Src_head_redex
thf(fact_1101_lambda__calculus_OArr__head__redex,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_head_redex @ T ) ) ) ).
% lambda_calculus.Arr_head_redex
thf(fact_1102_lambda__calculus_Ohead__redex_Osimps_I8_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Beta @ V @ Va2 ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Beta @ V @ Va2 ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(8)
thf(fact_1103_lambda__calculus_Ohead__redex_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_App @ ( lambda_Lam @ ( lambda_Src @ T ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(4)
thf(fact_1104_lambda__calculus_Oelementary__reduction__head__redex,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ~ ( lambda_Ide @ ( lambda_head_redex @ T ) )
=> ( lambda2670285437103676873uction @ ( lambda_head_redex @ T ) ) ) ) ).
% lambda_calculus.elementary_reduction_head_redex
thf(fact_1105_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_1106_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_1107__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_1108__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_1109__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_1110__092_060Lambda_062_Onull__eqI,axiom,
! [N: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( lambda_resid @ N @ T3 )
= N )
& ( ( lambda_resid @ T3 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% \<Lambda>.null_eqI
thf(fact_1111__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_1112__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_1113__092_060Lambda_062_Onot__arr__null,axiom,
~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.not_arr_null
thf(fact_1114_Trg_Osimps_I1_J,axiom,
( ( paths_2695323870737867955lambda @ lambda_resid @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Trg.simps(1)
thf(fact_1115_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_1116_join__of__un__upto__cong,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ V4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ V ) ) ) ) ) ).
% join_of_un_upto_cong
thf(fact_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125__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_1126__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_1127_joinable__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ? [X6: list_lambda_lambda] : ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ X6 ) ) ) ).
% joinable_def
thf(fact_1128_Trg_Oelims,axiom,
! [X2: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_2695323870737867955lambda @ lambda_resid @ X2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ( Y2
!= ( trg_lambda_lambda @ lambda_resid @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_2695323870737867955lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Trg.elims
thf(fact_1129__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_1130_rts_Ojoin__of_Ocong,axiom,
join_o6956785509312998601lambda = join_o6956785509312998601lambda ).
% rts.join_of.cong
thf(fact_1131_rts_Ojoin__of_Ocong,axiom,
join_o1386658102433174083lambda = join_o1386658102433174083lambda ).
% rts.join_of.cong
thf(fact_1132_lambda__calculus_Osseq_Oelims_I2_J,axiom,
! [X2: lambda_lambda,Xa: lambda_lambda] :
( ( lambda_sseq @ X2 @ Xa )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( lambda_Lam @ T3 ) )
=> ! [T9: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T9 ) )
=> ~ ( lambda_sseq @ T3 @ T9 ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_App @ T3 @ U5 ) )
=> ! [T9: lambda_lambda,U7: lambda_lambda] :
( ( Xa
= ( lambda_App @ T9 @ U7 ) )
=> ~ ( ( ( lambda_sseq @ T3 @ T9 )
& ( lambda_Ide @ U5 )
& ( U5 = U7 ) )
| ( ( lambda_Ide @ T3 )
& ( T3 = T9 )
& ( lambda_sseq @ U5 @ U7 ) )
| ( ( lambda2670285437103676873uction @ T3 )
& ( ( lambda_Trg @ T3 )
= T9 )
& ( U5
= ( lambda_Src @ U7 ) )
& ( lambda2670285437103676873uction @ U7 ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: nat] :
( ( Xa
= ( lambda_Var @ Va ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Var @ Va ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ Va ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Lam @ Va ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ Va @ Vb2 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Beta @ Va @ Vb2 ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: nat] :
( ( Xa
= ( lambda_Var @ Va ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Var @ Va ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ Va ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Lam @ Va ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_App @ Va @ Vb2 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_App @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_App @ Va @ Vb2 ) ) ) ) )
=> ~ ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ Va @ Vb2 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Beta @ Va @ Vb2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.sseq.elims(2)
thf(fact_1133_Resid_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(4)
thf(fact_1134__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_1135__092_060Lambda_062_Ojoin__of__un__upto__cong,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V4: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V4 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ V4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V4 @ V ) ) ) ) ) ).
% \<Lambda>.join_of_un_upto_cong
thf(fact_1136__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_1137__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_1138__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_1139__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_1140__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_1141__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_1142__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_1143__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_1144__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_1145__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_1146__092_060Lambda_062_Ojoinable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
= ( ? [X6: lambda_lambda] : ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ X6 ) ) ) ).
% \<Lambda>.joinable_def
thf(fact_1147_Resid1x_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) ) ).
% Resid1x.simps(3)
thf(fact_1148__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_1149__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_1150_Resid1x__null,axiom,
! [T4: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T4 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x_null
thf(fact_1151_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_1152_Resid1x_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x.simps(1)
thf(fact_1153_Resid1x__ide,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ( paths_3319247187298047121lambda @ lambda_resid @ A2 @ T4 )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
=> ( ide_lambda_lambda @ lambda_resid @ ( paths_3319247187298047121lambda @ lambda_resid @ A2 @ T4 ) ) ) ) ).
% Resid1x_ide
thf(fact_1154_development__append,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda] :
( ( reduct9208763521642324371opment @ T @ U2 )
=> ( ( reduct9208763521642324371opment @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 ) @ V3 )
=> ( reduct9208763521642324371opment @ T @ ( append_lambda_lambda @ U2 @ V3 ) ) ) ) ).
