TPTP Problem File: SLH0303^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : ResiduatedTransitionSystem/0001_LambdaCalculus/prob_03011_124685__14436390_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1411 ( 458 unt; 129 typ; 0 def)
% Number of atoms : 4520 (1619 equ; 0 cnn)
% Maximal formula atoms : 38 ( 3 avg)
% Number of connectives : 16612 ( 537 ~; 17 |; 718 &;13610 @)
% ( 0 <=>;1730 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 792 ( 792 >; 0 *; 0 +; 0 <<)
% Number of symbols : 127 ( 124 usr; 8 con; 0-5 aty)
% Number of variables : 3487 ( 384 ^;2976 !; 127 ?;3487 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:48.083
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
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% Explicit typings (124)
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thf(sy_c_ResiduatedTransitionSystem_Orts_Otargets_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
target4026644918049360146lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_ax7298520571313827089lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
rts_ax2283037410143324311lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
rts_wi3086218056445523938lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Orts__with__composites_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
rts_wi1553966559393445608lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osub__rts_Oresid_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
sub_re469828149216679723lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > $o ) > lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Osub__rts__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
sub_rt3575465509250962105lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > $o ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Osub__rts__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
sub_rt3099298114314975807lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > $o ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
transf356316610583137064lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > ( lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
transf4653517080217399086lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
transf3419659901140239394lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( lambda_lambda > lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > ( list_lambda_lambda > lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Otransformation__axioms_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
transf8838427370478101288lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda ) > ( list_lambda_lambda > list_lambda_lambda ) > $o ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
weakly4554751870655413501lambda: ( lambda_lambda > lambda_lambda > lambda_lambda ) > lambda_lambda > lambda_lambda ).
thf(sy_c_ResiduatedTransitionSystem_Oweakly__extensional__rts_Osrc_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
weakly2957279434024022659lambda: ( list_lambda_lambda > list_lambda_lambda > list_lambda_lambda ) > list_lambda_lambda > list_lambda_lambda ).
thf(sy_c_Set_OCollect_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
collec621444948832755541lambda: ( lambda_lambda > $o ) > set_lambda_lambda ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
collec8830145823237851355lambda: ( list_lambda_lambda > $o ) > set_li7878742357367608272lambda ).
thf(sy_c_Set_Oimage_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
image_2356535832909760517lambda: ( lambda_lambda > lambda_lambda ) > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Set_Oimage_001t__LambdaCalculus__Olambda____calculus__Olambda_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
image_3692823078739859851lambda: ( lambda_lambda > list_lambda_lambda ) > set_lambda_lambda > set_li7878742357367608272lambda ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
image_2458965899662700159lambda: ( list_lambda_lambda > lambda_lambda ) > set_li7878742357367608272lambda > set_lambda_lambda ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
image_5886195720093285125lambda: ( list_lambda_lambda > list_lambda_lambda ) > set_li7878742357367608272lambda > set_li7878742357367608272lambda ).
thf(sy_c_Set_Oinsert_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
insert_lambda_lambda: lambda_lambda > set_lambda_lambda > set_lambda_lambda ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
insert8465602035467394112lambda: list_lambda_lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda ).
thf(sy_c_Set_Ois__singleton_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
is_sin2795064709772026974lambda: set_lambda_lambda > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
is_sin4085330448496538980lambda: set_li7878742357367608272lambda > $o ).
thf(sy_c_Set_Othe__elem_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
the_el5431214018478934239lambda: set_lambda_lambda > lambda_lambda ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
the_el4548681628688475493lambda: set_li7878742357367608272lambda > list_lambda_lambda ).
thf(sy_c_member_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
member_lambda_lambda: lambda_lambda > set_lambda_lambda > $o ).
thf(sy_c_member_001t__List__Olist_It__LambdaCalculus__Olambda____calculus__Olambda_J,type,
member2667661554319814297lambda: list_lambda_lambda > set_li7878742357367608272lambda > $o ).
thf(sy_v_U,type,
u: list_lambda_lambda ).
thf(sy_v_t,type,
t: lambda_lambda ).
% Relevant facts (1278)
thf(fact_0__092_060Lambda_062_Ocube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) )
= ( lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.cube
thf(fact_1__092_060Lambda_062_Oex__un__null,axiom,
? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( lambda_resid @ X @ T2 )
= X )
& ( ( lambda_resid @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( lambda_resid @ Y @ T3 )
= Y )
& ( ( lambda_resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% \<Lambda>.ex_un_null
thf(fact_2_Trgs_Ocases,axiom,
! [X2: list_lambda_lambda] :
( ( X2 != nil_lambda_lambda )
=> ( ! [T3: lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ~ ! [T3: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ T3 @ ( cons_lambda_lambda @ V2 @ Va ) ) ) ) ) ).
% Trgs.cases
thf(fact_3_cube,axiom,
! [V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) )
= ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ).
% cube
thf(fact_4_ex__un__null,axiom,
? [X: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X @ T2 )
= X )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T2 @ X )
= X ) )
& ! [Y: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ Y @ T3 )
= Y )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% ex_un_null
thf(fact_5_Arr_Osimps_I1_J,axiom,
~ ( paths_366302053746739058lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Arr.simps(1)
thf(fact_6_lambda__calculus_Olambda_Oinject_I2_J,axiom,
! [X3: lambda_lambda,Y3: lambda_lambda] :
( ( ( lambda_Lam @ X3 )
= ( lambda_Lam @ Y3 ) )
= ( X3 = Y3 ) ) ).
% lambda_calculus.lambda.inject(2)
thf(fact_7_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_8_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_9_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_10_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_11_assms,axiom,
seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ ( lambda_Lam @ t ) @ nil_lambda_lambda ) @ u ).
% assms
thf(fact_12_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_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_paths__in__confluent__rts__axioms,axiom,
paths_9040249923598999359lambda @ lambda_resid ).
% paths_in_confluent_rts_axioms
thf(fact_27__092_060Lambda_062_Opartial__magma__axioms,axiom,
partia3924075026459072775lambda @ lambda_resid ).
% \<Lambda>.partial_magma_axioms
thf(fact_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_is__confluent__rts,axiom,
conflu4036547884742428384lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% is_confluent_rts
thf(fact_36_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_37_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_38_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_39_Ide__implies__Arr,axiom,
! [T4: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% Ide_implies_Arr
thf(fact_40_Ide_Osimps_I1_J,axiom,
~ ( paths_7717490669901197737lambda @ lambda_resid @ nil_lambda_lambda ) ).
% Ide.simps(1)
thf(fact_41_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_42_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_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
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: set_li7878742357367608272lambda] :
( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ 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_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_50_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_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_list_Osimps_I8_J,axiom,
! [F: lambda_lambda > lambda_lambda] :
( ( map_la336867333220232516lambda @ F @ nil_lambda_lambda )
= nil_lambda_lambda ) ).
% list.simps(8)
thf(fact_74_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_75_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 ) )
= ( ? [Z: lambda_lambda,Zs: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z @ Zs ) )
& ( ( F @ Z )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_76_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 ) )
= ( ? [Z: lambda_lambda,Zs: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_77_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 ) )
=> ? [Z2: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( cons_lambda_lambda @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y2 )
& ( ( map_la336867333220232516lambda @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_78_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 ) )
=> ? [Z2: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_la336867333220232516lambda @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_79_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_80_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_81_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 )
=> ( ! [Y4: lambda_lambda,Ys2: list_lambda_lambda] : ( P @ nil_lambda_lambda @ ( cons_lambda_lambda @ Y4 @ Ys2 ) )
=> ( ! [X: lambda_lambda,Xs2: list_lambda_lambda,Y4: lambda_lambda,Ys2: list_lambda_lambda] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_lambda_lambda @ X @ Xs2 ) @ ( cons_lambda_lambda @ Y4 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_82_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_83_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,Y4: lambda_lambda,Xs2: list_lambda_lambda] :
( X2
!= ( cons_lambda_lambda @ X @ ( cons_lambda_lambda @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_84_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_85_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_86_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_87_not__Cons__self2,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( cons_lambda_lambda @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_88_partial__magma__axioms,axiom,
partia748546660539056525lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% partial_magma_axioms
thf(fact_89_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_90_rts__with__composites__axioms,axiom,
rts_wi1553966559393445608lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% rts_with_composites_axioms
thf(fact_91_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_92_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_93_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_94_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_95_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_96_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_97_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_98_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_99__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_100__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_101__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_102__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_103__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_104__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_105__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_106__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_107__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_108__092_060Lambda_062_Otargets__are__cong,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B ) ) ) ) ) ).
% \<Lambda>.targets_are_cong
thf(fact_109__092_060Lambda_062_Otargets__cong__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B @ B2 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B2 @ B ) ) )
=> ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.targets_cong_closed
thf(fact_110__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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_paths__in__weakly__extensional__rts_OSrc_Ocong,axiom,
paths_1651982994781619320lambda = paths_1651982994781619320lambda ).
% paths_in_weakly_extensional_rts.Src.cong
thf(fact_122_residuation_Oide_Ocong,axiom,
ide_lambda_lambda = ide_lambda_lambda ).
% residuation.ide.cong
thf(fact_123_residuation_Oide_Ocong,axiom,
ide_li6773656263178250516lambda = ide_li6773656263178250516lambda ).
% residuation.ide.cong
thf(fact_124_rts_Ocomposable_Ocong,axiom,
compos3359172601056258045lambda = compos3359172601056258045lambda ).
% rts.composable.cong
thf(fact_125_rts_Ocomposable_Ocong,axiom,
compos7127019122090555255lambda = compos7127019122090555255lambda ).
% rts.composable.cong
thf(fact_126_rts_Ocoinitial_Ocong,axiom,
coinit1063259437155584316lambda = coinit1063259437155584316lambda ).
% rts.coinitial.cong
thf(fact_127_rts_Ocoinitial_Ocong,axiom,
coinit1392077014771647542lambda = coinit1392077014771647542lambda ).
% rts.coinitial.cong
thf(fact_128_rts_Otargets_Ocong,axiom,
target5490931411331770124lambda = target5490931411331770124lambda ).
% rts.targets.cong
thf(fact_129_rts_Otargets_Ocong,axiom,
target4026644918049360146lambda = target4026644918049360146lambda ).
% rts.targets.cong
thf(fact_130_paths__in__rts_OResid_Ocong,axiom,
paths_8607357831768207192lambda = paths_8607357831768207192lambda ).
% paths_in_rts.Resid.cong
thf(fact_131_residuation_Oarr_Ocong,axiom,
arr_lambda_lambda = arr_lambda_lambda ).
% residuation.arr.cong
thf(fact_132_residuation_Oarr_Ocong,axiom,
arr_li3691443969012319837lambda = arr_li3691443969012319837lambda ).
% residuation.arr.cong
thf(fact_133_paths__in__rts_OArr_Ocong,axiom,
paths_366302053746739058lambda = paths_366302053746739058lambda ).
% paths_in_rts.Arr.cong
thf(fact_134_paths__in__rts_OSrcs_Ocong,axiom,
paths_2897053694880971696lambda = paths_2897053694880971696lambda ).
% paths_in_rts.Srcs.cong
thf(fact_135_paths__in__rts_OIde_Ocong,axiom,
paths_7717490669901197737lambda = paths_7717490669901197737lambda ).
% paths_in_rts.Ide.cong
thf(fact_136_rts_Oseq_Ocong,axiom,
seq_li8093550137657879445lambda = seq_li8093550137657879445lambda ).
% rts.seq.cong
thf(fact_137_rts_Oseq_Ocong,axiom,
seq_lambda_lambda = seq_lambda_lambda ).
% rts.seq.cong
thf(fact_138_paths__in__rts_OResid1x_Ocong,axiom,
paths_3319247187298047121lambda = paths_3319247187298047121lambda ).
% paths_in_rts.Resid1x.cong
thf(fact_139_paths__in__weakly__extensional__rts_OTrg_Ocong,axiom,
paths_2695323870737867955lambda = paths_2695323870737867955lambda ).
% paths_in_weakly_extensional_rts.Trg.cong
thf(fact_140_partial__magma__def,axiom,
( partia3924075026459072775lambda
= ( ^ [OP: lambda_lambda > lambda_lambda > lambda_lambda] :
? [X4: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP @ X4 @ T5 )
= X4 )
& ( ( OP @ T5 @ X4 )
= X4 ) )
& ! [Y5: lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( ( OP @ Y5 @ T5 )
= Y5 )
& ( ( OP @ T5 @ Y5 )
= Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% partial_magma_def
thf(fact_141_partial__magma__def,axiom,
( partia748546660539056525lambda
= ( ^ [OP: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
? [X4: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( OP @ X4 @ T5 )
= X4 )
& ( ( OP @ T5 @ X4 )
= X4 ) )
& ! [Y5: list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( ( OP @ Y5 @ T5 )
= Y5 )
& ( ( OP @ T5 @ Y5 )
= Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ).
% partial_magma_def
thf(fact_142_partial__magma_Oex__un__null,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP2 @ X @ T2 )
= X )
& ( ( OP2 @ T2 @ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP2 @ Y @ T3 )
= Y )
& ( ( OP2 @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_143_partial__magma_Oex__un__null,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ? [X: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP2 @ X @ T2 )
= X )
& ( ( OP2 @ T2 @ X )
= X ) )
& ! [Y: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP2 @ Y @ T3 )
= Y )
& ( ( OP2 @ T3 @ Y )
= Y ) )
=> ( Y = X ) ) ) ) ).
% partial_magma.ex_un_null
thf(fact_144_partial__magma_Ointro,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ? [X5: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( OP2 @ X5 @ T3 )
= X5 )
& ( ( OP2 @ T3 @ X5 )
= X5 ) )
& ! [Y4: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( OP2 @ Y4 @ T2 )
= Y4 )
& ( ( OP2 @ T2 @ Y4 )
= Y4 ) )
=> ( Y4 = X5 ) ) )
=> ( partia3924075026459072775lambda @ OP2 ) ) ).
% partial_magma.intro
thf(fact_145_partial__magma_Ointro,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ? [X5: list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( ( OP2 @ X5 @ T3 )
= X5 )
& ( ( OP2 @ T3 @ X5 )
= X5 ) )
& ! [Y4: list_lambda_lambda] :
( ! [T2: list_lambda_lambda] :
( ( ( OP2 @ Y4 @ T2 )
= Y4 )
& ( ( OP2 @ T2 @ Y4 )
= Y4 ) )
=> ( Y4 = X5 ) ) )
=> ( partia748546660539056525lambda @ OP2 ) ) ).
% partial_magma.intro
thf(fact_146_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_147_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_148_red__iff,axiom,
( lambda_red
= ( ^ [A3: lambda_lambda,B3: lambda_lambda] :
? [T6: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T6 )
& ( ( paths_1651982994781619320lambda @ lambda_resid @ T6 )
= A3 )
& ( ( paths_2695323870737867955lambda @ lambda_resid @ T6 )
= B3 ) ) ) ) ).
% red_iff
thf(fact_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158__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_159_targets__cong__closed,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ B2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B2 @ B ) ) )
=> ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% targets_cong_closed
thf(fact_160_targets__are__cong,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B @ B2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B2 @ B ) ) ) ) ) ).
% targets_are_cong
thf(fact_161_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_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169__092_060Lambda_062_Ocong__implies__coinitial,axiom,
! [U: lambda_lambda,U3: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ U @ U3 ) ) ).
% \<Lambda>.cong_implies_coinitial
thf(fact_170__092_060Lambda_062_Ocoinitial__ide__are__eq,axiom,
! [A2: lambda_lambda,A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ A4 )
=> ( A2 = A4 ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_eq
thf(fact_171__092_060Lambda_062_Ocoinitial__ide__are__cong,axiom,
! [A2: lambda_lambda,A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( coinit1392077014771647542lambda @ lambda_resid @ A2 @ A4 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A4 @ A2 ) ) ) ) ) ) ).
% \<Lambda>.coinitial_ide_are_cong
thf(fact_172__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_173__092_060Lambda_062_Ocong__respects__seq,axiom,
! [T: lambda_lambda,U: lambda_lambda,T7: lambda_lambda,U3: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T7 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T7 @ T ) ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( seq_lambda_lambda @ lambda_resid @ T7 @ U3 ) ) ) ) ).
% \<Lambda>.cong_respects_seq
thf(fact_174__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_175__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_176_arr__char,axiom,
! [T4: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_366302053746739058lambda @ lambda_resid @ T4 ) ) ).
% arr_char
thf(fact_177_ide__char,axiom,
! [T4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( paths_7717490669901197737lambda @ lambda_resid @ T4 ) ) ).
% ide_char
thf(fact_178_cong__respects__seq_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T8: list_lambda_lambda,U4: 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 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U2 ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T8 @ U4 ) ) ) ) ).
% cong_respects_seq\<^sub>P
thf(fact_179_cong__respects__seq,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,T7: list_lambda_lambda,U3: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 @ U3 ) ) ) ) ).
% cong_respects_seq
thf(fact_180_cong__implies__coinitial,axiom,
! [U: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U3 ) ) ).
% cong_implies_coinitial
thf(fact_181_coinitial__ide__are__cong,axiom,
! [A2: list_lambda_lambda,A4: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A4 @ A2 ) ) ) ) ) ) ).
% coinitial_ide_are_cong
thf(fact_182_Resid__parallel,axiom,
! [T: list_lambda_lambda,T7: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T ) ) )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ U @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ U @ T7 ) ) ) ) ).
% Resid_parallel
thf(fact_183_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_184_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_185_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_186_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_187_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_188_cong__cons,axiom,
! [T: lambda_lambda,U2: list_lambda_lambda,T7: lambda_lambda,U4: 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 @ T7 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T7 @ T ) ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ U2 ) @ ( cons_lambda_lambda @ T7 @ U4 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T7 @ U4 ) @ ( cons_lambda_lambda @ T @ U2 ) ) ) ) ) ) ) ).
% cong_cons
thf(fact_189_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_190_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_191_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_192_rts_Ojoinable_Ocong,axiom,
joinab677607494065406278lambda = joinab677607494065406278lambda ).
% rts.joinable.cong
thf(fact_193_rts_Ojoinable_Ocong,axiom,
joinab2519788767067900224lambda = joinab2519788767067900224lambda ).
% rts.joinable.cong
thf(fact_194_lambda__calculus_Ored_Ointros_I2_J,axiom,
! [A2: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
( ( lambda_red @ A2 @ B )
=> ( ( lambda_red @ B @ C )
=> ( lambda_red @ A2 @ C ) ) ) ).
% lambda_calculus.red.intros(2)
thf(fact_195_lambda__calculus_ONF__unique,axiom,
! [T: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( lambda_red @ T @ U )
=> ( ( lambda_red @ T @ U3 )
=> ( ( lambda_NF @ U )
=> ( ( lambda_NF @ U3 )
=> ( U = U3 ) ) ) ) ) ).
% lambda_calculus.NF_unique
thf(fact_196_lambda__calculus_Oweak__diamond,axiom,
! [A2: lambda_lambda,B: lambda_lambda,B2: lambda_lambda] :
( ( lambda_red @ A2 @ B )
=> ( ( lambda_red @ A2 @ B2 )
=> ~ ! [C2: lambda_lambda] :
( ( lambda_red @ B @ C2 )
=> ~ ( lambda_red @ B2 @ C2 ) ) ) ) ).
% lambda_calculus.weak_diamond
thf(fact_197_lambda__calculus_ONF__reduct__is__trivial,axiom,
! [A2: lambda_lambda,B: lambda_lambda] :
( ( lambda_NF @ A2 )
=> ( ( lambda_red @ A2 @ B )
=> ( A2 = B ) ) ) ).
% lambda_calculus.NF_reduct_is_trivial
thf(fact_198_lambda__calculus_Ois__confluent__rts,axiom,
conflu4927472611300549594lambda @ lambda_resid ).
% lambda_calculus.is_confluent_rts
thf(fact_199_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_200_coterminal__iff,axiom,
! [T: list_lambda_lambda,T7: list_lambda_lambda] :
( ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T7 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 )
& ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 ) ) ) ) ).
% coterminal_iff
thf(fact_201_cong__implies__coterminal,axiom,
! [U: list_lambda_lambda,U3: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( coterm81126318419254786lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ U3 ) ) ).
% cong_implies_coterminal
thf(fact_202_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_203_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_204_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_205_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_206_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_207_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_208__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_209_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_210_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_211_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_212_append_Oassoc,axiom,
! [A2: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ A2 @ B ) @ C )
= ( append_lambda_lambda @ A2 @ ( append_lambda_lambda @ B @ C ) ) ) ).
% append.assoc
thf(fact_213_append__assoc,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( append_lambda_lambda @ ( append_lambda_lambda @ Xs @ Ys ) @ Zs3 )
= ( append_lambda_lambda @ Xs @ ( append_lambda_lambda @ Ys @ Zs3 ) ) ) ).
% append_assoc
thf(fact_214_append__same__eq,axiom,
! [Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Xs )
= ( append_lambda_lambda @ Zs3 @ Xs ) )
= ( Ys = Zs3 ) ) ).
% append_same_eq
thf(fact_215_same__append__eq,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Xs @ Zs3 ) )
= ( Ys = Zs3 ) ) ).
% same_append_eq
thf(fact_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_append__Nil2,axiom,
! [Xs: list_lambda_lambda] :
( ( append_lambda_lambda @ Xs @ nil_lambda_lambda )
= Xs ) ).
% append_Nil2
thf(fact_229_append_Oright__neutral,axiom,
! [A2: list_lambda_lambda] :
( ( append_lambda_lambda @ A2 @ nil_lambda_lambda )
= A2 ) ).
% append.right_neutral
thf(fact_230_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 ) )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_231_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_232_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_233_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_234_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_235_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_236_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_237_Resid__append2,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,V3: list_lambda_lambda,W: list_lambda_lambda] :
( ( T4 != nil_lambda_lambda )
=> ( ( U2 != nil_lambda_lambda )
=> ( ( V3 != nil_lambda_lambda )
=> ( ( W != nil_lambda_lambda )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V3 @ W ) )
!= nil_lambda_lambda )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ V3 @ W ) )
= ( append_lambda_lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) @ W ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ ( paths_8607357831768207192lambda @ lambda_resid @ V3 @ T4 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ V3 ) ) ) ) ) ) ) ) ) ) ).
% Resid_append2
thf(fact_238_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_239_cong__append,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda,T8: list_lambda_lambda,U4: 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 @ U4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U4 @ U2 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T4 @ U2 ) @ ( append_lambda_lambda @ T8 @ U4 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( append_lambda_lambda @ T8 @ U4 ) @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ) ) ) ) ).
% cong_append
thf(fact_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_append__eq__appendI,axiom,
! [Xs: list_lambda_lambda,Xs1: list_lambda_lambda,Zs3: list_lambda_lambda,Ys: list_lambda_lambda,Us: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Xs1 )
= Zs3 )
=> ( ( Ys
= ( append_lambda_lambda @ Xs1 @ Us ) )
=> ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs3 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_253_append__eq__append__conv2,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs3: list_lambda_lambda,Ts: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Xs @ Ys )
= ( append_lambda_lambda @ Zs3 @ Ts ) )
= ( ? [Us2: list_lambda_lambda] :
( ( ( Xs
= ( append_lambda_lambda @ Zs3 @ Us2 ) )
& ( ( append_lambda_lambda @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_lambda_lambda @ Xs @ Us2 )
= Zs3 )
& ( Ys
= ( append_lambda_lambda @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_254_split__list,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_255_split__list,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_256_split__list__last,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_257_split__list__last,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_258_split__list__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
& ( P @ X ) ) ) ).
% split_list_prop
thf(fact_259_split__list__first,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ? [Ys2: list_l5168835686021156086lambda,Zs2: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys2 @ ( cons_l4558873032622161958lambda @ X2 @ Zs2 ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_260_split__list__first,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
=> ? [Ys2: list_lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X2 @ Zs2 ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_261_split__list__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
=> ~ ( P @ X ) ) ) ).