% development_append
thf(fact_1155_lambda__calculus_Olambda_Oinject_I1_J,axiom,
! [X23: nat,Y23: nat] :
( ( ( lambda_Var @ X23 )
= ( lambda_Var @ Y23 ) )
= ( X23 = Y23 ) ) ).
% lambda_calculus.lambda.inject(1)
thf(fact_1156_Resid__cons_H,axiom,
! [T4: list_lambda_lambda,T: lambda_lambda,U2: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ T4 ) @ U2 )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) ) ) ) ) ) ) ).
% Resid_cons'
thf(fact_1157_Resid1x__as__Resid,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 ) @ nil_lambda_lambda ) ) ) ).
% Resid1x_as_Resid
thf(fact_1158_Resid1x_Oelims,axiom,
! [X2: lambda_lambda,Xa: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_3319247187298047121lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = nil_lambda_lambda )
=> ( Y2
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [U5: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) )
=> ( Y2
!= ( lambda_resid @ X2 @ U5 ) ) )
=> ~ ! [U5: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ( Y2
!= ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ X2 @ U5 ) @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ) ) ).
% Resid1x.elims
thf(fact_1159_Resid1x__as__Resid_H,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda] :
( ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
!= nil_lambda_lambda )
=> ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 )
= ( hd_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 ) ) ) )
& ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ U2 )
= nil_lambda_lambda )
=> ( ( paths_3319247187298047121lambda @ lambda_resid @ T @ U2 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ) ).
% Resid1x_as_Resid'
thf(fact_1160_lambda__calculus_Otargets__simps_I2_J,axiom,
! [X2: nat] :
( ( target5490931411331770124lambda @ lambda_resid @ ( lambda_Var @ X2 ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X2 ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.targets_simps(2)
thf(fact_1161_lambda__calculus_Osources__simps_I2_J,axiom,
! [X2: nat] :
( ( source5627170679743844886lambda @ lambda_resid @ ( lambda_Var @ X2 ) )
= ( insert_lambda_lambda @ ( lambda_Var @ X2 ) @ bot_bo1271416176687173302lambda ) ) ).
% lambda_calculus.sources_simps(2)
thf(fact_1162_reduction__paths_Ostandard__development_Osimps_I2_J,axiom,
! [Uu: nat] :
( ( reduct1603535164923377302opment @ ( lambda_Var @ Uu ) )
= nil_lambda_lambda ) ).
% reduction_paths.standard_development.simps(2)
thf(fact_1163_lambda__calculus_Olambda_Odistinct_I9_J,axiom,
! [X23: nat,X32: lambda_lambda] :
( ( lambda_Var @ X23 )
!= ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.distinct(9)
thf(fact_1164_lambda__calculus_Osseq_Osimps_I2_J,axiom,
! [Uv: nat,Uw: nat] :
~ ( lambda_sseq @ ( lambda_Var @ Uv ) @ ( lambda_Var @ Uw ) ) ).
% lambda_calculus.sseq.simps(2)
thf(fact_1165_lambda__calculus_Olambda_Odisc_I7_J,axiom,
! [X23: nat] :
~ ( lambda_is_Lam @ ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.disc(7)
thf(fact_1166_lambda__calculus_Olambda_Odistinct_I11_J,axiom,
! [X23: nat,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Var @ X23 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(11)
thf(fact_1167_lambda__calculus_Olambda_Odistinct_I13_J,axiom,
! [X23: nat,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Var @ X23 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(13)
thf(fact_1168_lambda__calculus_OIde_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Ide @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Ide.simps(2)
thf(fact_1169_lambda__calculus_OSrc__eq__iff_I1_J,axiom,
! [I: nat,I2: nat] :
( ( ( lambda_Src @ ( lambda_Var @ I ) )
= ( lambda_Src @ ( lambda_Var @ I2 ) ) )
= ( I = I2 ) ) ).
% lambda_calculus.Src_eq_iff(1)
thf(fact_1170_lambda__calculus_OSrc_Osimps_I2_J,axiom,
! [I: nat] :
( ( lambda_Src @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Src.simps(2)
thf(fact_1171_lambda__calculus_Oelementary__reduction_Osimps_I2_J,axiom,
! [Uu: nat] :
~ ( lambda2670285437103676873uction @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.elementary_reduction.simps(2)
thf(fact_1172_paths__in__rts_OResid1x_Ocong,axiom,
paths_3319247187298047121lambda = paths_3319247187298047121lambda ).
% paths_in_rts.Resid1x.cong
thf(fact_1173_lambda__calculus_OTrg_Osimps_I1_J,axiom,
! [I: nat] :
( ( lambda_Trg @ ( lambda_Var @ I ) )
= ( lambda_Var @ I ) ) ).
% lambda_calculus.Trg.simps(1)
thf(fact_1174_lambda__calculus_Osseq_Osimps_I14_J,axiom,
! [Va2: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_Lam @ Va2 ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(14)
thf(fact_1175_lambda__calculus_Osseq_Osimps_I20_J,axiom,
! [Va2: nat,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Var @ Va2 ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(20)
thf(fact_1176_lambda__calculus_Osseq_Osimps_I29_J,axiom,
! [Vb: nat,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va2 ) ) ).
% lambda_calculus.sseq.simps(29)
thf(fact_1177_lambda__calculus_Osseq_Osimps_I16_J,axiom,
! [Vc: nat,Vb: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Var @ Vc ) @ Vb ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(16)
thf(fact_1178_lambda__calculus_Osseq_Osimps_I18_J,axiom,
! [Vc: lambda_lambda,Vd: lambda_lambda,Vb: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_App @ Vc @ Vd ) @ Vb ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(18)
thf(fact_1179_lambda__calculus_Osseq_Osimps_I26_J,axiom,
! [Vb: nat,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_Var @ Vb ) @ ( lambda_App @ V @ Va2 ) ) ).
% lambda_calculus.sseq.simps(26)
thf(fact_1180_lambda__calculus_Oprfx__Var__iff,axiom,
! [U: lambda_lambda,I: nat] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_Var @ I ) ) )
= ( U
= ( lambda_Var @ I ) ) ) ).
% lambda_calculus.prfx_Var_iff
thf(fact_1181_lambda__calculus_OArr_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Arr @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Arr.simps(2)
thf(fact_1182_lambda__calculus_Ohead__redex_Osimps_I2_J,axiom,
! [X2: nat] :
( ( lambda_head_redex @ ( lambda_Var @ X2 ) )
= ( lambda_Var @ X2 ) ) ).
% lambda_calculus.head_redex.simps(2)
thf(fact_1183_lambda__calculus_OIde_Oelims_I2_J,axiom,
! [X2: lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ! [Uu2: nat] :
( X2
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Ide @ T3 ) )
=> ~ ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_App @ T3 @ U5 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(2)
thf(fact_1184_lambda__calculus_Ohead__redex_Osimps_I6_J,axiom,
! [V: nat,U: lambda_lambda] :
( ( lambda_head_redex @ ( lambda_App @ ( lambda_Var @ V ) @ U ) )
= ( lambda_App @ ( lambda_head_redex @ ( lambda_Var @ V ) ) @ ( lambda_Src @ U ) ) ) ).
% lambda_calculus.head_redex.simps(6)
thf(fact_1185_lambda__calculus_Osseq_Osimps_I32_J,axiom,
! [Vd: nat,Vc: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) @ ( lambda_Beta @ V @ Va2 ) ) ).
% lambda_calculus.sseq.simps(32)
thf(fact_1186_lambda__calculus_Osseq_Osimps_I22_J,axiom,
! [Vc: nat,Vb: lambda_lambda,V: lambda_lambda] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Var @ Vc ) @ Vb ) @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(22)
thf(fact_1187_lambda__calculus_Osseq_Osimps_I17_J,axiom,
! [Vc: lambda_lambda,Vb: lambda_lambda,V: nat] :
~ ( lambda_sseq @ ( lambda_App @ ( lambda_Lam @ Vc ) @ Vb ) @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(17)
thf(fact_1188_lambda__calculus_OArrE,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ! [I3: nat] :
( T
!= ( lambda_Var @ I3 ) )
=> ( ! [U5: lambda_lambda] :
( T
!= ( lambda_Lam @ U5 ) )
=> ( ! [U5: lambda_lambda,V2: lambda_lambda] :
( T
!= ( lambda_App @ U5 @ V2 ) )
=> ~ ! [U5: lambda_lambda,V2: lambda_lambda] :
( T
!= ( lambda_Beta @ U5 @ V2 ) ) ) ) ) ) ).
% lambda_calculus.ArrE
thf(fact_1189_lambda__calculus_OArr_Oelims_I2_J,axiom,
! [X2: lambda_lambda] :
( ( lambda_Arr @ X2 )
=> ( ! [Uu2: nat] :
( X2
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_App @ T3 @ U5 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U5 ) ) )
=> ~ ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U5 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(2)
thf(fact_1190_lambda__calculus_Osseq_Osimps_I9_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Va2: nat] :
( ( lambda_sseq @ ( lambda_Beta @ T @ U ) @ ( lambda_Var @ Va2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Var @ Va2 ) ) ) ) ).
% lambda_calculus.sseq.simps(9)
thf(fact_1191_lambda__calculus_Osseq_Osimps_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: nat] :
( ( lambda_sseq @ ( lambda_App @ ( lambda_Beta @ T @ U ) @ V ) @ ( lambda_Var @ Va2 ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U )
& ( lambda_Ide @ V )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T @ U ) @ V ) @ ( lambda_Var @ Va2 ) ) ) ) ).
% lambda_calculus.sseq.simps(6)
thf(fact_1192_Resid_Osimps_I7_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda,Vb: lambda_lambda,Vc: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(7)
thf(fact_1193_Resid_Osimps_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb: lambda_lambda,Vc: list_lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb @ Vc ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(6)
thf(fact_1194_Residx1_Osimps_I1_J,axiom,
! [U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ nil_lambda_lambda @ U )
= nil_lambda_lambda ) ).
% Residx1.simps(1)
thf(fact_1195_Residx1__as__Resid,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ T4 @ U )
= ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ).
% Residx1_as_Resid
thf(fact_1196_Residx1_Osimps_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ U )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ U )
= nil_lambda_lambda ) ) ) ).
% Residx1.simps(3)
thf(fact_1197_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_1198_Residx1_Oelims,axiom,
! [X2: list_lambda_lambda,Xa: lambda_lambda,Y2: list_lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
=> ( Y2 = nil_lambda_lambda ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ).
% Residx1.elims
thf(fact_1199_Con__sym1,axiom,
! [T4: list_lambda_lambda,U: lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ T4 @ U )
!= nil_lambda_lambda )
= ( ( paths_3319247187298047121lambda @ lambda_resid @ U @ T4 )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) ) ).