% split_list_propE
thf(fact_262_append__Cons__eq__iff,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda,Ys: list_l5168835686021156086lambda,Xs3: list_l5168835686021156086lambda,Ys4: list_l5168835686021156086lambda] :
( ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys ) )
=> ( ( ( append1017360404025248779lambda @ Xs @ ( cons_l4558873032622161958lambda @ X2 @ Ys ) )
= ( append1017360404025248779lambda @ Xs3 @ ( cons_l4558873032622161958lambda @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_263_append__Cons__eq__iff,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Xs3: list_lambda_lambda,Ys4: 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 @ Xs3 @ ( cons_lambda_lambda @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_264_in__set__conv__decomp,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_265_in__set__conv__decomp,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Xs ) )
= ( ? [Ys3: list_lambda_lambda,Zs: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_266_split__list__last__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa: lambda_lambda] :
( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_267_split__list__first__prop,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Xa: lambda_lambda] :
( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_268_split__list__last__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda,Zs2: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: lambda_lambda] :
( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_269_split__list__first__propE,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ? [X5: lambda_lambda] :
( ( member_lambda_lambda @ X5 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_lambda_lambda,X: lambda_lambda] :
( ? [Zs2: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ X @ Zs2 ) ) )
=> ( ( P @ X )
=> ~ ! [Xa: lambda_lambda] :
( ( member_lambda_lambda @ Xa @ ( set_lambda_lambda2 @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_270_in__set__conv__decomp__last,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_271_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,Zs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_272_in__set__conv__decomp__first,axiom,
! [X2: list_lambda_lambda,Xs: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Xs ) )
= ( ? [Ys3: list_l5168835686021156086lambda,Zs: list_l5168835686021156086lambda] :
( ( Xs
= ( append1017360404025248779lambda @ Ys3 @ ( cons_l4558873032622161958lambda @ X2 @ Zs ) ) )
& ~ ( member2667661554319814297lambda @ X2 @ ( set_li3846194476672328965lambda @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_273_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,Zs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X2 @ Zs ) ) )
& ~ ( member_lambda_lambda @ X2 @ ( set_lambda_lambda2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_274_split__list__last__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_lambda_lambda,X4: lambda_lambda,Zs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X4 @ Zs ) ) )
& ( P @ X4 )
& ! [Y5: lambda_lambda] :
( ( member_lambda_lambda @ Y5 @ ( set_lambda_lambda2 @ Zs ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_275_split__list__first__prop__iff,axiom,
! [Xs: list_lambda_lambda,P: lambda_lambda > $o] :
( ( ? [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [Ys3: list_lambda_lambda,X4: lambda_lambda] :
( ? [Zs: list_lambda_lambda] :
( Xs
= ( append_lambda_lambda @ Ys3 @ ( cons_lambda_lambda @ X4 @ Zs ) ) )
& ( P @ X4 )
& ! [Y5: lambda_lambda] :
( ( member_lambda_lambda @ Y5 @ ( set_lambda_lambda2 @ Ys3 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_276_rts_Ocoterminal_Ocong,axiom,
coterm81126318419254786lambda = coterm81126318419254786lambda ).
% rts.coterminal.cong
thf(fact_277_rts_Ocoterminal_Ocong,axiom,
coterm4142657152830665724lambda = coterm4142657152830665724lambda ).
% rts.coterminal.cong
thf(fact_278_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_279_Cons__eq__appendI,axiom,
! [X2: lambda_lambda,Xs1: list_lambda_lambda,Ys: list_lambda_lambda,Xs: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_lambda_lambda @ Xs1 @ Zs3 ) )
=> ( ( cons_lambda_lambda @ X2 @ Xs )
= ( append_lambda_lambda @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_280_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_281_append_Oleft__neutral,axiom,
! [A2: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_282_append__Nil,axiom,
! [Ys: list_lambda_lambda] :
( ( append_lambda_lambda @ nil_lambda_lambda @ Ys )
= Ys ) ).
% append_Nil
thf(fact_283_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_284_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_285_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_286_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_287_list_Oset__cases,axiom,
! [E: list_lambda_lambda,A2: list_l5168835686021156086lambda] :
( ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ A2 ) )
=> ( ! [Z22: list_l5168835686021156086lambda] :
( A2
!= ( cons_l4558873032622161958lambda @ E @ Z22 ) )
=> ~ ! [Z1: list_lambda_lambda,Z22: list_l5168835686021156086lambda] :
( ( A2
= ( cons_l4558873032622161958lambda @ Z1 @ Z22 ) )
=> ~ ( member2667661554319814297lambda @ E @ ( set_li3846194476672328965lambda @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_288_list_Oset__cases,axiom,
! [E: lambda_lambda,A2: list_lambda_lambda] :
( ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ A2 ) )
=> ( ! [Z22: list_lambda_lambda] :
( A2
!= ( cons_lambda_lambda @ E @ Z22 ) )
=> ~ ! [Z1: lambda_lambda,Z22: list_lambda_lambda] :
( ( A2
= ( cons_lambda_lambda @ Z1 @ Z22 ) )
=> ~ ( member_lambda_lambda @ E @ ( set_lambda_lambda2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_289_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_290_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_291_subset__code_I1_J,axiom,
! [Xs: list_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ Xs ) @ B4 )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Xs ) )
=> ( member_lambda_lambda @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_292_subset__code_I1_J,axiom,
! [Xs: list_l5168835686021156086lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( set_li3846194476672328965lambda @ Xs ) @ B4 )
= ( ! [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ ( set_li3846194476672328965lambda @ Xs ) )
=> ( member2667661554319814297lambda @ X4 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_293_map__eq__append__conv,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( ( map_la336867333220232516lambda @ F @ Xs )
= ( append_lambda_lambda @ Ys @ Zs3 ) )
= ( ? [Us2: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us2 ) )
& ( Zs3
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_294_append__eq__map__conv,axiom,
! [Ys: list_lambda_lambda,Zs3: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs3 )
= ( map_la336867333220232516lambda @ F @ Xs ) )
= ( ? [Us2: list_lambda_lambda,Vs: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Us2 @ Vs ) )
& ( Ys
= ( map_la336867333220232516lambda @ F @ Us2 ) )
& ( Zs3
= ( map_la336867333220232516lambda @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_295_ex__map__conv,axiom,
! [Ys: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( ? [Xs4: list_lambda_lambda] :
( Ys
= ( map_la336867333220232516lambda @ F @ Xs4 ) ) )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ ( set_lambda_lambda2 @ Ys ) )
=> ? [Y5: lambda_lambda] :
( X4
= ( F @ Y5 ) ) ) ) ) ).
% ex_map_conv
thf(fact_296_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_297_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_298_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_299_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_300_list_Omap__ident__strong,axiom,
! [T: list_l5168835686021156086lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ! [Z2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Z2 @ ( set_li3846194476672328965lambda @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_li385728805257398596lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_301_list_Omap__ident__strong,axiom,
! [T: list_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ! [Z2: lambda_lambda] :
( ( member_lambda_lambda @ Z2 @ ( set_lambda_lambda2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_la336867333220232516lambda @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_302_list_Oinj__map__strong,axiom,
! [X2: list_lambda_lambda,Xa2: list_lambda_lambda,F: lambda_lambda > lambda_lambda,Fa: lambda_lambda > lambda_lambda] :
( ! [Z2: lambda_lambda,Za: lambda_lambda] :
( ( member_lambda_lambda @ Z2 @ ( set_lambda_lambda2 @ X2 ) )
=> ( ( member_lambda_lambda @ Za @ ( set_lambda_lambda2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_303_list_Omap__cong0,axiom,
! [X2: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ! [Z2: lambda_lambda] :
( ( member_lambda_lambda @ Z2 @ ( set_lambda_lambda2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_304_list_Omap__cong,axiom,
! [X2: list_lambda_lambda,Ya: list_lambda_lambda,F: lambda_lambda > lambda_lambda,G: lambda_lambda > lambda_lambda] :
( ( X2 = Ya )
=> ( ! [Z2: lambda_lambda] :
( ( member_lambda_lambda @ Z2 @ ( set_lambda_lambda2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_la336867333220232516lambda @ F @ X2 )
= ( map_la336867333220232516lambda @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_305_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_306_rev__exhaust,axiom,
! [Xs: list_lambda_lambda] :
( ( Xs != nil_lambda_lambda )
=> ~ ! [Ys2: list_lambda_lambda,Y4: lambda_lambda] :
( Xs
!= ( append_lambda_lambda @ Ys2 @ ( cons_lambda_lambda @ Y4 @ nil_lambda_lambda ) ) ) ) ).
% rev_exhaust
thf(fact_307_Cons__eq__append__conv,axiom,
! [X2: lambda_lambda,Xs: list_lambda_lambda,Ys: list_lambda_lambda,Zs3: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Xs )
= ( append_lambda_lambda @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( ( cons_lambda_lambda @ X2 @ Xs )
= Zs3 ) )
| ? [Ys5: list_lambda_lambda] :
( ( ( cons_lambda_lambda @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_lambda_lambda @ Ys5 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_308_append__eq__Cons__conv,axiom,
! [Ys: list_lambda_lambda,Zs3: list_lambda_lambda,X2: lambda_lambda,Xs: list_lambda_lambda] :
( ( ( append_lambda_lambda @ Ys @ Zs3 )
= ( cons_lambda_lambda @ X2 @ Xs ) )
= ( ( ( Ys = nil_lambda_lambda )
& ( Zs3
= ( cons_lambda_lambda @ X2 @ Xs ) ) )
| ? [Ys5: list_lambda_lambda] :
( ( Ys
= ( cons_lambda_lambda @ X2 @ Ys5 ) )
& ( ( append_lambda_lambda @ Ys5 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_309_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_310_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_311_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_312_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_313_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_314__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_315_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_316_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_317_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_318_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_319__092_060Lambda_062_Ocoterminal__iff,axiom,
! [T: lambda_lambda,T7: lambda_lambda] :
( ( coterm4142657152830665724lambda @ lambda_resid @ T @ T7 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T7 )
& ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T7 ) ) ) ) ).
% \<Lambda>.coterminal_iff
thf(fact_320__092_060Lambda_062_Otrg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ T )
= ( lambda_resid @ T @ T ) ) ).
% \<Lambda>.trg_def
thf(fact_321__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_322_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_323_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_324__092_060Lambda_062_Ocong__implies__coterminal,axiom,
! [U: lambda_lambda,U3: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( coterm4142657152830665724lambda @ lambda_resid @ U @ U3 ) ) ).
% \<Lambda>.cong_implies_coterminal
thf(fact_325__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_326__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_327__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_328__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_329_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_330__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_331__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_332__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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340__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_341_set__empty2,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ Xs ) )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty2
thf(fact_342_set__empty2,axiom,
! [Xs: list_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ Xs ) )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty2
thf(fact_343_set__empty,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( ( set_li3846194476672328965lambda @ Xs )
= bot_bo1397037886704484924lambda )
= ( Xs = nil_li8379291187143545974lambda ) ) ).
% set_empty
thf(fact_344_set__empty,axiom,
! [Xs: list_lambda_lambda] :
( ( ( set_lambda_lambda2 @ Xs )
= bot_bo1271416176687173302lambda )
= ( Xs = nil_lambda_lambda ) ) ).
% set_empty
thf(fact_345__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_346__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_347_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_348_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_349__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_350__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_351_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_352_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_353_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_354_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_355_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_356_paths__in__rts_OTrgs_Ocong,axiom,
paths_7594662086398683701lambda = paths_7594662086398683701lambda ).
% paths_in_rts.Trgs.cong
thf(fact_357_residuation_Otrg_Ocong,axiom,
trg_lambda_lambda = trg_lambda_lambda ).
% residuation.trg.cong
thf(fact_358_residuation_Otrg_Ocong,axiom,
trg_li817795487149935061lambda = trg_li817795487149935061lambda ).
% residuation.trg.cong
thf(fact_359_lambda__calculus_Ois__Lam__def,axiom,
( lambda_is_Lam
= ( ^ [Lambda: lambda_lambda] :
? [X32: lambda_lambda] :
( Lambda
= ( lambda_Lam @ X32 ) ) ) ) ).
% lambda_calculus.is_Lam_def
thf(fact_360_lambda__calculus_Olambda_OdiscI_I2_J,axiom,
! [Lambda2: lambda_lambda,X3: lambda_lambda] :
( ( Lambda2
= ( lambda_Lam @ X3 ) )
=> ( lambda_is_Lam @ Lambda2 ) ) ).
% lambda_calculus.lambda.discI(2)
thf(fact_361_lambda__calculus_Olambda_Odisc_I8_J,axiom,
! [X3: lambda_lambda] : ( lambda_is_Lam @ ( lambda_Lam @ X3 ) ) ).
% lambda_calculus.lambda.disc(8)
thf(fact_362_empty__set,axiom,
( bot_bo1397037886704484924lambda
= ( set_li3846194476672328965lambda @ nil_li8379291187143545974lambda ) ) ).
% empty_set
thf(fact_363_empty__set,axiom,
( bot_bo1271416176687173302lambda
= ( set_lambda_lambda2 @ nil_lambda_lambda ) ) ).
% empty_set
thf(fact_364_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_365_subset__empty,axiom,
! [A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ bot_bo1271416176687173302lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% subset_empty
thf(fact_366_subset__empty,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ bot_bo1397037886704484924lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% subset_empty
thf(fact_367_empty__subsetI,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A ) ).
% empty_subsetI
thf(fact_368_empty__subsetI,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A ) ).
% empty_subsetI
thf(fact_369_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_370_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_371_targets__eqI,axiom,
! [T: list_lambda_lambda,T7: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 ) ) ) ).
% targets_eqI
thf(fact_372_trg__def,axiom,
! [T: list_lambda_lambda] :
( ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ).
% trg_def
thf(fact_373_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_374__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_375_Srcs_Osimps_I1_J,axiom,
( ( paths_2897053694880971696lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Srcs.simps(1)
thf(fact_376_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_377_Trgs_Osimps_I1_J,axiom,
( ( paths_7594662086398683701lambda @ lambda_resid @ nil_lambda_lambda )
= bot_bo1271416176687173302lambda ) ).
% Trgs.simps(1)
thf(fact_378_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_379_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_380_empty__Collect__eq,axiom,
! [P: list_lambda_lambda > $o] :
( ( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda @ P ) )
= ( ! [X4: list_lambda_lambda] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_381_empty__Collect__eq,axiom,
! [P: lambda_lambda > $o] :
( ( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ P ) )
= ( ! [X4: lambda_lambda] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_382_Collect__empty__eq,axiom,
! [P: list_lambda_lambda > $o] :
( ( ( collec8830145823237851355lambda @ P )
= bot_bo1397037886704484924lambda )
= ( ! [X4: list_lambda_lambda] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_383_Collect__empty__eq,axiom,
! [P: lambda_lambda > $o] :
( ( ( collec621444948832755541lambda @ P )
= bot_bo1271416176687173302lambda )
= ( ! [X4: lambda_lambda] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_384_all__not__in__conv,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ! [X4: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ X4 @ A ) )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% all_not_in_conv
thf(fact_385_all__not__in__conv,axiom,
! [A: set_lambda_lambda] :
( ( ! [X4: lambda_lambda] :
~ ( member_lambda_lambda @ X4 @ A ) )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% all_not_in_conv
thf(fact_386_empty__iff,axiom,
! [C: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ C @ bot_bo1397037886704484924lambda ) ).
% empty_iff
thf(fact_387_empty__iff,axiom,
! [C: lambda_lambda] :
~ ( member_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ).
% empty_iff
thf(fact_388_subset__antisym,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ( ord_le1926335974945931754lambda @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_389_subset__antisym,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ( ord_le3064231726521830256lambda @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_390_subsetI,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( member_lambda_lambda @ X @ B4 ) )
=> ( ord_le1926335974945931754lambda @ A @ B4 ) ) ).
% subsetI
thf(fact_391_subsetI,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( member2667661554319814297lambda @ X @ B4 ) )
=> ( ord_le3064231726521830256lambda @ A @ B4 ) ) ).
% subsetI
thf(fact_392_Int__subset__iff,axiom,
! [C3: set_lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C3 @ ( inf_in1766959559283430712lambda @ A @ B4 ) )
= ( ( ord_le1926335974945931754lambda @ C3 @ A )
& ( ord_le1926335974945931754lambda @ C3 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_393_Int__subset__iff,axiom,
! [C3: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C3 @ ( inf_in3354698378728726462lambda @ A @ B4 ) )
= ( ( ord_le3064231726521830256lambda @ C3 @ A )
& ( ord_le3064231726521830256lambda @ C3 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_394_lambda__calculus_Olambda_Ocollapse_I2_J,axiom,
! [Lambda2: lambda_lambda] :
( ( lambda_is_Lam @ Lambda2 )
=> ( ( lambda_Lam @ ( lambda_un_Lam @ Lambda2 ) )
= Lambda2 ) ) ).
% lambda_calculus.lambda.collapse(2)
thf(fact_395_Int__emptyI,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ~ ( member2667661554319814297lambda @ X @ B4 ) )
=> ( ( inf_in3354698378728726462lambda @ A @ B4 )
= bot_bo1397037886704484924lambda ) ) ).
% Int_emptyI
thf(fact_396_Int__emptyI,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ~ ( member_lambda_lambda @ X @ B4 ) )
=> ( ( inf_in1766959559283430712lambda @ A @ B4 )
= bot_bo1271416176687173302lambda ) ) ).
% Int_emptyI
thf(fact_397_disjoint__iff,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ A @ B4 )
= bot_bo1397037886704484924lambda )
= ( ! [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A )
=> ~ ( member2667661554319814297lambda @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_398_disjoint__iff,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A @ B4 )
= bot_bo1271416176687173302lambda )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
=> ~ ( member_lambda_lambda @ X4 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_399_Int__empty__left,axiom,
! [B4: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ B4 )
= bot_bo1397037886704484924lambda ) ).
% Int_empty_left
thf(fact_400_Int__empty__left,axiom,
! [B4: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ B4 )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_left
thf(fact_401_Int__empty__right,axiom,
! [A: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% Int_empty_right
thf(fact_402_Int__empty__right,axiom,
! [A: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% Int_empty_right
thf(fact_403_disjoint__iff__not__equal,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ A @ B4 )
= bot_bo1397037886704484924lambda )
= ( ! [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A )
=> ! [Y5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ Y5 @ B4 )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_404_disjoint__iff__not__equal,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ A @ B4 )
= bot_bo1271416176687173302lambda )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
=> ! [Y5: lambda_lambda] :
( ( member_lambda_lambda @ Y5 @ B4 )
=> ( X4 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_405_Int__Collect__mono,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda,P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ ( collec621444948832755541lambda @ P ) ) @ ( inf_in1766959559283430712lambda @ B4 @ ( collec621444948832755541lambda @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_406_Int__Collect__mono,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda,P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( ( P @ X )
=> ( Q @ X ) ) )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ ( collec8830145823237851355lambda @ P ) ) @ ( inf_in3354698378728726462lambda @ B4 @ ( collec8830145823237851355lambda @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_407_Int__greatest,axiom,
! [C3: set_lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C3 @ A )
=> ( ( ord_le1926335974945931754lambda @ C3 @ B4 )
=> ( ord_le1926335974945931754lambda @ C3 @ ( inf_in1766959559283430712lambda @ A @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_408_Int__greatest,axiom,
! [C3: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C3 @ A )
=> ( ( ord_le3064231726521830256lambda @ C3 @ B4 )
=> ( ord_le3064231726521830256lambda @ C3 @ ( inf_in3354698378728726462lambda @ A @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_409_Int__absorb2,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ( inf_in1766959559283430712lambda @ A @ B4 )
= A ) ) ).
% Int_absorb2
thf(fact_410_Int__absorb2,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ( inf_in3354698378728726462lambda @ A @ B4 )
= A ) ) ).
% Int_absorb2
thf(fact_411_Int__absorb1,axiom,
! [B4: set_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B4 @ A )
=> ( ( inf_in1766959559283430712lambda @ A @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_412_Int__absorb1,axiom,
! [B4: set_li7878742357367608272lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B4 @ A )
=> ( ( inf_in3354698378728726462lambda @ A @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_413_Int__lower2,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_414_Int__lower2,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_415_Int__lower1,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B4 ) @ A ) ).
% Int_lower1
thf(fact_416_Int__lower1,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B4 ) @ A ) ).
% Int_lower1
thf(fact_417_Int__mono,axiom,
! [A: set_lambda_lambda,C3: set_lambda_lambda,B4: set_lambda_lambda,D: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ C3 )
=> ( ( ord_le1926335974945931754lambda @ B4 @ D )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A @ B4 ) @ ( inf_in1766959559283430712lambda @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_418_Int__mono,axiom,
! [A: set_li7878742357367608272lambda,C3: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda,D: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ C3 )
=> ( ( ord_le3064231726521830256lambda @ B4 @ D )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A @ B4 ) @ ( inf_in3354698378728726462lambda @ C3 @ D ) ) ) ) ).
% Int_mono
thf(fact_419_lambda__calculus_Olambda_Osel_I2_J,axiom,
! [X3: lambda_lambda] :
( ( lambda_un_Lam @ ( lambda_Lam @ X3 ) )
= X3 ) ).
% lambda_calculus.lambda.sel(2)
thf(fact_420_ex__in__conv,axiom,
! [A: set_li7878742357367608272lambda] :
( ( ? [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ A ) )
= ( A != bot_bo1397037886704484924lambda ) ) ).
% ex_in_conv
thf(fact_421_ex__in__conv,axiom,
! [A: set_lambda_lambda] :
( ( ? [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ A ) )
= ( A != bot_bo1271416176687173302lambda ) ) ).
% ex_in_conv
thf(fact_422_equals0I,axiom,
! [A: set_li7878742357367608272lambda] :
( ! [Y4: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ Y4 @ A )
=> ( A = bot_bo1397037886704484924lambda ) ) ).
% equals0I
thf(fact_423_equals0I,axiom,
! [A: set_lambda_lambda] :
( ! [Y4: lambda_lambda] :
~ ( member_lambda_lambda @ Y4 @ A )
=> ( A = bot_bo1271416176687173302lambda ) ) ).
% equals0I
thf(fact_424_equals0D,axiom,
! [A: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
( ( A = bot_bo1397037886704484924lambda )
=> ~ ( member2667661554319814297lambda @ A2 @ A ) ) ).
% equals0D
thf(fact_425_equals0D,axiom,
! [A: set_lambda_lambda,A2: lambda_lambda] :
( ( A = bot_bo1271416176687173302lambda )
=> ~ ( member_lambda_lambda @ A2 @ A ) ) ).
% equals0D
thf(fact_426_emptyE,axiom,
! [A2: list_lambda_lambda] :
~ ( member2667661554319814297lambda @ A2 @ bot_bo1397037886704484924lambda ) ).
% emptyE
thf(fact_427_emptyE,axiom,
! [A2: lambda_lambda] :
~ ( member_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ).
% emptyE
thf(fact_428_Collect__mono__iff,axiom,
! [P: lambda_lambda > $o,Q: lambda_lambda > $o] :
( ( ord_le1926335974945931754lambda @ ( collec621444948832755541lambda @ P ) @ ( collec621444948832755541lambda @ Q ) )
= ( ! [X4: lambda_lambda] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_429_Collect__mono__iff,axiom,
! [P: list_lambda_lambda > $o,Q: list_lambda_lambda > $o] :
( ( ord_le3064231726521830256lambda @ ( collec8830145823237851355lambda @ P ) @ ( collec8830145823237851355lambda @ Q ) )
= ( ! [X4: list_lambda_lambda] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_430_set__eq__subset,axiom,
( ( ^ [Y6: set_lambda_lambda,Z3: set_lambda_lambda] : ( Y6 = Z3 ) )
= ( ^ [A5: set_lambda_lambda,B5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A5 @ B5 )
& ( ord_le1926335974945931754lambda @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_431_set__eq__subset,axiom,
( ( ^ [Y6: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] : ( Y6 = Z3 ) )
= ( ^ [A5: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A5 @ B5 )
& ( ord_le3064231726521830256lambda @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_432_subset__trans,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda,C3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ( ord_le1926335974945931754lambda @ B4 @ C3 )
=> ( ord_le1926335974945931754lambda @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_433_subset__trans,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda,C3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ( ord_le3064231726521830256lambda @ B4 @ C3 )
=> ( ord_le3064231726521830256lambda @ A @ C3 ) ) ) ).
% subset_trans
thf(fact_434_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_435_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_436_subset__refl,axiom,
! [A: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A @ A ) ).
% subset_refl
thf(fact_437_subset__refl,axiom,
! [A: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ A @ A ) ).
% subset_refl
thf(fact_438_subset__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A5: set_lambda_lambda,B5: set_lambda_lambda] :
! [T5: lambda_lambda] :
( ( member_lambda_lambda @ T5 @ A5 )
=> ( member_lambda_lambda @ T5 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_439_subset__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A5: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
! [T5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ T5 @ A5 )
=> ( member2667661554319814297lambda @ T5 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_440_equalityD2,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( A = B4 )
=> ( ord_le1926335974945931754lambda @ B4 @ A ) ) ).
% equalityD2
thf(fact_441_equalityD2,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( A = B4 )
=> ( ord_le3064231726521830256lambda @ B4 @ A ) ) ).
% equalityD2
thf(fact_442_equalityD1,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( A = B4 )
=> ( ord_le1926335974945931754lambda @ A @ B4 ) ) ).
% equalityD1
thf(fact_443_equalityD1,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( A = B4 )
=> ( ord_le3064231726521830256lambda @ A @ B4 ) ) ).
% equalityD1
thf(fact_444_subset__eq,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A5: set_lambda_lambda,B5: set_lambda_lambda] :
! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A5 )
=> ( member_lambda_lambda @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_445_subset__eq,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A5: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
! [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A5 )
=> ( member2667661554319814297lambda @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_446_equalityE,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( A = B4 )
=> ~ ( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ~ ( ord_le1926335974945931754lambda @ B4 @ A ) ) ) ).
% equalityE
thf(fact_447_equalityE,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( A = B4 )
=> ~ ( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ~ ( ord_le3064231726521830256lambda @ B4 @ A ) ) ) ).
% equalityE
thf(fact_448_subsetD,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda,C: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ( member_lambda_lambda @ C @ A )
=> ( member_lambda_lambda @ C @ B4 ) ) ) ).
% subsetD
thf(fact_449_subsetD,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ( member2667661554319814297lambda @ C @ A )
=> ( member2667661554319814297lambda @ C @ B4 ) ) ) ).