% Con_sym1
thf(fact_1200_Resid_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= ( cons_lambda_lambda @ ( lambda_resid @ T @ U ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(5)
thf(fact_1201_Resid_Oelims,axiom,
! [X2: list_lambda_lambda,Xa: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X2 @ Xa )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) )
=> ( ( ? [V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
= ( cons_lambda_lambda @ V2 @ Va ) )
=> ( ( Xa = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U5: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) )
=> ~ ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U5 ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U5: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ V2 @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y2
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ V2 @ Va ) ) @ nil_lambda_lambda ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ V2 @ Va ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ( ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ! [U5: lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U5 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ U5 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U5 @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U5 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) )
=> ! [U5: lambda_lambda,Vb2: lambda_lambda,Vc2: list_lambda_lambda] :
( ( Xa
= ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U5 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U5 ) ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U5 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U5 ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U5 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T3 @ U5 ) ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ) ) ) ) ).
% Resid.elims
thf(fact_1202_paths__in__rts_OResidx1_Ocong,axiom,
paths_2112734093361750239lambda = paths_2112734093361750239lambda ).
% paths_in_rts.Residx1.cong
thf(fact_1203_lambda__calculus_Osseq_Oelims_I3_J,axiom,
! [X2: lambda_lambda,Xa: lambda_lambda] :
( ~ ( lambda_sseq @ X2 @ Xa )
=> ( ( Xa != lambda_Nil )
=> ( ( ? [Uv2: nat] :
( X2
= ( lambda_Var @ Uv2 ) )
=> ! [Uw2: nat] :
( Xa
!= ( lambda_Var @ Uw2 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( lambda_Lam @ T3 ) )
=> ! [T9: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T9 ) )
=> ( lambda_sseq @ T3 @ T9 ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_App @ T3 @ U5 ) )
=> ! [T9: lambda_lambda,U7: lambda_lambda] :
( ( Xa
= ( lambda_App @ T9 @ U7 ) )
=> ( ( ( lambda_sseq @ T3 @ T9 )
& ( lambda_Ide @ U5 )
& ( U5 = U7 ) )
| ( ( lambda_Ide @ T3 )
& ( T3 = T9 )
& ( lambda_sseq @ U5 @ U7 ) )
| ( ( lambda2670285437103676873uction @ T3 )
& ( ( lambda_Trg @ T3 )
= T9 )
& ( U5
= ( lambda_Src @ U7 ) )
& ( lambda2670285437103676873uction @ U7 ) ) ) ) )
=> ( ( ? [T3: lambda_lambda,U5: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Lam @ T3 ) @ U5 ) )
=> ! [T9: lambda_lambda,U7: lambda_lambda] :
( Xa
!= ( lambda_Beta @ T9 @ U7 ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: nat] :
( ( Xa
= ( lambda_Var @ Va ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Var @ Va ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ Va ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Lam @ Va ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ Va @ Vb2 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Beta @ Va @ Vb2 ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: nat] :
( ( Xa
= ( lambda_Var @ Va ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Var @ Va ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ Va ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Lam @ Va ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_App @ Va @ Vb2 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_App @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_App @ Va @ Vb2 ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ Va @ Vb2 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Beta @ Va @ Vb2 ) ) ) ) )
=> ( ( ( X2 = lambda_Nil )
=> ! [V2: nat] :
( Xa
!= ( lambda_Var @ V2 ) ) )
=> ( ( ? [Va: lambda_lambda] :
( X2
= ( lambda_Lam @ Va ) )
=> ! [V2: nat] :
( Xa
!= ( lambda_Var @ V2 ) ) )
=> ( ( ? [Vb2: lambda_lambda] :
( X2
= ( lambda_App @ lambda_Nil @ Vb2 ) )
=> ! [V2: nat] :
( Xa
!= ( lambda_Var @ V2 ) ) )
=> ( ( ? [Vc2: nat,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Var @ Vc2 ) @ Vb2 ) )
=> ! [V2: nat] :
( Xa
!= ( lambda_Var @ V2 ) ) )
=> ( ( ? [Vc2: lambda_lambda,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Lam @ Vc2 ) @ Vb2 ) )
=> ! [V2: nat] :
( Xa
!= ( lambda_Var @ V2 ) ) )
=> ( ( ? [Vc2: lambda_lambda,Vd2: lambda_lambda,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_App @ Vc2 @ Vd2 ) @ Vb2 ) )
=> ! [V2: nat] :
( Xa
!= ( lambda_Var @ V2 ) ) )
=> ( ( ( X2 = lambda_Nil )
=> ! [V2: lambda_lambda] :
( Xa
!= ( lambda_Lam @ V2 ) ) )
=> ( ( ? [Va: nat] :
( X2
= ( lambda_Var @ Va ) )
=> ! [V2: lambda_lambda] :
( Xa
!= ( lambda_Lam @ V2 ) ) )
=> ( ( ? [Vb2: lambda_lambda] :
( X2
= ( lambda_App @ lambda_Nil @ Vb2 ) )
=> ! [V2: lambda_lambda] :
( Xa
!= ( lambda_Lam @ V2 ) ) )
=> ( ( ? [Vc2: nat,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Var @ Vc2 ) @ Vb2 ) )
=> ! [V2: lambda_lambda] :
( Xa
!= ( lambda_Lam @ V2 ) ) )
=> ( ( ? [Vc2: lambda_lambda,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Lam @ Vc2 ) @ Vb2 ) )
=> ! [V2: lambda_lambda] :
( Xa
!= ( lambda_Lam @ V2 ) ) )
=> ( ( ? [Vc2: lambda_lambda,Vd2: lambda_lambda,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_App @ Vc2 @ Vd2 ) @ Vb2 ) )
=> ! [V2: lambda_lambda] :
( Xa
!= ( lambda_Lam @ V2 ) ) )
=> ( ( ( X2 = lambda_Nil )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_App @ V2 @ Va ) ) )
=> ( ( ? [Vb2: nat] :
( X2
= ( lambda_Var @ Vb2 ) )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_App @ V2 @ Va ) ) )
=> ( ( ? [Vb2: lambda_lambda] :
( X2
= ( lambda_Lam @ Vb2 ) )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_App @ V2 @ Va ) ) )
=> ( ( ( X2 = lambda_Nil )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_Beta @ V2 @ Va ) ) )
=> ( ( ? [Vb2: nat] :
( X2
= ( lambda_Var @ Vb2 ) )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_Beta @ V2 @ Va ) ) )
=> ( ( ? [Vb2: lambda_lambda] :
( X2
= ( lambda_Lam @ Vb2 ) )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_Beta @ V2 @ Va ) ) )
=> ( ( ? [Vc2: lambda_lambda] :
( X2
= ( lambda_App @ lambda_Nil @ Vc2 ) )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_Beta @ V2 @ Va ) ) )
=> ( ( ? [Vd2: nat,Vc2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_Beta @ V2 @ Va ) ) )
=> ~ ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) )
=> ! [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
!= ( lambda_Beta @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.sseq.elims(3)
thf(fact_1204_lambda__calculus_Osseq_Oelims_I1_J,axiom,
! [X2: lambda_lambda,Xa: lambda_lambda,Y2: $o] :
( ( ( lambda_sseq @ X2 @ Xa )
= Y2 )
=> ( ( ( Xa = lambda_Nil )
=> Y2 )
=> ( ( ? [Uv2: nat] :
( X2
= ( lambda_Var @ Uv2 ) )
=> ( ? [Uw2: nat] :
( Xa
= ( lambda_Var @ Uw2 ) )
=> Y2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( lambda_Lam @ T3 ) )
=> ! [T9: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ T9 ) )
=> ( Y2
= ( ~ ( lambda_sseq @ T3 @ T9 ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_App @ T3 @ U5 ) )
=> ! [T9: lambda_lambda,U7: lambda_lambda] :
( ( Xa
= ( lambda_App @ T9 @ U7 ) )
=> ( Y2
= ( ~ ( ( ( lambda_sseq @ T3 @ T9 )
& ( lambda_Ide @ U5 )
& ( U5 = U7 ) )
| ( ( lambda_Ide @ T3 )
& ( T3 = T9 )
& ( lambda_sseq @ U5 @ U7 ) )
| ( ( lambda2670285437103676873uction @ T3 )
& ( ( lambda_Trg @ T3 )
= T9 )
& ( U5
= ( lambda_Src @ U7 ) )
& ( lambda2670285437103676873uction @ U7 ) ) ) ) ) ) )
=> ( ( ? [T3: lambda_lambda,U5: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Lam @ T3 ) @ U5 ) )
=> ( ? [T9: lambda_lambda,U7: lambda_lambda] :
( Xa
= ( lambda_Beta @ T9 @ U7 ) )
=> Y2 ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: nat] :
( ( Xa
= ( lambda_Var @ Va ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Var @ Va ) ) ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ Va ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Lam @ Va ) ) ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( X2
= ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ Va @ Vb2 ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda_Ide @ V2 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_App @ ( lambda_Beta @ T3 @ U5 ) @ V2 ) @ ( lambda_Beta @ Va @ Vb2 ) ) ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: nat] :
( ( Xa
= ( lambda_Var @ Va ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Var @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Var @ Va ) ) ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda] :
( ( Xa
= ( lambda_Lam @ Va ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Lam @ Va ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Lam @ Va ) ) ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_App @ Va @ Vb2 ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_App @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_App @ Va @ Vb2 ) ) ) ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( X2
= ( lambda_Beta @ T3 @ U5 ) )
=> ! [Va: lambda_lambda,Vb2: lambda_lambda] :
( ( Xa
= ( lambda_Beta @ Va @ Vb2 ) )
=> ( Y2
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U5 )
& ( lambda2670285437103676873uction @ ( lambda_Beta @ Va @ Vb2 ) )
& ( seq_lambda_lambda @ lambda_resid @ ( lambda_Beta @ T3 @ U5 ) @ ( lambda_Beta @ Va @ Vb2 ) ) ) ) ) ) )
=> ( ( ( X2 = lambda_Nil )