% subsetD
thf(fact_450_in__mono,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda,X2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ( member_lambda_lambda @ X2 @ A )
=> ( member_lambda_lambda @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_451_in__mono,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda,X2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ( member2667661554319814297lambda @ X2 @ A )
=> ( member2667661554319814297lambda @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_452_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_453_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_454_inf__bot__left,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X2 )
= bot_bo1397037886704484924lambda ) ).
% inf_bot_left
thf(fact_455_inf__bot__left,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X2 )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_left
thf(fact_456_inf__bot__right,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% inf_bot_right
thf(fact_457_inf__bot__right,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% inf_bot_right
thf(fact_458_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ bot_bo1397037886704484924lambda @ X2 )
= bot_bo1397037886704484924lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_459_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ bot_bo1271416176687173302lambda @ X2 )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_left
thf(fact_460_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X2 @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_461_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X2 @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% boolean_algebra.conj_zero_right
thf(fact_462_inf_Obounded__iff,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B @ C ) )
= ( ( ord_le1926335974945931754lambda @ A2 @ B )
& ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_463_inf_Obounded__iff,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B @ C ) )
= ( ( ord_le3064231726521830256lambda @ A2 @ B )
& ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_464__092_060Lambda_062_Otargets__eqI,axiom,
! [T: lambda_lambda,T7: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( target5490931411331770124lambda @ lambda_resid @ T ) @ ( target5490931411331770124lambda @ lambda_resid @ T7 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( target5490931411331770124lambda @ lambda_resid @ T7 ) ) ) ).
% \<Lambda>.targets_eqI
thf(fact_465_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_466_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_467__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_468_le__inf__iff,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z4 ) )
= ( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
& ( ord_le1926335974945931754lambda @ X2 @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_469_le__inf__iff,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z4 ) )
= ( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
& ( ord_le3064231726521830256lambda @ X2 @ Z4 ) ) ) ).
% le_inf_iff
thf(fact_470_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_471_bot__set__def,axiom,
( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda @ bot_bo430095131446760353mbda_o ) ) ).
% bot_set_def
thf(fact_472_bot__set__def,axiom,
( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda @ bot_bo3555140721347798439mbda_o ) ) ).
% bot_set_def
thf(fact_473_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_474_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_475_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_476_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_477_inf__le1,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_478_inf__le1,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_479_inf__le2,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_480_inf__le2,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_481_le__infE,axiom,
! [X2: set_lambda_lambda,A2: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ A2 @ B ) )
=> ~ ( ( ord_le1926335974945931754lambda @ X2 @ A2 )
=> ~ ( ord_le1926335974945931754lambda @ X2 @ B ) ) ) ).
% le_infE
thf(fact_482_le__infE,axiom,
! [X2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ A2 @ B ) )
=> ~ ( ( ord_le3064231726521830256lambda @ X2 @ A2 )
=> ~ ( ord_le3064231726521830256lambda @ X2 @ B ) ) ) ).
% le_infE
thf(fact_483_le__infI,axiom,
! [X2: set_lambda_lambda,A2: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ A2 )
=> ( ( ord_le1926335974945931754lambda @ X2 @ B )
=> ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_484_le__infI,axiom,
! [X2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ A2 )
=> ( ( ord_le3064231726521830256lambda @ X2 @ B )
=> ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_485_inf__mono,axiom,
! [A2: set_lambda_lambda,C: set_lambda_lambda,B: set_lambda_lambda,D2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ( ord_le1926335974945931754lambda @ B @ D2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B ) @ ( inf_in1766959559283430712lambda @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_486_inf__mono,axiom,
! [A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,D2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ( ord_le3064231726521830256lambda @ B @ D2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B ) @ ( inf_in3354698378728726462lambda @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_487_le__infI1,axiom,
! [A2: set_lambda_lambda,X2: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ X2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_488_le__infI1,axiom,
! [A2: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ X2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_489_le__infI2,axiom,
! [B: set_lambda_lambda,X2: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ X2 )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_490_le__infI2,axiom,
! [B: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ X2 )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_491_inf_OorderE,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( A2
= ( inf_in1766959559283430712lambda @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_492_inf_OorderE,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( A2
= ( inf_in3354698378728726462lambda @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_493_inf_OorderI,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda] :
( ( A2
= ( inf_in1766959559283430712lambda @ A2 @ B ) )
=> ( ord_le1926335974945931754lambda @ A2 @ B ) ) ).
% inf.orderI
thf(fact_494_inf_OorderI,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( A2
= ( inf_in3354698378728726462lambda @ A2 @ B ) )
=> ( ord_le3064231726521830256lambda @ A2 @ B ) ) ).
% inf.orderI
thf(fact_495_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,Y4: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda,Z2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ( ord_le1926335974945931754lambda @ X @ Z2 )
=> ( ord_le1926335974945931754lambda @ X @ ( F @ Y4 @ Z2 ) ) ) )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_496_inf__unique,axiom,
! [F: set_li7878742357367608272lambda > set_li7878742357367608272lambda > set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda,Z2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ( ord_le3064231726521830256lambda @ X @ Z2 )
=> ( ord_le3064231726521830256lambda @ X @ ( F @ Y4 @ Z2 ) ) ) )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_497_le__iff__inf,axiom,
( ord_le1926335974945931754lambda
= ( ^ [X4: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_498_le__iff__inf,axiom,
( ord_le3064231726521830256lambda
= ( ^ [X4: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ X4 @ Y5 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_499_inf_Oabsorb1,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_500_inf_Oabsorb1,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_501_inf_Oabsorb2,axiom,
! [B: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A2 )
=> ( ( inf_in1766959559283430712lambda @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_502_inf_Oabsorb2,axiom,
! [B: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A2 )
=> ( ( inf_in3354698378728726462lambda @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_503_inf__absorb1,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_504_inf__absorb1,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_505_inf__absorb2,axiom,
! [Y2: set_lambda_lambda,X2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ Y2 @ X2 )
=> ( ( inf_in1766959559283430712lambda @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_506_inf__absorb2,axiom,
! [Y2: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ Y2 @ X2 )
=> ( ( inf_in3354698378728726462lambda @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_507_inf_OboundedE,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B @ C ) )
=> ~ ( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ~ ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_508_inf_OboundedE,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B @ C ) )
=> ~ ( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ~ ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_509_inf_OboundedI,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ ( inf_in1766959559283430712lambda @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_510_inf_OboundedI,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ord_le3064231726521830256lambda @ A2 @ ( inf_in3354698378728726462lambda @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_511_inf__greatest,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ X2 @ Z4 )
=> ( ord_le1926335974945931754lambda @ X2 @ ( inf_in1766959559283430712lambda @ Y2 @ Z4 ) ) ) ) ).
% inf_greatest
thf(fact_512_inf__greatest,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ X2 @ Z4 )
=> ( ord_le3064231726521830256lambda @ X2 @ ( inf_in3354698378728726462lambda @ Y2 @ Z4 ) ) ) ) ).
% inf_greatest
thf(fact_513_inf_Oorder__iff,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( A3
= ( inf_in1766959559283430712lambda @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_514_inf_Oorder__iff,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A3: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( A3
= ( inf_in3354698378728726462lambda @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_515_inf_Ocobounded1,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_516_inf_Ocobounded1,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_517_inf_Ocobounded2,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_518_inf_Ocobounded2,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_519_inf_Oabsorb__iff1,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_520_inf_Oabsorb__iff1,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A3: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_521_inf_Oabsorb__iff2,axiom,
( ord_le1926335974945931754lambda
= ( ^ [B3: set_lambda_lambda,A3: set_lambda_lambda] :
( ( inf_in1766959559283430712lambda @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_522_inf_Oabsorb__iff2,axiom,
( ord_le3064231726521830256lambda
= ( ^ [B3: set_li7878742357367608272lambda,A3: set_li7878742357367608272lambda] :
( ( inf_in3354698378728726462lambda @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_523_inf_OcoboundedI1,axiom,
! [A2: set_lambda_lambda,C: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_524_inf_OcoboundedI1,axiom,
! [A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_525_inf_OcoboundedI2,axiom,
! [B: set_lambda_lambda,C: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ ( inf_in1766959559283430712lambda @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_526_inf_OcoboundedI2,axiom,
! [B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ ( inf_in3354698378728726462lambda @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_527_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_528_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_529_in__targetsE,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ~ ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B ) ) ) ).
% in_targetsE
thf(fact_530_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_531_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_532__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_533__092_060Lambda_062_Osources__are__cong,axiom,
! [A2: lambda_lambda,T: lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A4 @ A2 ) ) ) ) ) ).
% \<Lambda>.sources_are_cong
thf(fact_534__092_060Lambda_062_Osources__cong__closed,axiom,
! [A2: lambda_lambda,T: lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A2 @ A4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ A4 @ A2 ) ) )
=> ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.sources_cong_closed
thf(fact_535_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_536_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_537__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_538__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_539_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_540_cong__subst__left_I2_J,axiom,
! [T: list_lambda_lambda,T7: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ 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 @ T7 @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_left(2)
thf(fact_541_cong__subst__left_I1_J,axiom,
! [T: list_lambda_lambda,T7: list_lambda_lambda,U: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T7 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T7 @ T ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 @ U ) ) ) ).
% cong_subst_left(1)
thf(fact_542_cong__subst__right_I2_J,axiom,
! [U: list_lambda_lambda,U3: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U3 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U3 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) ) ) ) ) ).
% cong_subst_right(2)
thf(fact_543_cong__subst__right_I1_J,axiom,
! [U: list_lambda_lambda,U3: list_lambda_lambda,T: list_lambda_lambda] :
( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U3 ) ) ) ).
% cong_subst_right(1)
thf(fact_544_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_545_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_546_con__transitive__on__ide,axiom,
! [A2: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ C )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ B )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ C )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ C ) ) ) ) ) ) ).
% con_transitive_on_ide
thf(fact_547_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_548_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_549_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_550_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_551_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_552_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_553_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_554_sources__are__cong,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda,A4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A4 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A4 @ A2 ) ) ) ) ) ).
% sources_are_cong
thf(fact_555_sources__cong__closed,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda,A4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A2 @ A4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A4 @ A2 ) ) )
=> ( member2667661554319814297lambda @ A4 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% sources_cong_closed
thf(fact_556_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_557__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_558_sources__are__con,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda,A4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ A4 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A4 ) ) ) ).
% sources_are_con
thf(fact_559_targets__are__con,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ B2 ) ) ) ).
% targets_are_con
thf(fact_560_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_561_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_562_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_563_arrE,axiom,
! [T: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T ) ) ).
% arrE
thf(fact_564_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_565_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_566_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_567_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_568__092_060Lambda_062_Ocoinitial__iff,axiom,
! [T: lambda_lambda,T7: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ T7 )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ T7 )
& ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T7 ) ) ) ) ).
% \<Lambda>.coinitial_iff
thf(fact_569__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_570_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_571_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_572_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_573__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_574__092_060Lambda_062_Osources__eqI,axiom,
! [T: lambda_lambda,T7: lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( source5627170679743844886lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T7 ) )
!= bot_bo1271416176687173302lambda )
=> ( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( source5627170679743844886lambda @ lambda_resid @ T7 ) ) ) ).
% \<Lambda>.sources_eqI
thf(fact_575_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_576_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_577_sources__con__closed,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda,A4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A4 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A4 )
=> ( member2667661554319814297lambda @ A4 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% sources_con_closed
thf(fact_578_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_579_targets__con__closed,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda,B2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B @ B2 )
=> ( member2667661554319814297lambda @ B2 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ) ).
% targets_con_closed
thf(fact_580__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_581__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_582_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_583_coinitial__iff,axiom,
! [T: list_lambda_lambda,T7: list_lambda_lambda] :
( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T7 )
= ( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 )
& ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 ) ) ) ) ).
% coinitial_iff
thf(fact_584_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_585__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_586_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_587_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_588_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_589_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_590_sources__eqI,axiom,
! [T: list_lambda_lambda,T7: list_lambda_lambda] :
( ( ( inf_in3354698378728726462lambda @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 ) )
!= bot_bo1397037886704484924lambda )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 ) ) ) ).
% sources_eqI
thf(fact_591_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_592_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_593_singletonI,axiom,
! [A2: list_lambda_lambda] : ( member2667661554319814297lambda @ A2 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).
% singletonI
thf(fact_594_singletonI,axiom,
! [A2: lambda_lambda] : ( member_lambda_lambda @ A2 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).
% singletonI
thf(fact_595_insert__subset,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ X2 @ A ) @ B4 )
= ( ( member_lambda_lambda @ X2 @ B4 )
& ( ord_le1926335974945931754lambda @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_596_insert__subset,axiom,
! [X2: list_lambda_lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( insert8465602035467394112lambda @ X2 @ A ) @ B4 )
= ( ( member2667661554319814297lambda @ X2 @ B4 )
& ( ord_le3064231726521830256lambda @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_597_list_Osimps_I15_J,axiom,
! [X21: lambda_lambda,X22: list_lambda_lambda] :
( ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X21 @ X22 ) )
= ( insert_lambda_lambda @ X21 @ ( set_lambda_lambda2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_598_singleton__insert__inj__eq,axiom,
! [B: lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
( ( ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_599_singleton__insert__inj__eq,axiom,
! [B: list_lambda_lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda )
= ( insert8465602035467394112lambda @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_600_singleton__insert__inj__eq_H,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ A )
= ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
= ( ( A2 = B )
& ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_601_singleton__insert__inj__eq_H,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ A )
= ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
= ( ( A2 = B )
& ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_602_disjoint__insert_I2_J,axiom,
! [A: set_li7878742357367608272lambda,B: list_lambda_lambda,B4: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A @ ( insert8465602035467394112lambda @ B @ B4 ) ) )
= ( ~ ( member2667661554319814297lambda @ B @ A )
& ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_603_disjoint__insert_I2_J,axiom,
! [A: set_lambda_lambda,B: lambda_lambda,B4: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A @ ( insert_lambda_lambda @ B @ B4 ) ) )
= ( ~ ( member_lambda_lambda @ B @ A )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_604_disjoint__insert_I1_J,axiom,
! [B4: set_li7878742357367608272lambda,A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ B4 @ ( insert8465602035467394112lambda @ A2 @ A ) )
= bot_bo1397037886704484924lambda )
= ( ~ ( member2667661554319814297lambda @ A2 @ B4 )
& ( ( inf_in3354698378728726462lambda @ B4 @ A )
= bot_bo1397037886704484924lambda ) ) ) ).
% disjoint_insert(1)
thf(fact_605_disjoint__insert_I1_J,axiom,
! [B4: set_lambda_lambda,A2: lambda_lambda,A: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ B4 @ ( insert_lambda_lambda @ A2 @ A ) )
= bot_bo1271416176687173302lambda )
= ( ~ ( member_lambda_lambda @ A2 @ B4 )
& ( ( inf_in1766959559283430712lambda @ B4 @ A )
= bot_bo1271416176687173302lambda ) ) ) ).
% disjoint_insert(1)
thf(fact_606_insert__disjoint_I2_J,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ A ) @ B4 ) )
= ( ~ ( member2667661554319814297lambda @ A2 @ B4 )
& ( bot_bo1397037886704484924lambda
= ( inf_in3354698378728726462lambda @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_607_insert__disjoint_I2_J,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ A ) @ B4 ) )
= ( ~ ( member_lambda_lambda @ A2 @ B4 )
& ( bot_bo1271416176687173302lambda
= ( inf_in1766959559283430712lambda @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_608_insert__disjoint_I1_J,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ( ( inf_in3354698378728726462lambda @ ( insert8465602035467394112lambda @ A2 @ A ) @ B4 )
= bot_bo1397037886704484924lambda )
= ( ~ ( member2667661554319814297lambda @ A2 @ B4 )
& ( ( inf_in3354698378728726462lambda @ A @ B4 )
= bot_bo1397037886704484924lambda ) ) ) ).
% insert_disjoint(1)
thf(fact_609_insert__disjoint_I1_J,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ( inf_in1766959559283430712lambda @ ( insert_lambda_lambda @ A2 @ A ) @ B4 )
= bot_bo1271416176687173302lambda )
= ( ~ ( member_lambda_lambda @ A2 @ B4 )
& ( ( inf_in1766959559283430712lambda @ A @ B4 )
= bot_bo1271416176687173302lambda ) ) ) ).
% insert_disjoint(1)
thf(fact_610_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_611_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_612_arrI,axiom,
! [T: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ).
% arrI
thf(fact_613__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_614_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_615_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_616__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_617_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_618_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_619_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_620_in__targetsI,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B )
=> ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ) ).
% in_targetsI
thf(fact_621_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_622_singleton__inject,axiom,
! [A2: list_lambda_lambda,B: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda )
= ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_623_singleton__inject,axiom,
! [A2: lambda_lambda,B: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda )
= ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_624_insert__not__empty,axiom,
! [A2: list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( insert8465602035467394112lambda @ A2 @ A )
!= bot_bo1397037886704484924lambda ) ).
% insert_not_empty
thf(fact_625_insert__not__empty,axiom,
! [A2: lambda_lambda,A: set_lambda_lambda] :
( ( insert_lambda_lambda @ A2 @ A )
!= bot_bo1271416176687173302lambda ) ).
% insert_not_empty
thf(fact_626_doubleton__eq__iff,axiom,
! [A2: list_lambda_lambda,B: list_lambda_lambda,C: list_lambda_lambda,D2: list_lambda_lambda] :
( ( ( insert8465602035467394112lambda @ A2 @ ( insert8465602035467394112lambda @ B @ bot_bo1397037886704484924lambda ) )
= ( insert8465602035467394112lambda @ C @ ( insert8465602035467394112lambda @ D2 @ bot_bo1397037886704484924lambda ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_627_doubleton__eq__iff,axiom,
! [A2: lambda_lambda,B: lambda_lambda,C: lambda_lambda,D2: lambda_lambda] :
( ( ( insert_lambda_lambda @ A2 @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
= ( insert_lambda_lambda @ C @ ( insert_lambda_lambda @ D2 @ bot_bo1271416176687173302lambda ) ) )
= ( ( ( A2 = C )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_628_singleton__iff,axiom,
! [B: list_lambda_lambda,A2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_629_singleton__iff,axiom,
! [B: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_630_singletonD,axiom,
! [B: list_lambda_lambda,A2: list_lambda_lambda] :
( ( member2667661554319814297lambda @ B @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_631_singletonD,axiom,
! [B: lambda_lambda,A2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_632_rts_Osources_Ocong,axiom,
source6119337877621565212lambda = source6119337877621565212lambda ).
% rts.sources.cong
thf(fact_633_rts_Osources_Ocong,axiom,
source5627170679743844886lambda = source5627170679743844886lambda ).
% rts.sources.cong
thf(fact_634_residuation_Ocon_Ocong,axiom,
con_li79537494059641180lambda = con_li79537494059641180lambda ).
% residuation.con.cong
thf(fact_635_residuation_Ocon_Ocong,axiom,
con_lambda_lambda = con_lambda_lambda ).
% residuation.con.cong
thf(fact_636_subset__insertI2,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda,B: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_637_subset__insertI2,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda,B: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_638_subset__insertI,axiom,
! [B4: set_lambda_lambda,A2: lambda_lambda] : ( ord_le1926335974945931754lambda @ B4 @ ( insert_lambda_lambda @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_639_subset__insertI,axiom,
! [B4: set_li7878742357367608272lambda,A2: list_lambda_lambda] : ( ord_le3064231726521830256lambda @ B4 @ ( insert8465602035467394112lambda @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_640_subset__insert,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
( ~ ( member_lambda_lambda @ X2 @ A )
=> ( ( ord_le1926335974945931754lambda @ A @ ( insert_lambda_lambda @ X2 @ B4 ) )
= ( ord_le1926335974945931754lambda @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_641_subset__insert,axiom,
! [X2: list_lambda_lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] :
( ~ ( member2667661554319814297lambda @ X2 @ A )
=> ( ( ord_le3064231726521830256lambda @ A @ ( insert8465602035467394112lambda @ X2 @ B4 ) )
= ( ord_le3064231726521830256lambda @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_642_insert__mono,axiom,
! [C3: set_lambda_lambda,D: set_lambda_lambda,A2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C3 @ D )
=> ( ord_le1926335974945931754lambda @ ( insert_lambda_lambda @ A2 @ C3 ) @ ( insert_lambda_lambda @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_643_insert__mono,axiom,
! [C3: set_li7878742357367608272lambda,D: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ C3 @ D )
=> ( ord_le3064231726521830256lambda @ ( insert8465602035467394112lambda @ A2 @ C3 ) @ ( insert8465602035467394112lambda @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_644_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_645_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_646_subset__singleton__iff,axiom,
! [X6: set_lambda_lambda,A2: lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X6 @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
= ( ( X6 = bot_bo1271416176687173302lambda )
| ( X6
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_647_subset__singleton__iff,axiom,
! [X6: set_li7878742357367608272lambda,A2: list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ X6 @ ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) )
= ( ( X6 = bot_bo1397037886704484924lambda )
| ( X6
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% subset_singleton_iff
thf(fact_648_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,V: list_lambda_lambda,V4: list_lambda_lambda,W2: list_lambda_lambda,W3: list_lambda_lambda,T: list_lambda_lambda,T7: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( member2667661554319814297lambda @ V @ NN )
=> ( ( member2667661554319814297lambda @ V4 @ NN )
=> ( ( member2667661554319814297lambda @ W2 @ NN )
=> ( ( member2667661554319814297lambda @ W3 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ V )
= ( source6119337877621565212lambda @ Resid @ W2 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ V4 )
= ( source6119337877621565212lambda @ Resid @ W3 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ W2 )
= ( target4026644918049360146lambda @ Resid @ W3 ) )
=> ( ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T7 @ V4 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T7 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T7 @ W3 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T7 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_649_coherent__normal__sub__rts_Ocoherent_H,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,V: lambda_lambda,V4: lambda_lambda,W2: lambda_lambda,W3: lambda_lambda,T: lambda_lambda,T7: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( member_lambda_lambda @ V @ NN )
=> ( ( member_lambda_lambda @ V4 @ NN )
=> ( ( member_lambda_lambda @ W2 @ NN )
=> ( ( member_lambda_lambda @ W3 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ V )
= ( source5627170679743844886lambda @ Resid @ W2 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ V4 )
= ( source5627170679743844886lambda @ Resid @ W3 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ W2 )
= ( target5490931411331770124lambda @ Resid @ W3 ) )
=> ( ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ V ) @ ( Resid @ T7 @ V4 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T7 @ V4 ) @ ( Resid @ T @ V ) ) @ NN ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ W2 ) @ ( Resid @ T7 @ W3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T7 @ W3 ) @ ( Resid @ T @ W2 ) ) @ NN ) ) ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent'
thf(fact_650_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_651_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_652_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,T: list_lambda_lambda,U: list_lambda_lambda,U3: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( ( member2667661554319814297lambda @ U @ NN )
=> ( ( member2667661554319814297lambda @ U3 @ NN )
=> ( ( ( source6119337877621565212lambda @ Resid @ U )
= ( source6119337877621565212lambda @ Resid @ U3 ) )
=> ( ( ( target4026644918049360146lambda @ Resid @ U )
= ( target4026644918049360146lambda @ Resid @ U3 ) )
=> ( ( ( source6119337877621565212lambda @ Resid @ T )
= ( source6119337877621565212lambda @ Resid @ U ) )
=> ( ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U3 ) ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_653_coherent__normal__sub__rts_Ocoherent,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,U3: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ U3 @ NN )
=> ( ( ( source5627170679743844886lambda @ Resid @ U )
= ( source5627170679743844886lambda @ Resid @ U3 ) )
=> ( ( ( target5490931411331770124lambda @ Resid @ U )
= ( target5490931411331770124lambda @ Resid @ U3 ) )
=> ( ( ( source5627170679743844886lambda @ Resid @ T )
= ( source5627170679743844886lambda @ Resid @ U ) )
=> ( ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U3 ) ) @ NN )
& ( member_lambda_lambda @ ( Resid @ ( Resid @ T @ U3 ) @ ( Resid @ T @ U ) ) @ NN ) ) ) ) ) ) ) ) ) ).
% coherent_normal_sub_rts.coherent
thf(fact_654_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_655_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_656_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_657_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_658_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_659_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_660_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_661__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_662__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_663__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_664__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_665__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_666__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_667__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_668__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_669__092_060Lambda_062_Ocon__transitive__on__ide,axiom,
! [A2: lambda_lambda,B: lambda_lambda,C: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( ide_lambda_lambda @ lambda_resid @ C )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ B )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ C )
=> ( con_lambda_lambda @ lambda_resid @ A2 @ C ) ) ) ) ) ) ).
% \<Lambda>.con_transitive_on_ide
thf(fact_670__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_671__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_672__092_060Lambda_062_Ocon__ide__are__eq,axiom,
! [A2: lambda_lambda,A4: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ A4 )
=> ( A2 = A4 ) ) ) ) ).
% \<Lambda>.con_ide_are_eq
thf(fact_673__092_060Lambda_062_Ocong__subst__right_I1_J,axiom,
! [U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T @ U3 ) ) ) ).