=> ( ? [V2: nat] :
( Xa
= ( lambda_Var @ V2 ) )
=> Y2 ) )
=> ( ( ? [Va: lambda_lambda] :
( X2
= ( lambda_Lam @ Va ) )
=> ( ? [V2: nat] :
( Xa
= ( lambda_Var @ V2 ) )
=> Y2 ) )
=> ( ( ? [Vb2: lambda_lambda] :
( X2
= ( lambda_App @ lambda_Nil @ Vb2 ) )
=> ( ? [V2: nat] :
( Xa
= ( lambda_Var @ V2 ) )
=> Y2 ) )
=> ( ( ? [Vc2: nat,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Var @ Vc2 ) @ Vb2 ) )
=> ( ? [V2: nat] :
( Xa
= ( lambda_Var @ V2 ) )
=> Y2 ) )
=> ( ( ? [Vc2: lambda_lambda,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Lam @ Vc2 ) @ Vb2 ) )
=> ( ? [V2: nat] :
( Xa
= ( lambda_Var @ V2 ) )
=> Y2 ) )
=> ( ( ? [Vc2: lambda_lambda,Vd2: lambda_lambda,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_App @ Vc2 @ Vd2 ) @ Vb2 ) )
=> ( ? [V2: nat] :
( Xa
= ( lambda_Var @ V2 ) )
=> Y2 ) )
=> ( ( ( X2 = lambda_Nil )
=> ( ? [V2: lambda_lambda] :
( Xa
= ( lambda_Lam @ V2 ) )
=> Y2 ) )
=> ( ( ? [Va: nat] :
( X2
= ( lambda_Var @ Va ) )
=> ( ? [V2: lambda_lambda] :
( Xa
= ( lambda_Lam @ V2 ) )
=> Y2 ) )
=> ( ( ? [Vb2: lambda_lambda] :
( X2
= ( lambda_App @ lambda_Nil @ Vb2 ) )
=> ( ? [V2: lambda_lambda] :
( Xa
= ( lambda_Lam @ V2 ) )
=> Y2 ) )
=> ( ( ? [Vc2: nat,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Var @ Vc2 ) @ Vb2 ) )
=> ( ? [V2: lambda_lambda] :
( Xa
= ( lambda_Lam @ V2 ) )
=> Y2 ) )
=> ( ( ? [Vc2: lambda_lambda,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Lam @ Vc2 ) @ Vb2 ) )
=> ( ? [V2: lambda_lambda] :
( Xa
= ( lambda_Lam @ V2 ) )
=> Y2 ) )
=> ( ( ? [Vc2: lambda_lambda,Vd2: lambda_lambda,Vb2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_App @ Vc2 @ Vd2 ) @ Vb2 ) )
=> ( ? [V2: lambda_lambda] :
( Xa
= ( lambda_Lam @ V2 ) )
=> Y2 ) )
=> ( ( ( X2 = lambda_Nil )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_App @ V2 @ Va ) )
=> Y2 ) )
=> ( ( ? [Vb2: nat] :
( X2
= ( lambda_Var @ Vb2 ) )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_App @ V2 @ Va ) )
=> Y2 ) )
=> ( ( ? [Vb2: lambda_lambda] :
( X2
= ( lambda_Lam @ Vb2 ) )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_App @ V2 @ Va ) )
=> Y2 ) )
=> ( ( ( X2 = lambda_Nil )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_Beta @ V2 @ Va ) )
=> Y2 ) )
=> ( ( ? [Vb2: nat] :
( X2
= ( lambda_Var @ Vb2 ) )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_Beta @ V2 @ Va ) )
=> Y2 ) )
=> ( ( ? [Vb2: lambda_lambda] :
( X2
= ( lambda_Lam @ Vb2 ) )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_Beta @ V2 @ Va ) )
=> Y2 ) )
=> ( ( ? [Vc2: lambda_lambda] :
( X2
= ( lambda_App @ lambda_Nil @ Vc2 ) )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_Beta @ V2 @ Va ) )
=> Y2 ) )
=> ( ( ? [Vd2: nat,Vc2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_Var @ Vd2 ) @ Vc2 ) )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_Beta @ V2 @ Va ) )
=> Y2 ) )
=> ~ ( ? [Vd2: lambda_lambda,Ve2: lambda_lambda,Vc2: lambda_lambda] :
( X2
= ( lambda_App @ ( lambda_App @ Vd2 @ Ve2 ) @ Vc2 ) )
=> ( ? [V2: lambda_lambda,Va: lambda_lambda] :
( Xa
= ( lambda_Beta @ V2 @ Va ) )
=> Y2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.sseq.elims(1)
thf(fact_1205_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_1206_lambda__calculus_Onull__char,axiom,
( ( partia1461432607296028304lambda @ lambda_resid )
= lambda_Nil ) ).
% lambda_calculus.null_char
thf(fact_1207_lambda__calculus_OConI,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Arr @ U )
=> ( ( ( lambda_Src @ T )
= ( lambda_Src @ U ) )
=> ( ( lambda_resid @ T @ U )
!= lambda_Nil ) ) ) ) ).
% lambda_calculus.ConI
thf(fact_1208_lambda__calculus_Otargets__simps_I1_J,axiom,
( ( target5490931411331770124lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.targets_simps(1)
thf(fact_1209_lambda__calculus_Osources__simps_I1_J,axiom,
( ( source5627170679743844886lambda @ lambda_resid @ lambda_Nil )
= bot_bo1271416176687173302lambda ) ).
% lambda_calculus.sources_simps(1)
thf(fact_1210_lambda__calculus_Otrg__char,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_Trg @ T ) ) )
& ( ~ ( lambda_Arr @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
= lambda_Nil ) ) ) ).
% lambda_calculus.trg_char
thf(fact_1211_reduction__paths_Ostandard__development_Osimps_I1_J,axiom,
( ( reduct1603535164923377302opment @ lambda_Nil )
= nil_lambda_lambda ) ).
% reduction_paths.standard_development.simps(1)
thf(fact_1212_lambda__calculus_Osseq_Osimps_I25_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_sseq @ lambda_Nil @ ( lambda_App @ V @ Va2 ) ) ).
% lambda_calculus.sseq.simps(25)
thf(fact_1213_lambda__calculus_Osseq_Osimps_I28_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda] :
~ ( lambda_sseq @ lambda_Nil @ ( lambda_Beta @ V @ Va2 ) ) ).
% lambda_calculus.sseq.simps(28)
thf(fact_1214_lambda__calculus_Osseq_Osimps_I19_J,axiom,
! [V: lambda_lambda] :
~ ( lambda_sseq @ lambda_Nil @ ( lambda_Lam @ V ) ) ).