% \<Lambda>.cong_subst_right(1)
thf(fact_674__092_060Lambda_062_Ocong__subst__right_I2_J,axiom,
! [U: lambda_lambda,U3: lambda_lambda,T: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T @ U3 ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U3 ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_right(2)
thf(fact_675__092_060Lambda_062_Ocong__subst__left_I1_J,axiom,
! [T: lambda_lambda,T7: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T7 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T7 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( con_lambda_lambda @ lambda_resid @ T7 @ U ) ) ) ).
% \<Lambda>.cong_subst_left(1)
thf(fact_676__092_060Lambda_062_Ocong__subst__left_I2_J,axiom,
! [T: lambda_lambda,T7: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T7 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T7 @ T ) ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ U ) @ ( lambda_resid @ T7 @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T7 @ U ) @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>.cong_subst_left(2)
thf(fact_677__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_678__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_679__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_680__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_681__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_682__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_683__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_684__092_060Lambda_062_Otargets__are__con,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ B @ B2 ) ) ) ).
% \<Lambda>.targets_are_con
thf(fact_685__092_060Lambda_062_Osources__are__con,axiom,
! [A2: lambda_lambda,T: lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( con_lambda_lambda @ lambda_resid @ A2 @ A4 ) ) ) ).
% \<Lambda>.sources_are_con
thf(fact_686__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_687_Srcs__are__con,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( member_lambda_lambda @ A4 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( con_lambda_lambda @ lambda_resid @ A2 @ A4 ) ) ) ).
% Srcs_are_con
thf(fact_688_Trgs__are__con,axiom,
! [B: lambda_lambda,T4: list_lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( ( member_lambda_lambda @ B2 @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( con_lambda_lambda @ lambda_resid @ B @ B2 ) ) ) ).
% Trgs_are_con
thf(fact_689__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_690__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_691_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_692__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_693__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_694__092_060Lambda_062_Otargets__con__closed,axiom,
! [B: lambda_lambda,T: lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B2 )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ B2 )
=> ( member_lambda_lambda @ B2 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.targets_con_closed
thf(fact_695__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_696__092_060Lambda_062_Osources__con__closed,axiom,
! [A2: lambda_lambda,T: lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ A4 )
=> ( member_lambda_lambda @ A4 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>.sources_con_closed
thf(fact_697_Srcs__con__closed,axiom,
! [A2: lambda_lambda,T4: list_lambda_lambda,A4: lambda_lambda] :
( ( member_lambda_lambda @ A2 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ A4 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ A4 )
=> ( member_lambda_lambda @ A4 @ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Srcs_con_closed
thf(fact_698_Trgs__con__closed,axiom,
! [B: lambda_lambda,T4: list_lambda_lambda,B2: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) )
=> ( ( ide_lambda_lambda @ lambda_resid @ B2 )
=> ( ( con_lambda_lambda @ lambda_resid @ B @ B2 )
=> ( member_lambda_lambda @ B2 @ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Trgs_con_closed
thf(fact_699_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_700_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_701_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_702_join__of__resid,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V @ W2 )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ V ) ) ) ) ).
% join_of_resid
thf(fact_703_con__with__join__of__iff_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ V )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) ) )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V ) ) ) ).
% con_with_join_of_iff(1)
thf(fact_704_con__with__join__of__iff_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ V )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) ) ) ) ) ).
% con_with_join_of_iff(2)
thf(fact_705_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_706_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_707__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_708_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_709_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_710_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_711_order__refl,axiom,
! [X2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ X2 @ X2 ) ).
% order_refl
thf(fact_712_order__refl,axiom,
! [X2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ X2 @ X2 ) ).
% order_refl
thf(fact_713_dual__order_Orefl,axiom,
! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_714_dual__order_Orefl,axiom,
! [A2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_715_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_716_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_717_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_718_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_719_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_720_joinable__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( joinab677607494065406278lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ? [X7: list_lambda_lambda] : ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ X7 ) ) ) ).
% joinable_def
thf(fact_721__092_060Lambda_062_Oin__targetsE,axiom,
! [B: lambda_lambda,T: lambda_lambda] :
( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ B )
=> ~ ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B ) ) ) ).
% \<Lambda>.in_targetsE
thf(fact_722_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_723__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_724_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_725__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_726__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_727_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_728__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_729__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_730__092_060Lambda_062_Oin__targetsI,axiom,
! [B: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B )
=> ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ).
% \<Lambda>.in_targetsI
thf(fact_731_rts_Ojoin__of_Ocong,axiom,
join_o6956785509312998601lambda = join_o6956785509312998601lambda ).
% rts.join_of.cong
thf(fact_732_rts_Ojoin__of_Ocong,axiom,
join_o1386658102433174083lambda = join_o1386658102433174083lambda ).
% rts.join_of.cong
thf(fact_733_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_734_hd__in__set,axiom,
! [Xs: list_l5168835686021156086lambda] :
( ( Xs != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( hd_lis2183642875561147995lambda @ Xs ) @ ( set_li3846194476672328965lambda @ Xs ) ) ) ).
% hd_in_set
thf(fact_735_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_736_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_737_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_738_longest__common__prefix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ps: list_lambda_lambda,Xs5: list_lambda_lambda,Ys6: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Ps @ Xs5 ) )
& ( Ys
= ( append_lambda_lambda @ Ps @ Ys6 ) )
& ( ( Xs5 = nil_lambda_lambda )
| ( Ys6 = nil_lambda_lambda )
| ( ( hd_lambda_lambda @ Xs5 )
!= ( hd_lambda_lambda @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_739_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_740_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_741_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_742_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_lambda_lambda,Z3: set_lambda_lambda] : ( Y6 = Z3 ) )
= ( ^ [X4: set_lambda_lambda,Y5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X4 @ Y5 )
& ( ord_le1926335974945931754lambda @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_743_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] : ( Y6 = Z3 ) )
= ( ^ [X4: set_li7878742357367608272lambda,Y5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X4 @ Y5 )
& ( ord_le3064231726521830256lambda @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_744_ord__eq__le__trans,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A2 = B )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_745_ord__eq__le__trans,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A2 = B )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_746_ord__le__eq__trans,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( B = C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_747_ord__le__eq__trans,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( B = C )
=> ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_748_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_749_order__antisym,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_750_order_Otrans,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ord_le1926335974945931754lambda @ A2 @ C ) ) ) ).
% order.trans
thf(fact_751_order_Otrans,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ord_le3064231726521830256lambda @ A2 @ C ) ) ) ).
% order.trans
thf(fact_752_order__trans,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda,Z4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X2 @ Y2 )
=> ( ( ord_le1926335974945931754lambda @ Y2 @ Z4 )
=> ( ord_le1926335974945931754lambda @ X2 @ Z4 ) ) ) ).
% order_trans
thf(fact_753_order__trans,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda,Z4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
=> ( ( ord_le3064231726521830256lambda @ Y2 @ Z4 )
=> ( ord_le3064231726521830256lambda @ X2 @ Z4 ) ) ) ).
% order_trans
thf(fact_754_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_lambda_lambda,Z3: set_lambda_lambda] : ( Y6 = Z3 ) )
= ( ^ [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B3 @ A3 )
& ( ord_le1926335974945931754lambda @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_755_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] : ( Y6 = Z3 ) )
= ( ^ [A3: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B3 @ A3 )
& ( ord_le3064231726521830256lambda @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_756_dual__order_Oantisym,axiom,
! [B: set_lambda_lambda,A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A2 )
=> ( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_757_dual__order_Oantisym,axiom,
! [B: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A2 )
=> ( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_758_dual__order_Otrans,axiom,
! [B: set_lambda_lambda,A2: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B @ A2 )
=> ( ( ord_le1926335974945931754lambda @ C @ B )
=> ( ord_le1926335974945931754lambda @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_759_dual__order_Otrans,axiom,
! [B: set_li7878742357367608272lambda,A2: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B @ A2 )
=> ( ( ord_le3064231726521830256lambda @ C @ B )
=> ( ord_le3064231726521830256lambda @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_760_antisym,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( ord_le1926335974945931754lambda @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_761_antisym,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( ord_le3064231726521830256lambda @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_762_order__eq__iff,axiom,
( ( ^ [Y6: set_lambda_lambda,Z3: set_lambda_lambda] : ( Y6 = Z3 ) )
= ( ^ [A3: set_lambda_lambda,B3: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A3 @ B3 )
& ( ord_le1926335974945931754lambda @ B3 @ A3 ) ) ) ) ).
% order_eq_iff
thf(fact_763_order__eq__iff,axiom,
( ( ^ [Y6: set_li7878742357367608272lambda,Z3: set_li7878742357367608272lambda] : ( Y6 = Z3 ) )
= ( ^ [A3: set_li7878742357367608272lambda,B3: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A3 @ B3 )
& ( ord_le3064231726521830256lambda @ B3 @ A3 ) ) ) ) ).
% order_eq_iff
thf(fact_764_order__subst1,axiom,
! [A2: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_765_order__subst1,axiom,
! [A2: set_lambda_lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_766_order__subst1,axiom,
! [A2: set_li7878742357367608272lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3064231726521830256lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_767_order__subst1,axiom,
! [A2: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3064231726521830256lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_768_order__subst2,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( ord_le1926335974945931754lambda @ ( F @ B ) @ C )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_769_order__subst2,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( ord_le3064231726521830256lambda @ ( F @ B ) @ C )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_770_order__subst2,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( ord_le1926335974945931754lambda @ ( F @ B ) @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_771_order__subst2,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( ord_le3064231726521830256lambda @ ( F @ B ) @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_772_order__eq__refl,axiom,
! [X2: set_lambda_lambda,Y2: set_lambda_lambda] :
( ( X2 = Y2 )
=> ( ord_le1926335974945931754lambda @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_773_order__eq__refl,axiom,
! [X2: set_li7878742357367608272lambda,Y2: set_li7878742357367608272lambda] :
( ( X2 = Y2 )
=> ( ord_le3064231726521830256lambda @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_774_ord__eq__le__subst,axiom,
! [A2: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_775_ord__eq__le__subst,axiom,
! [A2: set_li7878742357367608272lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,B: set_lambda_lambda,C: set_lambda_lambda] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le1926335974945931754lambda @ B @ C )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3064231726521830256lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_776_ord__eq__le__subst,axiom,
! [A2: set_lambda_lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1926335974945931754lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_777_ord__eq__le__subst,axiom,
! [A2: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le3064231726521830256lambda @ B @ C )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3064231726521830256lambda @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_778_ord__le__eq__subst,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_779_ord__le__eq__subst,axiom,
! [A2: set_lambda_lambda,B: set_lambda_lambda,F: set_lambda_lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_lambda_lambda,Y4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ X @ Y4 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_780_ord__le__eq__subst,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_lambda_lambda,C: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ord_le1926335974945931754lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le1926335974945931754lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_781_ord__le__eq__subst,axiom,
! [A2: set_li7878742357367608272lambda,B: set_li7878742357367608272lambda,F: set_li7878742357367608272lambda > set_li7878742357367608272lambda,C: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: set_li7878742357367608272lambda,Y4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ X @ Y4 )
=> ( ord_le3064231726521830256lambda @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le3064231726521830256lambda @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_782_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_783_order__antisym__conv,axiom,
! [Y2: set_li7878742357367608272lambda,X2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ Y2 @ X2 )
=> ( ( ord_le3064231726521830256lambda @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_784_bot_Oextremum__uniqueI,axiom,
! [A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
=> ( A2 = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_785_bot_Oextremum__uniqueI,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
=> ( A2 = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_uniqueI
thf(fact_786_bot_Oextremum__unique,axiom,
! [A2: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A2 @ bot_bo1271416176687173302lambda )
= ( A2 = bot_bo1271416176687173302lambda ) ) ).
% bot.extremum_unique
thf(fact_787_bot_Oextremum__unique,axiom,
! [A2: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ A2 @ bot_bo1397037886704484924lambda )
= ( A2 = bot_bo1397037886704484924lambda ) ) ).
% bot.extremum_unique
thf(fact_788_bot_Oextremum,axiom,
! [A2: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ bot_bo1271416176687173302lambda @ A2 ) ).
% bot.extremum
thf(fact_789_bot_Oextremum,axiom,
! [A2: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ bot_bo1397037886704484924lambda @ A2 ) ).
% bot.extremum
thf(fact_790_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_791_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_792_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_793_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_794__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_795__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_796__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_797__092_060Lambda_062_Ocon__with__join__of__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ U @ T ) ) ) ) ) ).
% \<Lambda>.con_with_join_of_iff(2)
thf(fact_798__092_060Lambda_062_Ocon__with__join__of__iff_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( ( con_lambda_lambda @ lambda_resid @ U @ V )
& ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ U ) @ ( lambda_resid @ T @ U ) ) )
=> ( con_lambda_lambda @ lambda_resid @ W2 @ V ) ) ) ).
% \<Lambda>.con_with_join_of_iff(1)
thf(fact_799__092_060Lambda_062_Ojoin__of__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ V @ W2 )
=> ( join_o1386658102433174083lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ V ) @ ( lambda_resid @ W2 @ V ) ) ) ) ).
% \<Lambda>.join_of_resid
thf(fact_800__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_801__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_802__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_803__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_804__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_805__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_806__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_807__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_808__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_809__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_810__092_060Lambda_062_Ojoinable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( joinab2519788767067900224lambda @ lambda_resid @ T @ U )
= ( ? [X7: lambda_lambda] : ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ X7 ) ) ) ).
% \<Lambda>.joinable_def
thf(fact_811__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_812__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_813__092_060Lambda_062_Onot__arr__null,axiom,
~ ( arr_lambda_lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% \<Lambda>.not_arr_null
thf(fact_814__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_815__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_816_Resid1x__null,axiom,
! [T4: list_lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ ( partia1461432607296028304lambda @ lambda_resid ) @ T4 )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x_null
thf(fact_817_Resid1x_Osimps_I1_J,axiom,
! [T: lambda_lambda] :
( ( paths_3319247187298047121lambda @ lambda_resid @ T @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Resid1x.simps(1)
thf(fact_818_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_819_Trg_Osimps_I1_J,axiom,
( ( paths_2695323870737867955lambda @ lambda_resid @ nil_lambda_lambda )
= ( partia1461432607296028304lambda @ lambda_resid ) ) ).
% Trg.simps(1)
thf(fact_820_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_821_Resid1x_Oelims,axiom,
! [X2: lambda_lambda,Xa2: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_3319247187298047121lambda @ lambda_resid @ X2 @ Xa2 )
= Y2 )
=> ( ( ( Xa2 = nil_lambda_lambda )
=> ( Y2
!= ( partia1461432607296028304lambda @ lambda_resid ) ) )
=> ( ! [U5: lambda_lambda] :
( ( Xa2
= ( cons_lambda_lambda @ U5 @ nil_lambda_lambda ) )
=> ( Y2
!= ( lambda_resid @ X2 @ U5 ) ) )
=> ~ ! [U5: lambda_lambda,V2: lambda_lambda,Va: list_lambda_lambda] :
( ( Xa2
= ( 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_822_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_823_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_824__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_825__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_826__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_827_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_828_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_829_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ( ( OP2 @ T @ ( partia1461432607296028304lambda @ OP2 ) )
= ( partia1461432607296028304lambda @ OP2 ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_830_partial__magma_Onull__is__zero_I2_J,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ( ( OP2 @ T @ ( partia6873046715171720854lambda @ OP2 ) )
= ( partia6873046715171720854lambda @ OP2 ) ) ) ).
% partial_magma.null_is_zero(2)
thf(fact_831_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ( ( OP2 @ ( partia1461432607296028304lambda @ OP2 ) @ T )
= ( partia1461432607296028304lambda @ OP2 ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_832_partial__magma_Onull__is__zero_I1_J,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ( ( OP2 @ ( partia6873046715171720854lambda @ OP2 ) @ T )
= ( partia6873046715171720854lambda @ OP2 ) ) ) ).
% partial_magma.null_is_zero(1)
thf(fact_833_partial__magma_Onull__eqI,axiom,
! [OP2: lambda_lambda > lambda_lambda > lambda_lambda,N: lambda_lambda] :
( ( partia3924075026459072775lambda @ OP2 )
=> ( ! [T3: lambda_lambda] :
( ( ( OP2 @ N @ T3 )
= N )
& ( ( OP2 @ T3 @ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_834_partial__magma_Onull__eqI,axiom,
! [OP2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,N: list_lambda_lambda] :
( ( partia748546660539056525lambda @ OP2 )
=> ( ! [T3: list_lambda_lambda] :
( ( ( OP2 @ N @ T3 )
= N )
& ( ( OP2 @ T3 @ N )
= N ) )
=> ( N
= ( partia6873046715171720854lambda @ OP2 ) ) ) ) ).
% partial_magma.null_eqI
thf(fact_835_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_836_partial__magma_Onull_Ocong,axiom,
partia1461432607296028304lambda = partia1461432607296028304lambda ).
% partial_magma.null.cong
thf(fact_837_partial__magma_Onull_Ocong,axiom,
partia6873046715171720854lambda = partia6873046715171720854lambda ).
% partial_magma.null.cong
thf(fact_838_Resid_Oelims,axiom,
! [X2: list_lambda_lambda,Xa2: list_lambda_lambda,Y2: list_lambda_lambda] :
( ( ( paths_8607357831768207192lambda @ lambda_resid @ X2 @ Xa2 )
= Y2 )
=> ( ( ( X2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) )
=> ( ( ? [V2: lambda_lambda,Va: list_lambda_lambda] :
( X2
= ( cons_lambda_lambda @ V2 @ Va ) )
=> ( ( Xa2 = nil_lambda_lambda )
=> ( Y2 != nil_lambda_lambda ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X2
= ( cons_lambda_lambda @ T3 @ nil_lambda_lambda ) )
=> ! [U5: lambda_lambda] :
( ( Xa2
= ( 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] :
( ( Xa2
= ( 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] :
( ( Xa2
= ( 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,Vb: lambda_lambda,Vc: list_lambda_lambda] :
( ( Xa2
= ( cons_lambda_lambda @ U5 @ ( cons_lambda_lambda @ Vb @ Vc ) ) )
=> ~ ( ( ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( 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 @ Vb @ Vc ) @ ( lambda_resid @ T3 @ U5 ) ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb @ Vc ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ U5 @ T3 ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb @ Vc ) @ ( lambda_resid @ T3 @ U5 ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ U5 )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T3 @ U5 ) @ ( cons_lambda_lambda @ Vb @ Vc ) )
!= ( 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 @ Vb @ Vc ) @ ( lambda_resid @ T3 @ U5 ) ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ) ) ) ) ).
% Resid.elims
thf(fact_839_Resid_Osimps_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,Vb2: lambda_lambda,Vc2: list_lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(6)
thf(fact_840_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_841_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_842_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_843_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_844_null__char,axiom,
( ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) )
= nil_lambda_lambda ) ).
% null_char
thf(fact_845_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_846_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_847_Residx1_Osimps_I1_J,axiom,
! [U: lambda_lambda] :
( ( paths_2112734093361750239lambda @ lambda_resid @ nil_lambda_lambda @ U )
= nil_lambda_lambda ) ).
% Residx1.simps(1)
thf(fact_848_not__arr__null,axiom,
~ ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( partia6873046715171720854lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ) ) ).
% not_arr_null
thf(fact_849_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_850_Residx1_Oelims,axiom,
! [X2: list_lambda_lambda,Xa2: lambda_lambda,Y2: list_lambda_lambda] :
( ( ( paths_2112734093361750239lambda @ lambda_resid @ X2 @ Xa2 )
= 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 @ Xa2 )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa2 ) @ nil_lambda_lambda ) ) )
& ( ~ ( con_lambda_lambda @ lambda_resid @ T3 @ Xa2 )
=> ( 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 @ Xa2 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa2 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2
= ( cons_lambda_lambda @ ( lambda_resid @ T3 @ Xa2 ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa2 @ T3 ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T3 @ Xa2 )
& ( ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V2 @ Va ) @ ( lambda_resid @ Xa2 @ T3 ) )
!= nil_lambda_lambda ) )
=> ( Y2 = nil_lambda_lambda ) ) ) ) ) ) ) ).
% Residx1.elims
thf(fact_851_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_852_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_853_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_854_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_855_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_856_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_857_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_858_Resid_Osimps_I7_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,Va2: list_lambda_lambda,Vb2: lambda_lambda,Vc2: list_lambda_lambda] :
( ( ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= ( cons_lambda_lambda @ ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) ) ) ) )
& ( ~ ( ( con_lambda_lambda @ lambda_resid @ T @ U )
& ( ( paths_3319247187298047121lambda @ lambda_resid @ ( lambda_resid @ T @ U ) @ ( cons_lambda_lambda @ V @ Va2 ) )
!= ( partia1461432607296028304lambda @ lambda_resid ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) @ ( lambda_resid @ U @ T ) ) @ ( paths_2112734093361750239lambda @ lambda_resid @ ( cons_lambda_lambda @ V @ Va2 ) @ ( lambda_resid @ T @ U ) ) )
!= nil_lambda_lambda ) )
=> ( ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ T @ ( cons_lambda_lambda @ Vb2 @ Vc2 ) ) @ ( cons_lambda_lambda @ U @ ( cons_lambda_lambda @ V @ Va2 ) ) )
= nil_lambda_lambda ) ) ) ).
% Resid.simps(7)
thf(fact_859_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_860_paths__in__rts_OResidx1_Ocong,axiom,
paths_2112734093361750239lambda = paths_2112734093361750239lambda ).
% paths_in_rts.Residx1.cong
thf(fact_861_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_862_rts__axioms_Ointro,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( arr_lambda_lambda @ Resid @ T3 )
=> ( ide_lambda_lambda @ Resid @ ( trg_lambda_lambda @ Resid @ T3 ) ) )
=> ( ! [A6: lambda_lambda,T3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A6 )
=> ( ( con_lambda_lambda @ Resid @ T3 @ A6 )
=> ( ( Resid @ T3 @ A6 )
= T3 ) ) )
=> ( ! [A6: lambda_lambda,T3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A6 )
=> ( ( con_lambda_lambda @ Resid @ A6 @ T3 )
=> ( ide_lambda_lambda @ Resid @ ( Resid @ A6 @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( con_lambda_lambda @ Resid @ T3 @ U5 )
=> ? [A7: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A7 )
& ( con_lambda_lambda @ Resid @ A7 @ T3 )
& ( con_lambda_lambda @ Resid @ A7 @ U5 ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda,V2: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ ( Resid @ T3 @ U5 ) )
=> ( ( con_lambda_lambda @ Resid @ U5 @ V2 )
=> ( con_lambda_lambda @ Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ V2 @ U5 ) ) ) )
=> ( rts_ax7298520571313827089lambda @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_863_rts__axioms_Ointro,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ Resid @ T3 )
=> ( ide_li6773656263178250516lambda @ Resid @ ( trg_li817795487149935061lambda @ Resid @ T3 ) ) )
=> ( ! [A6: list_lambda_lambda,T3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A6 )
=> ( ( con_li79537494059641180lambda @ Resid @ T3 @ A6 )
=> ( ( Resid @ T3 @ A6 )
= T3 ) ) )
=> ( ! [A6: list_lambda_lambda,T3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A6 )
=> ( ( con_li79537494059641180lambda @ Resid @ A6 @ T3 )
=> ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ A6 @ T3 ) ) ) )
=> ( ! [T3: list_lambda_lambda,U5: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ Resid @ T3 @ U5 )
=> ? [A7: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A7 )
& ( con_li79537494059641180lambda @ Resid @ A7 @ T3 )
& ( con_li79537494059641180lambda @ Resid @ A7 @ U5 ) ) )
=> ( ! [T3: list_lambda_lambda,U5: list_lambda_lambda,V2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ T3 @ U5 ) )
=> ( ( con_li79537494059641180lambda @ Resid @ U5 @ V2 )
=> ( con_li79537494059641180lambda @ Resid @ ( Resid @ T3 @ U5 ) @ ( Resid @ V2 @ U5 ) ) ) )
=> ( rts_ax2283037410143324311lambda @ Resid ) ) ) ) ) ) ).
% rts_axioms.intro
thf(fact_864_rts__axioms__def,axiom,
( rts_ax7298520571313827089lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( arr_lambda_lambda @ Resid2 @ T5 )
=> ( ide_lambda_lambda @ Resid2 @ ( trg_lambda_lambda @ Resid2 @ T5 ) ) )
& ! [A3: lambda_lambda,T5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A3 )
=> ( ( con_lambda_lambda @ Resid2 @ T5 @ A3 )
=> ( ( Resid2 @ T5 @ A3 )
= T5 ) ) )
& ! [A3: lambda_lambda,T5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A3 )
=> ( ( con_lambda_lambda @ Resid2 @ A3 @ T5 )
=> ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ A3 @ T5 ) ) ) )
& ! [T5: lambda_lambda,U6: lambda_lambda] :
( ( con_lambda_lambda @ Resid2 @ T5 @ U6 )
=> ? [A3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A3 )
& ( con_lambda_lambda @ Resid2 @ A3 @ T5 )
& ( con_lambda_lambda @ Resid2 @ A3 @ U6 ) ) )
& ! [T5: lambda_lambda,U6: lambda_lambda,V5: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ ( Resid2 @ T5 @ U6 ) )
=> ( ( con_lambda_lambda @ Resid2 @ U6 @ V5 )
=> ( con_lambda_lambda @ Resid2 @ ( Resid2 @ T5 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_865_rts__axioms__def,axiom,
( rts_ax2283037410143324311lambda
= ( ^ [Resid2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ Resid2 @ T5 )
=> ( ide_li6773656263178250516lambda @ Resid2 @ ( trg_li817795487149935061lambda @ Resid2 @ T5 ) ) )
& ! [A3: list_lambda_lambda,T5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A3 )
=> ( ( con_li79537494059641180lambda @ Resid2 @ T5 @ A3 )
=> ( ( Resid2 @ T5 @ A3 )
= T5 ) ) )
& ! [A3: list_lambda_lambda,T5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A3 )
=> ( ( con_li79537494059641180lambda @ Resid2 @ A3 @ T5 )
=> ( ide_li6773656263178250516lambda @ Resid2 @ ( Resid2 @ A3 @ T5 ) ) ) )
& ! [T5: list_lambda_lambda,U6: list_lambda_lambda] :
( ( con_li79537494059641180lambda @ Resid2 @ T5 @ U6 )
=> ? [A3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A3 )
& ( con_li79537494059641180lambda @ Resid2 @ A3 @ T5 )
& ( con_li79537494059641180lambda @ Resid2 @ A3 @ U6 ) ) )
& ! [T5: list_lambda_lambda,U6: list_lambda_lambda,V5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ ( Resid2 @ T5 @ U6 ) )
=> ( ( con_li79537494059641180lambda @ Resid2 @ U6 @ V5 )
=> ( con_li79537494059641180lambda @ Resid2 @ ( Resid2 @ T5 @ U6 ) @ ( Resid2 @ V5 @ U6 ) ) ) ) ) ) ) ).
% rts_axioms_def
thf(fact_866_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_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_last__in__set,axiom,
! [As: list_l5168835686021156086lambda] :
( ( As != nil_li8379291187143545974lambda )
=> ( member2667661554319814297lambda @ ( last_l1874196399739661071lambda @ As ) @ ( set_li3846194476672328965lambda @ As ) ) ) ).
% last_in_set
thf(fact_876_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_877_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_878_longest__common__suffix,axiom,
! [Xs: list_lambda_lambda,Ys: list_lambda_lambda] :
? [Ss: list_lambda_lambda,Xs5: list_lambda_lambda,Ys6: list_lambda_lambda] :
( ( Xs
= ( append_lambda_lambda @ Xs5 @ Ss ) )
& ( Ys
= ( append_lambda_lambda @ Ys6 @ Ss ) )
& ( ( Xs5 = nil_lambda_lambda )
| ( Ys6 = nil_lambda_lambda )
| ( ( last_lambda_lambda @ Xs5 )
!= ( last_lambda_lambda @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_879_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_880_hd__Nil__eq__last,axiom,
( ( hd_lambda_lambda @ nil_lambda_lambda )
= ( last_lambda_lambda @ nil_lambda_lambda ) ) ).
% hd_Nil_eq_last
thf(fact_881_in__targets__iff,axiom,
! [T4: list_lambda_lambda,B4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( member2667661554319814297lambda @ B4 @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 ) )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B4 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ B4 ) ) ) ) ) ).
% in_targets_iff
thf(fact_882_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_883_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_884_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_885_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_886_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_887_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_888_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_889_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_890_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_891_sub__rts__axioms__def,axiom,
( sub_rt3099298114314975807lambda
= ( ^ [R: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Arr: list_lambda_lambda > $o] :
( ! [T5: list_lambda_lambda] :
( ( Arr @ T5 )
=> ( arr_li3691443969012319837lambda @ R @ T5 ) )
& ! [T5: list_lambda_lambda] :
( ( Arr @ T5 )
=> ( ord_le3064231726521830256lambda @ ( source6119337877621565212lambda @ R @ T5 ) @ ( collec8830145823237851355lambda @ Arr ) ) )
& ! [T5: list_lambda_lambda,U6: list_lambda_lambda] :
( ( Arr @ T5 )
=> ( ( Arr @ U6 )
=> ( ( con_li79537494059641180lambda @ R @ T5 @ U6 )
=> ( Arr @ ( R @ T5 @ U6 ) ) ) ) ) ) ) ) ).
% sub_rts_axioms_def
thf(fact_892_sub__rts__axioms__def,axiom,
( sub_rt3575465509250962105lambda
= ( ^ [R: lambda_lambda > lambda_lambda > lambda_lambda,Arr: lambda_lambda > $o] :
( ! [T5: lambda_lambda] :
( ( Arr @ T5 )
=> ( arr_lambda_lambda @ R @ T5 ) )
& ! [T5: lambda_lambda] :
( ( Arr @ T5 )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R @ T5 ) @ ( collec621444948832755541lambda @ Arr ) ) )
& ! [T5: lambda_lambda,U6: lambda_lambda] :
( ( Arr @ T5 )
=> ( ( Arr @ U6 )
=> ( ( con_lambda_lambda @ R @ T5 @ U6 )
=> ( Arr @ ( R @ T5 @ U6 ) ) ) ) ) ) ) ) ).
% sub_rts_axioms_def
thf(fact_893_sub__rts__axioms_Ointro,axiom,
! [Arr2: list_lambda_lambda > $o,R2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( Arr2 @ T3 )
=> ( arr_li3691443969012319837lambda @ R2 @ T3 ) )
=> ( ! [T3: list_lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ord_le3064231726521830256lambda @ ( source6119337877621565212lambda @ R2 @ T3 ) @ ( collec8830145823237851355lambda @ Arr2 ) ) )
=> ( ! [T3: list_lambda_lambda,U5: list_lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ( Arr2 @ U5 )
=> ( ( con_li79537494059641180lambda @ R2 @ T3 @ U5 )
=> ( Arr2 @ ( R2 @ T3 @ U5 ) ) ) ) )
=> ( sub_rt3099298114314975807lambda @ R2 @ Arr2 ) ) ) ) ).
% sub_rts_axioms.intro
thf(fact_894_sub__rts__axioms_Ointro,axiom,
! [Arr2: lambda_lambda > $o,R2: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( arr_lambda_lambda @ R2 @ T3 ) )
=> ( ! [T3: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ R2 @ T3 ) @ ( collec621444948832755541lambda @ Arr2 ) ) )
=> ( ! [T3: lambda_lambda,U5: lambda_lambda] :
( ( Arr2 @ T3 )
=> ( ( Arr2 @ U5 )
=> ( ( con_lambda_lambda @ R2 @ T3 @ U5 )
=> ( Arr2 @ ( R2 @ T3 @ U5 ) ) ) ) )
=> ( sub_rt3575465509250962105lambda @ R2 @ Arr2 ) ) ) ) ).
% sub_rts_axioms.intro
thf(fact_895_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_896_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_897_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_898_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_899_lambda__calculus_Olambda_Odistinct_I15_J,axiom,
! [X3: lambda_lambda,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Lam @ X3 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(15)
thf(fact_900_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_901_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_902_lambda__calculus_OSrc__eq__iff_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T7: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ T @ U ) )
= ( lambda_Src @ ( lambda_App @ T7 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T7 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(2)
thf(fact_903_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_904_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_905_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_906_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_907_lambda__calculus_Oide__char,axiom,
! [T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ T )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.ide_char
thf(fact_908_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_909_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_910_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_911_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_912_lambda__calculus_Onormalizable__def,axiom,
( lambda_normalizable
= ( ^ [A3: lambda_lambda] :
( ( lambda_Ide @ A3 )
& ? [B3: lambda_lambda] :
( ( lambda_red @ A3 @ B3 )
& ( lambda_NF @ B3 ) ) ) ) ) ).
% lambda_calculus.normalizable_def
thf(fact_913_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_914_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_915__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oinclusion,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_is_Lam @ T ) )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.inclusion
thf(fact_916__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oresid__closed,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_is_Lam @ T ) )
=> ( ( ( lambda_Arr @ U )
& ( lambda_is_Lam @ U ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( lambda_Arr @ ( lambda_resid @ T @ U ) )
& ( lambda_is_Lam @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.resid_closed
thf(fact_917_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_918_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_919_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_920_lambda__calculus_Oarr__char,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.arr_char
thf(fact_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_lambda__calculus_OIde__implies__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( lambda_Arr @ T ) ) ).
% lambda_calculus.Ide_implies_Arr
thf(fact_934_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_935_lambda__calculus_OArr__Trg,axiom,
! [X6: lambda_lambda] :
( ( lambda_Arr @ X6 )
=> ( lambda_Arr @ ( lambda_Trg @ X6 ) ) ) ).
% lambda_calculus.Arr_Trg
thf(fact_936_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_937_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_938_lambda__calculus_Olambda_Odistinct_I17_J,axiom,
! [X3: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Lam @ X3 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(17)
thf(fact_939_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_940_lambda__calculus_OSrc__eq__iff_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T7: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_Beta @ T @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T7 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T7 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(3)
thf(fact_941_lambda__calculus_OArr__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Arr_Src
thf(fact_942_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_943_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_944_lambda__calculus_OIde__Src,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Src @ T ) ) ) ).
% lambda_calculus.Ide_Src
thf(fact_945_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_946_lambda__calculus_OIde__Trg,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( lambda_Ide @ ( lambda_Trg @ T ) ) ) ).
% lambda_calculus.Ide_Trg
thf(fact_947_lambda__calculus_OIde__iff__Trg__self,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ( lambda_Ide @ T )
= ( ( lambda_Trg @ T )
= T ) ) ) ).
% lambda_calculus.Ide_iff_Trg_self
thf(fact_948_lambda__calculus_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_949_lambda__calculus_OSrc__eq__iff_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T7: lambda_lambda,U3: lambda_lambda] :
( ( ( lambda_Src @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
= ( lambda_Src @ ( lambda_Beta @ T7 @ U3 ) ) )
= ( ( ( lambda_Src @ T )
= ( lambda_Src @ T7 ) )
& ( ( lambda_Src @ U )
= ( lambda_Src @ U3 ) ) ) ) ).
% lambda_calculus.Src_eq_iff(4)
thf(fact_950_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_951_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_952_lambda__calculus_Ored_Osimps,axiom,
( lambda_red
= ( ^ [A1: lambda_lambda,A22: lambda_lambda] :
( ? [T5: lambda_lambda] :
( ( A1
= ( lambda_Src @ T5 ) )
& ( A22
= ( lambda_Trg @ T5 ) )
& ( lambda_Arr @ T5 ) )
| ? [A3: lambda_lambda,B3: lambda_lambda,C4: lambda_lambda] :
( ( A1 = A3 )
& ( A22 = C4 )
& ( lambda_red @ A3 @ B3 )
& ( lambda_red @ B3 @ C4 ) ) ) ) ) ).
% lambda_calculus.red.simps
thf(fact_953_lambda__calculus_Ored_Ocases,axiom,
! [A12: lambda_lambda,A23: lambda_lambda] :
( ( lambda_red @ A12 @ A23 )
=> ( ! [T3: lambda_lambda] :
( ( A12
= ( lambda_Src @ T3 ) )
=> ( ( A23
= ( lambda_Trg @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) ) )
=> ~ ! [B6: lambda_lambda] :
( ( lambda_red @ A12 @ B6 )
=> ~ ( lambda_red @ B6 @ A23 ) ) ) ) ).
% lambda_calculus.red.cases
thf(fact_954_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_955_lambda__calculus_ONF__def,axiom,
( lambda_NF
= ( ^ [A3: lambda_lambda] :
( ( lambda_Ide @ A3 )
& ! [T5: lambda_lambda] :
( ( ( lambda_Arr @ T5 )
& ( ( lambda_Src @ T5 )
= A3 ) )
=> ( lambda_Ide @ T5 ) ) ) ) ) ).
% lambda_calculus.NF_def
thf(fact_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_the__elem__eq,axiom,
! [X2: list_lambda_lambda] :
( ( the_el4548681628688475493lambda @ ( insert8465602035467394112lambda @ X2 @ bot_bo1397037886704484924lambda ) )
= X2 ) ).
% the_elem_eq
thf(fact_964_the__elem__eq,axiom,
! [X2: lambda_lambda] :
( ( the_el5431214018478934239lambda @ ( insert_lambda_lambda @ X2 @ bot_bo1271416176687173302lambda ) )
= X2 ) ).
% the_elem_eq
thf(fact_965__092_060Lambda_062_OseqE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( seq_lambda_lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( trg_lambda_lambda @ lambda_resid @ T )
!= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ) ).
% \<Lambda>.seqE\<^sub>W\<^sub>E
thf(fact_966__092_060Lambda_062_Ocon__imp__eq__src,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.con_imp_eq_src
thf(fact_967__092_060Lambda_062_Osrc__join__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.src_join_of(1)
thf(fact_968__092_060Lambda_062_Osrc__join__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ U )
= ( weakly4554751870655413501lambda @ lambda_resid @ V ) ) ) ).
% \<Lambda>.src_join_of(2)
thf(fact_969__092_060Lambda_062_Oide__iff__src__self,axiom,
! [A2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ A2 )
=> ( ( ide_lambda_lambda @ lambda_resid @ A2 )
= ( ( weakly4554751870655413501lambda @ lambda_resid @ A2 )
= A2 ) ) ) ).
% \<Lambda>.ide_iff_src_self
thf(fact_970__092_060Lambda_062_Oide__src,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ide_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.ide_src
thf(fact_971__092_060Lambda_062_Osrc__eqI,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( con_lambda_lambda @ lambda_resid @ A2 @ T )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= A2 ) ) ) ).
% \<Lambda>.src_eqI
thf(fact_972__092_060Lambda_062_Osrc__in__sources,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( member_lambda_lambda @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.src_in_sources
thf(fact_973__092_060Lambda_062_OcoinitialE_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
=> ~ ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
!= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ) ).
% \<Lambda>.coinitialE\<^sub>W\<^sub>E
thf(fact_974__092_060Lambda_062_Ocoinitial__iff_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coinit1392077014771647542lambda @ lambda_resid @ T @ U )
= ( ( arr_lambda_lambda @ lambda_resid @ T )
& ( arr_lambda_lambda @ lambda_resid @ U )
& ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ) ).
% \<Lambda>.coinitial_iff\<^sub>W\<^sub>E
thf(fact_975_Src_Oelims,axiom,
! [X2: list_lambda_lambda,Y2: lambda_lambda] :
( ( ( paths_1651982994781619320lambda @ lambda_resid @ X2 )
= Y2 )
=> ( Y2
= ( weakly4554751870655413501lambda @ lambda_resid @ ( hd_lambda_lambda @ X2 ) ) ) ) ).
% Src.elims
thf(fact_976_Src_Osimps,axiom,
! [T4: list_lambda_lambda] :
( ( paths_1651982994781619320lambda @ lambda_resid @ T4 )
= ( weakly4554751870655413501lambda @ lambda_resid @ ( hd_lambda_lambda @ T4 ) ) ) ).
% Src.simps
thf(fact_977__092_060Lambda_062_Osrc__src,axiom,
! [T: lambda_lambda] :
( ( weakly4554751870655413501lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.src_src
thf(fact_978__092_060Lambda_062_Osrc__ide,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ A2 )
= A2 ) ) ).
% \<Lambda>.src_ide
thf(fact_979__092_060Lambda_062_Oarr__src__iff__arr,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.arr_src_iff_arr
thf(fact_980__092_060Lambda_062_Otrg__src,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda @ lambda_resid @ ( weakly4554751870655413501lambda @ lambda_resid @ T ) )
= ( weakly4554751870655413501lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.trg_src
thf(fact_981__092_060Lambda_062_Osrc__trg,axiom,
! [T: lambda_lambda] :
( ( weakly4554751870655413501lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>.src_trg
thf(fact_982__092_060Lambda_062_Osrc__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
= ( trg_lambda_lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.src_resid
thf(fact_983__092_060Lambda_062_OcoinitialI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) )
=> ( coinit1392077014771647542lambda @ lambda_resid @ T @ U ) ) ) ).
% \<Lambda>.coinitialI\<^sub>W\<^sub>E
thf(fact_984__092_060Lambda_062_OseqI_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [U: lambda_lambda,T: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ U )
=> ( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( ( trg_lambda_lambda @ lambda_resid @ T )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) )
=> ( seq_lambda_lambda @ lambda_resid @ T @ U ) ) ) ) ).
% \<Lambda>.seqI\<^sub>W\<^sub>E
thf(fact_985_weakly__extensional__rts_Osrc_Ocong,axiom,
weakly4554751870655413501lambda = weakly4554751870655413501lambda ).
% weakly_extensional_rts.src.cong
thf(fact_986_the__elem__set,axiom,
! [X2: lambda_lambda] :
( ( the_el5431214018478934239lambda @ ( set_lambda_lambda2 @ ( cons_lambda_lambda @ X2 @ nil_lambda_lambda ) ) )
= X2 ) ).
% the_elem_set
thf(fact_987_transformation__axioms_Ointro,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Tau: list_lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: list_lambda_lambda > list_lambda_lambda,G2: list_lambda_lambda > list_lambda_lambda] :
( ! [F3: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A @ F3 )
=> ( ( Tau @ F3 )
= ( partia6873046715171720854lambda @ B4 ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A @ F3 )
=> ( ( weakly2957279434024022659lambda @ B4 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly2957279434024022659lambda @ A @ F3 ) ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A @ F3 )
=> ( ( trg_li817795487149935061lambda @ B4 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_li817795487149935061lambda @ A @ F3 ) ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F3 )
=> ( ( B4 @ ( Tau @ ( weakly2957279434024022659lambda @ A @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_li817795487149935061lambda @ A @ F3 ) ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F3 )
=> ( ( B4 @ ( F2 @ F3 ) @ ( Tau @ ( weakly2957279434024022659lambda @ A @ F3 ) ) )
= ( G2 @ F3 ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F3 )
=> ( join_o6956785509312998601lambda @ B4 @ ( Tau @ ( weakly2957279434024022659lambda @ A @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf8838427370478101288lambda @ A @ B4 @ F2 @ G2 @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_988_transformation__axioms_Ointro,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,Tau: lambda_lambda > list_lambda_lambda,B4: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F2: lambda_lambda > list_lambda_lambda,G2: lambda_lambda > list_lambda_lambda] :
( ! [F3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A @ F3 )
=> ( ( Tau @ F3 )
= ( partia6873046715171720854lambda @ B4 ) ) )
=> ( ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A @ F3 )
=> ( ( weakly2957279434024022659lambda @ B4 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly4554751870655413501lambda @ A @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A @ F3 )
=> ( ( trg_li817795487149935061lambda @ B4 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_lambda_lambda @ A @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F3 )
=> ( ( B4 @ ( Tau @ ( weakly4554751870655413501lambda @ A @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_lambda_lambda @ A @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F3 )
=> ( ( B4 @ ( F2 @ F3 ) @ ( Tau @ ( weakly4554751870655413501lambda @ A @ F3 ) ) )
= ( G2 @ F3 ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F3 )
=> ( join_o6956785509312998601lambda @ B4 @ ( Tau @ ( weakly4554751870655413501lambda @ A @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf4653517080217399086lambda @ A @ B4 @ F2 @ G2 @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_989_transformation__axioms_Ointro,axiom,
! [A: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,Tau: list_lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F2: list_lambda_lambda > lambda_lambda,G2: list_lambda_lambda > lambda_lambda] :
( ! [F3: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A @ F3 )
=> ( ( Tau @ F3 )
= ( partia1461432607296028304lambda @ B4 ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A @ F3 )
=> ( ( weakly4554751870655413501lambda @ B4 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly2957279434024022659lambda @ A @ F3 ) ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A @ F3 )
=> ( ( trg_lambda_lambda @ B4 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_li817795487149935061lambda @ A @ F3 ) ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F3 )
=> ( ( B4 @ ( Tau @ ( weakly2957279434024022659lambda @ A @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_li817795487149935061lambda @ A @ F3 ) ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F3 )
=> ( ( B4 @ ( F2 @ F3 ) @ ( Tau @ ( weakly2957279434024022659lambda @ A @ F3 ) ) )
= ( G2 @ F3 ) ) )
=> ( ! [F3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A @ F3 )
=> ( join_o1386658102433174083lambda @ B4 @ ( Tau @ ( weakly2957279434024022659lambda @ A @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf3419659901140239394lambda @ A @ B4 @ F2 @ G2 @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_990_transformation__axioms_Ointro,axiom,
! [A: lambda_lambda > lambda_lambda > lambda_lambda,Tau: lambda_lambda > lambda_lambda,B4: lambda_lambda > lambda_lambda > lambda_lambda,F2: lambda_lambda > lambda_lambda,G2: lambda_lambda > lambda_lambda] :
( ! [F3: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A @ F3 )
=> ( ( Tau @ F3 )
= ( partia1461432607296028304lambda @ B4 ) ) )
=> ( ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A @ F3 )
=> ( ( weakly4554751870655413501lambda @ B4 @ ( Tau @ F3 ) )
= ( F2 @ ( weakly4554751870655413501lambda @ A @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( ide_lambda_lambda @ A @ F3 )
=> ( ( trg_lambda_lambda @ B4 @ ( Tau @ F3 ) )
= ( G2 @ ( trg_lambda_lambda @ A @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F3 )
=> ( ( B4 @ ( Tau @ ( weakly4554751870655413501lambda @ A @ F3 ) ) @ ( F2 @ F3 ) )
= ( Tau @ ( trg_lambda_lambda @ A @ F3 ) ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F3 )
=> ( ( B4 @ ( F2 @ F3 ) @ ( Tau @ ( weakly4554751870655413501lambda @ A @ F3 ) ) )
= ( G2 @ F3 ) ) )
=> ( ! [F3: lambda_lambda] :
( ( arr_lambda_lambda @ A @ F3 )
=> ( join_o1386658102433174083lambda @ B4 @ ( Tau @ ( weakly4554751870655413501lambda @ A @ F3 ) ) @ ( F2 @ F3 ) @ ( Tau @ F3 ) ) )
=> ( transf356316610583137064lambda @ A @ B4 @ F2 @ G2 @ Tau ) ) ) ) ) ) ) ).
% transformation_axioms.intro
thf(fact_991_transformation__axioms__def,axiom,
( transf8838427370478101288lambda
= ( ^ [A5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: list_lambda_lambda > list_lambda_lambda,G3: list_lambda_lambda > list_lambda_lambda,Tau2: list_lambda_lambda > list_lambda_lambda] :
( ! [F5: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A5 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia6873046715171720854lambda @ B5 ) ) )
& ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A5 @ F5 )
=> ( ( weakly2957279434024022659lambda @ B5 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly2957279434024022659lambda @ A5 @ F5 ) ) ) )
& ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A5 @ F5 )
=> ( ( trg_li817795487149935061lambda @ B5 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_li817795487149935061lambda @ A5 @ F5 ) ) ) )
& ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A5 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly2957279434024022659lambda @ A5 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_li817795487149935061lambda @ A5 @ F5 ) ) ) )
& ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A5 @ F5 )
=> ( ( B5 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly2957279434024022659lambda @ A5 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A5 @ F5 )
=> ( join_o6956785509312998601lambda @ B5 @ ( Tau2 @ ( weakly2957279434024022659lambda @ A5 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_992_transformation__axioms__def,axiom,
( transf4653517080217399086lambda
= ( ^ [A5: lambda_lambda > lambda_lambda > lambda_lambda,B5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,F4: lambda_lambda > list_lambda_lambda,G3: lambda_lambda > list_lambda_lambda,Tau2: lambda_lambda > list_lambda_lambda] :
( ! [F5: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A5 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia6873046715171720854lambda @ B5 ) ) )
& ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A5 @ F5 )
=> ( ( weakly2957279434024022659lambda @ B5 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly4554751870655413501lambda @ A5 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A5 @ F5 )
=> ( ( trg_li817795487149935061lambda @ B5 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_lambda_lambda @ A5 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A5 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A5 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_lambda_lambda @ A5 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A5 @ F5 )
=> ( ( B5 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly4554751870655413501lambda @ A5 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A5 @ F5 )
=> ( join_o6956785509312998601lambda @ B5 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A5 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_993_transformation__axioms__def,axiom,
( transf3419659901140239394lambda
= ( ^ [A5: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F4: list_lambda_lambda > lambda_lambda,G3: list_lambda_lambda > lambda_lambda,Tau2: list_lambda_lambda > lambda_lambda] :
( ! [F5: list_lambda_lambda] :
( ~ ( arr_li3691443969012319837lambda @ A5 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia1461432607296028304lambda @ B5 ) ) )
& ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A5 @ F5 )
=> ( ( weakly4554751870655413501lambda @ B5 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly2957279434024022659lambda @ A5 @ F5 ) ) ) )
& ! [F5: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ A5 @ F5 )
=> ( ( trg_lambda_lambda @ B5 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_li817795487149935061lambda @ A5 @ F5 ) ) ) )
& ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A5 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly2957279434024022659lambda @ A5 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_li817795487149935061lambda @ A5 @ F5 ) ) ) )
& ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A5 @ F5 )
=> ( ( B5 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly2957279434024022659lambda @ A5 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ A5 @ F5 )
=> ( join_o1386658102433174083lambda @ B5 @ ( Tau2 @ ( weakly2957279434024022659lambda @ A5 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_994_transformation__axioms__def,axiom,
( transf356316610583137064lambda
= ( ^ [A5: lambda_lambda > lambda_lambda > lambda_lambda,B5: lambda_lambda > lambda_lambda > lambda_lambda,F4: lambda_lambda > lambda_lambda,G3: lambda_lambda > lambda_lambda,Tau2: lambda_lambda > lambda_lambda] :
( ! [F5: lambda_lambda] :
( ~ ( arr_lambda_lambda @ A5 @ F5 )
=> ( ( Tau2 @ F5 )
= ( partia1461432607296028304lambda @ B5 ) ) )
& ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A5 @ F5 )
=> ( ( weakly4554751870655413501lambda @ B5 @ ( Tau2 @ F5 ) )
= ( F4 @ ( weakly4554751870655413501lambda @ A5 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( ide_lambda_lambda @ A5 @ F5 )
=> ( ( trg_lambda_lambda @ B5 @ ( Tau2 @ F5 ) )
= ( G3 @ ( trg_lambda_lambda @ A5 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A5 @ F5 )
=> ( ( B5 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A5 @ F5 ) ) @ ( F4 @ F5 ) )
= ( Tau2 @ ( trg_lambda_lambda @ A5 @ F5 ) ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A5 @ F5 )
=> ( ( B5 @ ( F4 @ F5 ) @ ( Tau2 @ ( weakly4554751870655413501lambda @ A5 @ F5 ) ) )
= ( G3 @ F5 ) ) )
& ! [F5: lambda_lambda] :
( ( arr_lambda_lambda @ A5 @ F5 )
=> ( join_o1386658102433174083lambda @ B5 @ ( Tau2 @ ( weakly4554751870655413501lambda @ A5 @ F5 ) ) @ ( F4 @ F5 ) @ ( Tau2 @ F5 ) ) ) ) ) ) ).