% lambda_calculus.sseq.simps(19)
thf(fact_1215_lambda__calculus_OCon__implies__is__Lam__iff__is__Lam,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( ( lambda_is_Lam @ T )
= ( lambda_is_Lam @ U ) ) ) ).
% lambda_calculus.Con_implies_is_Lam_iff_is_Lam
thf(fact_1216_lambda__calculus_Olambda_Odistinct_I3_J,axiom,
! [X32: lambda_lambda] :
( lambda_Nil
!= ( lambda_Lam @ X32 ) ) ).
% lambda_calculus.lambda.distinct(3)
thf(fact_1217_lambda__calculus_Osseq_Osimps_I1_J,axiom,
! [Uu: lambda_lambda] :
~ ( lambda_sseq @ Uu @ lambda_Nil ) ).
% lambda_calculus.sseq.simps(1)
thf(fact_1218_lambda__calculus_Olambda_Odistinct__disc_I3_J,axiom,
! [Lambda: lambda_lambda] :
( ( Lambda = lambda_Nil )
=> ~ ( lambda_is_Lam @ Lambda ) ) ).
% lambda_calculus.lambda.distinct_disc(3)
thf(fact_1219_lambda__calculus_Olambda_Odistinct__disc_I4_J,axiom,
! [Lambda: lambda_lambda] :
( ( lambda_is_Lam @ Lambda )
=> ( Lambda != lambda_Nil ) ) ).
% lambda_calculus.lambda.distinct_disc(4)
thf(fact_1220_lambda__calculus_Olambda_Odisc_I6_J,axiom,
~ ( lambda_is_Lam @ lambda_Nil ) ).
% lambda_calculus.lambda.disc(6)
thf(fact_1221_lambda__calculus_Oresid_Osimps_I26_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_resid @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(26)
thf(fact_1222_lambda__calculus_Oresid_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(7)
thf(fact_1223_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_1224_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_1225_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_1226_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_1227_lambda__calculus_OIde_Osimps_I1_J,axiom,
~ ( lambda_Ide @ lambda_Nil ) ).
% lambda_calculus.Ide.simps(1)
thf(fact_1228_lambda__calculus_OSrc_Osimps_I1_J,axiom,
( ( lambda_Src @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Src.simps(1)
thf(fact_1229_lambda__calculus_Oelementary__reduction_Osimps_I1_J,axiom,
~ ( lambda2670285437103676873uction @ lambda_Nil ) ).
% lambda_calculus.elementary_reduction.simps(1)
thf(fact_1230_lambda__calculus_OTrg_Osimps_I5_J,axiom,
( ( lambda_Trg @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Trg.simps(5)
thf(fact_1231_lambda__calculus_Oresid_Osimps_I2_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda] :
( ( ( ( lambda_resid @ T @ T6 )
= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T6 ) )
= lambda_Nil ) )
& ( ( ( lambda_resid @ T @ T6 )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T6 ) )
= ( lambda_Lam @ ( lambda_resid @ T @ T6 ) ) ) ) ) ).
% lambda_calculus.resid.simps(2)
thf(fact_1232_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_1233_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_1234_lambda__calculus_Oresid_Osimps_I38_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Beta @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(38)
thf(fact_1235_lambda__calculus_Oresid_Osimps_I19_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va2 ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(19)
thf(fact_1236_lambda__calculus_OConD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T6: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T6 @ U4 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ T6 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U4 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(1)
thf(fact_1237_lambda__calculus_Oresid_Osimps_I31_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_App @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(31)
thf(fact_1238_lambda__calculus_Oresid_Osimps_I12_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va2 ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(12)
thf(fact_1239_lambda__calculus_Oresid_Osimps_I3_J,axiom,
! [T: lambda_lambda,T6: lambda_lambda,U: lambda_lambda,U4: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U4 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T6 @ U4 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T6 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U4 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T6 @ U4 ) )
= ( lambda_App @ ( lambda_resid @ T @ T6 ) @ ( lambda_resid @ U @ U4 ) ) ) ) ) ).
% lambda_calculus.resid.simps(3)
thf(fact_1240_lambda__calculus_OArr__not__Nil,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( T != lambda_Nil ) ) ).
% lambda_calculus.Arr_not_Nil
thf(fact_1241_lambda__calculus_OArr_Osimps_I1_J,axiom,
~ ( lambda_Arr @ lambda_Nil ) ).
% lambda_calculus.Arr.simps(1)
thf(fact_1242_lambda__calculus_OCon__implies__Arr2,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ U ) ) ).
% lambda_calculus.Con_implies_Arr2
thf(fact_1243_lambda__calculus_OCon__implies__Arr1,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Con_implies_Arr1
thf(fact_1244_lambda__calculus_OArr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ T @ U ) ) ) ).
% lambda_calculus.Arr_resid
thf(fact_1245_lambda__calculus_Ohead__redex_Osimps_I1_J,axiom,
( ( lambda_head_redex @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.head_redex.simps(1)
thf(fact_1246_lambda__calculus_Olambda_Odistinct_I1_J,axiom,
! [X23: nat] :
( lambda_Nil
!= ( lambda_Var @ X23 ) ) ).
% lambda_calculus.lambda.distinct(1)
thf(fact_1247_lambda__calculus_Oresid_Osimps_I1_J,axiom,
! [I: nat,I2: nat] :
( ( ( I = I2 )
=> ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I2 ) )
= ( lambda_Var @ I ) ) )
& ( ( I != I2 )
=> ( ( lambda_resid @ ( lambda_Var @ I ) @ ( lambda_Var @ I2 ) )
= lambda_Nil ) ) ) ).
% lambda_calculus.resid.simps(1)
thf(fact_1248_lambda__calculus_Osseq_Osimps_I13_J,axiom,
! [V: nat] :
~ ( lambda_sseq @ lambda_Nil @ ( lambda_Var @ V ) ) ).
% lambda_calculus.sseq.simps(13)
thf(fact_1249_lambda__calculus_Oresid_Osimps_I15_J,axiom,
! [Va2: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Va2 ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(15)
thf(fact_1250_lambda__calculus_Oresid_Osimps_I17_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va2 ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(17)
thf(fact_1251_lambda__calculus_Oresid_Osimps_I18_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va2 ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(18)
thf(fact_1252_lambda__calculus_Oresid_Osimps_I22_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va2 ) @ ( lambda_App @ lambda_Nil @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(22)
thf(fact_1253_lambda__calculus_Oresid_Osimps_I24_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va2 ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(24)
thf(fact_1254_lambda__calculus_Oresid_Osimps_I25_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va2 ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(25)
thf(fact_1255_lambda__calculus_Oresid_Osimps_I34_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ lambda_Nil @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(34)
thf(fact_1256_lambda__calculus_Oresid_Osimps_I36_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(36)
thf(fact_1257_lambda__calculus_Oresid_Osimps_I37_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(37)
thf(fact_1258_lambda__calculus_Oresid_Osimps_I41_J,axiom,
! [Vc: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Vc ) @ ( lambda_Beta @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(41)
thf(fact_1259_lambda__calculus_Oresid_Osimps_I43_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(43)
thf(fact_1260_lambda__calculus_Oresid_Osimps_I44_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(44)
thf(fact_1261_lambda__calculus_Oresid_Osimps_I32_J,axiom,
! [Vb: nat,V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_App @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(32)
thf(fact_1262_lambda__calculus_Oresid_Osimps_I13_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb: nat] :
( ( lambda_resid @ ( lambda_App @ V @ Va2 ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(13)
thf(fact_1263_lambda__calculus_Oresid_Osimps_I33_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(33)
thf(fact_1264_lambda__calculus_Oresid_Osimps_I29_J,axiom,
! [Va2: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ Va2 @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(29)
thf(fact_1265_lambda__calculus_Oresid_Osimps_I14_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va2 ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(14)
thf(fact_1266_lambda__calculus_Oresid_Osimps_I10_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_App @ Va2 @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(10)
thf(fact_1267_lambda__calculus_Oresid_Osimps_I39_J,axiom,
! [Vb: nat,V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(39)
thf(fact_1268_lambda__calculus_Oresid_Osimps_I20_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb: nat] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va2 ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(20)
thf(fact_1269_lambda__calculus_OConD_I2_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,V4: lambda_lambda,U4: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_Beta @ V4 @ U4 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Lam @ V4 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U4 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(2)
thf(fact_1270_lambda__calculus_Oresid_Osimps_I40_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va2: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(40)
thf(fact_1271_lambda__calculus_Oresid_Osimps_I30_J,axiom,
! [Va2: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Va2 @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(30)
thf(fact_1272_lambda__calculus_Oresid_Osimps_I21_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va2 ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(21)
thf(fact_1273_lambda__calculus_Oresid_Osimps_I11_J,axiom,
! [V: lambda_lambda,Va2: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Beta @ Va2 @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(11)
thf(fact_1274_lambda__calculus_Oresid_Osimps_I9_J,axiom,
! [V: lambda_lambda,Va2: nat] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Var @ Va2 ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(9)
thf(fact_1275_lambda__calculus_Oresid_Osimps_I28_J,axiom,
! [Va2: nat,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Va2 ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(28)
thf(fact_1276_reduction__paths_Ostdz__insert_Osimps_I14_J,axiom,
! [V: lambda_lambda,Va2: list_lambda_lambda] :
( ( reduct5390047737487997688insert @ lambda_Nil @ ( cons_lambda_lambda @ V @ Va2 ) )
= nil_lambda_lambda ) ).
% reduction_paths.stdz_insert.simps(14)
% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X2: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_T,axiom,
! [X2: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( if_lis7195930416592267574lambda @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (5)
thf(conj_0,hypothesis,
! [T2: lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ T2 @ ua ) )
=> ( ( reduct5390047737487997688insert @ T2 @ ua )
= nil_lambda_lambda ) ) ).
thf(conj_1,hypothesis,
( ( lambda_Ide @ x41 )
& ( lambda_Ide @ x42 )
& ( paths_7717490669901197737lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ x51 @ x52 ) @ ua ) )
& ( ord_le1926335974945931754lambda @ ( target5490931411331770124lambda @ lambda_resid @ ( lambda_App @ x41 @ x42 ) ) @ ( paths_2897053694880971696lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Beta @ x51 @ x52 ) @ ua ) ) ) ) ).
thf(conj_2,hypothesis,
( ta
= ( lambda_App @ x41 @ x42 ) ) ).
thf(conj_3,hypothesis,
( u
= ( lambda_Beta @ x51 @ x52 ) ) ).
thf(conj_4,conjecture,
( ( reduct5390047737487997688insert @ ( lambda_App @ x41 @ x42 ) @ ( cons_lambda_lambda @ ( lambda_Beta @ x51 @ x52 ) @ ua ) )
= nil_lambda_lambda ) ).
%------------------------------------------------------------------------------