% transformation_axioms_def
thf(fact_995_Collect__empty__eq__bot,axiom,
! [P: list_lambda_lambda > $o] :
( ( ( collec8830145823237851355lambda @ P )
= bot_bo1397037886704484924lambda )
= ( P = bot_bo430095131446760353mbda_o ) ) ).
% Collect_empty_eq_bot
thf(fact_996_Collect__empty__eq__bot,axiom,
! [P: lambda_lambda > $o] :
( ( ( collec621444948832755541lambda @ P )
= bot_bo1271416176687173302lambda )
= ( P = bot_bo3555140721347798439mbda_o ) ) ).
% Collect_empty_eq_bot
thf(fact_997_bot__empty__eq,axiom,
( bot_bo430095131446760353mbda_o
= ( ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ bot_bo1397037886704484924lambda ) ) ) ).
% bot_empty_eq
thf(fact_998_bot__empty__eq,axiom,
( bot_bo3555140721347798439mbda_o
= ( ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ bot_bo1271416176687173302lambda ) ) ) ).
% bot_empty_eq
thf(fact_999_is__singleton__the__elem,axiom,
( is_sin4085330448496538980lambda
= ( ^ [A5: set_li7878742357367608272lambda] :
( A5
= ( insert8465602035467394112lambda @ ( the_el4548681628688475493lambda @ A5 ) @ bot_bo1397037886704484924lambda ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1000_is__singleton__the__elem,axiom,
( is_sin2795064709772026974lambda
= ( ^ [A5: set_lambda_lambda] :
( A5
= ( insert_lambda_lambda @ ( the_el5431214018478934239lambda @ A5 ) @ bot_bo1271416176687173302lambda ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1001_residuation__axioms,axiom,
residu7698045915734027664lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) ).
% residuation_axioms
thf(fact_1002_is__singletonI,axiom,
! [X2: list_lambda_lambda] : ( is_sin4085330448496538980lambda @ ( insert8465602035467394112lambda @ X2 @ bot_bo1397037886704484924lambda ) ) ).
% is_singletonI
thf(fact_1003_is__singletonI,axiom,
! [X2: lambda_lambda] : ( is_sin2795064709772026974lambda @ ( insert_lambda_lambda @ X2 @ bot_bo1271416176687173302lambda ) ) ).
% is_singletonI
thf(fact_1004_residuation_Ocon__sym,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( con_li79537494059641180lambda @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_1005_residuation_Ocon__sym,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( con_lambda_lambda @ Resid @ U @ T ) ) ) ).
% residuation.con_sym
thf(fact_1006_residuation_Oresid__arr__self,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_1007_residuation_Oresid__arr__self,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( Resid @ T @ T )
= ( trg_li817795487149935061lambda @ Resid @ T ) ) ) ).
% residuation.resid_arr_self
thf(fact_1008_residuation_Otrg__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( trg_lambda_lambda @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_1009_residuation_Otrg__def,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( trg_li817795487149935061lambda @ Resid @ T )
= ( Resid @ T @ T ) ) ) ).
% residuation.trg_def
thf(fact_1010_residuation_Ocube,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_1011_residuation_Ocube,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ).
% residuation.cube
thf(fact_1012_residuation_Oaxioms_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( partia3924075026459072775lambda @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_1013_residuation_Oaxioms_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( partia748546660539056525lambda @ Resid ) ) ).
% residuation.axioms(1)
thf(fact_1014_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_1015_residuation_Ocon__imp__arr__resid,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ T @ U ) @ ( Resid @ T @ U ) )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% residuation.con_imp_arr_resid
thf(fact_1016_residuation_Ocon__sym__ax,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_1017_residuation_Ocon__sym__ax,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( ( Resid @ U @ T )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% residuation.con_sym_ax
thf(fact_1018_residuation_Ocube__ax,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_1019_residuation_Ocube__ax,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,V: list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( ( Resid @ ( Resid @ V @ T ) @ ( Resid @ U @ T ) )
= ( Resid @ ( Resid @ V @ U ) @ ( Resid @ T @ U ) ) ) ) ) ).
% residuation.cube_ax
thf(fact_1020_residuation_Onot__arr__null,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ~ ( arr_lambda_lambda @ Resid @ ( partia1461432607296028304lambda @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_1021_residuation_Onot__arr__null,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ~ ( arr_li3691443969012319837lambda @ Resid @ ( partia6873046715171720854lambda @ Resid ) ) ) ).
% residuation.not_arr_null
thf(fact_1022_residuation_Ocon__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_1023_residuation_Ocon__def,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
= ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% residuation.con_def
thf(fact_1024_residuation_OconI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) )
=> ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_1025_residuation_OconI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) )
=> ( con_li79537494059641180lambda @ Resid @ T @ U ) ) ) ).
% residuation.conI
thf(fact_1026_residuation_OconE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partia1461432607296028304lambda @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_1027_residuation_OconE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( ( Resid @ T @ U )
!= ( partia6873046715171720854lambda @ Resid ) ) ) ) ).
% residuation.conE
thf(fact_1028_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( arr_li3691443969012319837lambda @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_1029_residuation_Ocon__implies__arr_I2_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ U ) ) ) ).
% residuation.con_implies_arr(2)
thf(fact_1030_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( arr_li3691443969012319837lambda @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_1031_residuation_Ocon__implies__arr_I1_J,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.con_implies_arr(1)
thf(fact_1032_residuation_OarrE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
=> ( con_li79537494059641180lambda @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_1033_residuation_OarrE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
=> ( con_lambda_lambda @ Resid @ T @ T ) ) ) ).
% residuation.arrE
thf(fact_1034_residuation_OarrI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ T )
=> ( arr_li3691443969012319837lambda @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_1035_residuation_OarrI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ T )
=> ( arr_lambda_lambda @ Resid @ T ) ) ) ).
% residuation.arrI
thf(fact_1036_residuation_Oarr__def,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( arr_li3691443969012319837lambda @ Resid @ T )
= ( con_li79537494059641180lambda @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_1037_residuation_Oarr__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ T )
= ( con_lambda_lambda @ Resid @ T @ T ) ) ) ).
% residuation.arr_def
thf(fact_1038_residuation_Oarr__resid,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ T @ U )
=> ( arr_li3691443969012319837lambda @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_1039_residuation_Oarr__resid,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ T @ U )
=> ( arr_lambda_lambda @ Resid @ ( Resid @ T @ U ) ) ) ) ).
% residuation.arr_resid
thf(fact_1040_residuation_Oarr__resid__iff__con,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( arr_li3691443969012319837lambda @ Resid @ ( Resid @ T @ U ) )
= ( con_li79537494059641180lambda @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_1041_residuation_Oarr__resid__iff__con,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( arr_lambda_lambda @ Resid @ ( Resid @ T @ U ) )
= ( con_lambda_lambda @ Resid @ T @ U ) ) ) ).
% residuation.arr_resid_iff_con
thf(fact_1042_residuation_Oide__implies__arr,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A2: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A2 )
=> ( arr_lambda_lambda @ Resid @ A2 ) ) ) ).
% residuation.ide_implies_arr
thf(fact_1043_residuation_Oide__implies__arr,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A2: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A2 )
=> ( arr_li3691443969012319837lambda @ Resid @ A2 ) ) ) ).
% residuation.ide_implies_arr
thf(fact_1044_residuation_Oide__def,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A2: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A2 )
= ( ( con_lambda_lambda @ Resid @ A2 @ A2 )
& ( ( Resid @ A2 @ A2 )
= A2 ) ) ) ) ).
% residuation.ide_def
thf(fact_1045_residuation_Oide__def,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A2: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A2 )
= ( ( con_li79537494059641180lambda @ Resid @ A2 @ A2 )
& ( ( Resid @ A2 @ A2 )
= A2 ) ) ) ) ).
% residuation.ide_def
thf(fact_1046_residuation_OideI,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A2: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( con_lambda_lambda @ Resid @ A2 @ A2 )
=> ( ( ( Resid @ A2 @ A2 )
= A2 )
=> ( ide_lambda_lambda @ Resid @ A2 ) ) ) ) ).
% residuation.ideI
thf(fact_1047_residuation_OideI,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A2: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( con_li79537494059641180lambda @ Resid @ A2 @ A2 )
=> ( ( ( Resid @ A2 @ A2 )
= A2 )
=> ( ide_li6773656263178250516lambda @ Resid @ A2 ) ) ) ) ).
% residuation.ideI
thf(fact_1048_residuation_OideE,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,A2: lambda_lambda] :
( ( residu6197316955523523210lambda @ Resid )
=> ( ( ide_lambda_lambda @ Resid @ A2 )
=> ~ ( ( con_lambda_lambda @ Resid @ A2 @ A2 )
=> ( ( Resid @ A2 @ A2 )
!= A2 ) ) ) ) ).
% residuation.ideE
thf(fact_1049_residuation_OideE,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,A2: list_lambda_lambda] :
( ( residu7698045915734027664lambda @ Resid )
=> ( ( ide_li6773656263178250516lambda @ Resid @ A2 )
=> ~ ( ( con_li79537494059641180lambda @ Resid @ A2 @ A2 )
=> ( ( Resid @ A2 @ A2 )
!= A2 ) ) ) ) ).
% residuation.ideE
thf(fact_1050_is__singletonI_H,axiom,
! [A: set_li7878742357367608272lambda] :
( ( A != bot_bo1397037886704484924lambda )
=> ( ! [X: list_lambda_lambda,Y4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( ( member2667661554319814297lambda @ Y4 @ A )
=> ( X = Y4 ) ) )
=> ( is_sin4085330448496538980lambda @ A ) ) ) ).
% is_singletonI'
thf(fact_1051_is__singletonI_H,axiom,
! [A: set_lambda_lambda] :
( ( A != bot_bo1271416176687173302lambda )
=> ( ! [X: lambda_lambda,Y4: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( ( member_lambda_lambda @ Y4 @ A )
=> ( X = Y4 ) ) )
=> ( is_sin2795064709772026974lambda @ A ) ) ) ).
% is_singletonI'
thf(fact_1052_is__singletonE,axiom,
! [A: set_li7878742357367608272lambda] :
( ( is_sin4085330448496538980lambda @ A )
=> ~ ! [X: list_lambda_lambda] :
( A
!= ( insert8465602035467394112lambda @ X @ bot_bo1397037886704484924lambda ) ) ) ).
% is_singletonE
thf(fact_1053_is__singletonE,axiom,
! [A: set_lambda_lambda] :
( ( is_sin2795064709772026974lambda @ A )
=> ~ ! [X: lambda_lambda] :
( A
!= ( insert_lambda_lambda @ X @ bot_bo1271416176687173302lambda ) ) ) ).
% is_singletonE
thf(fact_1054_is__singleton__def,axiom,
( is_sin4085330448496538980lambda
= ( ^ [A5: set_li7878742357367608272lambda] :
? [X4: list_lambda_lambda] :
( A5
= ( insert8465602035467394112lambda @ X4 @ bot_bo1397037886704484924lambda ) ) ) ) ).
% is_singleton_def
thf(fact_1055_is__singleton__def,axiom,
( is_sin2795064709772026974lambda
= ( ^ [A5: set_lambda_lambda] :
? [X4: lambda_lambda] :
( A5
= ( insert_lambda_lambda @ X4 @ bot_bo1271416176687173302lambda ) ) ) ) ).
% is_singleton_def
thf(fact_1056_append__is__composite__of,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 @ ( append_lambda_lambda @ T4 @ U2 ) ) ) ).
% append_is_composite_of
thf(fact_1057__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oresid__closed,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_is_App @ T ) )
=> ( ( ( lambda_Arr @ U )
& ( lambda_is_App @ U ) )
=> ( ( con_lambda_lambda @ lambda_resid @ T @ U )
=> ( ( lambda_Arr @ ( lambda_resid @ T @ U ) )
& ( lambda_is_App @ ( lambda_resid @ T @ U ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.resid_closed
thf(fact_1058__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oinclusion,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_is_App @ T ) )
=> ( arr_lambda_lambda @ lambda_resid @ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.inclusion
thf(fact_1059_diamond__commutes__upto__cong,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ V4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ V4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ V ) ) ) ) ) ).
% diamond_commutes_upto_cong
thf(fact_1060_composite__of__unq__upto__cong,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V4 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ V4 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V4 @ V ) ) ) ) ) ).
% composite_of_unq_upto_cong
thf(fact_1061_composite__of__ide__self,axiom,
! [A2: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ A2 @ A2 ) ) ).
% composite_of_ide_self
thf(fact_1062_composite__of__def,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
= ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) ) ) ) ).
% composite_of_def
thf(fact_1063_composite__of__cancel__left,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,U3: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U3 @ V )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ U3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U3 @ U ) ) ) ) ) ).
% composite_of_cancel_left
thf(fact_1064_composite__ofE,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
=> ~ ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) ) ) ) ) ).
% composite_ofE
thf(fact_1065_con__composite__of__iff,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,W2: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ T ) @ U ) ) ) ).
% con_composite_of_iff
thf(fact_1066_bounded__imp__con,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,T7: list_lambda_lambda,U3: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T7 @ U3 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ T7 ) ) ) ).
% bounded_imp_con
thf(fact_1067_resid__composite__of_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ T ) ) ) ) ).
% resid_composite_of(1)
thf(fact_1068_resid__composite__of_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ U ) ) ) ).
% resid_composite_of(2)
thf(fact_1069_resid__composite__of_I4_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ V ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) ) @ ( paths_8607357831768207192lambda @ lambda_resid @ W2 @ V ) ) ) ) ).
% resid_composite_of(4)
thf(fact_1070_con__prfx__composite__of_I1_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ W2 ) ) ).
% con_prfx_composite_of(1)
thf(fact_1071_con__prfx__composite__of_I2_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ V ) ) ) ).
% con_prfx_composite_of(2)
thf(fact_1072_sources__composite__of,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V )
= ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U ) ) ) ).
% sources_composite_of
thf(fact_1073_targets__composite__of,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V )
= ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) ) ) ).
% targets_composite_of
thf(fact_1074_arr__composite__of,axiom,
! [U: list_lambda_lambda,T: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V )
=> ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ V ) ) ).
% arr_composite_of
thf(fact_1075_obtains__composite__of,axiom,
! [G: list_lambda_lambda,F: list_lambda_lambda] :
( ( seq_li8093550137657879445lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F )
=> ~ ! [H: list_lambda_lambda] :
~ ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ G @ F @ H ) ) ).
% obtains_composite_of
thf(fact_1076_join__of__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
= ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) @ V )
& ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ V ) ) ) ).
% join_of_def
thf(fact_1077_join__ofE,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V )
=> ~ ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) @ V )
=> ~ ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ V ) ) ) ).
% join_ofE
thf(fact_1078_composable__def,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda] :
( ( compos3359172601056258045lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U )
= ( ? [X7: list_lambda_lambda] : ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ X7 ) ) ) ).
% composable_def
thf(fact_1079_resid__composite__of_I3_J,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,W2: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ W2 )
=> ( ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ W2 @ V )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ W2 ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ U ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ T ) @ U ) @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ W2 ) ) ) ) ) ) ).
% resid_composite_of(3)
thf(fact_1080_composite__of__arr__ide,axiom,
! [B: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ B @ T )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) @ B ) ) ) ).
% composite_of_arr_ide
thf(fact_1081_composite__of__ide__arr,axiom,
! [A2: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ T @ T )
= ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A2 ) ) ) ).
% composite_of_ide_arr
thf(fact_1082_composite__of__source__arr,axiom,
! [T: list_lambda_lambda,A2: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ A2 @ ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A2 @ T @ T ) ) ) ).
% composite_of_source_arr
thf(fact_1083_composite__of__arr__target,axiom,
! [T: list_lambda_lambda,B: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
=> ( ( member2667661554319814297lambda @ B @ ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ B @ T ) ) ) ).
% composite_of_arr_target
thf(fact_1084_composite__ofI,axiom,
! [U: list_lambda_lambda,V: list_lambda_lambda,T: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ V ) )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) @ T ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ V @ U ) ) ) )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ T @ V ) ) ) ).
% composite_ofI
thf(fact_1085_join__ofI,axiom,
! [T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda] :
( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ ( paths_8607357831768207192lambda @ lambda_resid @ U @ T ) @ V )
=> ( ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ U @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ U ) @ V )
=> ( join_o6956785509312998601lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ U @ V ) ) ) ).
% join_ofI
thf(fact_1086_lambda__calculus_Oresid__is__residuation,axiom,
residu6197316955523523210lambda @ lambda_resid ).
% lambda_calculus.resid_is_residuation
thf(fact_1087_lambda__calculus_Olambda_Odisc_I15_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
~ ( lambda_is_App @ ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.disc(15)
thf(fact_1088_lambda__calculus_Ois__App__def,axiom,
( lambda_is_App
= ( ^ [Lambda: lambda_lambda] :
? [X412: lambda_lambda,X422: lambda_lambda] :
( Lambda
= ( lambda_App @ X412 @ X422 ) ) ) ) ).
% lambda_calculus.is_App_def
thf(fact_1089_lambda__calculus_Olambda_OdiscI_I3_J,axiom,
! [Lambda2: lambda_lambda,X41: lambda_lambda,X42: lambda_lambda] :
( ( Lambda2
= ( lambda_App @ X41 @ X42 ) )
=> ( lambda_is_App @ Lambda2 ) ) ).
% lambda_calculus.lambda.discI(3)
thf(fact_1090_lambda__calculus_Olambda_Odisc_I14_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] : ( lambda_is_App @ ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.disc(14)
thf(fact_1091_lambda__calculus_Olambda_Odistinct__disc_I15_J,axiom,
! [Lambda2: lambda_lambda] :
( ( lambda_is_Lam @ Lambda2 )
=> ~ ( lambda_is_App @ Lambda2 ) ) ).
% lambda_calculus.lambda.distinct_disc(15)
thf(fact_1092_lambda__calculus_Olambda_Odistinct__disc_I16_J,axiom,
! [Lambda2: lambda_lambda] :
( ( lambda_is_App @ Lambda2 )
=> ~ ( lambda_is_Lam @ Lambda2 ) ) ).
% lambda_calculus.lambda.distinct_disc(16)
thf(fact_1093_lambda__calculus_Olambda_Odisc_I13_J,axiom,
! [X3: lambda_lambda] :
~ ( lambda_is_App @ ( lambda_Lam @ X3 ) ) ).
% lambda_calculus.lambda.disc(13)
thf(fact_1094_rts_Ocomposite__of_Ocong,axiom,
compos7114946393191943382lambda = compos7114946393191943382lambda ).
% rts.composite_of.cong
thf(fact_1095_rts_Ocomposite__of_Ocong,axiom,
compos9183582412961930448lambda = compos9183582412961930448lambda ).
% rts.composite_of.cong
thf(fact_1096_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN: set_li7878742357367608272lambda,T: list_lambda_lambda,U: list_lambda_lambda,T7: list_lambda_lambda] :
( ( cohere6622258358606131289lambda @ Resid @ NN )
=> ( ( compos7114946393191943382lambda @ Resid @ T @ U @ T7 )
=> ( ( member2667661554319814297lambda @ U @ NN )
=> ( ( member2667661554319814297lambda @ ( Resid @ T7 @ T ) @ NN )
& ( member2667661554319814297lambda @ ( Resid @ T @ T7 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_1097_coherent__normal__sub__rts_OCong_092_060_094sub_0620__composite__of__arr__normal,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,NN: set_lambda_lambda,T: lambda_lambda,U: lambda_lambda,T7: lambda_lambda] :
( ( cohere1690309856004283859lambda @ Resid @ NN )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ U @ T7 )
=> ( ( member_lambda_lambda @ U @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T7 @ T ) @ NN )
& ( member_lambda_lambda @ ( Resid @ T @ T7 ) @ NN ) ) ) ) ) ).
% coherent_normal_sub_rts.Cong\<^sub>0_composite_of_arr_normal
thf(fact_1098_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,V4: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( compos9183582412961930448lambda @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( compos9183582412961930448lambda @ Resid @ U @ ( Resid @ T @ U ) @ V4 )
=> ( ( ide_lambda_lambda @ Resid @ ( Resid @ V @ V4 ) )
& ( ide_lambda_lambda @ Resid @ ( Resid @ V4 @ V ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_1099_rts__with__composites_Odiamond__commutes__upto__cong,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,T: list_lambda_lambda,U: list_lambda_lambda,V: list_lambda_lambda,V4: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( compos7114946393191943382lambda @ Resid @ T @ ( Resid @ U @ T ) @ V )
=> ( ( compos7114946393191943382lambda @ Resid @ U @ ( Resid @ T @ U ) @ V4 )
=> ( ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ V @ V4 ) )
& ( ide_li6773656263178250516lambda @ Resid @ ( Resid @ V4 @ V ) ) ) ) ) ) ).
% rts_with_composites.diamond_commutes_upto_cong
thf(fact_1100_rts__with__composites_Oobtains__composite__of,axiom,
! [Resid: lambda_lambda > lambda_lambda > lambda_lambda,G: lambda_lambda,F: lambda_lambda] :
( ( rts_wi3086218056445523938lambda @ Resid )
=> ( ( seq_lambda_lambda @ Resid @ G @ F )
=> ~ ! [H: lambda_lambda] :
~ ( compos9183582412961930448lambda @ Resid @ G @ F @ H ) ) ) ).
% rts_with_composites.obtains_composite_of
thf(fact_1101_rts__with__composites_Oobtains__composite__of,axiom,
! [Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,G: list_lambda_lambda,F: list_lambda_lambda] :
( ( rts_wi1553966559393445608lambda @ Resid )
=> ( ( seq_li8093550137657879445lambda @ Resid @ G @ F )
=> ~ ! [H: list_lambda_lambda] :
~ ( compos7114946393191943382lambda @ Resid @ G @ F @ H ) ) ) ).
% rts_with_composites.obtains_composite_of
thf(fact_1102_Arr__map__un__App2,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_App2 @ T4 ) ) ) ) ).
% Arr_map_un_App2
thf(fact_1103_Arr__map__un__App1,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ T4 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( paths_366302053746739058lambda @ lambda_resid @ ( map_la336867333220232516lambda @ lambda_un_App1 @ T4 ) ) ) ) ).
% Arr_map_un_App1
thf(fact_1104__092_060Lambda_062_Ocomposite__ofE,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ~ ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) ) ) ) ).
% \<Lambda>.composite_ofE
thf(fact_1105__092_060Lambda_062_Ocomposite__of__cancel__left,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,U3: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T @ U3 @ V )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ U3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U3 @ U ) ) ) ) ) ).
% \<Lambda>.composite_of_cancel_left
thf(fact_1106__092_060Lambda_062_Ocomposite__of__def,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
= ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) ) ) ).
% \<Lambda>.composite_of_def
thf(fact_1107__092_060Lambda_062_Ocomposite__of__ide__self,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( compos9183582412961930448lambda @ lambda_resid @ A2 @ A2 @ A2 ) ) ).
% \<Lambda>.composite_of_ide_self
thf(fact_1108__092_060Lambda_062_Ocomposite__of__unq__upto__cong,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda,V4: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V4 )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ V4 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ V4 @ V ) ) ) ) ) ).
% \<Lambda>.composite_of_unq_upto_cong
thf(fact_1109__092_060Lambda_062_Oarr__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( arr_lambda_lambda @ lambda_resid @ V ) ) ).
% \<Lambda>.arr_composite_of
thf(fact_1110__092_060Lambda_062_Ocon__prfx__composite__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( con_lambda_lambda @ lambda_resid @ T @ V ) ) ) ).
% \<Lambda>.con_prfx_composite_of(2)
thf(fact_1111__092_060Lambda_062_Ocon__prfx__composite__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( con_lambda_lambda @ lambda_resid @ T @ W2 ) ) ).
% \<Lambda>.con_prfx_composite_of(1)
thf(fact_1112__092_060Lambda_062_Oresid__composite__of_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( compos9183582412961930448lambda @ lambda_resid @ ( lambda_resid @ T @ V ) @ ( lambda_resid @ U @ ( lambda_resid @ V @ T ) ) @ ( lambda_resid @ W2 @ V ) ) ) ) ).
% \<Lambda>.resid_composite_of(4)
thf(fact_1113__092_060Lambda_062_Oresid__composite__of_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ U ) ) ) ).
% \<Lambda>.resid_composite_of(2)
thf(fact_1114__092_060Lambda_062_Oresid__composite__of_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ V @ T ) @ ( lambda_resid @ W2 @ T ) ) ) ) ).
% \<Lambda>.resid_composite_of(1)
thf(fact_1115__092_060Lambda_062_Obounded__imp__con,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,T7: lambda_lambda,U3: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T7 @ U3 @ V )
=> ( con_lambda_lambda @ lambda_resid @ T @ T7 ) ) ) ).
% \<Lambda>.bounded_imp_con
thf(fact_1116__092_060Lambda_062_Ocon__composite__of__iff,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda,W2: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
= ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ W2 @ T ) @ U ) ) ) ).
% \<Lambda>.con_composite_of_iff
thf(fact_1117__092_060Lambda_062_Otargets__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( target5490931411331770124lambda @ lambda_resid @ V )
= ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.targets_composite_of
thf(fact_1118__092_060Lambda_062_Osources__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( source5627170679743844886lambda @ lambda_resid @ V )
= ( source5627170679743844886lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.sources_composite_of
thf(fact_1119__092_060Lambda_062_Otrg__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( trg_lambda_lambda @ lambda_resid @ V )
= ( trg_lambda_lambda @ lambda_resid @ T ) ) ) ).
% \<Lambda>.trg_composite_of
thf(fact_1120__092_060Lambda_062_Ojoin__ofE,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
=> ~ ( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
=> ~ ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V ) ) ) ).
% \<Lambda>.join_ofE
thf(fact_1121__092_060Lambda_062_Ojoin__of__def,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V )
= ( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
& ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V ) ) ) ).
% \<Lambda>.join_of_def
thf(fact_1122__092_060Lambda_062_Osrc__composite__of,axiom,
! [U: lambda_lambda,T: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V )
=> ( ( weakly4554751870655413501lambda @ lambda_resid @ V )
= ( weakly4554751870655413501lambda @ lambda_resid @ U ) ) ) ).
% \<Lambda>.src_composite_of
thf(fact_1123__092_060Lambda_062_Ocomposable__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( compos7127019122090555255lambda @ lambda_resid @ T @ U )
= ( ? [X7: lambda_lambda] : ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ X7 ) ) ) ).
% \<Lambda>.composable_def
thf(fact_1124__092_060Lambda_062_Ocomposite__of__ide__arr,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A2 )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ A2 @ T @ T )
= ( con_lambda_lambda @ lambda_resid @ T @ A2 ) ) ) ).
% \<Lambda>.composite_of_ide_arr
thf(fact_1125__092_060Lambda_062_Ocomposite__of__arr__ide,axiom,
! [B: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ T @ B @ T )
= ( con_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ T ) @ B ) ) ) ).
% \<Lambda>.composite_of_arr_ide
thf(fact_1126__092_060Lambda_062_Oresid__composite__of_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,W2: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ W2 )
=> ( ( con_lambda_lambda @ lambda_resid @ W2 @ V )
=> ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ W2 ) @ ( lambda_resid @ ( lambda_resid @ V @ T ) @ U ) ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ T ) @ U ) @ ( lambda_resid @ V @ W2 ) ) ) ) ) ) ).
% \<Lambda>.resid_composite_of(3)
thf(fact_1127__092_060Lambda_062_Ocomposite__of__arr__target,axiom,
! [T: lambda_lambda,B: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ B @ ( target5490931411331770124lambda @ lambda_resid @ T ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ T @ B @ T ) ) ) ).
% \<Lambda>.composite_of_arr_target
thf(fact_1128__092_060Lambda_062_Ocomposite__of__source__arr,axiom,
! [T: lambda_lambda,A2: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
=> ( ( member_lambda_lambda @ A2 @ ( source5627170679743844886lambda @ lambda_resid @ T ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ A2 @ T @ T ) ) ) ).
% \<Lambda>.composite_of_source_arr
thf(fact_1129_composite__of__single__single,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ U @ V )
=> ( compos7114946393191943382lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) @ ( append_lambda_lambda @ ( cons_lambda_lambda @ T @ nil_lambda_lambda ) @ ( cons_lambda_lambda @ U @ nil_lambda_lambda ) ) ) ) ).
% composite_of_single_single
thf(fact_1130__092_060Lambda_062_Ocomposite__ofI,axiom,
! [U: lambda_lambda,V: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ V ) )
=> ( ( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ V @ U ) @ T ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ ( lambda_resid @ V @ U ) ) ) )
=> ( compos9183582412961930448lambda @ lambda_resid @ U @ T @ V ) ) ) ).
% \<Lambda>.composite_ofI
thf(fact_1131__092_060Lambda_062_Ojoin__ofI,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( compos9183582412961930448lambda @ lambda_resid @ T @ ( lambda_resid @ U @ T ) @ V )
=> ( ( compos9183582412961930448lambda @ lambda_resid @ U @ ( lambda_resid @ T @ U ) @ V )
=> ( join_o1386658102433174083lambda @ lambda_resid @ T @ U @ V ) ) ) ).
% \<Lambda>.join_ofI
thf(fact_1132_lambda__calculus_Olambda_Ocollapse_I3_J,axiom,
! [Lambda2: lambda_lambda] :
( ( lambda_is_App @ Lambda2 )
=> ( ( lambda_App @ ( lambda_un_App1 @ Lambda2 ) @ ( lambda_un_App2 @ Lambda2 ) )
= Lambda2 ) ) ).
% lambda_calculus.lambda.collapse(3)
thf(fact_1133_lambda__calculus_Oprfx__App__iff,axiom,
! [U: lambda_lambda,T1: lambda_lambda,T22: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ ( lambda_App @ T1 @ T22 ) ) )
= ( ( lambda_is_App @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_App1 @ U ) @ T1 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_un_App2 @ U ) @ T22 ) ) ) ) ).
% lambda_calculus.prfx_App_iff
thf(fact_1134_lambda__calculus_Olambda_Osel_I4_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_un_App2 @ ( lambda_App @ X41 @ X42 ) )
= X42 ) ).
% lambda_calculus.lambda.sel(4)
thf(fact_1135_lambda__calculus_Olambda_Osel_I3_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_un_App1 @ ( lambda_App @ X41 @ X42 ) )
= X41 ) ).
% lambda_calculus.lambda.sel(3)
thf(fact_1136_map__App__map__un__App2,axiom,
! [U2: list_lambda_lambda,A2: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( lambda_Ide @ A2 )
=> ( ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ lambda_un_App1 @ ( set_lambda_lambda2 @ U2 ) ) @ ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) )
=> ( ( map_la336867333220232516lambda @ ( lambda_App @ A2 ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U2 ) )
= U2 ) ) ) ) ) ).
% map_App_map_un_App2
thf(fact_1137_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_li7878742357367608272lambda,Resid: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda] :
( ! [T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ T3 @ NN )
=> ( arr_li3691443969012319837lambda @ Resid @ T3 ) )
=> ( ! [A6: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid @ A6 )
=> ( member2667661554319814297lambda @ A6 @ NN ) )
=> ( ! [U5: list_lambda_lambda,T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U5 @ NN )
=> ( ( coinit1063259437155584316lambda @ Resid @ T3 @ U5 )
=> ( member2667661554319814297lambda @ ( Resid @ U5 @ T3 ) @ NN ) ) )
=> ( ! [U5: list_lambda_lambda,T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U5 @ NN )
=> ( ( member2667661554319814297lambda @ ( Resid @ T3 @ U5 ) @ NN )
=> ( member2667661554319814297lambda @ T3 @ NN ) ) )
=> ( ! [U5: list_lambda_lambda,T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U5 @ NN )
=> ( ( seq_li8093550137657879445lambda @ Resid @ U5 @ T3 )
=> ? [X_1: list_lambda_lambda] : ( compos7114946393191943382lambda @ Resid @ U5 @ T3 @ X_1 ) ) )
=> ( ! [U5: list_lambda_lambda,T3: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U5 @ NN )
=> ( ( seq_li8093550137657879445lambda @ Resid @ T3 @ U5 )
=> ? [X_1: list_lambda_lambda] : ( compos7114946393191943382lambda @ Resid @ T3 @ U5 @ X_1 ) ) )
=> ( normal6810188024974762232lambda @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_1138_normal__sub__rts__axioms_Ointro,axiom,
! [NN: set_lambda_lambda,Resid: lambda_lambda > lambda_lambda > lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( member_lambda_lambda @ T3 @ NN )
=> ( arr_lambda_lambda @ Resid @ T3 ) )
=> ( ! [A6: lambda_lambda] :
( ( ide_lambda_lambda @ Resid @ A6 )
=> ( member_lambda_lambda @ A6 @ NN ) )
=> ( ! [U5: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U5 @ NN )
=> ( ( coinit1392077014771647542lambda @ Resid @ T3 @ U5 )
=> ( member_lambda_lambda @ ( Resid @ U5 @ T3 ) @ NN ) ) )
=> ( ! [U5: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U5 @ NN )
=> ( ( member_lambda_lambda @ ( Resid @ T3 @ U5 ) @ NN )
=> ( member_lambda_lambda @ T3 @ NN ) ) )
=> ( ! [U5: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U5 @ NN )
=> ( ( seq_lambda_lambda @ Resid @ U5 @ T3 )
=> ? [X_1: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ U5 @ T3 @ X_1 ) ) )
=> ( ! [U5: lambda_lambda,T3: lambda_lambda] :
( ( member_lambda_lambda @ U5 @ NN )
=> ( ( seq_lambda_lambda @ Resid @ T3 @ U5 )
=> ? [X_1: lambda_lambda] : ( compos9183582412961930448lambda @ Resid @ T3 @ U5 @ X_1 ) ) )
=> ( normal5162919049924913650lambda @ Resid @ NN ) ) ) ) ) ) ) ).
% normal_sub_rts_axioms.intro
thf(fact_1139_image__empty,axiom,
! [F: list_lambda_lambda > list_lambda_lambda] :
( ( image_5886195720093285125lambda @ F @ bot_bo1397037886704484924lambda )
= bot_bo1397037886704484924lambda ) ).
% image_empty
thf(fact_1140_image__empty,axiom,
! [F: list_lambda_lambda > lambda_lambda] :
( ( image_2458965899662700159lambda @ F @ bot_bo1397037886704484924lambda )
= bot_bo1271416176687173302lambda ) ).
% image_empty
thf(fact_1141_image__empty,axiom,
! [F: lambda_lambda > list_lambda_lambda] :
( ( image_3692823078739859851lambda @ F @ bot_bo1271416176687173302lambda )
= bot_bo1397037886704484924lambda ) ).
% image_empty
thf(fact_1142_image__empty,axiom,
! [F: lambda_lambda > lambda_lambda] :
( ( image_2356535832909760517lambda @ F @ bot_bo1271416176687173302lambda )
= bot_bo1271416176687173302lambda ) ).
% image_empty
thf(fact_1143_empty__is__image,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( bot_bo1397037886704484924lambda
= ( image_5886195720093285125lambda @ F @ A ) )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% empty_is_image
thf(fact_1144_empty__is__image,axiom,
! [F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( bot_bo1397037886704484924lambda
= ( image_3692823078739859851lambda @ F @ A ) )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% empty_is_image
thf(fact_1145_empty__is__image,axiom,
! [F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( bot_bo1271416176687173302lambda
= ( image_2458965899662700159lambda @ F @ A ) )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% empty_is_image
thf(fact_1146_empty__is__image,axiom,
! [F: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( bot_bo1271416176687173302lambda
= ( image_2356535832909760517lambda @ F @ A ) )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% empty_is_image
thf(fact_1147_image__is__empty,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( image_5886195720093285125lambda @ F @ A )
= bot_bo1397037886704484924lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% image_is_empty
thf(fact_1148_image__is__empty,axiom,
! [F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( ( image_3692823078739859851lambda @ F @ A )
= bot_bo1397037886704484924lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% image_is_empty
thf(fact_1149_image__is__empty,axiom,
! [F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ( image_2458965899662700159lambda @ F @ A )
= bot_bo1271416176687173302lambda )
= ( A = bot_bo1397037886704484924lambda ) ) ).
% image_is_empty
thf(fact_1150_image__is__empty,axiom,
! [F: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( ( image_2356535832909760517lambda @ F @ A )
= bot_bo1271416176687173302lambda )
= ( A = bot_bo1271416176687173302lambda ) ) ).
% image_is_empty
thf(fact_1151_list_Oset__map,axiom,
! [F: lambda_lambda > list_lambda_lambda,V: list_lambda_lambda] :
( ( set_li3846194476672328965lambda @ ( map_la6434941737452557130lambda @ F @ V ) )
= ( image_3692823078739859851lambda @ F @ ( set_lambda_lambda2 @ V ) ) ) ).
% list.set_map
thf(fact_1152_list_Oset__map,axiom,
! [F: lambda_lambda > lambda_lambda,V: list_lambda_lambda] :
( ( set_lambda_lambda2 @ ( map_la336867333220232516lambda @ F @ V ) )
= ( image_2356535832909760517lambda @ F @ ( set_lambda_lambda2 @ V ) ) ) ).
% list.set_map
thf(fact_1153_the__elem__image__unique,axiom,
! [A: set_lambda_lambda,F: lambda_lambda > lambda_lambda,X2: lambda_lambda] :
( ( A != bot_bo1271416176687173302lambda )
=> ( ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_el5431214018478934239lambda @ ( image_2356535832909760517lambda @ F @ A ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1154_the__elem__image__unique,axiom,
! [A: set_lambda_lambda,F: lambda_lambda > list_lambda_lambda,X2: lambda_lambda] :
( ( A != bot_bo1271416176687173302lambda )
=> ( ! [Y4: lambda_lambda] :
( ( member_lambda_lambda @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_el4548681628688475493lambda @ ( image_3692823078739859851lambda @ F @ A ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1155_image__Int__subset,axiom,
! [F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] : ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F @ ( inf_in3354698378728726462lambda @ A @ B4 ) ) @ ( inf_in1766959559283430712lambda @ ( image_2458965899662700159lambda @ F @ A ) @ ( image_2458965899662700159lambda @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_1156_image__Int__subset,axiom,
! [F: lambda_lambda > lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] : ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ ( inf_in1766959559283430712lambda @ A @ B4 ) ) @ ( inf_in1766959559283430712lambda @ ( image_2356535832909760517lambda @ F @ A ) @ ( image_2356535832909760517lambda @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_1157_image__Int__subset,axiom,
! [F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda] : ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F @ ( inf_in3354698378728726462lambda @ A @ B4 ) ) @ ( inf_in3354698378728726462lambda @ ( image_5886195720093285125lambda @ F @ A ) @ ( image_5886195720093285125lambda @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_1158_image__Int__subset,axiom,
! [F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] : ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ ( inf_in1766959559283430712lambda @ A @ B4 ) ) @ ( inf_in3354698378728726462lambda @ ( image_3692823078739859851lambda @ F @ A ) @ ( image_3692823078739859851lambda @ F @ B4 ) ) ) ).
% image_Int_subset
thf(fact_1159_image__set,axiom,
! [F: lambda_lambda > list_lambda_lambda,Xs: list_lambda_lambda] :
( ( image_3692823078739859851lambda @ F @ ( set_lambda_lambda2 @ Xs ) )
= ( set_li3846194476672328965lambda @ ( map_la6434941737452557130lambda @ F @ Xs ) ) ) ).
% image_set
thf(fact_1160_image__set,axiom,
! [F: lambda_lambda > lambda_lambda,Xs: list_lambda_lambda] :
( ( image_2356535832909760517lambda @ F @ ( set_lambda_lambda2 @ Xs ) )
= ( set_lambda_lambda2 @ ( map_la336867333220232516lambda @ F @ Xs ) ) ) ).
% image_set
thf(fact_1161_image__mono,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda,F: lambda_lambda > lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ A ) @ ( image_2356535832909760517lambda @ F @ B4 ) ) ) ).
% image_mono
thf(fact_1162_image__mono,axiom,
! [A: set_lambda_lambda,B4: set_lambda_lambda,F: lambda_lambda > list_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ A @ B4 )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ A ) @ ( image_3692823078739859851lambda @ F @ B4 ) ) ) ).
% image_mono
thf(fact_1163_image__mono,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda,F: list_lambda_lambda > lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F @ A ) @ ( image_2458965899662700159lambda @ F @ B4 ) ) ) ).
% image_mono
thf(fact_1164_image__mono,axiom,
! [A: set_li7878742357367608272lambda,B4: set_li7878742357367608272lambda,F: list_lambda_lambda > list_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ A @ B4 )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F @ A ) @ ( image_5886195720093285125lambda @ F @ B4 ) ) ) ).
% image_mono
thf(fact_1165_image__subsetI,axiom,
! [A: set_lambda_lambda,F: lambda_lambda > lambda_lambda,B4: set_lambda_lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( member_lambda_lambda @ ( F @ X ) @ B4 ) )
=> ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1166_image__subsetI,axiom,
! [A: set_li7878742357367608272lambda,F: list_lambda_lambda > lambda_lambda,B4: set_lambda_lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( member_lambda_lambda @ ( F @ X ) @ B4 ) )
=> ( ord_le1926335974945931754lambda @ ( image_2458965899662700159lambda @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1167_image__subsetI,axiom,
! [A: set_lambda_lambda,F: lambda_lambda > list_lambda_lambda,B4: set_li7878742357367608272lambda] :
( ! [X: lambda_lambda] :
( ( member_lambda_lambda @ X @ A )
=> ( member2667661554319814297lambda @ ( F @ X ) @ B4 ) )
=> ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1168_image__subsetI,axiom,
! [A: set_li7878742357367608272lambda,F: list_lambda_lambda > list_lambda_lambda,B4: set_li7878742357367608272lambda] :
( ! [X: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X @ A )
=> ( member2667661554319814297lambda @ ( F @ X ) @ B4 ) )
=> ( ord_le3064231726521830256lambda @ ( image_5886195720093285125lambda @ F @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1169_subset__imageE,axiom,
! [B4: set_lambda_lambda,F: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B4 @ ( image_2356535832909760517lambda @ F @ A ) )
=> ~ ! [C5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C5 @ A )
=> ( B4
!= ( image_2356535832909760517lambda @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1170_subset__imageE,axiom,
! [B4: set_lambda_lambda,F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ B4 @ ( image_2458965899662700159lambda @ F @ A ) )
=> ~ ! [C5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C5 @ A )
=> ( B4
!= ( image_2458965899662700159lambda @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1171_subset__imageE,axiom,
! [B4: set_li7878742357367608272lambda,F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ B4 @ ( image_3692823078739859851lambda @ F @ A ) )
=> ~ ! [C5: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ C5 @ A )
=> ( B4
!= ( image_3692823078739859851lambda @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1172_subset__imageE,axiom,
! [B4: set_li7878742357367608272lambda,F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B4 @ ( image_5886195720093285125lambda @ F @ A ) )
=> ~ ! [C5: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ C5 @ A )
=> ( B4
!= ( image_5886195720093285125lambda @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1173_image__subset__iff,axiom,
! [F: lambda_lambda > lambda_lambda,A: set_lambda_lambda,B4: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ F @ A ) @ B4 )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
=> ( member_lambda_lambda @ ( F @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_1174_image__subset__iff,axiom,
! [F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda,B4: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ ( image_3692823078739859851lambda @ F @ A ) @ B4 )
= ( ! [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
=> ( member2667661554319814297lambda @ ( F @ X4 ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_1175_subset__image__iff,axiom,
! [B4: set_lambda_lambda,F: lambda_lambda > lambda_lambda,A: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ B4 @ ( image_2356535832909760517lambda @ F @ A ) )
= ( ? [AA: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ AA @ A )
& ( B4
= ( image_2356535832909760517lambda @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1176_subset__image__iff,axiom,
! [B4: set_lambda_lambda,F: list_lambda_lambda > lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ord_le1926335974945931754lambda @ B4 @ ( image_2458965899662700159lambda @ F @ A ) )
= ( ? [AA: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ AA @ A )
& ( B4
= ( image_2458965899662700159lambda @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1177_subset__image__iff,axiom,
! [B4: set_li7878742357367608272lambda,F: lambda_lambda > list_lambda_lambda,A: set_lambda_lambda] :
( ( ord_le3064231726521830256lambda @ B4 @ ( image_3692823078739859851lambda @ F @ A ) )
= ( ? [AA: set_lambda_lambda] :
( ( ord_le1926335974945931754lambda @ AA @ A )
& ( B4
= ( image_3692823078739859851lambda @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1178_subset__image__iff,axiom,
! [B4: set_li7878742357367608272lambda,F: list_lambda_lambda > list_lambda_lambda,A: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ B4 @ ( image_5886195720093285125lambda @ F @ A ) )
= ( ? [AA: set_li7878742357367608272lambda] :
( ( ord_le3064231726521830256lambda @ AA @ A )
& ( B4
= ( image_5886195720093285125lambda @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1179_normal__sub__rts__axioms__def,axiom,
( normal6810188024974762232lambda
= ( ^ [Resid2: list_lambda_lambda > list_lambda_lambda > list_lambda_lambda,NN2: set_li7878742357367608272lambda] :
( ! [T5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ T5 @ NN2 )
=> ( arr_li3691443969012319837lambda @ Resid2 @ T5 ) )
& ! [A3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ Resid2 @ A3 )
=> ( member2667661554319814297lambda @ A3 @ NN2 ) )
& ! [U6: list_lambda_lambda,T5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( coinit1063259437155584316lambda @ Resid2 @ T5 @ U6 )
=> ( member2667661554319814297lambda @ ( Resid2 @ U6 @ T5 ) @ NN2 ) ) )
& ! [U6: list_lambda_lambda,T5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( member2667661554319814297lambda @ ( Resid2 @ T5 @ U6 ) @ NN2 )
=> ( member2667661554319814297lambda @ T5 @ NN2 ) ) )
& ! [U6: list_lambda_lambda,T5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( seq_li8093550137657879445lambda @ Resid2 @ U6 @ T5 )
=> ? [X7: list_lambda_lambda] : ( compos7114946393191943382lambda @ Resid2 @ U6 @ T5 @ X7 ) ) )
& ! [U6: list_lambda_lambda,T5: list_lambda_lambda] :
( ( member2667661554319814297lambda @ U6 @ NN2 )
=> ( ( seq_li8093550137657879445lambda @ Resid2 @ T5 @ U6 )
=> ? [X7: list_lambda_lambda] : ( compos7114946393191943382lambda @ Resid2 @ T5 @ U6 @ X7 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_1180_normal__sub__rts__axioms__def,axiom,
( normal5162919049924913650lambda
= ( ^ [Resid2: lambda_lambda > lambda_lambda > lambda_lambda,NN2: set_lambda_lambda] :
( ! [T5: lambda_lambda] :
( ( member_lambda_lambda @ T5 @ NN2 )
=> ( arr_lambda_lambda @ Resid2 @ T5 ) )
& ! [A3: lambda_lambda] :
( ( ide_lambda_lambda @ Resid2 @ A3 )
=> ( member_lambda_lambda @ A3 @ NN2 ) )
& ! [U6: lambda_lambda,T5: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( coinit1392077014771647542lambda @ Resid2 @ T5 @ U6 )
=> ( member_lambda_lambda @ ( Resid2 @ U6 @ T5 ) @ NN2 ) ) )
& ! [U6: lambda_lambda,T5: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( member_lambda_lambda @ ( Resid2 @ T5 @ U6 ) @ NN2 )
=> ( member_lambda_lambda @ T5 @ NN2 ) ) )
& ! [U6: lambda_lambda,T5: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( seq_lambda_lambda @ Resid2 @ U6 @ T5 )
=> ? [X7: lambda_lambda] : ( compos9183582412961930448lambda @ Resid2 @ U6 @ T5 @ X7 ) ) )
& ! [U6: lambda_lambda,T5: lambda_lambda] :
( ( member_lambda_lambda @ U6 @ NN2 )
=> ( ( seq_lambda_lambda @ Resid2 @ T5 @ U6 )
=> ? [X7: lambda_lambda] : ( compos9183582412961930448lambda @ Resid2 @ T5 @ U6 @ X7 ) ) ) ) ) ) ).
% normal_sub_rts_axioms_def
thf(fact_1181_map__App__decomp,axiom,
! [U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ U2 ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U2 ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ U2 ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U2 ) ) )
@ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid @ U2
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Src @ ( lambda_un_App2 @ ( hd_lambda_lambda @ U2 ) ) ) )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U2 ) )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( lambda_un_App1 @ ( last_lambda_lambda @ U2 ) ) ) ) @ ( map_la336867333220232516lambda @ lambda_un_App2 @ U2 ) ) ) ) ) ) ) ) ).
% map_App_decomp
thf(fact_1182_map__App__map__un__App1,axiom,
! [U2: list_lambda_lambda,B: lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ord_le1926335974945931754lambda @ ( set_lambda_lambda2 @ U2 ) @ ( collec621444948832755541lambda @ lambda_is_App ) )
=> ( ( lambda_Ide @ B )
=> ( ( ord_le1926335974945931754lambda @ ( image_2356535832909760517lambda @ lambda_un_App2 @ ( set_lambda_lambda2 @ U2 ) ) @ ( insert_lambda_lambda @ B @ bot_bo1271416176687173302lambda ) )
=> ( ( map_la336867333220232516lambda
@ ^ [T5: lambda_lambda] : ( lambda_App @ T5 @ B )
@ ( map_la336867333220232516lambda @ lambda_un_App1 @ U2 ) )
= U2 ) ) ) ) ) ).
% map_App_map_un_App1
thf(fact_1183__092_060Lambda_062_Otargets__char,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B3: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ ( lambda_resid @ T @ T ) @ B3 ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ B3 @ ( lambda_resid @ T @ T ) ) ) ) ) ) ).
% \<Lambda>.targets_char
thf(fact_1184__092_060Lambda_062_Osources__def,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [A3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ A3 )
& ( con_lambda_lambda @ lambda_resid @ T @ A3 ) ) ) ) ).
% \<Lambda>.sources_def
thf(fact_1185__092_060Lambda_062_Otargets__char_092_060_094sub_062W_092_060_094sub_062E,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B3: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ( trg_lambda_lambda @ lambda_resid @ T )
= B3 ) ) ) ) ).
% \<Lambda>.targets_char\<^sub>W\<^sub>E
thf(fact_1186__092_060Lambda_062_Osources__char,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [A3: lambda_lambda] :
( ( arr_lambda_lambda @ lambda_resid @ T )
& ( ( weakly4554751870655413501lambda @ lambda_resid @ T )
= A3 ) ) ) ) ).
% \<Lambda>.sources_char
thf(fact_1187__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__closed,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_is_App @ T ) )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ lambda_resid @ T )
@ ( collec621444948832755541lambda
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_closed
thf(fact_1188__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Osources__closed,axiom,
! [T: lambda_lambda] :
( ( ( lambda_Arr @ T )
& ( lambda_is_Lam @ T ) )
=> ( ord_le1926335974945931754lambda @ ( source5627170679743844886lambda @ lambda_resid @ T )
@ ( collec621444948832755541lambda
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.sources_closed
thf(fact_1189__092_060Lambda_062_Otargets__def,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda @ lambda_resid @ T )
= ( collec621444948832755541lambda
@ ^ [B3: lambda_lambda] :
( ( ide_lambda_lambda @ lambda_resid @ B3 )
& ( con_lambda_lambda @ lambda_resid @ ( trg_lambda_lambda @ lambda_resid @ T ) @ B3 ) ) ) ) ).
% \<Lambda>.targets_def
thf(fact_1190_sources__def,axiom,
! [T: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [A3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ A3 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T @ A3 ) ) ) ) ).
% sources_def
thf(fact_1191_Arr__map__App1,axiom,
! [B: lambda_lambda,T4: list_lambda_lambda] :
( ( lambda_Ide @ B )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( paths_366302053746739058lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T5: lambda_lambda] : ( lambda_App @ T5 @ B )
@ T4 ) ) ) ) ).
% Arr_map_App1
thf(fact_1192_targets__char,axiom,
! [T: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [B3: list_lambda_lambda] :
( ( arr_li3691443969012319837lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) @ B3 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B3 @ ( paths_8607357831768207192lambda @ lambda_resid @ T @ T ) ) ) ) ) ) ).
% targets_char
thf(fact_1193_map__ident,axiom,
( ( map_la336867333220232516lambda
@ ^ [X4: lambda_lambda] : X4 )
= ( ^ [Xs4: list_lambda_lambda] : Xs4 ) ) ).
% map_ident
thf(fact_1194_Resid__single__ide_I1_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 )
=> ( member2667661554319814297lambda @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A2 @ nil_lambda_lambda ) @ T4 )
@ ( image_3692823078739859851lambda
@ ^ [B3: lambda_lambda] : ( cons_lambda_lambda @ B3 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% Resid_single_ide(1)
thf(fact_1195_cong__map__App2,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ U2 @ T4 ) ) )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ X2 )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ X2 )
@ U2 ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ X2 )
@ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ X2 )
@ T4 ) ) ) ) ) ) ).
% cong_map_App2
thf(fact_1196_sources__char_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [A5: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ A5 )
& ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ A5 )
= ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% sources_char\<^sub>P
thf(fact_1197_map__App2__Resid,axiom,
! [X2: lambda_lambda,T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( lambda_Ide @ X2 )
=> ( ( coinit1063259437155584316lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 @ U2 )
=> ( ( map_la336867333220232516lambda
@ ^ [T5: lambda_lambda] : ( lambda_App @ T5 @ X2 )
@ ( paths_8607357831768207192lambda @ lambda_resid @ T4 @ U2 ) )
= ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [T5: lambda_lambda] : ( lambda_App @ T5 @ X2 )
@ T4 )
@ ( map_la336867333220232516lambda
@ ^ [T5: lambda_lambda] : ( lambda_App @ T5 @ X2 )
@ U2 ) ) ) ) ) ).
% map_App2_Resid
thf(fact_1198_targets__def,axiom,
! [T: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T )
= ( collec8830145823237851355lambda
@ ^ [B3: list_lambda_lambda] :
( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ B3 )
& ( con_li79537494059641180lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( trg_li817795487149935061lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T ) @ B3 ) ) ) ) ).
% targets_def
thf(fact_1199_img__Resid__Srcs,axiom,
! [T4: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ord_le3064231726521830256lambda
@ ( image_3692823078739859851lambda
@ ^ [A3: lambda_lambda] : ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ A3 @ nil_lambda_lambda ) @ T4 )
@ ( paths_2897053694880971696lambda @ lambda_resid @ T4 ) )
@ ( image_3692823078739859851lambda
@ ^ [B3: lambda_lambda] : ( cons_lambda_lambda @ B3 @ nil_lambda_lambda )
@ ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ).
% img_Resid_Srcs
thf(fact_1200_targets__char_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [B5: list_lambda_lambda] :
( ( paths_7717490669901197737lambda @ lambda_resid @ B5 )
& ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ( paths_2897053694880971696lambda @ lambda_resid @ B5 )
= ( paths_7594662086398683701lambda @ lambda_resid @ T4 ) ) ) ) ) ).
% targets_char\<^sub>P
thf(fact_1201_singleton__conv,axiom,
! [A2: list_lambda_lambda] :
( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] : ( X4 = A2 ) )
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).
% singleton_conv
thf(fact_1202_singleton__conv,axiom,
! [A2: lambda_lambda] :
( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] : ( X4 = A2 ) )
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).
% singleton_conv
thf(fact_1203_singleton__conv2,axiom,
! [A2: list_lambda_lambda] :
( ( collec8830145823237851355lambda
@ ( ^ [Y6: list_lambda_lambda,Z3: list_lambda_lambda] : ( Y6 = Z3 )
@ A2 ) )
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) ).
% singleton_conv2
thf(fact_1204_singleton__conv2,axiom,
! [A2: lambda_lambda] :
( ( collec621444948832755541lambda
@ ( ^ [Y6: lambda_lambda,Z3: lambda_lambda] : ( Y6 = Z3 )
@ A2 ) )
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) ).
% singleton_conv2
thf(fact_1205_sources__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( source6119337877621565212lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [A5: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ A5 @ ( 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 ) @ A5 ) ) ) ) ) ).
% sources_char\<^sub>\<Lambda>\<^sub>P
thf(fact_1206_targets__char_092_060_094sub_062_092_060Lambda_062_092_060_094sub_062P,axiom,
! [T4: list_lambda_lambda] :
( ( target4026644918049360146lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ T4 )
= ( collec8830145823237851355lambda
@ ^ [B5: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ B5 @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid ) @ ( paths_8607357831768207192lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) @ nil_lambda_lambda ) @ B5 ) ) ) ) ) ).
% targets_char\<^sub>\<Lambda>\<^sub>P
thf(fact_1207_orthogonal__App__Arr__Arr,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ( paths_8607357831768207192lambda @ lambda_resid @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 ) )
= ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) )
& ( ( paths_8607357831768207192lambda @ lambda_resid
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 ) )
= ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) ) ) ) ) ).
% orthogonal_App_Arr_Arr
thf(fact_1208_orthogonal__App__cong,axiom,
! [T4: list_lambda_lambda,U2: list_lambda_lambda] :
( ( paths_366302053746739058lambda @ lambda_resid @ T4 )
=> ( ( paths_366302053746739058lambda @ lambda_resid @ U2 )
=> ( ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) )
@ ( append_lambda_lambda @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) ) ) )
& ( ide_li6773656263178250516lambda @ ( paths_8607357831768207192lambda @ lambda_resid )
@ ( paths_8607357831768207192lambda @ lambda_resid
@ ( append_lambda_lambda @ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Src @ ( hd_lambda_lambda @ T4 ) ) ) @ U2 )
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Trg @ ( last_lambda_lambda @ U2 ) ) )
@ T4 ) )
@ ( append_lambda_lambda
@ ( map_la336867333220232516lambda
@ ^ [X7: lambda_lambda] : ( lambda_App @ X7 @ ( lambda_Src @ ( hd_lambda_lambda @ U2 ) ) )
@ T4 )
@ ( map_la336867333220232516lambda @ ( lambda_App @ ( lambda_Trg @ ( last_lambda_lambda @ T4 ) ) ) @ U2 ) ) ) ) ) ) ) ).
% orthogonal_App_cong
thf(fact_1209_image__constant__conv,axiom,
! [A: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( ( A = bot_bo1397037886704484924lambda )
=> ( ( image_5886195720093285125lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= bot_bo1397037886704484924lambda ) )
& ( ( A != bot_bo1397037886704484924lambda )
=> ( ( image_5886195720093285125lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ) ).
% image_constant_conv
thf(fact_1210_image__constant__conv,axiom,
! [A: set_li7878742357367608272lambda,C: lambda_lambda] :
( ( ( A = bot_bo1397037886704484924lambda )
=> ( ( image_2458965899662700159lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= bot_bo1271416176687173302lambda ) )
& ( ( A != bot_bo1397037886704484924lambda )
=> ( ( image_2458965899662700159lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ) ).
% image_constant_conv
thf(fact_1211_image__constant__conv,axiom,
! [A: set_lambda_lambda,C: list_lambda_lambda] :
( ( ( A = bot_bo1271416176687173302lambda )
=> ( ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= bot_bo1397037886704484924lambda ) )
& ( ( A != bot_bo1271416176687173302lambda )
=> ( ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ) ).
% image_constant_conv
thf(fact_1212_image__constant__conv,axiom,
! [A: set_lambda_lambda,C: lambda_lambda] :
( ( ( A = bot_bo1271416176687173302lambda )
=> ( ( image_2356535832909760517lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= bot_bo1271416176687173302lambda ) )
& ( ( A != bot_bo1271416176687173302lambda )
=> ( ( image_2356535832909760517lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ) ).
% image_constant_conv
thf(fact_1213_image__constant,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda,C: list_lambda_lambda] :
( ( member_lambda_lambda @ X2 @ A )
=> ( ( image_3692823078739859851lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ).
% image_constant
thf(fact_1214_image__constant,axiom,
! [X2: list_lambda_lambda,A: set_li7878742357367608272lambda,C: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ A )
=> ( ( image_5886195720093285125lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= ( insert8465602035467394112lambda @ C @ bot_bo1397037886704484924lambda ) ) ) ).
% image_constant
thf(fact_1215_image__constant,axiom,
! [X2: lambda_lambda,A: set_lambda_lambda,C: lambda_lambda] :
( ( member_lambda_lambda @ X2 @ A )
=> ( ( image_2356535832909760517lambda
@ ^ [X4: lambda_lambda] : C
@ A )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ).
% image_constant
thf(fact_1216_image__constant,axiom,
! [X2: list_lambda_lambda,A: set_li7878742357367608272lambda,C: lambda_lambda] :
( ( member2667661554319814297lambda @ X2 @ A )
=> ( ( image_2458965899662700159lambda
@ ^ [X4: list_lambda_lambda] : C
@ A )
= ( insert_lambda_lambda @ C @ bot_bo1271416176687173302lambda ) ) ) ).
% image_constant
thf(fact_1217_pred__subset__eq,axiom,
! [R2: set_lambda_lambda,S: set_lambda_lambda] :
( ( ord_le5488116190248881011mbda_o
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ R2 )
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ S ) )
= ( ord_le1926335974945931754lambda @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1218_pred__subset__eq,axiom,
! [R2: set_li7878742357367608272lambda,S: set_li7878742357367608272lambda] :
( ( ord_le4030711025213088621mbda_o
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ R2 )
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ S ) )
= ( ord_le3064231726521830256lambda @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1219_empty__def,axiom,
( bot_bo1397037886704484924lambda
= ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] : $false ) ) ).
% empty_def
thf(fact_1220_empty__def,axiom,
( bot_bo1271416176687173302lambda
= ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] : $false ) ) ).
% empty_def
thf(fact_1221_Collect__conv__if,axiom,
! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
( ( ( P @ A2 )
=> ( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bo1397037886704484924lambda ) ) ) ).
% Collect_conv_if
thf(fact_1222_Collect__conv__if,axiom,
! [P: lambda_lambda > $o,A2: lambda_lambda] :
( ( ( P @ A2 )
=> ( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bo1271416176687173302lambda ) ) ) ).
% Collect_conv_if
thf(fact_1223_Collect__conv__if2,axiom,
! [P: list_lambda_lambda > $o,A2: list_lambda_lambda] :
( ( ( P @ A2 )
=> ( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert8465602035467394112lambda @ A2 @ bot_bo1397037886704484924lambda ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bo1397037886704484924lambda ) ) ) ).
% Collect_conv_if2
thf(fact_1224_Collect__conv__if2,axiom,
! [P: lambda_lambda > $o,A2: lambda_lambda] :
( ( ( P @ A2 )
=> ( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert_lambda_lambda @ A2 @ bot_bo1271416176687173302lambda ) ) )
& ( ~ ( P @ A2 )
=> ( ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bo1271416176687173302lambda ) ) ) ).
% Collect_conv_if2
thf(fact_1225_list_Omap__ident,axiom,
! [T: list_lambda_lambda] :
( ( map_la336867333220232516lambda
@ ^ [X4: lambda_lambda] : X4
@ T )
= T ) ).
% list.map_ident
thf(fact_1226_Collect__subset,axiom,
! [A: set_lambda_lambda,P: lambda_lambda > $o] :
( ord_le1926335974945931754lambda
@ ( collec621444948832755541lambda
@ ^ [X4: lambda_lambda] :
( ( member_lambda_lambda @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1227_Collect__subset,axiom,
! [A: set_li7878742357367608272lambda,P: list_lambda_lambda > $o] :
( ord_le3064231726521830256lambda
@ ( collec8830145823237851355lambda
@ ^ [X4: list_lambda_lambda] :
( ( member2667661554319814297lambda @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_1228_less__eq__set__def,axiom,
( ord_le1926335974945931754lambda
= ( ^ [A5: set_lambda_lambda,B5: set_lambda_lambda] :
( ord_le5488116190248881011mbda_o
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ A5 )
@ ^ [X4: lambda_lambda] : ( member_lambda_lambda @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1229_less__eq__set__def,axiom,
( ord_le3064231726521830256lambda
= ( ^ [A5: set_li7878742357367608272lambda,B5: set_li7878742357367608272lambda] :
( ord_le4030711025213088621mbda_o
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ A5 )
@ ^ [X4: list_lambda_lambda] : ( member2667661554319814297lambda @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_1230__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocoterminal__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T
@ U )
= ( ( inf_in1766959559283430712lambda
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T )
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.coterminal_def
thf(fact_1231__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocoterminal__def,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( coterm4142657152830665724lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T
@ U )
= ( ( inf_in1766959559283430712lambda
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T )
@ ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ U ) )
!= bot_bo1271416176687173302lambda ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.coterminal_def
thf(fact_1232__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ T ) )
= ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ V
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cube
thf(fact_1233__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oex__un__null,axiom,
? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) )
@ X
@ T2 )
= X )
& ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) )
@ T2
@ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) )
@ Y
@ T3 )
= Y )
& ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) )
@ T3
@ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ex_un_null
thf(fact_1234__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocube,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ T ) )
= ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ V
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.cube
thf(fact_1235__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oex__un__null,axiom,
? [X: lambda_lambda] :
( ! [T2: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_Lam @ U6 ) )
@ X
@ T2 )
= X )
& ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_Lam @ U6 ) )
@ T2
@ X )
= X ) )
& ! [Y: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_Lam @ U6 ) )
@ Y
@ T3 )
= Y )
& ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_Lam @ U6 ) )
@ T3
@ Y )
= Y ) )
=> ( Y = X ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.ex_un_null
thf(fact_1236__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oprfx__transitive,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ V ) )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ V ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.prfx_transitive
thf(fact_1237__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oprfx__char_092_060_094sub_062S_092_060_094sub_062R_092_060_094sub_062T_092_060_094sub_062S,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( lambda_Arr @ U )
& ( lambda_is_App @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.prfx_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1238__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oide__backward__stable,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ A2 )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ A2 ) )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ide_backward_stable
thf(fact_1239__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__transitive,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ T ) ) )
=> ( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ V ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ V
@ U ) ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ V ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ V
@ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_transitive
thf(fact_1240__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__symmetric,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ T ) ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ T ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_symmetric
thf(fact_1241__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__char_092_060_094sub_062S_092_060_094sub_062R_092_060_094sub_062T_092_060_094sub_062S,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ T ) ) )
= ( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( lambda_Arr @ U )
& ( lambda_is_App @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1242__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oprfx__transitive,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ V ) )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ V ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.prfx_transitive
thf(fact_1243__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oprfx__char_092_060_094sub_062S_092_060_094sub_062R_092_060_094sub_062T_092_060_094sub_062S,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_is_Lam @ T )
& ( lambda_Arr @ U )
& ( lambda_is_Lam @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.prfx_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1244__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oide__backward__stable,axiom,
! [A2: lambda_lambda,T: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ A2 )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ A2 ) )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.ide_backward_stable
thf(fact_1245__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocong__transitive,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ T ) ) )
=> ( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ V ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ V
@ U ) ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ V ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ V
@ T ) ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.cong_transitive
thf(fact_1246__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocong__symmetric,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ T ) ) )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ T ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.cong_symmetric
thf(fact_1247__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocong__char_092_060_094sub_062S_092_060_094sub_062R_092_060_094sub_062T_092_060_094sub_062S,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ T ) ) )
= ( ( lambda_Arr @ T )
& ( lambda_is_Lam @ T )
& ( lambda_Arr @ U )
& ( lambda_is_Lam @ U )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ T @ U ) )
& ( ide_lambda_lambda @ lambda_resid @ ( lambda_resid @ U @ T ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.cong_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1248__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T
@ U )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ U
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_sym
thf(fact_1249__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oresid__reflects__con,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T
@ V )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ U
@ V )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ V )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ V ) )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.resid_reflects_con
thf(fact_1250__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocon__sym,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T
@ U )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ U
@ T ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.con_sym
thf(fact_1251__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oresid__reflects__con,axiom,
! [T: lambda_lambda,V: lambda_lambda,U: lambda_lambda] :
( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T
@ V )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ U
@ V )
=> ( ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ V )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ V ) )
=> ( con_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.resid_reflects_con
thf(fact_1252__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otargets__char_092_060_094sub_062S_092_060_094sub_062R_092_060_094sub_062T_092_060_094sub_062S,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T )
= ( collec621444948832755541lambda
@ ^ [B3: lambda_lambda] :
( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.targets_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1253__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Osources__char_092_060_094sub_062S_092_060_094sub_062R_092_060_094sub_062T_092_060_094sub_062S,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T )
= ( collec621444948832755541lambda
@ ^ [A3: lambda_lambda] :
( ( lambda_Arr @ T )
& ( lambda_is_App @ T )
& ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.sources_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1254__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Otargets__char_092_060_094sub_062S_092_060_094sub_062R_092_060_094sub_062T_092_060_094sub_062S,axiom,
! [T: lambda_lambda] :
( ( target5490931411331770124lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T )
= ( collec621444948832755541lambda
@ ^ [B3: lambda_lambda] :
( ( lambda_Arr @ T )
& ( lambda_is_Lam @ T )
& ( member_lambda_lambda @ B3 @ ( target5490931411331770124lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.targets_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1255__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Osources__char_092_060_094sub_062S_092_060_094sub_062R_092_060_094sub_062T_092_060_094sub_062S,axiom,
! [T: lambda_lambda] :
( ( source5627170679743844886lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T )
= ( collec621444948832755541lambda
@ ^ [A3: lambda_lambda] :
( ( lambda_Arr @ T )
& ( lambda_is_Lam @ T )
& ( member_lambda_lambda @ A3 @ ( source5627170679743844886lambda @ lambda_resid @ T ) ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.sources_char\<^sub>S\<^sub>R\<^sub>T\<^sub>S
thf(fact_1256__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Onull__eqI,axiom,
! [N: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) )
@ N
@ T3 )
= N )
& ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_App @ U6 ) )
@ T3
@ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.null_eqI
thf(fact_1257__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocube__ax,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ T ) )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ T ) )
= ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ V
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cube_ax
thf(fact_1258__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__sym__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ U
@ T )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_sym_ax
thf(fact_1259__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocon__imp__arr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ U ) )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.con_imp_arr_resid
thf(fact_1260__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Onull__eqI,axiom,
! [N: lambda_lambda] :
( ! [T3: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_Lam @ U6 ) )
@ N
@ T3 )
= N )
& ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [U6: lambda_lambda] :
( ( lambda_Arr @ U6 )
& ( lambda_is_Lam @ U6 ) )
@ T3
@ N )
= N ) )
=> ( N
= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.null_eqI
thf(fact_1261__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocube__ax,axiom,
! [V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ T ) )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ V
@ T )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ T ) )
= ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ V
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.cube_ax
thf(fact_1262__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocon__sym__ax,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ U
@ T )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.con_sym_ax
thf(fact_1263__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocon__imp__arr__resid,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) )
=> ( ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ U ) )
!= ( partia1461432607296028304lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.con_imp_arr_resid
thf(fact_1264__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Otrg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T )
= ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ T ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.trg_def
thf(fact_1265__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Otrg__def,axiom,
! [T: lambda_lambda] :
( ( trg_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T )
= ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ T ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.trg_def
thf(fact_1266__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ojoin__of__symmetric,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T
@ U
@ V )
=> ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ U
@ T
@ V ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.join_of_symmetric
thf(fact_1267__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ojoin__of__symmetric,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T
@ U
@ V )
=> ( join_o1386658102433174083lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ U
@ T
@ V ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.join_of_symmetric
thf(fact_1268__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oresiduation__axioms,axiom,
( residu6197316955523523210lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.residuation_axioms
thf(fact_1269__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Opartial__magma__axioms,axiom,
( partia3924075026459072775lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.partial_magma_axioms
thf(fact_1270__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oresiduation__axioms,axiom,
( residu6197316955523523210lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.residuation_axioms
thf(fact_1271__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Opartial__magma__axioms,axiom,
( partia3924075026459072775lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.partial_magma_axioms
thf(fact_1272__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oprfx__reflexive,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ T ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.prfx_reflexive
thf(fact_1273__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Oide__implies__arr,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ A2 )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ A2 ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.ide_implies_arr
thf(fact_1274__092_060Lambda_062_092_060_094sub_062A_092_060_094sub_062p_092_060_094sub_062p_Ocong__reflexive,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ T )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ T ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_App @ T5 ) )
@ T
@ T ) ) ) ) ).
% \<Lambda>\<^sub>A\<^sub>p\<^sub>p.cong_reflexive
thf(fact_1275__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oprfx__reflexive,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T )
=> ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ T ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.prfx_reflexive
thf(fact_1276__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Oide__implies__arr,axiom,
! [A2: lambda_lambda] :
( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ A2 )
=> ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ A2 ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.ide_implies_arr
thf(fact_1277__092_060Lambda_062_092_060_094sub_062L_092_060_094sub_062a_092_060_094sub_062m_Ocong__reflexive,axiom,
! [T: lambda_lambda] :
( ( arr_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ T )
=> ( ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ T ) )
& ( ide_lambda_lambda
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) ) )
@ ( sub_re469828149216679723lambda @ lambda_resid
@ ^ [T5: lambda_lambda] :
( ( lambda_Arr @ T5 )
& ( lambda_is_Lam @ T5 ) )
@ T
@ T ) ) ) ) ).
% \<Lambda>\<^sub>L\<^sub>a\<^sub>m.cong_reflexive
% 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 (1)
thf(conj_0,conjecture,
paths_366302053746739058lambda @ lambda_resid @ ( cons_lambda_lambda @ ( lambda_Lam @ t ) @ u ) ).
%------------------------------------------------------------------------------